Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A coordinated scheduling of delivery and inventory in a multi-location hospital supplied with a central pharmacy

A coordinated scheduling of delivery and inventory in a multi-location hospital supplied with a... Logist. Res. (2016) 9:18 DOI 10.1007/s12159-016-0145-8 OR IGINAL PAPER A coordinated scheduling of delivery and inventory in a multi- location hospital supplied with a central pharmacy 1 1 2 1 • • • Zakaria Hammoudan Olivier Grunder Toufik Boudouh Abdellah El Moudni Received: 25 March 2015 / Accepted: 8 August 2016 / Published online: 25 August 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In today’s manufacturing outlook, coordinated last, the experimental results show the efficiency of the scheduling of delivery and inventory represents a leading proposed solving methods, based on the two following leverage to enhance the competitiveness of firms which criteria: solution quality and processing time. aims to address the new challenge coming from scheduling problems. Though in the last decades this kind of issue has Keywords Case study  Coordinated scheduling been extensively approached in the literature, a set of Production and transportation  Mixed-integer constraints and compulsory dispositions strongly increases programming  Branch and bound  Heuristic algorithm the complexity of the considered problem. Actors of the pharmaceutical supply chain have to meet various global regulatory requirements while handling, storing and dis- 1 Introduction and related literature tributing environmentally sensitive products. The studied problem in this paper focuses on a real-case scheduling Today, the expansion of suppliers to accommodate the problem in a multi-location hospital supplied with a central maximum number of customers is considered as a key pharmacy. The objective of this work is to find a coordi- factor in the evolution of companies, in order to increase nated production and delivery schedule such that the sum their profits. Industrial companies are continuously of delivery and inventory costs is minimized. A mixed- assessing their operations with the objective of increasing integer programming formulation is first detailed to con- the overall effectiveness of manufacturing systems. Mar- sider the problem under study. Then, a branch-and-bound kets, where these organizations operate, tend to become algorithm is proposed as an exact method and a dedicated more complex over time, forcing companies to increase heuristic algorithm is highlighted to solve the problem. At their responsiveness, both in terms of time and cost. The case of the pharmaceutical industry is a good example of how market is driving the change on product development & Zakaria Hammoudan cycles and manufacturing activities. Delivery and inven- zakaria.hamoudan@utbm.fr tory scheduling stages are systematically considered to be Olivier Grunder very difficult functions. They are intended to produce olivier.grunder@utbm.fr operational plans dealing with several potential conflicting Toufik Boudouh objectives, namely minimizing costs, completion times, toufik.boudouh@utbm.fr and delays or maximizing profit. One important benefit of Abdellah El Moudni this coordination is a more efficient management of abdellah.elmoudni@utbm.fr inventories across the entire supply chain. In traditional inventory management, the optimal production and ship- IRTES-SET, Universite de Technologie de Belfort- ment policies for vendors and customers in a two-echelon Montbeliard (UTBM), rue Thierry Mieg, 90010 Belfort Cedex, France supply chain are managed independently. Additionally, these functions are closely related to other areas such as ´ ´ M3M, Universite de Technologie de Belfort-Montbeliard sales, procurement, production execution and control; (UTBM), rue Thierry Mieg, 90010 Belfort Cedex, France 123 18 Page 2 of 13 Logist. Res. (2016) 9:18 Table 1 A numerical example Jobs 1 2 3.5 4 Hospital’s 1 2 1 2 Due date Thursday 8:00 am Thursday 8:00 am Thursday 8:00 am Thursday 8:00 am Departure time Wednesday 06:00 am Wednesday 06:00 pm Thursday 2:00 am Wednesday 06:00 pm Arrival time Wednesday 12:00 pm Wednesday 10:00 pm Thursday 8:00 am Wednesday 10:00 pm Inventory cost 600 200 0 200 hence, they may interface with decisions at the strategic consideration holding, transportation, fixed ordering and and operational levels. For this reason, the integrated stock out costs. Viswanathan and Mathur [31] have studied a vendor–customer model is developed where the total rel- distribution systems with a central warehouse and many evant costs for the customers as well as the vendors have to retailers that stock a number of different products, where the be minimized. Consequently, determining the production products are delivered from the warehouse to the retailers by and shipment policies based on an integrated total cost vehicles that combine the deliveries to several retailers into function, rather than several customer’s or vendor’s indi- efficient vehicle routes. They have proposed a heuristic that vidual cost functions, results in the reduction of the develops a stationary nested joint replenishment policy. inventory costs of the system. These results showed that the proposed heuristic is capable of The system under study in this paper is composed of a solving problems involving distribution systems with multi- central pharmacy from which sterilized medical devices ple products. Sindhuchao et al. [29] have considered a system have to be delivered before given due dates, to different that consists of a set of geographically dispersed suppliers hospitals located around the central pharmacy. This supply that manufacture one or more non-identical items, and a chain process incurs both delivery costs and earliness central warehouse that stocks these items. The warehouse penalty costs in case the devices are delivered too early. faces a constant and deterministic demand for the items from Therefore, the considered problem is an integrated outside retailers. The items are collected by a fleet of vehicles delivery and inventory problem with due dates constraints, that are dispatched from the central warehouse. The vehicles for which we have to minimize the total delivery and are capacitated and must also satisfy a frequency constraint. holding costs. Therefore, the problem can be formulated They studied the case where each vehicle always collects the from a batch scheduling point of view with a cost objective same set of items. They have formulated and solved the function or from a lot sizing problem point of view with a problem by using a branch-and-price algorithm, and then time horizon. These two classes of problem have been they have proposed a greedy constructive heuristic and a very proven to be equivalent under given conditions [19]. In our large-scale neighborhood search algorithm. These results case, a batch scheduling approach seems to be more indicate that the constructive heuristic used in conjunction appropriate in the context of the study of the healthcare with one of the proposed very large-scale neighborhood system with specific constraints for the due dates. algorithms can find near-optimal solutions very efficient. The delivery-inventory problem is denoted as Vendor- Recently, Archetti et al. [2] have studied a distribution Managed Inventory (VMI) problem. The VMI problem is a problem in which a product has to be shipped from a supplier widely used collaborative inventory management policy in to several retailers over a given time horizon. Each retailer which manufacturers manage the inventory of retailer and defines a maximum inventory level. The supplier monitors take responsibility for making decisions related to the timing the inventory of each retailer and determines its replenish- and extent of inventory replenishment [7]. VMI partnerships ment policy, guaranteeing that no stock out occurs at the help organizations to reduce demand variability, inventory retailer (supplier-managed inventory policy). Every time a holding and distribution costs. A pioneering paper is due to retailer is visited, the quantity delivered by the supplier is Bertazzi et al. [6], where a given set of shipping frequencies is such that the maximum inventory level is reached (deter- allowed and different products may be shipped at different ministic order-up-to level policy). Shipments from the sup- frequencies. Herer and Levy [14] have considered a system of plier to the retailers are performed by a vehicle of given a central warehouse, a fleet of trucks with a finite capacity, capacity. They presented a mixed-integer linear program- and a set of customers, for each of whom there is an estimated ming model, and they derived new additional valid inequal- consumption rate, and a known storage capacity. The ities used to strengthen the linear relaxation of the model. objective is to determine when to service each customer, as They implemented a branch-and-cut algorithm to solve the well as the way to be performed by each truck, in order to model optimally. Then, they have studied two different types minimize the total discounted costs. To solve the problem, of replenishment policies in [3]. The first one is the well- they have proposed a rolling horizon approach that takes into known order-up-to level (OU) policy, where the quantity 123 Logist. Res. (2016) 9:18 Page 3 of 13 18 shipped to each retailer is such that the level of its inventory et al. [10] studied the problem of coordinated scheduling of reaches the maximum level. The second one is the maximum production and delivery subject to the production window level (ML) policy, where the quantity shipped to each retailer constraints and delivery capacity constraints. They consid- is such that the inventory is not greater than the maximum ered both a single delivery time case and multiple delivery level. In this study, Archetti et al. [3] have focused on the ML time case. Chen [8] reviewed the production and distribution policy and the design of a hybrid heuristic, and they imple- scheduling models and classified these problems in five mented an exact algorithm for the solution of the problem groups. Problems addressing an objective function that with one vehicle and designed a hybrid heuristic for the combines machine scheduling with the delivery costs are multi-vehicle case. Most recently, Archetti et al. [4], have rather complex. However, they are more practical than those studied the previous problem with a single vehicle which has involving just one of the two factors, since these combined- a given capacity. The transportation cost is proportional to the optimization problems are often encountered when real-world distance traveled, whereas the inventory holding cost is supply chain management is considered. proportional to the level of the inventory at the customers and The number of customers and products has been a topic at the supplier. They have proposed a heuristic that combines of intense investigation for decades in the integrated supply a tabu search scheme with ad hoc designed mixed-integer chain. Although researchers have given a considerable programming models. The effectiveness of the heuristic was attention on the synchronization of the single-vendor single- proved over a set of benchmark instances for which the customer integrated inventory system, the single-vendor optimal solution was known. multi-customer integrated inventory case has gotten little There are numerous researches on batch scheduling of attention in regard. Lu [23] developed a one-vendor multi- delivery-inventory problem. Scheduling problems arise in customer integrated inventory model, while Parija and Sar- almost any type of industrial production facilities (Pulp and ker [25] extended their published work on single-vendor, Paper, Metals, Oil and gas, Chemicals, Food and Beverages, single-customer, integrated production-inventory problems Pharmaceuticals, Transportation, Service, Military, etc.) with lumpy delivery systems under perfect and imperfect where given operations need to be processed on specified production cycle situations [27]. Lu [23] argued that all the resources. The corresponding scheduling problems are previous studies assumed that the vendor must know the already very difficult to solve [20]. Much research has customer’s holding and ordering costs, which are quite dif- focused on the same area under various assumptions and ficult to estimate unless the customer is willing to reveal the objective measures that differ from the considered problem in true values. Therefore, he considered another circumstance, this paper. Potts [12], Hall[ 26] and Zhang et al. [35]have in which the objective is to minimize the vendor’s total cost studied scheduling problems with non-identical job release per year, subject to the maximum cost that the customer may be prepared to incur. Parija and Sarker [25 times and delivery times, under the assumption that a suffi- ] introduced the cient number of vehicles is available to deliver the jobs. problem of determining the production start time and pro- Kimms [21] has examined the problem of single-machine and posed a method that determines the cycle length and raw proposed two heuristic approaches: randomized regrets based material ordering frequency for a long-range planning and tabu search approaches. Each production plan is gener- horizon. The cycle length is restricted to be an integer- ated without using any information obtained from previous multiple of all shipment intervals to the customers as an plans. This work has been extended by Kimms [22]witha ideal situation, the solution to which may be sub-optimal. proposition of a genetic algorithm that dominates the tabu Viswanathan and Piplani [32] proposed a model to study and search procedure, both in terms of run-time performance and analyze the benefit of coordinating supply chain inventories the ability to find feasible solutions. Pinedo and Michael [24] by means of common replenishment epochs or time periods. reviewed different models and solution approaches, and then A one-vendor multi-customer supply chain is considered for they explained the complexity of scheduling problems. a single product. Under their strategies, the vendor specifies Multi-echelon inventory models have attracted much common replenishment periods and requires all customers to attention, and the integrated approach has been extensively replenish only at pre-determined time periods. However, the studied. In this way, Grunder [11] considered a single-product authors did not include any inventory cost of the vendor in batch scheduling problem with the objective of minimizing the model. In most papers dealing with integrated inventory the sum of production, transportation and inventory cost. models, the transportation cost is considered only as a part of Particularly, he assumed that the delivery time depends on the fixed setup or replenishment cost. Ertogral et al. [9]studied batch sizes and proposes a dynamic programming approach how the results of incorporating transportation cost into the based on a dominance relation property. Wang et al. [33] model influence the decision-making process under equal extended this study with an integrated scheduling problem for size shipment policies. A fundamental advance in the two- single-item supply chain involving due date considerations side cost structure is in recognizing how delivery-trans- and an objective of minimizing the total logistics cost. Fu portation costs apply to both sides. 123 18 Page 4 of 13 Logist. Res. (2016) 9:18 Hoque [15] proposed three models for supplying a single- of Hoque’s models. Hariga et al. [13] compared the cost item from a single-vendor to multiple customers under between the results of the models in Hoque [16] and deterministic demand by synchronizing the production flow Zavanella and Zanoni [34], and then they concluded that with equal-sized batch transfer in the first two and unequal- both models are not appropriate as they are using different sized batches transfer in the third. In the first two models, all functional forms of the total setup and ordering costs. batches forwarded are of exactly the same size but the Moreover, it is shown that Hoque’s model yields imprac- timing of their shipment is different. In the first of these, the tical solutions for zero transportation costs. When the total manufacturer transfers a batch to a customer as soon as its setup and ordering cost was adjusted to be similar to the processing is finished, whereas in the second a batch is one in Zavanella and Zanoni’s model, Hoque’s model transferred to a customer as soon as the previously sent resulted in a larger total cost. batch to the customer is finished. In the third model, the Existing inventory models for multi-customers are not subsequent shipment lot sizes increase by the ratio of pro- applicable to pharmaceutical products for several reasons. duction rate and sum of demand rates on all the customers. Pharmaceutical products can be more expensive than other Zavanella and Zanoni [34] proposed a model for a single- products to purchase and distribute, and shortages and vendor multi-customer system, integrated in a shared man- improper use of essential medicines can have a high cost in agement of the customers’ inventory, so as to pursue a terms of wasted resources and preventable diseases and reduction or the stability of the holding costs while death. Therefore, special care should be taken in pharma- descending the chain. Hoque [16] transferred the lot from a ceutical inventory decisions to ensure 100 % product vendor to multiple customers with l number of unequal- availability at the right time, at the right cost, and in good sized batches first; where the next one is a multiple of the condition to the right customers. The quality of health care previous one by the ratio ðk [ 1Þ of the production and the industries strongly depends on the availability of pharma- total demand rates, followed by ðn  lÞ number of equal- ceuticals on time. If a shortage occurs at a hospital, an sized batches. The equal-sized batches are restricted to be emergency delivery is necessary, which is very costly and less than or equal to the lth batch (the largest unequal-sized can affect the patient health. Inventory management batch) multiplied by k. The models developed were solved strategies that are unsuitable for health care industries may by applying Lagrangian Multiplier method. However, in lead to large financial losses and a significant impact on cases of single-vendor single-customer or single-vendor patients. Hence, inventory strategies for pharmaceutical multi-customer or multi-stage production, synchronization products are more critical than those for other products. of the production flow by transferring the lot with equal and/ Thus, a specific inventory model is necessary to control or unequal-sized batches was found to lead to the least total pharmaceutical products, to save patient lives and reduce cost for some numerical problems. Although Hoque [16] unnecessary inventory costs. served that purpose, he did not cope with the relaxation of Here we investigate a delivery-inventory supply chain the discussed impractical assumptions. Following this trend composed of a central pharmacy which has to deliver of synchronization, Hoque [17] developed two generalized pharmaceutical supplies to distant hospitals with a single single-vendor multi-customer integrated inventory models transporter at given due dates. The objective is to reduce by accumulating the inventory at the vendor’s and cus- the overall cost which includes the delivery costs and an tomer’s independently, but with the traditional trend of earliness penalty cost. ignoring the cost of benefit sharing. Transportation of each The contributions of this paper are twofolds. First, we of the batches incurs a transportation cost. In order to propose a MIP model to minimize the total delivery and implement the models by taking into account the industry inventory costs for the considered supply chain under the reality, he also incorporates them with the relaxation of the constraints of healthcare systems. Second, we propose an discussed impractical assumptions. Battini [5] developed a efficient solving algorithm which is compared with two single-vendor and multi-customer consignment stock exact methods. inventory model in which many clients can establish a The outline of the remainder of the paper is organized in consignment stock inventory policy with the same vendor. seven sections. In Sect. 2, the problem definition and for- Recently, Jha and Shanker [18] studied an integrated mulation is introduced. In Sect. 3, the problem is formu- production-inventory model in a single-vendor multi-cus- lated as a mixed-integer programming (MIP) model. Then, tomer supply chain with lead time reduction under inde- we describe the proposed branch-and-bound algorithm pendent normally distributed demand on the customers. (B&B) as an exact method of resolution in Sect. 4.We They assume a non-identical lead time for the customers develop a heuristic algorithm in Sect. 5 for solving the and that customers’ inventory is reviewed using continuous problem. In Sects. 6 and 7, we eventually provide the review policy. Hariga et al. [13] analyzed Hoque’s models experimental results and draw some conclusions and sug- I and II studied in Hoque [16], and then they modified some gest the future research directions. 123 Logist. Res. (2016) 9:18 Page 5 of 13 18 2.1 Notations The following notations are used in developing the math- ematical model: Parameters • J ¼ 1; 2;...; n: set of all jobs, where n is the total number of jobs, • H ¼ 1; 2;...; m: set of all hospitals, • j: index for jobs, j 2 J, • k: index for batches, • h: index for hospitals, h 2 H, • d : due date of job j, • cl : destination of job j, cl 2 H, j j • c : capacity of the transporter, • s : time for the vehicle to deliver a batch to hospital h and to return to the central pharmacy location, Fig. 1 Central pharmacy and multi-location hospital model • g : delivery cost to deliver a batch to hospital h and to return to the central pharmacy location, • b : hospital earliness penalty function for hospital h. 2 Problem definition and formulation Primary variables We consider a supply chain scheduling problem where 1 • d ¼ 1 if the job j belongs to the kth batch, 0 otherwise, jk there is one central pharmacy which has to deliver medical • d ¼ 1 if the batch k belongs to the customer h,0 kh supplies, or jobs, to m hospital sites, which are the final otherwise. customers (Fig. 1). Each hospital h orders a finite number Secondary variables of jobs from the central pharmacy. The following assumptions are considered for this study. • y ¼ 1 if the batch k exists and is not empty, 0 First, we will consider a single transporter to deliver the otherwise, sterilized medical devices as the number of distant hospi- • C : the arrival time of the job j at the hospital, tals is reduced in practice (less than 4) and the distances • B : the arrival time of the batch k at the hospital, with the central pharmacy are quite short. Second, we will • u : number of delivered batches for hospital h. only consider direct shipping (i.e., commuter tours), with- out considering routing considerations between customers 2.2 Numerical example [8]. This assumption is explained by the fact that the pharmacy is located in the center of the distant hospitals. To clarify the problem, we consider a simple numerical Moreover, the road network is centralized on the main town of the central pharmacy; hence, travel times are example in Table (1) as follows. Two hospitals ordered five longer between distant hospitals. jobs at the same time (Monday at 8:00 am) and they would Each round trip between the pharmacy and a hospital h receive their products at the same time (Thursday at 8:00 requires a delivery cost g as well as a delivery time s . am), that means all the products have the same due date equal to 72 h. The central pharmacy and its hospital cus- The batches delivered from the central pharmacy to the hospitals can be of different sizes. tomers open 24 h/day. Three jobs (j ¼ 1; 3and5) for hos- pital 1 and two jobs (j ¼ 2 and 4) for hospital 2. The The total number of jobs belonging to the same batch cannot exceed the capacity c of the transporter. Each job j vehicle capacity is c ¼ 2. The transporter delivery cost and time depend on the hospitals’ positions with (g ¼ 1000 has a due date d specified by the hospitals and each job has to arrive to the hospital site before its due date. If job j of Euro, s ¼ 6 h, and g ¼ 750 Euro, s ¼ 4 h) belongs to 1 2 hospital 1 and 2, respectively, (b ¼ b ¼ b ¼ 30 Euro/h hospital h arrives before its due date d , it will incur as an 1 3 5 and b ¼ b ¼ 20 Euro/h) belongs to hospital 1 and 2, earliness penalty b . Batching and sending several jobs in 2 4 respectively. the batches will reduce the transportation costs. The solution is shown in Table (1) for this problem. As The objective is to determine the sequence of batches it is shown, the vehicle makes three round trips among that has to be processed, so that the expected total cost of them two to hospital 1 and one to hospital 2. Three batches both central pharmacy and hospitals sites is minimized. 123 18 Page 6 of 13 Logist. Res. (2016) 9:18 k ¼ 1; k ¼ 2 and k ¼ 2 are denoted. The products arrive The objective function (1) minimizes the sum of the 1 2 3 at the customers in the batch to which they belong to in the delivery costs, through the g u term, and the customers completion time cited in Table (1). The total delivery cost earliness penalty, through b ðd  C Þ. Constraint (2) cl j j equals g  2 þ g  1 ¼ 2750 Euro and the total storage guarantees that each job must be scheduled exactly in one 1 2 cost at the hospitals equals b ½ðd  C Þþðd  C Þþ 1 1 3 3 1 batch. In this constraint, the jobs will be batched only in the ðd  C Þ þ b ½ðd  C Þþðd  C Þ ¼ 30 ½20 þ 5 5 2 2 2 4 4 batch which it belongs to. Constraints (3 and 4) force each 0 þ 0þ 20 ½10 þ 10¼ 1000 Euro. The amount of the batch to be delivered to the customer it belongs to. Con- objective function is 3750 Euro. straint (5) calculates the number of batches delivered to each customer. Constraint (6) guarantees that no empty batch is allowed. Constraint (7) prevents the number of 3 The mixed-integer programming model jobs scheduled in one delivery batch to exceed the capacity of the vehicle. Constraint (8) indicates that arrival time of The pharmaceutical supply chain has many aspects that each job is at least equal to the contracted due date for each need to be considered in a supply chain model. However, customer. Constraint (9) orders the batches in the by taking all concerned factors into account, the model increasing order of their arrival times. Constraint (10) would be of so high complexity that it would be extremely expresses the minimum interval duration between the hard for analysis. In this section, the mathematical pro- arrivals of two consecutive batches has to be greater than gramming model of the above-mentioned problem is pre- the delivery time of the transporter. Constraint (11) rep- sented. Using the structural properties, we develop a MIP resents the relation between the completion time of the jobs model for the mentioned problem as follows: and the arrival time of the batch they belong to. This m n X X constraint is represented in a nonlinear way in this math- Min Z ¼ g u þ b ðd  C Þ h j j ð1Þ h cl ematical representation to facilitate the understanding of h¼1 j¼1 the problem. Constraints (12) and (13) define the range of Subject to : the variables. For ease of reference, we denote this problem: Multiple Customers Batching Delivery Scheduling Problem d ¼ 1; j ¼ 1;...; n ð2Þ jk k¼1 (MCBDSP). The complexity of the MCBDSP is still an open ques- d  1; k ¼ 1;...; n ð3Þ kh tion. To the best of our knowledge, no polynomial algo- h¼1 rithm can solve this problem. However, from simulation 2 1 experiments, we observe that the problem is still d  d ; j; k ¼ 1;...; n and k  j ð4Þ k;cl jk intractable on an empirical basis. In the next section, a B&B with a lower bound is described to solve the problem u ¼ d ; h ¼ 1;...; m ð5Þ kh as an exact method. k¼1 y  y ; k ¼ 1;...; n  1 ð6Þ k kþ1 X 4 Branch-and-bound algorithm d  c; k ¼ 1;...; n ð7Þ jk j¼k In this section, we describe the B&B algorithm that we have developed to solve the MCBDSP. The objective of C  d ; j ¼ 1;...; n ð8Þ j j this B&B is to solve small to medium-sized instances, and B  B ; k ¼ 1;...; n  1 ð9Þ kþ1 k to be a reference for validating the efficiency of the pro- posed heuristic algorithm. This B&B algorithm maintains a 2 2 B  B  s ðd þ d Þ; kþ1 k h kþ1;h kh list of subproblems (nodes) whose union of feasible solu- ð10Þ h¼1 tions contains all feasible solutions of the original problem. k ¼ 1;...; n  1 and h 2 H The list is initialized with the original problem itself. In n each major iteration, the algorithm selects a current sub- C ¼ B  d ; j ¼ 1;...; n ð11Þ problem from the list of unevaluated nodes. This branching j k jk k¼1 seems to be natural; however, the number of branches will be very large for large problems. Consequently, if this C  0; j ¼ 1; 2;...; n ð12Þ method is used in the B&B algorithm, it may take too much 1 2 d ; d 2f0; 1g; j; k ¼ 1;...; n and h ¼ 1;...; m ð13Þ jk kh time to find optimal solutions, as redundant schedules 123 Logist. Res. (2016) 9:18 Page 7 of 13 18 would be checked repeatedly. Yet, several of the sub- solve large-sized instances grows exponentially in the problems would already have been eliminated upon the experimental results. Therefore, developing fast heuristic generation of nodes, since the search tree includes redun- algorithm to yield near-optimal solutions in a reasonable dant solutions. running time is still of great importance. In the next sec- At each node of the search tree, the number of products tion, a solving method is proposed to solve the problem. that still need to be delivered to each customer has to be updated. Iterations are performed until the list of sub- problems to be processed is empty. The crucial part of a 5 Heuristic algorithm successful B&B algorithm is the computation of the lower bounds. Therefore, we have developed a lower bound In this section, a heuristic algorithm, which is denoted described in the next part. Batching and Scheduling algorithm (B&S), is proposed. Efficient lower bound would significantly reduce the This algorithm is composed of two steps, the first one time and efforts needed for the B&B method. Based on the consists in defining the size of the batches and the second main feature of the problem, the lower bound value for the one will schedule them according to the different con- problem is the summation of lower bounds on the total straints of the problem. earliness cost and the transportation cost. We assume that The B&S algorithm starts by generating an initial w is a partial batch sequence solution, z(w) is the evaluation solution through the means of a progressive constructive of w, and r ðwÞ is the number of products remaining at the procedure. Then, the above-mentioned two-steps process is customer’s h for partial solution w. This notation will be applied until a predefined stop condition is satisfied. At used throughout this part. first, some elements of the current solution are constructed. In each node, the solutions are built from the last batch Then, a local improvement phase based on a swap operator to the first one and the evaluation of the partial or complete is applied to the reconstructed solution in order to improve solution is processed with backward equations. The its quality. Finally, B&S chooses the optimum solution research of a solution starts by constructing a partial between the current solution and the solution obtained from solution w. Then, the remainder of products is added in the improvement procedure. order to get a complete solution, with the objective of Let us denote that ðq ; cl ; B Þ is the notation which will k k k achieving a minimum delivery cost. Therefore, more the be used for a solution of a batch k, where the first term q transporter will be loaded, more this lower bound will be describes the number of jobs in this batch, the second term efficient. cl describes the customers destination of batch k and the third term B is the arrival time of this batch. For example, Proposition 1 For a partial solution w, a lower bound for a solution of three batches, which contains 2, 3 and 2 jobs, the delivery cost of the remaining products is given by: respectively, belongs to customers 2, 3 and 1, respectively, m lm and arrives at due dates 1000, 1015 and 1020, respectively, ð14Þ h will be written as follows: h¼1 ½ð2 ; cl ; 1000Þ; ð3 ; cl ; 1015Þ; ð2 ; cl ; 1020Þ 1 2 2 3 3 1 Proof For each customer h,if r ðwÞ is the number of Based on the prune rule, the following heuristic algorithm products remaining to be delivered, the number of round is proposed as follows: for level 0, there is no job. For the trips will be equal to , and the delivery cost of the first level, which includes only last job n, there is only one remaining products is as denoted in Eq. (14). h possible joint solution which is ð1; cl ; B Þ. For level k n n We add the partial solution w to the solution found in (includes q jobs), all ‘‘good’’ solutions for a number of k equation (14) to get the lower bound of the current node jobs will be kept. The process to build ‘‘good’’ solutions for under study. level k is described as follows: (1) build solutions of level k by considering all the solutions in the retained ‘‘good’’ Corollary 1 The lower bound LB(w) of the partial solu- solutions of all the previous levels from 1 to ðk  1Þ. For tion w is given as follows: each retained solution of level k  k, a new solution of lm h 0 level k is built by simply adding a batch of (k  k Þ jobs, if LBðwÞ¼ zðwÞþ  g ð15Þ this is possible. Then, this procedure is repeated until the h¼1 level n is reached. Proof Straightforward. h The details of the algorithm (1) are presented as follows: The mathematical model and the B&B algorithm The generation of the initial solution and the construction developed in the previous sections could solve small to procedure is represented from line 1 to 4. Then, the batch medium-sized instances; however, the time of resolution to sizing procedure is represented from line 5 to 17 according 123 18 Page 8 of 13 Logist. Res. (2016) 9:18 to a scattering/gathering procedure. The improvement not, a new swap operation is generated. The improvement procedure is called in line 14, and then it is described in operation stops when the index of batches equals 0. Algorithm 2. The batch sizing procedure, performed in an iterative way, extends a partial solution by adding one job from a set J of all jobs. The construction of the good solution advances progressively and in a hierarchical manner. The process starts from the last job and arrives recursively to the first one. The jobs are distributed to the customers to whom they belong, and the batches sizes are defined according to a scattering/gathering procedure described in Algorithm 1. Let’s take an example to explain the application of the B&S in Algorithm 1, to illustrate the MCBDSP. We con- sider a problem of three jobs and two customers. The due dates associated with these jobs equal 1000; 1100; 1150, where jobs 1 and 3 have due dates 1000 and 1150 and belong to customer 1 and job 2 has due date 1100 and belongs to customer 2. The transport delivery cost and time depend upon the customer’s location with (g ¼ 20; s ¼ 60 u:t and g ¼ 15; s ¼ 40 u:t) belonging 1 2 1 2 to customer 1 and 2, respectively. The customers’ holding costs are defined as follows: (b ¼ b ¼ 30 and b ¼ cl cl cl 1 3 2 15 ), belonging to customer 1 and 2, respectively. The B&S process is described in detail as follows: the process starts by the last job recursively to arrive to the first one. 1. For j ¼ 1; currentJob = 3, there is only one possible joint solution which is ð1; cl ; 1150Þ. 2. For j ¼ 2; currentJob j ¼ 3 or 2, there are different In this algorithm, the number of delivered jobs j possible solutions. Firstly, a complementary solution is varies from 0 to n (line 1). For each level of j delivered built by simply adding a batch of (2  1) job to the jobs, the different partial solutions are built from the previous delivery solution. The potential delivery solution of previous levels (\j). Moreover, the neces- scheme is equal to 2 batches: sary number of batches to these solutions is added, to ½ð1; cl ; 1150Þ; ð1; cl ; 1100Þ complete the partial solution of level j. After every 1 2 product addition to level j, the partial solution of this Due to the improvement phase, two solutions could be level is completed by adding the necessary delivery obtained according to the swap operation which are: scheme to the considered solution in the list of all kept ½ð1; cl ; 1100Þ; ð1; cl ; 1150Þ and solutions from 0 to (j  1), to obtain the new list of 2 1 solutions of level j. A test of verification of the capacity ½ð1; cl ; 1150Þ; ð1; cl ; 1100Þ 1 2 of the transporter used is done directly after each Then the two potential joint solutions are compared advancement in level (See line 6 in Algorithm 1). and the good solution is kept, which is: The final step of each level j is denoted in line 15, which is mentioned in Algorithm 1. In this phase, the good ½ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 2 1 solution is memorized and inscribed to level j. After that, a 3. For j ¼ 3. currentJob j ¼ 3; 2 or 1, based on the new level ðj þ 1; j þ 2...Þ is started till reaching level n. delivery scheme of the first step, the new delivery In the improvement phase, all consecutive batches are schemes are: swapped, by starting from the last batch recursively to the first one, while the index of the batch is positive (See line 5 ½ð1; cl ; 1150Þ; ð1; cl ; 1100Þ; ð1; cl ; 1000Þ and 1 2 1 in Algorithm 2). After every swap operation, the new ½ð1; cl ; 1150Þ; ð1; cl ; 1000Þ; ð1; cl ; 1100Þ: 1 1 2 solution is kept if it is better than the current solution. If 123 Logist. Res. (2016) 9:18 Page 9 of 13 18 Due to the improvement phase, two solutions could be The maximum solving time allowed for these instances obtained according to the swap operation which are: is 1 h. The reference of time limit of resolution is based on the real time of the preparation of a schedule in the ½ð1; cl ; 1100Þ; ð1; cl ; 1000Þ; ð1; cl ; 1150Þ and 2 1 1 actual case. As a comparison, CPLEX solver is used to ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 exactly solve the model with small scale random instances. Some adjustments are done on the parameters Then the four potential joint solutions are compared of research in CPLEX order to accelerate the research of and the good solution is kept, which is: solutions. ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 According to the confidentiality of the data base of the central pharmacy under study, several cases of problem After that, based on the delivery scheme of the second were considered for which several instances were gener- step, the delivery scheme on this step is: ated randomly. ½ð1; cl ; 1100Þ; ð1; cl ; 1150Þ; ð1; cl ; 1000Þ 2 1 1 6.1 Test cases In the improvement phase, a new solution could be obtained which is: The characteristics of orders to schedule differ by cus- ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 tomers, transporter capacity, quantity delivered, due date, transporter time, transporter cost and the storage cost at The two potential joint solutions are compared and the each customer. Three cases are considered to test the good solution is kept, which is: proposed methods. The characteristics of the case are listed ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 in Table 2. For each case {A, B and C}, the number of products n, the number of customers h, the transporter time Finally, the two potential joint solutions kept in each s , the transporter cost (g ), and the storage cost at each level are compared and the best final solution is customer (b ) are displayed. recovered, which is: In the first case, g is higher than b ,where g and b h h h h ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 are randomly generated from the uniform distribution with ranges [1000, 1500] and [1, 5], respectively. In the second case, g is generated in the same way and with the same distribution with ranges [1000, 1500] and b of 6 Experimental results the first case is multiplied by 10, where b is randomly generated from the uniform distribution with ranges In this section, a set of problems taken from the central [10, 50]. In the third case, b is calculated by multi- pharmacy data with different sizes are used for this study. plying the ranges of the first case by 100, where b is The computational experiments are carried out to test the randomly generated from the uniform distribution with performance of the three techniques of resolution used to ranges [100, 500]. solve the problem under study: the MIP model solved by CPLEX, the proposed B&B algorithm and the developed B&S heuristic algorithm. Table 2 Main characteristics of the test cases The performance of B&S was measured by the Case nh b (Euro) g (Euro) s (Hours) average error gap compared to the fast exact method h h h (which is the developed B&B algorithm in this study) A 10 2, 3, 4 [1; 5] [1000; 1500] [3; 5] and was defined as ER(B&S/B&B)=(E -E )/E B&S B&B B&B 20 2, 3, 4 where E denotes the best evaluation found by the B&S 30 2, 3, 4 heuristic algorithm and E the best evaluation of the B&B 40 2, 3, 4 branch-and-bound algorithm. The performance of the B 10 2, 3, 4 [10; 50] [1000; 1500] [3; 5] proposed B&B procedure was measured by its Central 20 2, 3, 4 processing unit (Cpu) time needed to find the optimal 30 2, 3, 4 solutions and was compared with the CPLEX solver that 40 2, 3, 4 solves the MIP model directly. Both the B&B procedure C 10 2, 3, 4 [100; 500] [1000; 1500] [3; 5] and the B&S were programmed in JAVA language and 20 2, 3, 4 implemented through a desktop Intel core 2 processor 30 2, 3, 4 operatingat2.67GHz clockspeed and4 GB RAM. The 40 2, 3, 4 MIP model was solved by CPLEX on the same machine. 123 18 Page 10 of 13 Logist. Res. (2016) 9:18 6.2 Comparison of the performance of the B&B number of batches is increased and the number of products algorithm and the MIP model by batch is decreased gradually. In the third class of the problem C in Table 5,we Both methods of resolution, B&B algorithm and MIP observe that the B&B algorithm runs much faster than the model solved by CPLEX, find optimal solutions. Their CPLEX solver when the number of products is more than performances are measured by their Cpu time, then the fast 10 products. Interestingly, the efficiency of CPLEX method will be compared to the developed heuristic algo- decreases drastically, where the MIP model solves only the rithm B&S . instances of ten products with 2, 3 and 4 customers. In this The CPLEX solver, which is used to solve the MIP case, the computational time of the two methods becomes model, finds the optimal solution. However, its computa- very large so that the variation of b equals ½100; 500, tional time grows exponentially as the instance size where in the optimal solution the batches are very lightly increases, regardless of the parameters of the studied loaded, but the proposed B&B algorithm is still more problem. In contrast, the proposed B&B algorithm is efficient than the results of the MIP model. influenced by the value of the parameters used and the These results show the efficiency of the proposed B&B increase of the complexity of the problem. With small to method to give the optimal solution from small to medium- medium-sized instances, the computational time of the sized instances. In the next section, the performance of the proposed B&B algorithm will never exceed one hour. The B&B algorithm will be compared to the proposed heuristic results show that the B&B algorithm runs much faster than algorithm B&S. the CPLEX solver. For the problem of the class A in Table 3, we notice that 6.3 Comparison of the quality of solutions B&B which is supported by the lower bound runs faster than the CPLEX solver. The CPLEX solver finds the In this section, the performance of the proposed B&S optimal solution but its computational time grows rapidly heuristic algorithm is analyzed thoroughly by comparing as the size of the instance and the number of customers these results with the performance of the proposed exact increase. Conversely, the computational time of the B&B methods. algorithm is very short, which explains the efficiency of the The three considered cases are found in Table 6. For lower bound used in the B&B method to give the optimal each case, three scenarios are considered beginning with solution from small to medium-sized instances. In this three hospitals in use, then four and five. Moreover, for case, the two methods solve the problem rapidly. In these each case, the number of products sets as 10, 20, 30 and 40, experiments, the optimal solution corresponds to fully respectively. In each case, the customer storage cost b is loaded batches. In this case, the total holding cost is less generated from a discrete uniform distribution in the than the total transporter cost. Consequently, this configu- interval ½1; 5; ½10; 50; and ½100; 500 Euro for the three ration is the least complex to solve, because the batches cases, respectively. have to be fully loaded in order to minimize the delivery In the computational study, the following parameters are cost. used: the vehicle’s capacity is randomly generated from the In the second class of the problem B in Table 4, the uniform distribution with range [n/5, 2n/5]; further, its problem becomes harder to solve with CPLEX onset from round-trip delivery time for each customer is randomly 30 products regardless of the number of transporters. The generated from the uniform distribution with range ½3; 5 h. B&B algorithm runs faster than the CPLEX solver, but the The due dates ðd Þ are uniformly separated with val- j¼1...n gap between the two methods becomes significantly ues randomly generated. prominent as the number of products and transporters Considering the different parameters, 36 situations of increases. In this case, the time of resolution of the CPLEX the problem are tested. For each situation, 25 problem solver starts to increase rapidly according to the variation instances are generated to study the performance of the of b in ½10; 50. In the optimal solution we noticed that the B&S. Based on the results of the exact methods, the error Table 3 Computational results Class of problem# Size(n)! 10 20 30 of instances with b ¼½1; 5 CPLEX B&B CPLEX B&B CPLEX B&B (h)# CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) 2 0.64 0.006 53.67 0.046 144.842 0.221 Case A 3 0.505 0.005 45.31 0.385 197.274 1.749 4 1.660 0.019 43.25 0.891 288.585 308.997 123 Logist. Res. (2016) 9:18 Page 11 of 13 18 Table 4 Computational results Class of problem# Size(n)! 10 20 30 of instances with b ¼½10; 50 (h)# CPLEX B&B CPLEX B&B CPLEX B&B CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) 2 3.439 0.006 462.669 0.149 [3600 1.471 Case B 3 2.106 0.006 515.522 1.133 [3600 63.433 4 4.753 0.023 474.654 6.161 [3600 [3600 Table 5 Computational results Class of problem# Size(n)! 10 20 30 of instances with b ¼½100; 500 (h)# CPLEX B&B CPLEX B&B CPLEX B&B CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) 2 15.208 0.008 [3600 0.301 [3600 6.602 Case C 3 18.749 0.008 [3600 2.882 [3600 241.455 4 29.442 0.020 [3600 32.497 [3600 [3600 6.4 Comparison of the computational time Table 6 The error ratio results for g 2½1000; 1500 Euro and b 2 h h of solving methods ½1; 5; ½10; 50; ½100; 500 Euro for three cases, respectively Class of problem# The error ratio Tables (7, 8, 9) show the solution time obtained for each ER = (E -E )/E B&S B&B B&B method. In the computational study, the following param- eters are used: the vehicle’s capacity is randomly generated (h)! 345 (n)# from the uniform distribution with range [n/5, 2n / 5]; further, its round-trip delivery time for each hospital is Case A 10 9.11 % 2.17 % 0.21 % randomly generated from the uniform distribution with 20 7.13 % 7.67 % 0.35 % range ½3; 5 h. The due dates ðd Þ are uniformly sep- j¼1...n 30 9.52 % 2.32 % 0.83 % arated with values randomly generated. 40 8.80 % 5.27 % 7.99 % Moreover, for each case, the number of jobs set as Case B 10 0.00 % 1.15 % 0.04 % 10; 20; 30 and 40, and the number of hospitals as 2; 3 and 20 8.62 % 3.68 % 0.14 % 4 for each case. 30 8.19 % 4.03 % 0.35 % The parameters are generated with a magnitude order 40 12.43 % 12.28 % 4.60 % which is consistent with those of the central pharmacy. For Case C 10 5.90 % 2.18 % 0.98 % each combination, 25 problem instances are randomly 20 5.10 % 2.89 % 1.59 % generated and the average Cpu time for each resolution 30 9.00 % 12.19 % 2.95 % method is collected. 40 12.12 % 10.19 % 7.58 % The results show that the heuristic algorithm runs much Overall average = 5.82 % faster than the B&B algorithm. In this class, the resolution of the B&B algorithm is acceptable, which explains the efficiency of the lower bound used in the B&B algorithm to ratio is defined as ER(B&S/B&B) = (E - E )/E , B&S B&B B&B give the optimal solution for small to medium-sized where E denotes the mean evaluation of the solution B&S instances. The computational time of the proposed B&S generated by the proposed B&S, and E denotes the B&B will never exceed 0.3 s; moreover, the B&S can give mean evaluation of the solution generated by exact meth- optimal or near-optimal solutions for all of the situations. ods. The results are displayed in Table 6. In Table 7, the results show that it was possible to solve Table 6 shows clearly that the overall average equals 5:82 % which demonstrates that the proposed B&S is all the instances with the three proposed methods. In this case, the total storage cost at the customer’s capable of generating near-optimal solutions within a rea- sonable amount of Cpu time. One of the reasons may be the b ðd  C Þ, which constitutes the second part of the j j cl j¼1 j improvement phase which is presented in the heuristic objective function (1), will be less than those of the total algorithm (Sect. 5). In each case, we observe that the transporter cost g u , which constitutes the first part h h h¼1 average error ratio appears in an increasing trend as the of the objective function (1). This configuration is the least value of n increases. complex to solve, because the vehicle is fully loaded 123 18 Page 12 of 13 Logist. Res. (2016) 9:18 Table 7 Computational results for g 2½1000; 1500 Euro and b 2 Table 9 Computational results for g 2½1000; 1500 Euro and b 2 h h h h ½1; 5 Euro ½100; 500 Euro Class of problem! Class A Class of problem! Class C (h)! 23 4 (h)! 23 4 (n)# CpuT (s) CpuT (s) CpuT (s) (n)# CpuT (s) CpuT (s) CpuT (s) B&B 10 0.006 0.005 0.019 B&B 10 0.008 0.008 0.020 B&S 0.015 0.013 0.012 B&S 0.014 0.013 0.013 B&B 20 0.046 0.385 0.891 B&B 20 0.301 2.882 32.497 B&S 0.026 0.031 0.041 B&S 0.026 0.031 0.036 B&B 30 0.221 1.749 20.934 B&B 30 6.602 241.455 [3600 B&S 0.044 0.064 0.103 B&S 0.046 0.069 0.110 B&B 40 1.785 40.361 308.997 B&B 40 78.240 [3600 [3600 B&S 0.069 0.121 0.193 B&S 0.074 0.135 0.213 7 Conclusion Table 8 Computational results for g 2½1000; 1500 Euro and b 2 h h ½10; 50 Euro In this paper a real-life delivery and inventory problem Class of problem! Class B from the pharmaceutical industry is addressed. A central pharmacy delivers products to multiple heterogeneous (h)! 23 4 (n)# CpuT (s) CpuT (s) CpuT (s) hospitals sites with a single transporter. The transporter serves every hospital separately. It is supposed that each B&B 10 0.006 0.006 0.023 job that arrives in the hospital before its due date will incur B&S 0.015 0.013 0.012 an earliness penalty cost. The objective is to minimize the B&B 20 0.149 1.133 6.161 total cost defined by the weighted sum of the delivery cost B&S 0.025 0.031 0.042 and the earliness cost. B&B 30 1.471 63.433 [3600 Firstly, we focused on the development of a complete B&S 0.045 0.061 0.101 deterministic model formulated as a mixed-integer pro- B&B 40 24.229 [3600 [3600 gramming model. Then, in a subsequent step, a branch and B&S 0.068 0.110 0.186 bound based on a lower bound is developed. Secondly, we described an effective heuristic algorithm based on the determination of the batch sizing and the batch scheduling of the problem. The efficiency of the proposed heuristic according to the cheapness of the storage cost at the hospital. algorithm guarantees the determination of a feasible schedule for any given set of requests of the central In Table 8, the problem becomes harder to solve for the pharmacy. proposed B&B algorithm. In this case, the time of resolu- The proposed heuristic algorithm is compared with the tion of the B&B algorithm exceeds the proposed time limit proposed exact methods. The results illustrate the inter- onset from four hospitals if the number of products equals esting potential of the proposed approach. The branch and 30, and onset from three hospitals when the number of bound proved to be very efficient. Indeed, it proved to be products equals 40. Here, the number of batches is far more efficient than the existing MIP model for solving increased and the number of products by batch is the problem. The efficiency of the branch-and-bound decreased. In the third class of problem, in Table 9, the B&B algorithm is attributable to the tightness of the lower bounds derived. Moreover, efficiency of branch and bound algorithm solves instances until 30 products with four hospitals. Its processing time grows progressively when the increases for problem instances with a medium number of products. A very effective heuristic algorithm procedure is number of hospitals and products increase. In this case, the developed. The results show clearly that the proposed vehicle is very lightly loaded. heuristic algorithm is capable of generating near-optimal These results show that the proposed B&B algorithm is solutions within a reasonable Cpu time. efficient for small to medium-sized instances and finds There are several directions for future research. Firstly, optimal solutions, and the B&S proposed algorithm gives the model could be advanced by allowing the vehicle an optimal or a near-optimal solutions for small to large- routing with integrated delivery and storage cost. Secondly, sized instances. 123 Logist. Res. (2016) 9:18 Page 13 of 13 18 15. Hoque M (2008) Synchronization in the single-manufacturer the setup time and cost could be integrated into the pro- multi-buyer integrated inventory supply chain. Eur J Oper Res duction stage, and the volume of products into the delivery 188(3):811–825 stage. Finally, we aim to extend the considered model to 16. Hoque M (2011a) Generalized single-vendor multi-buyer inte- the multi-transporters case, where each transporter could be grated inventory supply chain models with a better synchro- nization. Int J Prod Econ 131(2):463–472 assigned to one customer. 17. Hoque M (2011b) An optimal solution technique to the single- vendor multi-buyer integrated inventory supply chain by incor- Open Access This article is distributed under the terms of the porating some realistic factors. Eur J Oper Res 215(1):80–88 Creative Commons Attribution 4.0 International License (http://crea 18. Jha J, Shanker K (2012) Single-vendor multi-buyer integrated tivecommons.org/licenses/by/4.0/), which permits unrestricted use, production-inventory model with controllable lead time and ser- distribution, and reproduction in any medium, provided you give vice level constraints. Appl Math Model 37(4):1753–1767 appropriate credit to the original author(s) and the source, provide a 19. Jordan C, Drexl A (1998) Discrete lotsizing and scheduling by link to the Creative Commons license, and indicate if changes were batch sequencing. Manag Sci 44(5):698–713 made. 20. Kallrath J (2002) Planning and scheduling in the process industry. OR Spectrum 24(3):219–250 21. Kimms A (1996) Competitive methods for multi-level lot sizing References and scheduling: Tabu search and randomized regrets. Int J Prod Res 34(8):2279–2298 22. Kimms A (1999) A genetic algorithm for multi-level, multi-ma- 1. Aptel O, Pourjalali H (2001) Improving activities and decreasing chine lot sizing and scheduling. Comput Oper Res 26(8):829–848 costs of logistics in hospitals: a comparison of US and French 23. Lu L (1995) A one-vendor multi-buyer integrated inventory hospitals. Int J Account 36(1):65–90 model. Eur J Oper Res 81(2):312–323 2. Archetti C, Bertazzi L, Laporte G, Speranza MG (2007) A 24. Pinedo ML (2012) Scheduling: theory, algorithms, and systems. branch-and-cut algorithm for a vendor-managed inventory-rout- Springer, New York ing problem. Transp Sci 41(3):382–391 25. Parija GR, Sarker BR (1999) Operations planning in a supply 3. Archetti C, Bertazzi L, Paletta G, Speranza MG (2011) Analysis chain system with fixed-interval deliveries of finished goods to of the maximum level policy in a production–distribution system. Comput Oper Res 38(12):1731–1746 multiple customers. IIE Trans 31(11):1075–1082 4. Archetti C, Bertazzi L, Hertz A, Speranza MG (2012) A hybrid 26. Potts CN (1980) Technical analysis of a heuristic for one machine heuristic for an inventory routing problem. INFORMS J Comput sequencing with release dates and delivery times. Oper Res 28(6):1436–1441 24(1):101–116 27. Sarker BR, Parija GR (1996) Optimal batch size and raw material 5. Battini D, Gunasekaran A, Faccio M, Persona A, Sgarbossa F ordering policy for a production system with a fixed-interval, (2010) Consignment stock inventory model in an integrated lumpy demand delivery system. Eur J Oper Res 89(3):593–608 supply chain. Int J Prod Res 48(2):477–500 28. Shah N (2004) Pharmaceutical supply chains: key issues and 6. Bertazzi L, Speranza MG, Ukovich W (1997) Minimization of strategies for optimisation. Comput Chem Eng 28(6):929–941 logistic costs with given frequencies. Transp Res B Methodol 29. Sindhuchao S, Romeijn HE, Akali E, Boondiskulchok R (2005) 31(4):327–340 An integrated inventory-routing system for multi-item joint 7. Borade AB, Sweeney E (2015) Decision support system for replenishment with limited vehicle capacity. J Glob Optim vendor managed inventory supply chain: a case study. Int J Prod 32(1):93–118 Res 53(16):4789–4818 30. Stefansson H, Sigmarsdottir S, Jensson P, Shah N (2011) Discrete 8. Chen Z-L (2010) Integrated production and outbound distribution and continuous time representations and mathematical models for scheduling: review and extensions. Oper Res 58(1):130–148 large production scheduling problems: a case study from the 9. Ertogral K, Darwish M, Ben-Daya M (2007) Production and pharmaceutical industry. Eur J Oper Res 215(2):383–392 shipment lot sizing in a vendor–buyer supply chain with trans- 31. Viswanathan S, Mathur K (1997) Integrating routing and inven- portation cost. Eur J Oper Res 176(3):1592–1606 tory decisions in one-warehouse multiretailer multiproduct dis- 10. Fu B, Huo Y, Zhao H (2010) Coordinated scheduling of pro- tribution systems. Manag Sci 43(3):294–312 duction and delivery with production window and delivery 32. Viswanathan S, Piplani R (2001) Coordinating supply chain capacity constraints. In: Chen B (ed) Algorithmic aspects in inventories through common replenishment epochs. Eur J Oper information and management. Springer, New York, pp 141–149 Res 129(2):277–286 11. Grunder O (2010) Lot sizing, delivery and scheduling of identical 33. Wang D, Grunder O, El Moudni A (2013) Single-item produc- jobs in a single-stage supply chain. Int J Innov Comput Inf tion-delivery scheduling problem with stage-dependent inventory Control 6(8):3657–3668 costs and due date considerations. Int J Prod Res 51(3):828–846 12. Hall LA, Shmoys DB (1992) Jackson’s rule for single-machine scheduling: making a good heuristic better. Math Oper Res 34. Zavanella L, Zanoni S (2009) A one-vendor multi-buyer inte- 17(1):22–35 grated production-inventory model: the consignment stock case. 13. Hariga M, Hassini E, Ben-Daya M (2013) A note on generalized Int J Prod Econ 118(1):225–232 single-vendor multi-buyer integrated inventory supply chain 35. Zhang Y, Cao Z (2008) An asymptotic PTAS for batch models with better synchronization. Int J Prod Econ 154:313–316 scheduling with nonidentical job sizes to minimize makespan. J Comb Optim 16(2):119–126 14. Herer YT, Levy R (1997) The metered inventory routing prob- lem, an integrative heuristic algorithm. Int J Prod Econ 51(1):69–81 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Logistics Research Springer Journals

A coordinated scheduling of delivery and inventory in a multi-location hospital supplied with a central pharmacy

Loading next page...
 
/lp/springer-journals/a-coordinated-scheduling-of-delivery-and-inventory-in-a-multi-location-mmeG2OFJqP
Publisher
Springer Journals
Copyright
Copyright © 2016 by The Author(s)
Subject
Engineering; Engineering Economics, Organization, Logistics, Marketing; Logistics; Industrial and Production Engineering; Simulation and Modeling; Operation Research/Decision Theory
ISSN
1865-035X
eISSN
1865-0368
DOI
10.1007/s12159-016-0145-8
Publisher site
See Article on Publisher Site

Abstract

Logist. Res. (2016) 9:18 DOI 10.1007/s12159-016-0145-8 OR IGINAL PAPER A coordinated scheduling of delivery and inventory in a multi- location hospital supplied with a central pharmacy 1 1 2 1 • • • Zakaria Hammoudan Olivier Grunder Toufik Boudouh Abdellah El Moudni Received: 25 March 2015 / Accepted: 8 August 2016 / Published online: 25 August 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In today’s manufacturing outlook, coordinated last, the experimental results show the efficiency of the scheduling of delivery and inventory represents a leading proposed solving methods, based on the two following leverage to enhance the competitiveness of firms which criteria: solution quality and processing time. aims to address the new challenge coming from scheduling problems. Though in the last decades this kind of issue has Keywords Case study  Coordinated scheduling been extensively approached in the literature, a set of Production and transportation  Mixed-integer constraints and compulsory dispositions strongly increases programming  Branch and bound  Heuristic algorithm the complexity of the considered problem. Actors of the pharmaceutical supply chain have to meet various global regulatory requirements while handling, storing and dis- 1 Introduction and related literature tributing environmentally sensitive products. The studied problem in this paper focuses on a real-case scheduling Today, the expansion of suppliers to accommodate the problem in a multi-location hospital supplied with a central maximum number of customers is considered as a key pharmacy. The objective of this work is to find a coordi- factor in the evolution of companies, in order to increase nated production and delivery schedule such that the sum their profits. Industrial companies are continuously of delivery and inventory costs is minimized. A mixed- assessing their operations with the objective of increasing integer programming formulation is first detailed to con- the overall effectiveness of manufacturing systems. Mar- sider the problem under study. Then, a branch-and-bound kets, where these organizations operate, tend to become algorithm is proposed as an exact method and a dedicated more complex over time, forcing companies to increase heuristic algorithm is highlighted to solve the problem. At their responsiveness, both in terms of time and cost. The case of the pharmaceutical industry is a good example of how market is driving the change on product development & Zakaria Hammoudan cycles and manufacturing activities. Delivery and inven- zakaria.hamoudan@utbm.fr tory scheduling stages are systematically considered to be Olivier Grunder very difficult functions. They are intended to produce olivier.grunder@utbm.fr operational plans dealing with several potential conflicting Toufik Boudouh objectives, namely minimizing costs, completion times, toufik.boudouh@utbm.fr and delays or maximizing profit. One important benefit of Abdellah El Moudni this coordination is a more efficient management of abdellah.elmoudni@utbm.fr inventories across the entire supply chain. In traditional inventory management, the optimal production and ship- IRTES-SET, Universite de Technologie de Belfort- ment policies for vendors and customers in a two-echelon Montbeliard (UTBM), rue Thierry Mieg, 90010 Belfort Cedex, France supply chain are managed independently. Additionally, these functions are closely related to other areas such as ´ ´ M3M, Universite de Technologie de Belfort-Montbeliard sales, procurement, production execution and control; (UTBM), rue Thierry Mieg, 90010 Belfort Cedex, France 123 18 Page 2 of 13 Logist. Res. (2016) 9:18 Table 1 A numerical example Jobs 1 2 3.5 4 Hospital’s 1 2 1 2 Due date Thursday 8:00 am Thursday 8:00 am Thursday 8:00 am Thursday 8:00 am Departure time Wednesday 06:00 am Wednesday 06:00 pm Thursday 2:00 am Wednesday 06:00 pm Arrival time Wednesday 12:00 pm Wednesday 10:00 pm Thursday 8:00 am Wednesday 10:00 pm Inventory cost 600 200 0 200 hence, they may interface with decisions at the strategic consideration holding, transportation, fixed ordering and and operational levels. For this reason, the integrated stock out costs. Viswanathan and Mathur [31] have studied a vendor–customer model is developed where the total rel- distribution systems with a central warehouse and many evant costs for the customers as well as the vendors have to retailers that stock a number of different products, where the be minimized. Consequently, determining the production products are delivered from the warehouse to the retailers by and shipment policies based on an integrated total cost vehicles that combine the deliveries to several retailers into function, rather than several customer’s or vendor’s indi- efficient vehicle routes. They have proposed a heuristic that vidual cost functions, results in the reduction of the develops a stationary nested joint replenishment policy. inventory costs of the system. These results showed that the proposed heuristic is capable of The system under study in this paper is composed of a solving problems involving distribution systems with multi- central pharmacy from which sterilized medical devices ple products. Sindhuchao et al. [29] have considered a system have to be delivered before given due dates, to different that consists of a set of geographically dispersed suppliers hospitals located around the central pharmacy. This supply that manufacture one or more non-identical items, and a chain process incurs both delivery costs and earliness central warehouse that stocks these items. The warehouse penalty costs in case the devices are delivered too early. faces a constant and deterministic demand for the items from Therefore, the considered problem is an integrated outside retailers. The items are collected by a fleet of vehicles delivery and inventory problem with due dates constraints, that are dispatched from the central warehouse. The vehicles for which we have to minimize the total delivery and are capacitated and must also satisfy a frequency constraint. holding costs. Therefore, the problem can be formulated They studied the case where each vehicle always collects the from a batch scheduling point of view with a cost objective same set of items. They have formulated and solved the function or from a lot sizing problem point of view with a problem by using a branch-and-price algorithm, and then time horizon. These two classes of problem have been they have proposed a greedy constructive heuristic and a very proven to be equivalent under given conditions [19]. In our large-scale neighborhood search algorithm. These results case, a batch scheduling approach seems to be more indicate that the constructive heuristic used in conjunction appropriate in the context of the study of the healthcare with one of the proposed very large-scale neighborhood system with specific constraints for the due dates. algorithms can find near-optimal solutions very efficient. The delivery-inventory problem is denoted as Vendor- Recently, Archetti et al. [2] have studied a distribution Managed Inventory (VMI) problem. The VMI problem is a problem in which a product has to be shipped from a supplier widely used collaborative inventory management policy in to several retailers over a given time horizon. Each retailer which manufacturers manage the inventory of retailer and defines a maximum inventory level. The supplier monitors take responsibility for making decisions related to the timing the inventory of each retailer and determines its replenish- and extent of inventory replenishment [7]. VMI partnerships ment policy, guaranteeing that no stock out occurs at the help organizations to reduce demand variability, inventory retailer (supplier-managed inventory policy). Every time a holding and distribution costs. A pioneering paper is due to retailer is visited, the quantity delivered by the supplier is Bertazzi et al. [6], where a given set of shipping frequencies is such that the maximum inventory level is reached (deter- allowed and different products may be shipped at different ministic order-up-to level policy). Shipments from the sup- frequencies. Herer and Levy [14] have considered a system of plier to the retailers are performed by a vehicle of given a central warehouse, a fleet of trucks with a finite capacity, capacity. They presented a mixed-integer linear program- and a set of customers, for each of whom there is an estimated ming model, and they derived new additional valid inequal- consumption rate, and a known storage capacity. The ities used to strengthen the linear relaxation of the model. objective is to determine when to service each customer, as They implemented a branch-and-cut algorithm to solve the well as the way to be performed by each truck, in order to model optimally. Then, they have studied two different types minimize the total discounted costs. To solve the problem, of replenishment policies in [3]. The first one is the well- they have proposed a rolling horizon approach that takes into known order-up-to level (OU) policy, where the quantity 123 Logist. Res. (2016) 9:18 Page 3 of 13 18 shipped to each retailer is such that the level of its inventory et al. [10] studied the problem of coordinated scheduling of reaches the maximum level. The second one is the maximum production and delivery subject to the production window level (ML) policy, where the quantity shipped to each retailer constraints and delivery capacity constraints. They consid- is such that the inventory is not greater than the maximum ered both a single delivery time case and multiple delivery level. In this study, Archetti et al. [3] have focused on the ML time case. Chen [8] reviewed the production and distribution policy and the design of a hybrid heuristic, and they imple- scheduling models and classified these problems in five mented an exact algorithm for the solution of the problem groups. Problems addressing an objective function that with one vehicle and designed a hybrid heuristic for the combines machine scheduling with the delivery costs are multi-vehicle case. Most recently, Archetti et al. [4], have rather complex. However, they are more practical than those studied the previous problem with a single vehicle which has involving just one of the two factors, since these combined- a given capacity. The transportation cost is proportional to the optimization problems are often encountered when real-world distance traveled, whereas the inventory holding cost is supply chain management is considered. proportional to the level of the inventory at the customers and The number of customers and products has been a topic at the supplier. They have proposed a heuristic that combines of intense investigation for decades in the integrated supply a tabu search scheme with ad hoc designed mixed-integer chain. Although researchers have given a considerable programming models. The effectiveness of the heuristic was attention on the synchronization of the single-vendor single- proved over a set of benchmark instances for which the customer integrated inventory system, the single-vendor optimal solution was known. multi-customer integrated inventory case has gotten little There are numerous researches on batch scheduling of attention in regard. Lu [23] developed a one-vendor multi- delivery-inventory problem. Scheduling problems arise in customer integrated inventory model, while Parija and Sar- almost any type of industrial production facilities (Pulp and ker [25] extended their published work on single-vendor, Paper, Metals, Oil and gas, Chemicals, Food and Beverages, single-customer, integrated production-inventory problems Pharmaceuticals, Transportation, Service, Military, etc.) with lumpy delivery systems under perfect and imperfect where given operations need to be processed on specified production cycle situations [27]. Lu [23] argued that all the resources. The corresponding scheduling problems are previous studies assumed that the vendor must know the already very difficult to solve [20]. Much research has customer’s holding and ordering costs, which are quite dif- focused on the same area under various assumptions and ficult to estimate unless the customer is willing to reveal the objective measures that differ from the considered problem in true values. Therefore, he considered another circumstance, this paper. Potts [12], Hall[ 26] and Zhang et al. [35]have in which the objective is to minimize the vendor’s total cost studied scheduling problems with non-identical job release per year, subject to the maximum cost that the customer may be prepared to incur. Parija and Sarker [25 times and delivery times, under the assumption that a suffi- ] introduced the cient number of vehicles is available to deliver the jobs. problem of determining the production start time and pro- Kimms [21] has examined the problem of single-machine and posed a method that determines the cycle length and raw proposed two heuristic approaches: randomized regrets based material ordering frequency for a long-range planning and tabu search approaches. Each production plan is gener- horizon. The cycle length is restricted to be an integer- ated without using any information obtained from previous multiple of all shipment intervals to the customers as an plans. This work has been extended by Kimms [22]witha ideal situation, the solution to which may be sub-optimal. proposition of a genetic algorithm that dominates the tabu Viswanathan and Piplani [32] proposed a model to study and search procedure, both in terms of run-time performance and analyze the benefit of coordinating supply chain inventories the ability to find feasible solutions. Pinedo and Michael [24] by means of common replenishment epochs or time periods. reviewed different models and solution approaches, and then A one-vendor multi-customer supply chain is considered for they explained the complexity of scheduling problems. a single product. Under their strategies, the vendor specifies Multi-echelon inventory models have attracted much common replenishment periods and requires all customers to attention, and the integrated approach has been extensively replenish only at pre-determined time periods. However, the studied. In this way, Grunder [11] considered a single-product authors did not include any inventory cost of the vendor in batch scheduling problem with the objective of minimizing the model. In most papers dealing with integrated inventory the sum of production, transportation and inventory cost. models, the transportation cost is considered only as a part of Particularly, he assumed that the delivery time depends on the fixed setup or replenishment cost. Ertogral et al. [9]studied batch sizes and proposes a dynamic programming approach how the results of incorporating transportation cost into the based on a dominance relation property. Wang et al. [33] model influence the decision-making process under equal extended this study with an integrated scheduling problem for size shipment policies. A fundamental advance in the two- single-item supply chain involving due date considerations side cost structure is in recognizing how delivery-trans- and an objective of minimizing the total logistics cost. Fu portation costs apply to both sides. 123 18 Page 4 of 13 Logist. Res. (2016) 9:18 Hoque [15] proposed three models for supplying a single- of Hoque’s models. Hariga et al. [13] compared the cost item from a single-vendor to multiple customers under between the results of the models in Hoque [16] and deterministic demand by synchronizing the production flow Zavanella and Zanoni [34], and then they concluded that with equal-sized batch transfer in the first two and unequal- both models are not appropriate as they are using different sized batches transfer in the third. In the first two models, all functional forms of the total setup and ordering costs. batches forwarded are of exactly the same size but the Moreover, it is shown that Hoque’s model yields imprac- timing of their shipment is different. In the first of these, the tical solutions for zero transportation costs. When the total manufacturer transfers a batch to a customer as soon as its setup and ordering cost was adjusted to be similar to the processing is finished, whereas in the second a batch is one in Zavanella and Zanoni’s model, Hoque’s model transferred to a customer as soon as the previously sent resulted in a larger total cost. batch to the customer is finished. In the third model, the Existing inventory models for multi-customers are not subsequent shipment lot sizes increase by the ratio of pro- applicable to pharmaceutical products for several reasons. duction rate and sum of demand rates on all the customers. Pharmaceutical products can be more expensive than other Zavanella and Zanoni [34] proposed a model for a single- products to purchase and distribute, and shortages and vendor multi-customer system, integrated in a shared man- improper use of essential medicines can have a high cost in agement of the customers’ inventory, so as to pursue a terms of wasted resources and preventable diseases and reduction or the stability of the holding costs while death. Therefore, special care should be taken in pharma- descending the chain. Hoque [16] transferred the lot from a ceutical inventory decisions to ensure 100 % product vendor to multiple customers with l number of unequal- availability at the right time, at the right cost, and in good sized batches first; where the next one is a multiple of the condition to the right customers. The quality of health care previous one by the ratio ðk [ 1Þ of the production and the industries strongly depends on the availability of pharma- total demand rates, followed by ðn  lÞ number of equal- ceuticals on time. If a shortage occurs at a hospital, an sized batches. The equal-sized batches are restricted to be emergency delivery is necessary, which is very costly and less than or equal to the lth batch (the largest unequal-sized can affect the patient health. Inventory management batch) multiplied by k. The models developed were solved strategies that are unsuitable for health care industries may by applying Lagrangian Multiplier method. However, in lead to large financial losses and a significant impact on cases of single-vendor single-customer or single-vendor patients. Hence, inventory strategies for pharmaceutical multi-customer or multi-stage production, synchronization products are more critical than those for other products. of the production flow by transferring the lot with equal and/ Thus, a specific inventory model is necessary to control or unequal-sized batches was found to lead to the least total pharmaceutical products, to save patient lives and reduce cost for some numerical problems. Although Hoque [16] unnecessary inventory costs. served that purpose, he did not cope with the relaxation of Here we investigate a delivery-inventory supply chain the discussed impractical assumptions. Following this trend composed of a central pharmacy which has to deliver of synchronization, Hoque [17] developed two generalized pharmaceutical supplies to distant hospitals with a single single-vendor multi-customer integrated inventory models transporter at given due dates. The objective is to reduce by accumulating the inventory at the vendor’s and cus- the overall cost which includes the delivery costs and an tomer’s independently, but with the traditional trend of earliness penalty cost. ignoring the cost of benefit sharing. Transportation of each The contributions of this paper are twofolds. First, we of the batches incurs a transportation cost. In order to propose a MIP model to minimize the total delivery and implement the models by taking into account the industry inventory costs for the considered supply chain under the reality, he also incorporates them with the relaxation of the constraints of healthcare systems. Second, we propose an discussed impractical assumptions. Battini [5] developed a efficient solving algorithm which is compared with two single-vendor and multi-customer consignment stock exact methods. inventory model in which many clients can establish a The outline of the remainder of the paper is organized in consignment stock inventory policy with the same vendor. seven sections. In Sect. 2, the problem definition and for- Recently, Jha and Shanker [18] studied an integrated mulation is introduced. In Sect. 3, the problem is formu- production-inventory model in a single-vendor multi-cus- lated as a mixed-integer programming (MIP) model. Then, tomer supply chain with lead time reduction under inde- we describe the proposed branch-and-bound algorithm pendent normally distributed demand on the customers. (B&B) as an exact method of resolution in Sect. 4.We They assume a non-identical lead time for the customers develop a heuristic algorithm in Sect. 5 for solving the and that customers’ inventory is reviewed using continuous problem. In Sects. 6 and 7, we eventually provide the review policy. Hariga et al. [13] analyzed Hoque’s models experimental results and draw some conclusions and sug- I and II studied in Hoque [16], and then they modified some gest the future research directions. 123 Logist. Res. (2016) 9:18 Page 5 of 13 18 2.1 Notations The following notations are used in developing the math- ematical model: Parameters • J ¼ 1; 2;...; n: set of all jobs, where n is the total number of jobs, • H ¼ 1; 2;...; m: set of all hospitals, • j: index for jobs, j 2 J, • k: index for batches, • h: index for hospitals, h 2 H, • d : due date of job j, • cl : destination of job j, cl 2 H, j j • c : capacity of the transporter, • s : time for the vehicle to deliver a batch to hospital h and to return to the central pharmacy location, Fig. 1 Central pharmacy and multi-location hospital model • g : delivery cost to deliver a batch to hospital h and to return to the central pharmacy location, • b : hospital earliness penalty function for hospital h. 2 Problem definition and formulation Primary variables We consider a supply chain scheduling problem where 1 • d ¼ 1 if the job j belongs to the kth batch, 0 otherwise, jk there is one central pharmacy which has to deliver medical • d ¼ 1 if the batch k belongs to the customer h,0 kh supplies, or jobs, to m hospital sites, which are the final otherwise. customers (Fig. 1). Each hospital h orders a finite number Secondary variables of jobs from the central pharmacy. The following assumptions are considered for this study. • y ¼ 1 if the batch k exists and is not empty, 0 First, we will consider a single transporter to deliver the otherwise, sterilized medical devices as the number of distant hospi- • C : the arrival time of the job j at the hospital, tals is reduced in practice (less than 4) and the distances • B : the arrival time of the batch k at the hospital, with the central pharmacy are quite short. Second, we will • u : number of delivered batches for hospital h. only consider direct shipping (i.e., commuter tours), with- out considering routing considerations between customers 2.2 Numerical example [8]. This assumption is explained by the fact that the pharmacy is located in the center of the distant hospitals. To clarify the problem, we consider a simple numerical Moreover, the road network is centralized on the main town of the central pharmacy; hence, travel times are example in Table (1) as follows. Two hospitals ordered five longer between distant hospitals. jobs at the same time (Monday at 8:00 am) and they would Each round trip between the pharmacy and a hospital h receive their products at the same time (Thursday at 8:00 requires a delivery cost g as well as a delivery time s . am), that means all the products have the same due date equal to 72 h. The central pharmacy and its hospital cus- The batches delivered from the central pharmacy to the hospitals can be of different sizes. tomers open 24 h/day. Three jobs (j ¼ 1; 3and5) for hos- pital 1 and two jobs (j ¼ 2 and 4) for hospital 2. The The total number of jobs belonging to the same batch cannot exceed the capacity c of the transporter. Each job j vehicle capacity is c ¼ 2. The transporter delivery cost and time depend on the hospitals’ positions with (g ¼ 1000 has a due date d specified by the hospitals and each job has to arrive to the hospital site before its due date. If job j of Euro, s ¼ 6 h, and g ¼ 750 Euro, s ¼ 4 h) belongs to 1 2 hospital 1 and 2, respectively, (b ¼ b ¼ b ¼ 30 Euro/h hospital h arrives before its due date d , it will incur as an 1 3 5 and b ¼ b ¼ 20 Euro/h) belongs to hospital 1 and 2, earliness penalty b . Batching and sending several jobs in 2 4 respectively. the batches will reduce the transportation costs. The solution is shown in Table (1) for this problem. As The objective is to determine the sequence of batches it is shown, the vehicle makes three round trips among that has to be processed, so that the expected total cost of them two to hospital 1 and one to hospital 2. Three batches both central pharmacy and hospitals sites is minimized. 123 18 Page 6 of 13 Logist. Res. (2016) 9:18 k ¼ 1; k ¼ 2 and k ¼ 2 are denoted. The products arrive The objective function (1) minimizes the sum of the 1 2 3 at the customers in the batch to which they belong to in the delivery costs, through the g u term, and the customers completion time cited in Table (1). The total delivery cost earliness penalty, through b ðd  C Þ. Constraint (2) cl j j equals g  2 þ g  1 ¼ 2750 Euro and the total storage guarantees that each job must be scheduled exactly in one 1 2 cost at the hospitals equals b ½ðd  C Þþðd  C Þþ 1 1 3 3 1 batch. In this constraint, the jobs will be batched only in the ðd  C Þ þ b ½ðd  C Þþðd  C Þ ¼ 30 ½20 þ 5 5 2 2 2 4 4 batch which it belongs to. Constraints (3 and 4) force each 0 þ 0þ 20 ½10 þ 10¼ 1000 Euro. The amount of the batch to be delivered to the customer it belongs to. Con- objective function is 3750 Euro. straint (5) calculates the number of batches delivered to each customer. Constraint (6) guarantees that no empty batch is allowed. Constraint (7) prevents the number of 3 The mixed-integer programming model jobs scheduled in one delivery batch to exceed the capacity of the vehicle. Constraint (8) indicates that arrival time of The pharmaceutical supply chain has many aspects that each job is at least equal to the contracted due date for each need to be considered in a supply chain model. However, customer. Constraint (9) orders the batches in the by taking all concerned factors into account, the model increasing order of their arrival times. Constraint (10) would be of so high complexity that it would be extremely expresses the minimum interval duration between the hard for analysis. In this section, the mathematical pro- arrivals of two consecutive batches has to be greater than gramming model of the above-mentioned problem is pre- the delivery time of the transporter. Constraint (11) rep- sented. Using the structural properties, we develop a MIP resents the relation between the completion time of the jobs model for the mentioned problem as follows: and the arrival time of the batch they belong to. This m n X X constraint is represented in a nonlinear way in this math- Min Z ¼ g u þ b ðd  C Þ h j j ð1Þ h cl ematical representation to facilitate the understanding of h¼1 j¼1 the problem. Constraints (12) and (13) define the range of Subject to : the variables. For ease of reference, we denote this problem: Multiple Customers Batching Delivery Scheduling Problem d ¼ 1; j ¼ 1;...; n ð2Þ jk k¼1 (MCBDSP). The complexity of the MCBDSP is still an open ques- d  1; k ¼ 1;...; n ð3Þ kh tion. To the best of our knowledge, no polynomial algo- h¼1 rithm can solve this problem. However, from simulation 2 1 experiments, we observe that the problem is still d  d ; j; k ¼ 1;...; n and k  j ð4Þ k;cl jk intractable on an empirical basis. In the next section, a B&B with a lower bound is described to solve the problem u ¼ d ; h ¼ 1;...; m ð5Þ kh as an exact method. k¼1 y  y ; k ¼ 1;...; n  1 ð6Þ k kþ1 X 4 Branch-and-bound algorithm d  c; k ¼ 1;...; n ð7Þ jk j¼k In this section, we describe the B&B algorithm that we have developed to solve the MCBDSP. The objective of C  d ; j ¼ 1;...; n ð8Þ j j this B&B is to solve small to medium-sized instances, and B  B ; k ¼ 1;...; n  1 ð9Þ kþ1 k to be a reference for validating the efficiency of the pro- posed heuristic algorithm. This B&B algorithm maintains a 2 2 B  B  s ðd þ d Þ; kþ1 k h kþ1;h kh list of subproblems (nodes) whose union of feasible solu- ð10Þ h¼1 tions contains all feasible solutions of the original problem. k ¼ 1;...; n  1 and h 2 H The list is initialized with the original problem itself. In n each major iteration, the algorithm selects a current sub- C ¼ B  d ; j ¼ 1;...; n ð11Þ problem from the list of unevaluated nodes. This branching j k jk k¼1 seems to be natural; however, the number of branches will be very large for large problems. Consequently, if this C  0; j ¼ 1; 2;...; n ð12Þ method is used in the B&B algorithm, it may take too much 1 2 d ; d 2f0; 1g; j; k ¼ 1;...; n and h ¼ 1;...; m ð13Þ jk kh time to find optimal solutions, as redundant schedules 123 Logist. Res. (2016) 9:18 Page 7 of 13 18 would be checked repeatedly. Yet, several of the sub- solve large-sized instances grows exponentially in the problems would already have been eliminated upon the experimental results. Therefore, developing fast heuristic generation of nodes, since the search tree includes redun- algorithm to yield near-optimal solutions in a reasonable dant solutions. running time is still of great importance. In the next sec- At each node of the search tree, the number of products tion, a solving method is proposed to solve the problem. that still need to be delivered to each customer has to be updated. Iterations are performed until the list of sub- problems to be processed is empty. The crucial part of a 5 Heuristic algorithm successful B&B algorithm is the computation of the lower bounds. Therefore, we have developed a lower bound In this section, a heuristic algorithm, which is denoted described in the next part. Batching and Scheduling algorithm (B&S), is proposed. Efficient lower bound would significantly reduce the This algorithm is composed of two steps, the first one time and efforts needed for the B&B method. Based on the consists in defining the size of the batches and the second main feature of the problem, the lower bound value for the one will schedule them according to the different con- problem is the summation of lower bounds on the total straints of the problem. earliness cost and the transportation cost. We assume that The B&S algorithm starts by generating an initial w is a partial batch sequence solution, z(w) is the evaluation solution through the means of a progressive constructive of w, and r ðwÞ is the number of products remaining at the procedure. Then, the above-mentioned two-steps process is customer’s h for partial solution w. This notation will be applied until a predefined stop condition is satisfied. At used throughout this part. first, some elements of the current solution are constructed. In each node, the solutions are built from the last batch Then, a local improvement phase based on a swap operator to the first one and the evaluation of the partial or complete is applied to the reconstructed solution in order to improve solution is processed with backward equations. The its quality. Finally, B&S chooses the optimum solution research of a solution starts by constructing a partial between the current solution and the solution obtained from solution w. Then, the remainder of products is added in the improvement procedure. order to get a complete solution, with the objective of Let us denote that ðq ; cl ; B Þ is the notation which will k k k achieving a minimum delivery cost. Therefore, more the be used for a solution of a batch k, where the first term q transporter will be loaded, more this lower bound will be describes the number of jobs in this batch, the second term efficient. cl describes the customers destination of batch k and the third term B is the arrival time of this batch. For example, Proposition 1 For a partial solution w, a lower bound for a solution of three batches, which contains 2, 3 and 2 jobs, the delivery cost of the remaining products is given by: respectively, belongs to customers 2, 3 and 1, respectively, m lm and arrives at due dates 1000, 1015 and 1020, respectively, ð14Þ h will be written as follows: h¼1 ½ð2 ; cl ; 1000Þ; ð3 ; cl ; 1015Þ; ð2 ; cl ; 1020Þ 1 2 2 3 3 1 Proof For each customer h,if r ðwÞ is the number of Based on the prune rule, the following heuristic algorithm products remaining to be delivered, the number of round is proposed as follows: for level 0, there is no job. For the trips will be equal to , and the delivery cost of the first level, which includes only last job n, there is only one remaining products is as denoted in Eq. (14). h possible joint solution which is ð1; cl ; B Þ. For level k n n We add the partial solution w to the solution found in (includes q jobs), all ‘‘good’’ solutions for a number of k equation (14) to get the lower bound of the current node jobs will be kept. The process to build ‘‘good’’ solutions for under study. level k is described as follows: (1) build solutions of level k by considering all the solutions in the retained ‘‘good’’ Corollary 1 The lower bound LB(w) of the partial solu- solutions of all the previous levels from 1 to ðk  1Þ. For tion w is given as follows: each retained solution of level k  k, a new solution of lm h 0 level k is built by simply adding a batch of (k  k Þ jobs, if LBðwÞ¼ zðwÞþ  g ð15Þ this is possible. Then, this procedure is repeated until the h¼1 level n is reached. Proof Straightforward. h The details of the algorithm (1) are presented as follows: The mathematical model and the B&B algorithm The generation of the initial solution and the construction developed in the previous sections could solve small to procedure is represented from line 1 to 4. Then, the batch medium-sized instances; however, the time of resolution to sizing procedure is represented from line 5 to 17 according 123 18 Page 8 of 13 Logist. Res. (2016) 9:18 to a scattering/gathering procedure. The improvement not, a new swap operation is generated. The improvement procedure is called in line 14, and then it is described in operation stops when the index of batches equals 0. Algorithm 2. The batch sizing procedure, performed in an iterative way, extends a partial solution by adding one job from a set J of all jobs. The construction of the good solution advances progressively and in a hierarchical manner. The process starts from the last job and arrives recursively to the first one. The jobs are distributed to the customers to whom they belong, and the batches sizes are defined according to a scattering/gathering procedure described in Algorithm 1. Let’s take an example to explain the application of the B&S in Algorithm 1, to illustrate the MCBDSP. We con- sider a problem of three jobs and two customers. The due dates associated with these jobs equal 1000; 1100; 1150, where jobs 1 and 3 have due dates 1000 and 1150 and belong to customer 1 and job 2 has due date 1100 and belongs to customer 2. The transport delivery cost and time depend upon the customer’s location with (g ¼ 20; s ¼ 60 u:t and g ¼ 15; s ¼ 40 u:t) belonging 1 2 1 2 to customer 1 and 2, respectively. The customers’ holding costs are defined as follows: (b ¼ b ¼ 30 and b ¼ cl cl cl 1 3 2 15 ), belonging to customer 1 and 2, respectively. The B&S process is described in detail as follows: the process starts by the last job recursively to arrive to the first one. 1. For j ¼ 1; currentJob = 3, there is only one possible joint solution which is ð1; cl ; 1150Þ. 2. For j ¼ 2; currentJob j ¼ 3 or 2, there are different In this algorithm, the number of delivered jobs j possible solutions. Firstly, a complementary solution is varies from 0 to n (line 1). For each level of j delivered built by simply adding a batch of (2  1) job to the jobs, the different partial solutions are built from the previous delivery solution. The potential delivery solution of previous levels (\j). Moreover, the neces- scheme is equal to 2 batches: sary number of batches to these solutions is added, to ½ð1; cl ; 1150Þ; ð1; cl ; 1100Þ complete the partial solution of level j. After every 1 2 product addition to level j, the partial solution of this Due to the improvement phase, two solutions could be level is completed by adding the necessary delivery obtained according to the swap operation which are: scheme to the considered solution in the list of all kept ½ð1; cl ; 1100Þ; ð1; cl ; 1150Þ and solutions from 0 to (j  1), to obtain the new list of 2 1 solutions of level j. A test of verification of the capacity ½ð1; cl ; 1150Þ; ð1; cl ; 1100Þ 1 2 of the transporter used is done directly after each Then the two potential joint solutions are compared advancement in level (See line 6 in Algorithm 1). and the good solution is kept, which is: The final step of each level j is denoted in line 15, which is mentioned in Algorithm 1. In this phase, the good ½ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 2 1 solution is memorized and inscribed to level j. After that, a 3. For j ¼ 3. currentJob j ¼ 3; 2 or 1, based on the new level ðj þ 1; j þ 2...Þ is started till reaching level n. delivery scheme of the first step, the new delivery In the improvement phase, all consecutive batches are schemes are: swapped, by starting from the last batch recursively to the first one, while the index of the batch is positive (See line 5 ½ð1; cl ; 1150Þ; ð1; cl ; 1100Þ; ð1; cl ; 1000Þ and 1 2 1 in Algorithm 2). After every swap operation, the new ½ð1; cl ; 1150Þ; ð1; cl ; 1000Þ; ð1; cl ; 1100Þ: 1 1 2 solution is kept if it is better than the current solution. If 123 Logist. Res. (2016) 9:18 Page 9 of 13 18 Due to the improvement phase, two solutions could be The maximum solving time allowed for these instances obtained according to the swap operation which are: is 1 h. The reference of time limit of resolution is based on the real time of the preparation of a schedule in the ½ð1; cl ; 1100Þ; ð1; cl ; 1000Þ; ð1; cl ; 1150Þ and 2 1 1 actual case. As a comparison, CPLEX solver is used to ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 exactly solve the model with small scale random instances. Some adjustments are done on the parameters Then the four potential joint solutions are compared of research in CPLEX order to accelerate the research of and the good solution is kept, which is: solutions. ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 According to the confidentiality of the data base of the central pharmacy under study, several cases of problem After that, based on the delivery scheme of the second were considered for which several instances were gener- step, the delivery scheme on this step is: ated randomly. ½ð1; cl ; 1100Þ; ð1; cl ; 1150Þ; ð1; cl ; 1000Þ 2 1 1 6.1 Test cases In the improvement phase, a new solution could be obtained which is: The characteristics of orders to schedule differ by cus- ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 tomers, transporter capacity, quantity delivered, due date, transporter time, transporter cost and the storage cost at The two potential joint solutions are compared and the each customer. Three cases are considered to test the good solution is kept, which is: proposed methods. The characteristics of the case are listed ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 in Table 2. For each case {A, B and C}, the number of products n, the number of customers h, the transporter time Finally, the two potential joint solutions kept in each s , the transporter cost (g ), and the storage cost at each level are compared and the best final solution is customer (b ) are displayed. recovered, which is: In the first case, g is higher than b ,where g and b h h h h ½ð1; cl ; 1000Þ; ð1; cl ; 1100Þ; ð1; cl ; 1150Þ 1 2 1 are randomly generated from the uniform distribution with ranges [1000, 1500] and [1, 5], respectively. In the second case, g is generated in the same way and with the same distribution with ranges [1000, 1500] and b of 6 Experimental results the first case is multiplied by 10, where b is randomly generated from the uniform distribution with ranges In this section, a set of problems taken from the central [10, 50]. In the third case, b is calculated by multi- pharmacy data with different sizes are used for this study. plying the ranges of the first case by 100, where b is The computational experiments are carried out to test the randomly generated from the uniform distribution with performance of the three techniques of resolution used to ranges [100, 500]. solve the problem under study: the MIP model solved by CPLEX, the proposed B&B algorithm and the developed B&S heuristic algorithm. Table 2 Main characteristics of the test cases The performance of B&S was measured by the Case nh b (Euro) g (Euro) s (Hours) average error gap compared to the fast exact method h h h (which is the developed B&B algorithm in this study) A 10 2, 3, 4 [1; 5] [1000; 1500] [3; 5] and was defined as ER(B&S/B&B)=(E -E )/E B&S B&B B&B 20 2, 3, 4 where E denotes the best evaluation found by the B&S 30 2, 3, 4 heuristic algorithm and E the best evaluation of the B&B 40 2, 3, 4 branch-and-bound algorithm. The performance of the B 10 2, 3, 4 [10; 50] [1000; 1500] [3; 5] proposed B&B procedure was measured by its Central 20 2, 3, 4 processing unit (Cpu) time needed to find the optimal 30 2, 3, 4 solutions and was compared with the CPLEX solver that 40 2, 3, 4 solves the MIP model directly. Both the B&B procedure C 10 2, 3, 4 [100; 500] [1000; 1500] [3; 5] and the B&S were programmed in JAVA language and 20 2, 3, 4 implemented through a desktop Intel core 2 processor 30 2, 3, 4 operatingat2.67GHz clockspeed and4 GB RAM. The 40 2, 3, 4 MIP model was solved by CPLEX on the same machine. 123 18 Page 10 of 13 Logist. Res. (2016) 9:18 6.2 Comparison of the performance of the B&B number of batches is increased and the number of products algorithm and the MIP model by batch is decreased gradually. In the third class of the problem C in Table 5,we Both methods of resolution, B&B algorithm and MIP observe that the B&B algorithm runs much faster than the model solved by CPLEX, find optimal solutions. Their CPLEX solver when the number of products is more than performances are measured by their Cpu time, then the fast 10 products. Interestingly, the efficiency of CPLEX method will be compared to the developed heuristic algo- decreases drastically, where the MIP model solves only the rithm B&S . instances of ten products with 2, 3 and 4 customers. In this The CPLEX solver, which is used to solve the MIP case, the computational time of the two methods becomes model, finds the optimal solution. However, its computa- very large so that the variation of b equals ½100; 500, tional time grows exponentially as the instance size where in the optimal solution the batches are very lightly increases, regardless of the parameters of the studied loaded, but the proposed B&B algorithm is still more problem. In contrast, the proposed B&B algorithm is efficient than the results of the MIP model. influenced by the value of the parameters used and the These results show the efficiency of the proposed B&B increase of the complexity of the problem. With small to method to give the optimal solution from small to medium- medium-sized instances, the computational time of the sized instances. In the next section, the performance of the proposed B&B algorithm will never exceed one hour. The B&B algorithm will be compared to the proposed heuristic results show that the B&B algorithm runs much faster than algorithm B&S. the CPLEX solver. For the problem of the class A in Table 3, we notice that 6.3 Comparison of the quality of solutions B&B which is supported by the lower bound runs faster than the CPLEX solver. The CPLEX solver finds the In this section, the performance of the proposed B&S optimal solution but its computational time grows rapidly heuristic algorithm is analyzed thoroughly by comparing as the size of the instance and the number of customers these results with the performance of the proposed exact increase. Conversely, the computational time of the B&B methods. algorithm is very short, which explains the efficiency of the The three considered cases are found in Table 6. For lower bound used in the B&B method to give the optimal each case, three scenarios are considered beginning with solution from small to medium-sized instances. In this three hospitals in use, then four and five. Moreover, for case, the two methods solve the problem rapidly. In these each case, the number of products sets as 10, 20, 30 and 40, experiments, the optimal solution corresponds to fully respectively. In each case, the customer storage cost b is loaded batches. In this case, the total holding cost is less generated from a discrete uniform distribution in the than the total transporter cost. Consequently, this configu- interval ½1; 5; ½10; 50; and ½100; 500 Euro for the three ration is the least complex to solve, because the batches cases, respectively. have to be fully loaded in order to minimize the delivery In the computational study, the following parameters are cost. used: the vehicle’s capacity is randomly generated from the In the second class of the problem B in Table 4, the uniform distribution with range [n/5, 2n/5]; further, its problem becomes harder to solve with CPLEX onset from round-trip delivery time for each customer is randomly 30 products regardless of the number of transporters. The generated from the uniform distribution with range ½3; 5 h. B&B algorithm runs faster than the CPLEX solver, but the The due dates ðd Þ are uniformly separated with val- j¼1...n gap between the two methods becomes significantly ues randomly generated. prominent as the number of products and transporters Considering the different parameters, 36 situations of increases. In this case, the time of resolution of the CPLEX the problem are tested. For each situation, 25 problem solver starts to increase rapidly according to the variation instances are generated to study the performance of the of b in ½10; 50. In the optimal solution we noticed that the B&S. Based on the results of the exact methods, the error Table 3 Computational results Class of problem# Size(n)! 10 20 30 of instances with b ¼½1; 5 CPLEX B&B CPLEX B&B CPLEX B&B (h)# CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) 2 0.64 0.006 53.67 0.046 144.842 0.221 Case A 3 0.505 0.005 45.31 0.385 197.274 1.749 4 1.660 0.019 43.25 0.891 288.585 308.997 123 Logist. Res. (2016) 9:18 Page 11 of 13 18 Table 4 Computational results Class of problem# Size(n)! 10 20 30 of instances with b ¼½10; 50 (h)# CPLEX B&B CPLEX B&B CPLEX B&B CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) 2 3.439 0.006 462.669 0.149 [3600 1.471 Case B 3 2.106 0.006 515.522 1.133 [3600 63.433 4 4.753 0.023 474.654 6.161 [3600 [3600 Table 5 Computational results Class of problem# Size(n)! 10 20 30 of instances with b ¼½100; 500 (h)# CPLEX B&B CPLEX B&B CPLEX B&B CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) CpuT (s) 2 15.208 0.008 [3600 0.301 [3600 6.602 Case C 3 18.749 0.008 [3600 2.882 [3600 241.455 4 29.442 0.020 [3600 32.497 [3600 [3600 6.4 Comparison of the computational time Table 6 The error ratio results for g 2½1000; 1500 Euro and b 2 h h of solving methods ½1; 5; ½10; 50; ½100; 500 Euro for three cases, respectively Class of problem# The error ratio Tables (7, 8, 9) show the solution time obtained for each ER = (E -E )/E B&S B&B B&B method. In the computational study, the following param- eters are used: the vehicle’s capacity is randomly generated (h)! 345 (n)# from the uniform distribution with range [n/5, 2n / 5]; further, its round-trip delivery time for each hospital is Case A 10 9.11 % 2.17 % 0.21 % randomly generated from the uniform distribution with 20 7.13 % 7.67 % 0.35 % range ½3; 5 h. The due dates ðd Þ are uniformly sep- j¼1...n 30 9.52 % 2.32 % 0.83 % arated with values randomly generated. 40 8.80 % 5.27 % 7.99 % Moreover, for each case, the number of jobs set as Case B 10 0.00 % 1.15 % 0.04 % 10; 20; 30 and 40, and the number of hospitals as 2; 3 and 20 8.62 % 3.68 % 0.14 % 4 for each case. 30 8.19 % 4.03 % 0.35 % The parameters are generated with a magnitude order 40 12.43 % 12.28 % 4.60 % which is consistent with those of the central pharmacy. For Case C 10 5.90 % 2.18 % 0.98 % each combination, 25 problem instances are randomly 20 5.10 % 2.89 % 1.59 % generated and the average Cpu time for each resolution 30 9.00 % 12.19 % 2.95 % method is collected. 40 12.12 % 10.19 % 7.58 % The results show that the heuristic algorithm runs much Overall average = 5.82 % faster than the B&B algorithm. In this class, the resolution of the B&B algorithm is acceptable, which explains the efficiency of the lower bound used in the B&B algorithm to ratio is defined as ER(B&S/B&B) = (E - E )/E , B&S B&B B&B give the optimal solution for small to medium-sized where E denotes the mean evaluation of the solution B&S instances. The computational time of the proposed B&S generated by the proposed B&S, and E denotes the B&B will never exceed 0.3 s; moreover, the B&S can give mean evaluation of the solution generated by exact meth- optimal or near-optimal solutions for all of the situations. ods. The results are displayed in Table 6. In Table 7, the results show that it was possible to solve Table 6 shows clearly that the overall average equals 5:82 % which demonstrates that the proposed B&S is all the instances with the three proposed methods. In this case, the total storage cost at the customer’s capable of generating near-optimal solutions within a rea- sonable amount of Cpu time. One of the reasons may be the b ðd  C Þ, which constitutes the second part of the j j cl j¼1 j improvement phase which is presented in the heuristic objective function (1), will be less than those of the total algorithm (Sect. 5). In each case, we observe that the transporter cost g u , which constitutes the first part h h h¼1 average error ratio appears in an increasing trend as the of the objective function (1). This configuration is the least value of n increases. complex to solve, because the vehicle is fully loaded 123 18 Page 12 of 13 Logist. Res. (2016) 9:18 Table 7 Computational results for g 2½1000; 1500 Euro and b 2 Table 9 Computational results for g 2½1000; 1500 Euro and b 2 h h h h ½1; 5 Euro ½100; 500 Euro Class of problem! Class A Class of problem! Class C (h)! 23 4 (h)! 23 4 (n)# CpuT (s) CpuT (s) CpuT (s) (n)# CpuT (s) CpuT (s) CpuT (s) B&B 10 0.006 0.005 0.019 B&B 10 0.008 0.008 0.020 B&S 0.015 0.013 0.012 B&S 0.014 0.013 0.013 B&B 20 0.046 0.385 0.891 B&B 20 0.301 2.882 32.497 B&S 0.026 0.031 0.041 B&S 0.026 0.031 0.036 B&B 30 0.221 1.749 20.934 B&B 30 6.602 241.455 [3600 B&S 0.044 0.064 0.103 B&S 0.046 0.069 0.110 B&B 40 1.785 40.361 308.997 B&B 40 78.240 [3600 [3600 B&S 0.069 0.121 0.193 B&S 0.074 0.135 0.213 7 Conclusion Table 8 Computational results for g 2½1000; 1500 Euro and b 2 h h ½10; 50 Euro In this paper a real-life delivery and inventory problem Class of problem! Class B from the pharmaceutical industry is addressed. A central pharmacy delivers products to multiple heterogeneous (h)! 23 4 (n)# CpuT (s) CpuT (s) CpuT (s) hospitals sites with a single transporter. The transporter serves every hospital separately. It is supposed that each B&B 10 0.006 0.006 0.023 job that arrives in the hospital before its due date will incur B&S 0.015 0.013 0.012 an earliness penalty cost. The objective is to minimize the B&B 20 0.149 1.133 6.161 total cost defined by the weighted sum of the delivery cost B&S 0.025 0.031 0.042 and the earliness cost. B&B 30 1.471 63.433 [3600 Firstly, we focused on the development of a complete B&S 0.045 0.061 0.101 deterministic model formulated as a mixed-integer pro- B&B 40 24.229 [3600 [3600 gramming model. Then, in a subsequent step, a branch and B&S 0.068 0.110 0.186 bound based on a lower bound is developed. Secondly, we described an effective heuristic algorithm based on the determination of the batch sizing and the batch scheduling of the problem. The efficiency of the proposed heuristic according to the cheapness of the storage cost at the hospital. algorithm guarantees the determination of a feasible schedule for any given set of requests of the central In Table 8, the problem becomes harder to solve for the pharmacy. proposed B&B algorithm. In this case, the time of resolu- The proposed heuristic algorithm is compared with the tion of the B&B algorithm exceeds the proposed time limit proposed exact methods. The results illustrate the inter- onset from four hospitals if the number of products equals esting potential of the proposed approach. The branch and 30, and onset from three hospitals when the number of bound proved to be very efficient. Indeed, it proved to be products equals 40. Here, the number of batches is far more efficient than the existing MIP model for solving increased and the number of products by batch is the problem. The efficiency of the branch-and-bound decreased. In the third class of problem, in Table 9, the B&B algorithm is attributable to the tightness of the lower bounds derived. Moreover, efficiency of branch and bound algorithm solves instances until 30 products with four hospitals. Its processing time grows progressively when the increases for problem instances with a medium number of products. A very effective heuristic algorithm procedure is number of hospitals and products increase. In this case, the developed. The results show clearly that the proposed vehicle is very lightly loaded. heuristic algorithm is capable of generating near-optimal These results show that the proposed B&B algorithm is solutions within a reasonable Cpu time. efficient for small to medium-sized instances and finds There are several directions for future research. Firstly, optimal solutions, and the B&S proposed algorithm gives the model could be advanced by allowing the vehicle an optimal or a near-optimal solutions for small to large- routing with integrated delivery and storage cost. Secondly, sized instances. 123 Logist. Res. (2016) 9:18 Page 13 of 13 18 15. Hoque M (2008) Synchronization in the single-manufacturer the setup time and cost could be integrated into the pro- multi-buyer integrated inventory supply chain. Eur J Oper Res duction stage, and the volume of products into the delivery 188(3):811–825 stage. Finally, we aim to extend the considered model to 16. Hoque M (2011a) Generalized single-vendor multi-buyer inte- the multi-transporters case, where each transporter could be grated inventory supply chain models with a better synchro- nization. Int J Prod Econ 131(2):463–472 assigned to one customer. 17. Hoque M (2011b) An optimal solution technique to the single- vendor multi-buyer integrated inventory supply chain by incor- Open Access This article is distributed under the terms of the porating some realistic factors. Eur J Oper Res 215(1):80–88 Creative Commons Attribution 4.0 International License (http://crea 18. Jha J, Shanker K (2012) Single-vendor multi-buyer integrated tivecommons.org/licenses/by/4.0/), which permits unrestricted use, production-inventory model with controllable lead time and ser- distribution, and reproduction in any medium, provided you give vice level constraints. Appl Math Model 37(4):1753–1767 appropriate credit to the original author(s) and the source, provide a 19. Jordan C, Drexl A (1998) Discrete lotsizing and scheduling by link to the Creative Commons license, and indicate if changes were batch sequencing. Manag Sci 44(5):698–713 made. 20. Kallrath J (2002) Planning and scheduling in the process industry. OR Spectrum 24(3):219–250 21. Kimms A (1996) Competitive methods for multi-level lot sizing References and scheduling: Tabu search and randomized regrets. Int J Prod Res 34(8):2279–2298 22. Kimms A (1999) A genetic algorithm for multi-level, multi-ma- 1. Aptel O, Pourjalali H (2001) Improving activities and decreasing chine lot sizing and scheduling. Comput Oper Res 26(8):829–848 costs of logistics in hospitals: a comparison of US and French 23. Lu L (1995) A one-vendor multi-buyer integrated inventory hospitals. Int J Account 36(1):65–90 model. Eur J Oper Res 81(2):312–323 2. Archetti C, Bertazzi L, Laporte G, Speranza MG (2007) A 24. Pinedo ML (2012) Scheduling: theory, algorithms, and systems. branch-and-cut algorithm for a vendor-managed inventory-rout- Springer, New York ing problem. Transp Sci 41(3):382–391 25. Parija GR, Sarker BR (1999) Operations planning in a supply 3. Archetti C, Bertazzi L, Paletta G, Speranza MG (2011) Analysis chain system with fixed-interval deliveries of finished goods to of the maximum level policy in a production–distribution system. Comput Oper Res 38(12):1731–1746 multiple customers. IIE Trans 31(11):1075–1082 4. Archetti C, Bertazzi L, Hertz A, Speranza MG (2012) A hybrid 26. Potts CN (1980) Technical analysis of a heuristic for one machine heuristic for an inventory routing problem. INFORMS J Comput sequencing with release dates and delivery times. Oper Res 28(6):1436–1441 24(1):101–116 27. Sarker BR, Parija GR (1996) Optimal batch size and raw material 5. Battini D, Gunasekaran A, Faccio M, Persona A, Sgarbossa F ordering policy for a production system with a fixed-interval, (2010) Consignment stock inventory model in an integrated lumpy demand delivery system. Eur J Oper Res 89(3):593–608 supply chain. Int J Prod Res 48(2):477–500 28. Shah N (2004) Pharmaceutical supply chains: key issues and 6. Bertazzi L, Speranza MG, Ukovich W (1997) Minimization of strategies for optimisation. Comput Chem Eng 28(6):929–941 logistic costs with given frequencies. Transp Res B Methodol 29. Sindhuchao S, Romeijn HE, Akali E, Boondiskulchok R (2005) 31(4):327–340 An integrated inventory-routing system for multi-item joint 7. Borade AB, Sweeney E (2015) Decision support system for replenishment with limited vehicle capacity. J Glob Optim vendor managed inventory supply chain: a case study. Int J Prod 32(1):93–118 Res 53(16):4789–4818 30. Stefansson H, Sigmarsdottir S, Jensson P, Shah N (2011) Discrete 8. Chen Z-L (2010) Integrated production and outbound distribution and continuous time representations and mathematical models for scheduling: review and extensions. Oper Res 58(1):130–148 large production scheduling problems: a case study from the 9. Ertogral K, Darwish M, Ben-Daya M (2007) Production and pharmaceutical industry. Eur J Oper Res 215(2):383–392 shipment lot sizing in a vendor–buyer supply chain with trans- 31. Viswanathan S, Mathur K (1997) Integrating routing and inven- portation cost. Eur J Oper Res 176(3):1592–1606 tory decisions in one-warehouse multiretailer multiproduct dis- 10. Fu B, Huo Y, Zhao H (2010) Coordinated scheduling of pro- tribution systems. Manag Sci 43(3):294–312 duction and delivery with production window and delivery 32. Viswanathan S, Piplani R (2001) Coordinating supply chain capacity constraints. In: Chen B (ed) Algorithmic aspects in inventories through common replenishment epochs. Eur J Oper information and management. Springer, New York, pp 141–149 Res 129(2):277–286 11. Grunder O (2010) Lot sizing, delivery and scheduling of identical 33. Wang D, Grunder O, El Moudni A (2013) Single-item produc- jobs in a single-stage supply chain. Int J Innov Comput Inf tion-delivery scheduling problem with stage-dependent inventory Control 6(8):3657–3668 costs and due date considerations. Int J Prod Res 51(3):828–846 12. Hall LA, Shmoys DB (1992) Jackson’s rule for single-machine scheduling: making a good heuristic better. Math Oper Res 34. Zavanella L, Zanoni S (2009) A one-vendor multi-buyer inte- 17(1):22–35 grated production-inventory model: the consignment stock case. 13. Hariga M, Hassini E, Ben-Daya M (2013) A note on generalized Int J Prod Econ 118(1):225–232 single-vendor multi-buyer integrated inventory supply chain 35. Zhang Y, Cao Z (2008) An asymptotic PTAS for batch models with better synchronization. Int J Prod Econ 154:313–316 scheduling with nonidentical job sizes to minimize makespan. J Comb Optim 16(2):119–126 14. Herer YT, Levy R (1997) The metered inventory routing prob- lem, an integrative heuristic algorithm. Int J Prod Econ 51(1):69–81

Journal

Logistics ResearchSpringer Journals

Published: Aug 25, 2016

References