Abstract
Fuzzy Inf. Eng. (2011) 4: 393-410 DOI 10.1007/s12543-011-0094-5 ORIGINAL ARTICLE An Interactive Fuzzy Goal Programming Approach for Multi-period Multi-product Production Planning Problem K. Taghizadeh· M. Bagherpour· I. Mahdavi Received: 30 January 2011/ Revised: 25 July 2011/ Accepted: 15 September 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract This study addresses an interactive multiple fuzzy goal programming (FGP) approach to the multi-period multi-product (MPMP) production planning prob- lem in an imprecise environment. The proposed model attempts to simultaneously minimize total production costs, rates of changes in labor levels, and maximizing machine utilization, while considering individual production routes of parts, inven- tory levels, labor levels, machine capacity, warehouse space, and the time value of money. Piecewise linear membership functions are utilized to represent decision maker’s (DM’s) overall satisfaction levels. A numerical example demonstrates the feasibility of applying the proposed model to the MPMP problem. Furthermore, the proposed interactive approach facilitates the DM with a systematic framework of de- cision making process which enables DM to modify the search direction to reach the most satisfactory results during solving process. Keywords Multi-period multi-product production planning · Fuzzy goal program- ming · Piecewise linear membership functions 1. Introduction An MPMP production planning problem is a well-known problem in production en- vironment, which has attracted considerable attention from both practitioners and academia [1]. The MPMP production planning typically encompasses a time horizon between 3 weeks to 3 months. From the planning point of view, MPMP produc- tion planning falls between decisions of medium-range planning and highly specif- ic, detailed short-range planning decisions. Planners in the process of MPMP plan- ning conventionally make decisions regarding production, inventory, and backorder K.Taghizadeh () · I. Mahdavi Department of Industrial Engineering, Mazandaran University of Science & Technology, Babol, Iran email: zehdarkt@gmail.com M. Bagherpour Department of Industrial Engineering, Iran University of Science & Technology, Tehran, Iran 394 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) volumes, considering issues such as ﬂuctuated market demand, available invento- ry space, production capacity of machines, and available labor levels. While the MPMP production planning problem attempts to match production rates of individ- ual products with ﬂuctuated demand of market, the aggregate production planning (APP) considers the planning problem from an aggregated viewpoint, which utilizes an aggregate production unit such as the “average” items or in term of weight, vol- ume, production time, and currency value to aggregate a family of similar products with small differences. Thus, both the MPMP and APP production planning problems share some identical concepts and notions. Since the pioneering work done by Holt, Modigliani, Muth, and Simon (HMM- S) in 1955, researchers have developed numerous crisp optimization models to help solving the MPMP and APP problems, each with their own pros and constraints, how- ever, in the real world MPMP production planning problems input data or parameters such as demand, resources, and also objective values cannot be expressed precisely due to incomplete or unobtainable information [3]. Fuzzy sets theory proposed by Zadeh [4] is a type of imprecision that has no well-deﬁned boundaries for its de- scription. Fuzzy sets theory is highly applicable in the area where human judgments, evaluations, and decisions are important [5]. Inclusion of fuzzy sets theory and linear programming (LP) systems can result in reduction of information costs and avoidance of unrealistic modeling [6]. Zimmermann [7] ﬁrst introduced fuzzy sets theory into conventional LP problem- s, considering fuzzy objective and fuzzy constraints. Then in 1978 Zimmermann [8] applied the fuzzy sets theory to the LP problems with several objective func- tions, in this study Zimmermann transformed the fuzzy multi objective linear pro- gramming (FMOLP) model into a standard single objective LP model utilizing fuzzy min-operator. Narasimhan [9] illustrated the application of “fuzzy subsets” concepts to goal programming in an imprecise environment. This study developed a solution approach for an FGP problem with multiple goals having equal weights. Then, the solution approach is extended to the case where unequal fuzzy weights are associat- ed with multiple goals. Hannan [10] utilized piecewise linear membership functions to quantify fuzzy and imprecise aspirations of decision makers in goal programming problems. He also utilized fuzzy minimum operator to aggregate aspiration levels of all goals. Subsequent researches on the FGP models include [11-19]. Saad [20] categorized the APP models into six categories namely: LP models, linear decision rule, transportation method, management coefﬁcient approach, search decision rule, and simulation. Bakir and Byrne [21] suggested a stochastic LP model to deal with uncertainty of market demand over planning horizon in the conventional MPMP production planning model. Byrne and Bakir [22] discussed, in real world, production systems consumption of resources has a complex behavior, and thus ana- lytical solutions of LP models may not yield feasibility. So, they developed a hybrid simulation-analytical approach to the MPMP problems. Then, Kim and Kim [23] showed that by introducing the actual workload of jobs in each simulation run and passing the information to the LP model, the optimal solution would be achieved in less iteration comparing with Byrne and Bakir [22] approach. Byrne and Hossain [24] extended models, proposed in [22], [23] by considering unit load notion and Fuzzy Inf. Eng. (2011) 4: 393-410 395 modiﬁcation of resource requirements and constraints in the LP formulation. Con- sidering just in time (JIT) concept as a main contribution of this research results in reduction of work in process (WIP) inventories and total ﬂow time. Chen and Lio [25] also discussed applying traditional single objective models to the MPMP pro- duction problems may lead to complexity and impracticality, thus they proposed a model consist of multiple objectives of production environment. Noori et al [26] reported, converting the MPMP production planning problem into a project network and assigning recourses to activities and consequently leveling re- source proﬁles may improve machine utilization. Bagherpour et al [27] implemented the earned value analysis as a project control mechanism, through the MPMP prob- lems. They utilized triangular fuzzy numbers to model uncertainties in activity du- rations; they also utilized forecasting features of earned value analysis for predicting completion time of production of individual products. Kazemi et al [28] developed a robust optimization model for the MPMP problems and highlighted the superiority of robust optimization in comparison with stochastic optimization. Many research have been dedicated to investigation of applying fuzzy set theory to the APP production planning problems. Rinks [29] utilized fuzzy logic and fuzzy linguistics through the APP production planning problems. In this regard he suggest- ed a series of 40 relational assignment rules. Gen et al [30] proposed a method to the multi objective APP problems with fuzzy parameters. Their proposed approach attempts to transform the FMOLP model to a crisp MOLP formulation and to solve it through an interactive solution procedure. Wang and Fang [31] introduced a novel optimization model for the MPMP problems where demand and capacities are im- precise. They considered minimization of total cost of quadratic production costs and linear inventory holding costs as the objective of their proposed model. Wang and Fang [32] proposed an FMOLP model for the APP multi objective problem, considering fuzzy product price, subcontracting price, work force level, production capacity, and market demand. Moreover, an interactive solution procedure was de- veloped to provide a compromise solution. Dai et al [33] presented a fuzzy linear programming approach to deal with uncertain/imprecise environmental parameters of the APP industrial applications. They suggested utilization of fuzzy linear program- ming provides a great advantage where parameters of the stochastic factors, involved in the production planning, are neither deﬁnitely reliable nor precise. Jamalnia and Soukhakian [34] developed a hybrid goal programming approach with different goal priorities for multi objective APP problems. In their proposed model products are produced in regular and overtime production shifts, they also considered the impact of learning curve and product life cycle on the optimum production plan costs. In this study a novel nonlinear fuzzy goal programming (NLFGP) model is devel- oped for the MPMP production planning problem. The proposed model attempts to simultaneously satisfy goals which are deﬁned to minimize production costs, rate of changes in work force levels between periods, and also to maximize machine utiliza- tion. Then piecewise linear membership functions are introduced to cater the uncer- tainty and vagueness of DM’s aspiration levels. Furthermore, a solving procedure is suggested to transform the original NLFGP model into a standard single objective LP model which can be efﬁciently solved by simplex method. Moreover, an interactive 396 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) decision making procedure is suggested which provides the DM with a more ﬂexible decision making framework. In this regard the DM can correct the search direction during solving procedure based on his/her idea, judgments, and experiments. Finally, a numerical example justiﬁes the feasibility of applying the proposed approach. The reminder of this paper is organized as follows. Section 2 describes the as- sumptions, notations, and formulation of the proposed NLFGP model. Section 3 shows defuzziﬁcation procedure and discusses solving procedure. Section 4 provides a numerical example to investigate the feasibility of applying the proposed procedure. Finally, conclusions and further recommendations are presented in Section 5. 2. Problem Formulation This section focuses on the formulation of the MPMP problem as an FNLGP model. The considered problem is stated based on following assumptions: • The production, inventory, backorder, subcontracting, hiring and dismissing costs are proportionally related to the production rates, inventory, backorder, and subcontracting levels, and rate of hiring and dismissing work force, re- spectively. • All parameters values of objectives and constraints are considered to be deter- ministically known over planning horizon. • Piecewise concave linear membership functions are introduced for aspiration levels of each goal and objectives are deﬁned using terms “The total production costs should be less than··· ”, “Rate of changes in labor levels in each period should be equal to··· ”, and “The percentage of machine utilization should be more than··· ”. • All decision variables related to production, inventory, backlogging, and sub- contracting are positive integer values and variables related to hiring and ﬁring man-hour are real positive values. • The forecasted demand for each product type is deterministically known over each particular period. A forecasted demand can be either satisﬁed or backo- rdered. • Different warehouse spaces are provided for ﬁnished goods and semi-produced- goods. 2.1. Notations • Subscripts N Number of products. P Number of processes of products. T Number of Periods. J Number of Machine Centers. G Number of goals. n Index for product type (n = 1, 2,··· ,N). i Index for processes (i = 1, 2,··· ,P). Fuzzy Inf. Eng. (2011) 4: 393-410 397 t, k Index for planning time period (t and k = 1, 2,··· ,T ). j Index for machine center ( j = 1, 2,··· ,J). g Index for goals (g = 1, 2,··· ,G). • Goals Z Total production costs. Z Rate of changes in labor levels. Z Total machine utilization. • Input parameters D Forecasted demand for product type n in period k (Units). nk C Operation cost of ith process of product type n in period t ($/U- nit nits). Ec Escalation factor for operation costs (%). Cs Inventory carrying cost per unit of product type n which has nit completed its ith process in period t ($/Unit). Es Escalation factor for inventory carrying costs (%). Cb Backorder cost of product type n in period t ($/Unit). nt Eb Escalation factor for backorder costs (%). Csu Subcontracting cost of ith process of product type n in period t nit ($/Unit). Esu Escalation factor for subcontracting costs (%). Ch Cost of hiring man-hour in period t ($/Man-hour). Eh Escalation factor for costs of hiring workers (%). Cf Cost of dismissing man-hour in period t ($/Man-hour). Ef Escalation factor for costs of dismissing workers (%). M Processing time of ith process of product type n on jth machine ni j (Machine-hour/Unit). Mh Man-hour usage for processing ith process of product type n ni (Man-hour/Unit). Ws Warehouse space per unit of product type n which has complet- ni ed its ith process ( ft /Unit). Mc Maximum available capacity of machine center j in period t jt (Machine-hour). Imax Maximum inventory space available for ﬁnished products in period t ( ft ). WIPmax Maximum inventory space available for work-in-process inven- tories in period t ( ft ). Lmax Maximum man-hour available in period t (Man-hour). Smax Maximum available subcontracting volume in period t (Unit). R An arbitrary large positive number. 398 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) • Decision variables Q The total production volume of the ith process of product type n nitt in period t (Unit). Q The quantity of product type n which has completed its ith pro- nitk cess in period t and will be stored to be used in period k; where k>t (Unit). Q The quantity of backordered demand of product n in period t that nptk will be fulﬁlled in period k; wherek<t (Unit). Su The quantity of subcontracted items of ith process of product nit type n in period t (Unit). H Man-hour hired in period t (Man-hour). F Man-hour dismissed in period t (Man-hour). Bc 1, if demand of product type n is planned to be backordered in nk period k; 0, otherwise. In 1, if ﬁnished products of type n are planned to be stored in period nk k; 0, otherwise. Bh 1, if new labors are hired in period t; 0, otherwise. 2.2. Objective Functions Generally, there are many functional departments in a manufacturing environment involved in production planning process which their expectations of production plan are supposed to be fulﬁlled. Some of these expectations include minimizing total cost/maximizing total proﬁt, minimizing variation in material and labor levels (quan- titative objectives), and maximizing customer satisfaction (qualitative objective) [3], [21], [35]. This study attempts to simultaneously minimize the net present value of production cost, and changes in labor levels between periods as well as maximizing machine utilization. • Minimize net present value of production costs N P T Min Z Q × C × (1+ Ec) + 1 nitt nit n=1 i=1 t=1 N P T T (k− t)× Q × Cs × (1+ Es) + nitk nit n=1 i=1 t=1 k=t+1 N T T (t− k)× Q × Cb × (1+ Eb) + (1) nptk nt n=1 k=1 t=k+1 N T T Su × CS u × (1+ Esu) + nit nit n=1 k=1 t=1 t t (H × Ch × (1+ Eh) + F × Cf × (1+ Ef ) ). t t t t=1 • Minimize rate of changes in labor levels Max Z { f (H − F )|∀t ∈ T}. (2) 2 2 t t Fuzzy Inf. Eng. (2011) 4: 393-410 399 • Maximize machine utilization N P T J (Q × M ) nitt ni j n=1 i=1 t=1 j=1 Max Z . (3) J T Mc jt j=1 t=1 The Symbol “” represents equivalency under fuzzy conditions and refers to the fuzziﬁcation of the aspiration levels. In real-world applications of the MPMP prob- lem, Equation (1)-(3) are often involved with variations in DM’s judgments. Thus, piecewise linear membership functions are introduced to embed DM’s ideas, judg- ments, and implicit knowledge within planning process. Equation (1) demonstrates the net present value of production plan, in this regard the time value of each cost category in each period is compared with ﬁrst period. The ﬁrst term in Equation (1) represents production related costs which may encom- pass machine costs, raw material, etc. Inventory carrying costs expressed by the sec- ond term involves holding periods of both semi-ﬁnished-products and ﬁnished goods. Backlogging costs as the third term takes into account the interval time between de- mand and fulﬁllment (shortage periods). Finally, the fourth and ﬁfth terms represent the cost associated with subcontracting and hiring/dismissing workforce respectively. Equation (2) is dedicated to minimize rates of changes in labor levels between pe- riods. In this regard, f represents the fuzzy membership function of rates of changes in labor levels at each period and Figure 1 illustrates its typical shape. The member- ship function which is depicted in Figure 1 assigns higher values to lower deviations from zero. Equation (2) attempts to maximize the membership value of the largest variation in labor levels over planning horizon. Equation (3) represents machine utilization which is calculated by dividing pro- duction time with total available machinery time. Fig. 1 A typical membership function for rate of changes in labor levels 2.3. Constraints 400 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) • Machine center capacity constraints N P Q × M ≤ Mc ,∀t,∀ j. (4) nitt ni j jt n=1 i=1 Equation (4) ensures total utilization of each machine in each period (left hand side) does not exceed the total available capacity of machine center (right hand side). • Material balance constraints k T Q + Su = Q − Q nikk nik n,i−1,t,k n,i−1,k,t t=1 t=k+1 +Su ,∀i ≥ 2,∀n,∀k. (5) n,i−1,k k T T Q + Q − Q + Su nPtk nPtk nPkt npk t=1 t=k+1 t=k+1 = D + Q ,∀n,∀k. (6) nk nPkt t=1 In a production system, materials in each period should be balanced from two re- spects: (i) inventory/shortage and subcontracting levels should be equal to those from previous periods, production quantity, and demand. (ii) The total number of parts at each process is restricted to the total number of available parts that have completed the previous processes beforehand. In this respect, Equation (5) is proposed here to make sure that the total number of produced and subcontracted parts in each process does not exceed the total production and inventory of the previous process (denoted by the ﬁrst term in the right hand side), and the number of parts stored for future periods (given by the second term in the right hand side), plus the subcontracted parts of the previous process (given by the third term in the right hand side). Similar explanation can be given for Equation (6), where the total produced and subcontracted parts and inventory of the completed parts should be equal to the fore- casted demand of each product and inventory level for future periods. T T Q × Q = 0,∀n,∀k. (7) nPtk nPkt t=k+1 t=k+1 Equation (7) is presented here to prevent occurrence of both carrying inventories and planning backorder for each ﬁnished product in any period. • Maximum inventory space constraints N k T Q × Ws ≤ Imax ,∀k, (8) nPtk nP k n=1 t=1 k =k+1 N P−1 k T Q × Ws ≤ WIPmax ,∀k. (9) nitk ni k n=1 i=1 t=1 k =k+1 Fuzzy Inf. Eng. (2011) 4: 393-410 401 Equation (8) and (9) are formulated to make sure the total inventory levels of ﬁn- ished and semi products do not exceed the maximum available inventory space for ﬁnished and WIP goods. • Workforce constraints N P Q × Mh ≤ Lmax ,∀t, (10) nitt ni t n=1 i=1 N P N P Q × Mh = Q × Mh + H − F ,∀t. (11) nitt ni n,i,t−1,t−1 ni t t n=1 i=1 n=1 i=1 Equation (10) limits production’s required man-hour to maximum available man- hour. Equation (11) attempts to balance labor levels between planning periods. • Subcontracting constraints N P Su ≤ Smax ,∀t. (12) nit t n=1 i=1 Equation (12) restrict maximum subcontracting items, the maximum subcontract- ing level is usually dictated by general managements based on subcontracting strate- gies. 2.4. Linearization This section demonstrates how nonlinear terms of model (Equation (2) and (4)) are transformed to linear form utilizing auxiliary variables and constraints. Max Z = μ (13) μ ≤ f (H − F ),∀t, 2 t t R× Bc ≥ Q ,∀n,∀k, (14) nk nPtk t=k+1 R× In ≥ Q ,∀n,∀k, (15) nk nPkt t=k+1 Bc + I ≤ 1,∀n,∀k. (16) nk nk By introducing the auxiliary variable μ, Equation (2) can be transformed to the Equation (13), indeed by substitution of Equation (4) with Equation (14)-(16) the proposed NLFGP model changes into a FGLP model. 3. Defuzziﬁcation 3.1. Objective Functions Both fuzzy decision making concept of Bellman and Zadeh [5] and fuzzy goal pro- gramming method presented by Hannan [10] are utilized to develop a procedure in 402 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) order to transform the proposed FMOLP model to a single objective LP model which can deﬁantly be solved by simplex method. In this regard, piecewise linear member- ship functions are adopted to represent fuzzy goals and the fuzzy minimum operator is utilized to aggregate all fuzzy sets. Following steps demonstrate the procedure of converting the FMOLP model to an LP one: Step 1: Specify the type of goal (inequality/equality) and degree of membership f (Z ) for several values of each objective function Z ,g = 1, 2,··· , G. g g g Step 2: Draw the piecewise linear membership functions for each objective based on data acquired in the ﬁrst step. Step 3: Break each goal of type equality ( f (Z )) into left and right membership fun- 2 2 − + ctions ( f (Z ), f (Z )) and modify table of membership function according 2 2 2 2 to Table 2. Table 1: Membership function of f (Z ). g g Z > X X X X ... X X < X 1 10 10 11 12 1P 1P+1 1P+1 f (Z)0 0 q q ... q 11 1 1 11 12 1P Z > X X X X X X X < X 2 20 20 21 22 2P 2P+1 2P+1 ij f (Z)0 0 q q 1 q q q 2 2 21 12 2P 2P+1 2P+1 Z > X X X X ... X X < X 3 30 30 31 32 3P 3P+1 3P+1 f (Z)0 0 q q ... q 11 3 3 31 32 3P Table 2: Modiﬁed membership function of f (Z ). g g Z > X X X X ... X X < X 1 10 10 11 12 1P 1P+1 1P+1 f (Z)0 0 q q ... q 11 1 1 11 12 1P − − − − − − − − Z > X X X X ... X X < X 2 20 20 21 22 2,P/2 2,P/2+1 2,P/2+1 − − − − f (Z)0 0 q q ... q 11 2 21 12 2,P/2 + + + + + + + + Z > X X X X ... X X < X 2 20 20 21 22 2,P/2 2,P/2+1 2,P/2+1 + + + + f (Z)0 0 q q ... q 11 2 21 12 2,P/2 Z > X X X X ... X X < X 3 30 30 31 32 3P 3P+1 3P+1 f (Z)0 0 q q ... q 11 3 3 31 32 3P Step 4: Assume f (Z ) = t Z + S to be linear function of each objectives seg- g g gr g gr ments X ≤ Z ≤ X in which t , and S are the slope and y-intercept g,r−1 g gr gr gr of the linear segment. t − t t + t S + t g,e+1 ge g,P+1 g1 g,P+1 g1 Step 5: Calculateα = | |, β = ,γ = and convert ge g g 2 2 2 the membership function f (Z ) in to the following form: g g f (Z ) = −α |z − X |+β Z +γ , g = 1, 2,··· , G. (17) g g ge g ge g g g e=1 Fuzzy Inf. Eng. (2011) 4: 393-410 403 Now by expanding Equation (17), Equation (18) would be formulated as fol- lows: t − t t − t g2 g1 g3 g2 f (Z ) = −( )|Z − X |− ( )|Z − X |−···− g g g g1 g g2 2 2 t − t t + t S + S gP+1 gP gP+1 g1 gP+1 g1 ( )|Z − X |+ ( )+ , (18) g gP 2 2 2 g = 1, 2,··· , G, e = 1, 2,··· , P, where P is the numbers of breakpoints of gth goal. Step 6: Introducing the non-negative divisional variables: − + Z + d − d = X , g = 1, 2,··· , G, e = 1, 2,··· , P. (19) g ge ge ge Equation (20) is derived from Equations (18), (19). t − t t − t g2 g1 g3 g2 − + − + f (Z )= −( )(d + d )− ( )(d + d )−···− g g g1 g1 g2 g2 2 2 t − t t + t gP+1 gP gP+1 g1 − + ( )(d − d )+ ( )+ (20) gP gP 2 2 S + S gP+1 g1 , g = 1, 2,··· , G, e = 1, 2,··· , P. Step 7: For objectives of type equality, add auxiliary variables and constraints (21): R× Bh ≥ (H − F ),∀t t t t −R× (1− Bh ) < (H − F ),∀t t t t f (Z ) ≥ f (Z )− R(1− Bh ),∀t (21) 2 2 2 t f (Z ) ≤ f (Z )+ R(1− Bh ),∀t 2 2 2 t f (Z ) ≥ f (Z )− R(Bh ),∀t 2 2 2 t f (Z ) ≤ f (Z )+ R(Bh ),∀t. 2 2 2 t Step 8: By introducing the auxiliary variableμ, as a minimum operator, the proposed FGP model can be converted to an ordinary single objective LP model (Equa- tion (22)): Maxμ t − t t − t g2 g1 g3 g2 + − + − s.t: μ≤−( )(d + d )− ( )(d + d )−···− g1 g1 g2 g2 2 2 t − t t + t g,P+1 gP g,P+1 g1 + − ( )(d + d )+ ( )Z + gP gP 2 2 S + S gP+1 g1 , g = 1, 2, 3, e = 1, 2,··· , P, μ ≥ f (Z )− R(1− Bh ),∀t, 2 t + (22) μ ≤ f (Z )+ R(1− Bh ),∀t, 2 t μ ≥ f (Z )− R(Bh ),∀t, 2 t μ ≤ f (Z )+ R(Bh ),∀t, 2 t R× Bh ≥ (H − F ),∀t, t t t −R× (1− Bh ) < (H − F ),∀t, t t t − + Z + d − d = X ,∀g,∀e, g ge ge ge Eqs.(4)− (6), (8)− (12), (14)− (16). 404 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) 3.2. Solving Procedure The block diagram of solving procedure is demonstrated in Figure 2 the proposed procedure enables the DM to adjust search direction through each run until satisfac- tory results obtained. Fig. 2 Interactive procedure of solving FGP 4. An Illustrative Numerical Example 4.1. Data description In order to implement the proposed model, a production system is generated including three products. The system encompasses ﬁve machine centers and each product goes through four machine centers according to its own production sequence. Figure 3 depicts operational sequence including machine processing time for each product. Table 3 - 6 summarized related problem parameters. Fig. 3 Machine order visit Fuzzy Inf. Eng. (2011) 4: 393-410 405 Table 3: Production data. Product Process C Cs CS u Mh Ws Cb ni ni ni ni ni ni 1 1 100 145 180 1.0 1.5 2 120 235 185 1.2 2.3 3 125 255 400 1.4 1.8 4 135 265 500 1.0 1.9 65 2 1 150 115 130 2.6 1.8 2 150 235 200 0.8 2.6 3 150 255 185 2.5 2.0 4 165 265 200 1.8 2.5 34 3 1 145 115 195 1.1 1.7 2 160 235 185 2.3 2.7 3 150 255 400 0.9 2.5 4 140 265 500 0.8 3.0 81 Table 4: Market demand. Products 1 2 3 Periods 1 105 110 155 2 110 100 90 3 755055 Table 5: Machine capacity. MC 123 Periods 1 2500 2500 2500 2 2500 2500 2500 3 2000 2000 2000 4 3000 3000 3000 5 2400 2400 2400 Other complementary information is given as follows: • Escalation factors assumed to be 0.1. • The initial available labor level in period 1 is 700 and maximum level of work- force in each period is 2000. • Costs associated with hiring and dismissing man-hour are equal to 10 ($/man- hour). 406 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) • There is no initial inventory, and no end inventory is required. • The available space of ﬁnished product’s warehouse and WIP warehouse are equal to 200 and 100 ft . • Based on subcontracting strategy plan, the maximum allowable numbers of subcontracted items are equal to 450, 200, and 200 in periods respectively. 4.2. Model Implementation Considering the aforementioned procedure, the proposed MPMP problem can be for- mulated as follows. First the degree of membership functions for different values of goals should be speciﬁed by DM, as listed in Table 6, Figure 1, 4, and 5 depict the corresponding shapes of goals. Table 6: Piecewise linear membership functions. Z ≥$700,917 $690,000 $679,200 $669,400 $642,698 $630,000 f (Z ) 0 0.4 0.7 0.9 1.0 1.0 1 1 Z 0 0.7 1 0.85 0 0 f (Z ) 0 0.7 11.0 0.85 0 0 2 2 Z ≤0.645 0.648 0.652 0.657 0.662 0.665 f (Z ) 0 0.45 0.69 0.88 1.0 1.0 3 3 Fig. 4 Piecewise linear membership function (Z , f (Z )) 1 1 1 Fig. 5 Piecewise linear membership function (Z , f (Z )) 3 3 3 Fuzzy Inf. Eng. (2011) 4: 393-410 407 Consequently, by applying the auxiliary variable μ, which speciﬁes the total sat- isfaction rate of DM, the FGP model for the MPMP problem can be converted to a single objective LP model. The complete LP model can be formulated as follow: Maxμ −6 + − −6 + − s.t: μ≤−4.43116× 10 (d + d )− 3.68481× 10 (d + d )− 11 11 12 12 −6 + − −5 8.33156× 10 × (d + d )− 2.01926× 10 × Z + 14.5443 13 13 μ ≤ f (H − F ),∀t, 2 t t + − + − μ≤−45.870195579× (d + d )− 10.6122449× (d + d )− 31 31 32 32 + − 8.205390373× (d + d )+ 87.05256052× Z + 55.8533, 33 33 − + Z + d − d = X , g = 1, 3,∀e, g ge ge ge − + H + dh − dh = 400,∀t, t t − + −F + df − df = −450,∀t, t t −4 + − f (H − F )≥−6.625× 10 (dh + dh )− 1.0375× (H − F ) 2 t t t t t t +1.265− R× (1− Bh ),∀t, −4 + − f (H − F )≤−6.625× 10 (dh + dh )− 1.0375× (H − F ) 2 t t t t t t +1.265− R× (1− Bh ),∀t, −3 + − −3 f (H − F )≥−1.06667× 10 (df + df )− 1.7333× 10 × 2 t t t t (H − F )+ 1.48− R× (Bh ),∀t, t t t −3 + − −3 f (H − F )≤−1.06667× 10 (df + df )− 1.7333× 10 × 2 t t t t (H − F )+ 1.48− R× (Bh ),∀t, t t t Eqs.(4)− (6), (8)− (12), (14)− (16). Finally, the crisp single objective LP model is solved by LINGO 9.0 and optimum solution is demonstrated in Table 7. Figure 6 depicts the trend of changes in objectives values where the model is solved as a single objective model for each goal as well as results obtained from solving FGP model. Fig. 6 Objectives values regarding single objective optimization 408 K. Taghizadeh· M. Bagherpour · I. Mahdavi (2011) Table 7: Optimum solution obtained from solving FGP. Total satisfactory degree = 0.139, NPV of production cost = $695, 558, The highest change in labor level = 818.2, Machine utilization = 0.646 Production Q =163, Q =151, Q =151, Q =0, Q =48, 1111 1211 1311 1411 2111 quantities Q =48, Q =0, Q =0, Q =3, Q =49, Q =49, 2211 2311 2411 3111 3211 3311 Q =0, Q = 116, Q =116, Q =104, Q =0, 3411 1122 1222 1322 1422 Q =74, Q =74, Q =50, Q =1, Q =42, Q =42, 2122 222 2322 2422 3122 3222 Q =6, Q =0, Q =63, Q =63, Q =75, Q =76, 3322 3422 1133 1233 1333 1433 Q =179, Q =179, Q =0, Q =172, Q =86, 2133 2233 2333 2433 3133 Q =89, Q =125, Q =120. 3233 3333 3433 Inventories Q =0, Q =0, Q =0, Q =0, Q =0, Q =0, 1112 1113 1212 1213 1312 1313 Q =7, Q =79, Q =0, Q =0, Q =0, Q =0, Q = 1412 1413 2112 2113 2212 2213 2312 0, Q =0, Q =0, Q =0, Q =0, Q =0, Q =0, 2313 2412 2413 3112 3113 3212 Q =0, Q =0, Q =0, Q =0, Q =0, Q =0, 3213 3312 3313 3412 3413 1123 Q =12, Q =1, Q =0, Q =0, Q =3, Q =0, 1223 1323 1423 2123 2223 2323 Q =0, Q =0, Q =36, Q =0, Q =18. 2423 3123 3223 3323 3423 Backorders Q =0, Q =0, Q =62, Q =26, Q =0, Q =0, 1421 1431 2421 2431 3421 3431 Q =0, Q =92, Q =44. 1432 2432 3432 Subcontracts SU =28, SU =0, SU =40, SU =191, SU =0, 111 121 131 141 211 SU =0, SU =48, SU =48, SU =46, SU =0, 221 231 241 311 321 SU =0, SU =49, SU =0, SU =0, SU =0, 331 341 112 122 132 SU =103, SU =0, SU =0, SU =21, SU =70, 142 212 222 232 242 SU =0, SU =0, SU =0, SU =6, SU =0, SU =0, 312 322 332 342 113 123 SU =0, SU =0, SU =0, SU =0, SU =182, 133 143 213 223 233 SU =10, SU =3, SU =0, SU =3, SU =5. 243 313 323 333 343 Labor hiring H =226.9, H =0.5, H =818.2, F =0, F =0, F =0. 1 2 3 1 2 3 and dismiss- ing 5. Conclusion This study presents a novel FNLGP model for the MPMP production planning prob- lem. The proposed model attempts to simultaneously minimize net present value of production costs, rate of changes in the workforce level, and also maximizing ma- chine utilization regarding issues such as production capacity, maximum inventory and labor levels. The original NLFGP model is initially transformed into an FLGP model by applying appropriate linearization of non-linear terms. Fuzzy piecewise lin- ear membership functions have been introduced to deal with imprecision and vague- ness of DM’s aspiration levels. Moreover, a solving procedure is suggested to adjust search direction during solving process in order to assist the DM to reach the most satisfactory results. Finally, a numerical example demonstrates the applicability of the proposed procedure. The proposed model and solving procedure is still open for future research. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Dec 1, 2011
Keywords: Multi-period multi-product production planning; Fuzzy goal programming; Piecewise linear membership functions