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Towards dynamic contract extension in supplier development

Towards dynamic contract extension in supplier development Logist. Res. (2016) 9:14 DOI 10.1007/s12159-016-0141-z ORIGINAL PAPER 1 2 3 2 • • • • Karl Worthmann Michael Proch Philipp Braun Jo¨rg Schlu¨chtermann Ju¨rgen Pannek Received: 4 January 2016 / Accepted: 13 July 2016 / Published online: 22 July 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We consider supplier development within a supply chain profit. Our findings are validated by a supply chain consisting of a single manufacturer and a numerical case study. single supplier. Because investments in supplier develop- ment are usually relationship-specific, safeguard mecha- Keywords Supply chain management  Supplier nisms against the hazards of partner opportunism have to development  Optimal control  Receding horizon be installed. Here, formal contracts are considered as the scheme  Dynamic systems primary measure to safeguard investments. However, for- mal contracts entail certain risks, e.g., a lack of flexibility, particular in an ambiguous environment. We propose a 1 Introduction receding horizon control scheme to mitigate possible con- tractual drawbacks while significantly enhancing the sup- Since manufacturing firms increasingly focus on their core plier development process and, thus, to increase the overall business activities, an efficient supply chain plays a major role in generating competitive advantages. However, sup- pliers too often lack the capability to perform adequately. In response, manufacturers across a wide range of indus- This article is part of a focus collection on ‘‘Dynamics in Logistics: tries are implementing supplier development programmes Digital Technologies and Related Management Methods’’. to improve supply chain performance [48]. According to & Ju¨rgen Pannek [22, p. 206], supplier development is defined as any effort pan@biba.uni-bremen.de by a buying firm to improve a supplier’s performance and/ Karl Worthmann or capabilities to meet the manufacturing firm’s short- and/ karl.worthmann@tu-ilmenau.de or long-term supply needs. Michael Proch In accordance with the relational view as proposed michael.proch@uni-bayreuth.de by [10], activities of supplier development, in which firms Philipp Braun convert general-purpose resources such as money, people philipp.braun@uni-bayreuth.de skills, or managerial knowledge into relationship-specific Jo¨rg Schlu¨chtermann resources, represent a rent-generating process. However, j.schluechtermann@uni-bayreuth.de relationship-specific resources are difficult or even impos- sible to redeploy outside the particular business relation- Institute for Mathematics, Technische Universita¨t Ilmenau, 98693 Ilmenau, Germany ship [54]. Thus, firms may see resources committed to supplier development as vulnerable to opportunistic expro- Faculty of Law, Business Administration and Economics, University of Bayreuth, 95440 Bayreuth, Germany priation [51]. Following this line of reasoning, supplier development activities with high levels of asset specificity Mathematical Institute, University of Bayreuth, 95440 Bayreuth, Germany should be safeguarded against the hazards of partner opportunism [27]. Here, contracts in terms of formalized, Dynamics in Logistics, BIBA, University of Bremen, legally binding agreements that explicitly specify the 28359 Bremen, Germany 123 14 Page 2 of 12 Logist. Res. (2016) 9:14 obligations of each firm, are usually viewed as the primary 2 Related literature means of safeguarding, particularly in a dynamically evolving environment [2, 7]. The drawback of long-term The topic of supplier development has received consider- contracts is, as the degree of uncertainty increases, both able attention from researchers in the past two decades. specifying ex ante all possible contingencies and verifying Previous research has provided good insights into the use ex post the performance of the business partner becomes of certain activities [47], the antecedents [22], critical increasingly difficult [54]. Therefore, firms might be reluc- success factors [27, 49], and the prevalence of supplier tant to sign long-term contracts, which potentially dimin- development in practice [24, 41]. ishes the firms’ propensity to invest in supplier development Supplier development has been applied in various fields activities and thus impedes the manufacturer’s initial strat- of application [44]. Within the automotive industry, Toy- egy to enhance supply chain performance [37]. ota initially began providing on-site assistance to help Given this background, the purpose of our research is to suppliers implement the Toyota Production System [39]. analyse the impact of dynamically extending a contract to Other manufacturers have followed this collaborative mitigate possible contractual hazards. In addition, we seek to approach to develop suppliers’ performance and/or capa- answer the following questions: How does the contract bilities, including Boeing, Chrysler, Daimler, Dell, Ford, period, i.e., planning horizon, affect firms’ willingness to General Motors, Honda, Nissan, Siemens, and Volkswagen commit relationship-specific resources to supplier develop- [34, 38]. Typically, manufacturing firms use a variety of ment? Does receding horizon control offer a straightforward supplier development activities, e.g., providing perfor- method for dynamically extending the planning horizon, mance feedback, training suppliers’ personnel, furnishing while simultaneously facilitating value generation within temporary on-site support to enhance further interaction, supplier development? Further, how should receding hori- providing equipment and tools, or even dedicating capital zon control be arranged to optimize supply chain profit? resources to suppliers [47, 50]. By answering these questions, the contribution of our Empirical studies support that supplier development is a paper is threefold. Firstly, we formulate a continuous-time key factor to attenuate inefficiencies within the supply optimal control problem characterizing the supplier devel- chain and, thus, strategically contributes to strengthen the opment investment decision. We conduct a detailed study, manufacturer’s competitiveness [28, 40]. Benefits resulting showing that the incentives for firms to participate in supplier from supplier development include, e.g., improvements in development critically depend on the contract period. Sec- cost efficiency, product quality and/or lead time [17, 25]. ondly, given the fact that long-term contracts entail certain However, [23] note that firms’ success in supplier devel- risks, e.g., a lack of flexibility, we utilize receding horizon opment varies. In particular, relationship-specific invest- control and show that the supplier development process can ments lead, in general, to a more satisfactory outcome. be enhanced by dynamically extending the contract, see [43] Further, [22] shows that the firms’ propensity to participate for the basic idea of prediction-based control. Based on this in supplier development activities is higher if a continua- result, a one-to-one map is derived linking the contract per- tion of the relationship is expected. Here, [49] adds that iod to the optimal level of supplier development (collabo- supplier development is more effective in mature as ration). The insight gained from these considerations allows opposed to initial phases of relationship life cycles. to either increase the supply chain efficiency or realize the According to [10], appropriate safeguard mechanisms same level of collaboration while being obliged to a shorter may influence both transaction costs and the willingness of contract period. Finally, we present a simple strategy slightly firms to commit relationship-specific resources to supplier modifying the proposed receding horizon control scheme in development, a condition that could be an important source order to avoid pathological behaviour of the supply chain. of competitive advantage. In the first case, firms achieve an This allows to realize the optimal level of collaboration while advantage by incurring lower transaction costs to realize a avoiding unnecessary transaction costs. The remainder of given level of supplier development specificity. In the this paper is structured as follows. Firstly, the related liter- second case, firms create relational rents by attaining a ature is briefly reviewed in Sect. 2. Then, in Sect. 3 the basic higher level of asset specificity [9, 46]. Following this line optimal control problem is described. In the subsequent of reasoning, the firms’ ability to align a considerable level Sect. 4, the dependence of the control policy on the contract of relationship-specific investments with an appropriate period is studied in detail. In Sect. 5, a receding horizon safeguard mechanism could enhance efficiency and effec- scheme is proposed and analysed before the effectiveness of tiveness of supplier development activities and thereby the developed methodology is demonstrated by means of a should be critical to the success of supplier development. numerical case study in Sect. 6 before conclusions are Scholars usually distinguish between two classes of drawn. governance mechanism: the first relies on third-party 123 Logist. Res. (2016) 9:14 Page 3 of 12 14 enforcement of agreements, e.g., legal contracts, whereas the coefficients a [ 0 and b [ 0 denote the prohibitive the second relies on self-enforcing agreements, e.g., rela- price and the price elasticity of the commodity, respec- tional norms, that make long-term gains from the ongoing tively. This market condition is comparable with an relationship exceed potential short-term payoffs from act- oligopolistic or monopolistic market structure, in which a ing opportunistically [8, 45]. Here, it has been suggested firm can increase market demand by lowering the sale that self-enforcing agreements are a less costly and more price. Similar approaches to specify the price distribution effective means of safeguarding relationship-specific curve have been proposed by [4, 20, 27]. investments in comparison with formal contracts [1, 35]. Despite the significant methodological and theoretical 3.1 Basic model contributions of these streams of research, empirical evi- dence shows that formal contracts are still viewed as the It is supposed that the decision-making process is struc- primary means of safeguarding against the hazards of tured such that M determines the quantity supplied to the partner opportunism, particular in an ambiguous environ- market obeying the paradigm of profit maximization. Note ment [2, 7]. However, contract research is moving away that we do not distinguish market demand from the pro- from a narrow focus on contract structure and its safe- duction quantity of the manufacturer because the market guarding function towards a broader focus that also price is endogenous to the quantity sold. Moreover, the highlights adaptation and coordination as shown in [42]. supplier produces the components to satisfy the demand d In [53] it is even suggested that contracts function as and thus does not decide on the production quantity. relationship management tools. Because the manufacturer’s goal is profit maximization, the Nevertheless, the application of formal decision-making production quantity d chosen by M is determined by models proposed for assisting firms in contract negotiations differentiating in order to adequately safeguard relationship-specific d ðpðdÞ c  cÞð1Þ M SC investments has received limited attention in the supplier development literature [3]. Without understanding the with respect to d and setting the resulting expression equal impact of the contract period on the firms’ incentives to to zero, i.e., commit relationship-specific resources to supplier devel- ð2Þ pðdÞ c  c  bd ¼ 0; M SC opment, its return will be negligible, perhaps even leading to the premature discontinuation of such collaborative cost- which yields the optimum production quantity d ¼ reduction efforts. ac c aþc þc M SC H M SC and the optimal sale price pðd Þ¼ . 2b 2 The trend to utilize mathematical models in general and Here, c and c denote the manufacturer’s unit produc- M SC control theory in particular in decision-making within tion costs and the supply costs per unit charged by S, supply chains is clearly visible [18] and [16]. Here, model respectively. We further assume that the supplier wants to predictive control (MPC), also termed receding (rolling) earn a fixed profit margin r. Thus, the supply costs c SC horizon control, plays a predominant role due to its ability consist of the supplier’s fixed profit margin r and the to deal with nonlinear constrained multi-input multi-output supplier’s unit production costs c , i.e., c ¼ r þ c .This S SC S systems on the one hand, see, e.g., [6, 14], and its inherent assumption is not completely new: Honda Motor Com- robustness on the other hand, see [31, 32, 57] for details. pany, e.g., first learns extensively about a suppliers cost Consequently, MPC is a well-established strategy to deal structure and then specifies a target price that combines with uncertainties in supply chains, see, e.g., [33, 52] and both the suppliers unit production cost and a percent [19]. In this paper, MPC is first used in supplier develop- margin [29]. Similar approaches to specify the supply costs ment to mitigate possible contractual hazards by dynamical have been proposed by [4, 21, 27]. Summing up, the supply extending the contract, see also our preliminary study [55]. chain profit is given by ða  c  c Þ a  c ðr þ c Þ M SC M S M S J ¼ J þ J ¼ þ r 3 Model description 4b 2b ða  c  c Þ  r M S We consider a particular supply chain consisting of a single ¼ : 4b manufacturer M and a single supplier S, in which M It is supposed that the manufacturer wants to decrease the assembles components from S and sells the final product to supplier’s unit production costs c by conducting supplier the market. We restrict ourselves to the linear price dis- S development projects to increase the market share if that tribution curve pðdÞ¼ a  bd, which establishes a con- increases the overall profit of the supply chain. To this end, nection between the production quantity d and the sale the sustainable effect of supplier development on the price p, in order to streamline the upcoming analysis. Here, 123 14 Page 4 of 12 Logist. Res. (2016) 9:14 Table 1 List of parameter 3.2 Solution of the optimal control problem Symbol Description Value Pontryagin’s maximum principle, see, e.g., [26], is used T Contract period 60 analogously to [20] to solve the optimal control problem a Prohibitive price 200 introduced in the preceding subsection. To formulate the b Price elasticity 0.01 necessary optimality conditions, we require the so-called c Variable cost per unit (M)70 Hamiltonian H, which is defined as c Variable cost per unit (S) 100 m 2 ða  c  c x Þ  r r Fixed profit margin (S)15 M 0 ð5Þ Hðx; u; kÞ :¼  c u þ ku: SD c Supplier development cost per unit 100,000 4b SD x Resource availability 1 From the necessary conditions, we obtain the system m Learning rate 0:1 dynamics H H H H x_ ðtÞ¼ H ðx ðtÞ; u ðtÞ; kðtÞÞ ¼ u ðtÞ; supplier’s unit production costs c is modelled by c ðxÞ¼ c x , where c [ 0 denotes the supplier’s unit S 0 0 the so-called adjoint k : ½0; T! R, which is characterized production cost at the outset, m\0 characterizes the sup- by plier’s learning rate, and x defines the cumulative number m1 m H H mc x ðtÞ ða  c  c x ðtÞ Þ 0 M 0 H H of realized supplier development projects. The latter is kðtÞ¼H ðx ðtÞ; u ðtÞ; kðtÞÞ ¼ ; 2b modelled as a time-dependent function x : ½0; T! R ð6Þ governed by the ordinary differential equation and the transversality condition _ ð3Þ xðtÞ :¼ xðtÞ¼ uðtÞ; xð0Þ¼ x ¼ 1; dt kðTÞ¼ 0: ð7Þ with u 2L ðR ; ½0; xÞ. Here, u(t) describes the number The solution u : ½0; TÞ! ½0; x of the optimal control of supplier development projects at time t; with capacity problem exhibits the structural property bound x [ 0 to reflect limited availability of resources in terms of time, manpower, or budget. Similar models of cost x if t\t reduction through learning have been proposed by u ðtÞ :¼ ð8Þ 0if t  t [4, 11, 20, 27, 56]. The costs of supplier development are integrated into the depending on the (optimal) switching time t 2½0; T, proposed model by a penalization term c uðtÞ, c  0. SD SD which is characterized by the equation Overall, this yields the supply chain’s profit function SC J : u 7! R m1 m H H mc ðx þ xt Þ ða  c  c ðx þ xt Þ Þ c 0 0 M 0 0 SD Z ¼ : m 2 T 2 ða  c  c xðtÞ Þ  r 2b ðt  TÞ M 0 J ðu; x Þ :¼  c uðtÞdt T 0 SD 4b ð9Þ ð4Þ In the following, (9) is called switching condition. Indeed, since the cost function is (strictly) convex and the system for a given time interval [0, T], which must be maximized subject to the control constraints 0  uðtÞ x, t 2½0; TÞ, dynamics are governed by a linear ordinary differential equation, it can be shown that this condition is necessary and the system dynamics (3). The contract period T is of particular interest since investments into the cost structure and sufficient for the considered problem, see [36] for a detailed derivation. We emphasize that the switching of the supply chain require their amortization during the runtime of the contractual agreement. A summary of the time t characterizes the optimal time of collaboration parameters is given in Table 1. since every investment in supplier development up to t results in an increased profit while expenditures spent after t do not amortize during the contract period and are, thus, not economically reasonable within the considered setting. The optimal value function V ðx Þ of the problem under T 0 Because supplier development is most often used as of the end of the growth stage as opposed to initial stages of a product’s life cycle, consideration reads we consider solely the learning that occurs through the cumulative SC V ðx Þ :¼ sup J ðu; x Þ T 0 0 number of realized supplier development projects without considering T u2L ð½0;TÞ;½0;xÞ further effects, e.g., total number of units produced [5, 30]. 123 Logist. Res. (2016) 9:14 Page 5 of 12 14 cooperate with a different supplier instead of adhere to the x 10 already existing business relation, see, e.g., [12] and [36] for the considered setting with multiple suppliers. Here, however, it is supposed that continuation of the collabo- ration is preferable since our focus is on the arrangement of the manufacturer/supplier cooperation. Hence, Option 1 corresponds to the scenario, in which supplier development cannot increase profitability within the supply chain and the cooperation with another supplier acting on the market is also not economically reasonable. Hence, we focus on the second case within this paper. Here, from the specific structure (8) of the optimal λ(t) SD control function we can conclude that all investments up to 0 10 20 30 40 50 60 H time t pay off during the contract period. Then, taking Time t into account the already reduced supply costs given by c ðtÞ¼ r þ c xðt Þ with SC 0 Fig. 1 The adjoint k : ½0; T! R computed based on the param- eters given in Table 1 Z m m H H H c xðt Þ ¼ c x þ u ðsÞdt ¼ c ð1 þ xt Þ ; 0 0 0 0 where the expression on the right-hand side is maximized further effort in terms of uðtÞ [ 0, t 2½t ; TÞ, does not lead subject to x_ðtÞ¼ uðtÞ, xð0Þ¼ x . V : R ! R maps the 0 T [ 0 to an increased profit. The latter holds true since cost-re- initial value x to the optimal value. The index T indicates duction efforts after t do not amortize within the the contract period and can be considered as a parameter— remaining time interval of at most length T  t and are, an interpretation, which is crucial for the upcoming thus, not economically reasonable. We show that a pro- analysis. longation of the contract period yields an augmentation of Evidently, investments (in the cost structure) pay off in the investments in supplier development, which corre- the long run: while all the effort is spent directly at the sponds to an increased switching time t . A proof of beginning of the collaboration, the resulting cost decreas- Lemma 1 is given in ‘‘Appendix 8’’. ing effect is exploited during the remainder of the contract period. Lemma 1 Suppose that the contract period T is chosen H H (long enough) such that t ¼ t ðTÞ [ 0 holds. In addition, Remark 1 At the switching time t , the marginal revenue let the condition of further investments in supplier development (given by the adjoint variable k) equals the marginal costs (given by ð1  mÞða  c  c Þþ c m  0 ð10Þ M 0 0 c ) as indicated in Fig. 1. This reasoning is expressed by SD hold. Then prolonging the contract period T, T [ T, im- the switching condition (9). H H plies a strictly larger switching time t ¼ t ðTÞ, H H i.e., t ðTÞ [ t ðTÞ. 4 Interplay of switching time and contract period Remark 2 The assumptions of Lemma 1 imply the inequality a  c  c  r [ 0 as a by-product because the M 0 If the desired contract between manufacturer M and sup- manufacturer cannot realize a profit per unit sold otherwise plier S ranges over the interval [0, T], two cases can be (prohibitive price is greater than the production cost per distinguished: unit at time t ¼ 0 from the manufacturer’s point of view). Hence, the seemingly technical Condition (10) links the 1. The (optimal) switching time is given by t ¼ 0 supplier’s production costs c with the difference of profit meaning that investments in supplier development do per unit a  c  c by the learning rate m. Note that the M 0 not pay off during the contract period. assumptions of Lemma 1 can be easily verified for a given 2. A switching time t [ 0 represents the scenario where dataset of parameters. investing into supplier development amortizes during Lemma 1 shows that investments in supplier develop- the contract period. ment are extended if the contract period is prolonged. After determining the outcome of a potential collaboration Hence, the collaboration continues after the previously over the interval [0, T], the overall market situation has to determined switching time t . As a result, the supplier’s be taken into account, e.g., does it make (more) sense to Adjoint variable λ(t) 14 Page 6 of 12 Logist. Res. (2016) 9:14 Fig. 2 Optimal switching x 10 H H 13 time t ¼ t ðTÞ in dependence of the length of the contract T ¼ 3.5 T þ i  DT (T ¼ 60, DT ¼ 3 and i ¼ 0; 1; .. .; 7) 2.5 1.5 60 65 70 75 80 0 5 10 15 20 Switching time t Contract period T SC H unit production costs are further decreased, the quantity i.e., J ð; xðDTÞÞ is considered. Since DT  t holds by offered is increased and the supply chain profit per time assumption, the new initial state xðDTÞ is given by unit grows. The argument that a longer contract period Z DT leads to larger switching times can also be validated ð11Þ xðDTÞ¼ xð0Þþ u ðsÞdt ¼ x þ DT  x numericallyasvisualizedinFig. 2. Here, we observe that the supply costs c ðtÞ¼ r þ c xðtÞ are further reduced SC 0 in view of Property (8). Hence, the profit on the new if both the manufacturer and the supplier agree on a contract period ½DT; T þ DT is determined by maximizing longer contract period. The relation between the contract m 2 T 2 ða  c  c x ~ðtÞ Þ  r H M 0 period T and the optimal switching time t ðTÞ is almost J ðu; xðDTÞÞ ¼  c uðtÞdt T SD 4b linear. In summary and according to the initial question how subject to uðtÞ2½0; x, t 2½0; TÞ and the differential does the contract period, i.e., planning horizon, affect equation (3) with initial condition x ~ð0Þ¼ xðDTÞ¼ firms’ willingness to commit relationship-specific resources x þ xDT. Here, we used the notation x ~ to distinguish the to supplier development, the findings show that the supply previously computed (state) trajectory xð; x Þ and its chain partners’ incentives to commit relationship-specific counterpart x ~ð; xðDTÞÞ depending on the new initial con- resources, i.e., to invest in cost-reduction efforts, critically dition xðDTÞ. Another option is to use the time invariance depend on the length of the contract period. of the linear differential equation x_ðtÞ¼ uðtÞ, which allows to rewrite the profit functional as TþDT m 2 5 Successive prolongation of the contract period ða  c  c xðtÞ Þ  r M 0 c uðtÞdt SD 4b DT The benefits of an increased switching time come along with initial value xðDTÞ given by (11) at initial time DT. with the inflexibility resulting from long-term contracts. In We point out that the resulting trajectory deviates from the this section, we propose a methodology for assisting supply previously computed one already before time T. In con- chain partners in contract negotiations to achieve the clusion, the implemented control strategy on ½0; T þ DTÞ is benefits of long-term contracts while committing them- given by selves only to agreements of a certain, prespecified (col- laboration) time period. To this end, it is assumed that the H SC u ðtÞ maximizing J ð; x Þ t 2½0; DTÞ manufacturer and the supplier are only content to make uðtÞ :¼ ; H SC u ðtÞ maximizing J ð; xðDTÞÞ t  DT contracts of length T. If the collaboration is successful for a ð12Þ certain amount of time ½0; DTÞ, DT  t , they might agree to renew the contract on the time interval ½DT; T þ DT. i.e., the first piece of the old policy concatenated with the Before we continue the discussion, let us briefly sketch newly negotiated strategy. This strategy yields an optimal the computation of the (optimal) control func- policy on the time span ½0; T þ DTÞ. Hence, the same tion u : ½DT; T þ DTÞ!½0; x. Here, the profit function overall supply chain profit is reached without the hazards has to be maximized based on the new (initial) state xðDTÞ, of being committed already at the beginning (time 0) as shown in the following corollary. Indeed, the slope of the curve is slightly increasing. SC J (u; x ) T +i·ΔT t (T ) Logist. Res. (2016) 9:14 Page 7 of 12 14 Corollary 1 Let the optimal switching time t deter- mined by Condition (9) be strictly greater than zero. Fur- thermore, let DT, DT\t , be given. Then, the control strategy defined in (12) and the corresponding supply chain profit on ½0; T þ DT equal their counterparts obtained by maximizing J ðu; x Þ with respect TþDT 0 to u : ½0; T þ DTÞ!½0; x Proof Since the profit J ðu; x Þ on the considered TþDT 0 time interval ½0; T þ DT with u from (12) is the sum of 0 3 6 9 12 15 m 2 DT 2 ða  c  c xðtÞ Þ  r M 0 Planed collaboration interval c x dt SD 4b Fig. 3 Application of Algorithm 1 to compute the optimal switching and times for T ¼ 60 and changing initial conditions x ^. The lengths of the collaboration intervals are decreasing m 2 TþDT 2 ða  c  c xðtÞ Þ  r M 0 þ  c uðtÞdt; SD 4b DT previously described steps are repeated, which is referred the dynamic programming principle yields the equality to as receding horizon principle. Note that since the underlying system dynamics are time invariant, the newly J ðu; x Þ¼ V ðx Þ; TþDT 0 TþDT 0 (measured) initial state x ^ represents all information which completes the proof. h required. In particular, no knowledge regarding the previ- ously applied control is needed to solve the adapted 5.1 Receding horizon control switching condition of Step (2) with respect to t . Figure 3 illustrates the outcome of Algorithm 1 with prediction The idea of an iterative prolongation of collaboration horizon T ¼ 60 (contract period) and control hori- contracts can be algorithmically formalized as receding zon DT ¼ 3 (time step) based on the parameters given in horizon control (RHC) also known as model predictive Table 1. control. Upon start, the manufacturer M and the supplier S agree Firstly (t ¼ 0), the original optimal control problem is H H on a collaboration for a given contract period of length solved resulting in t  9:21. Then, u  x is applied on T. Firstly, the status quo—represented by x ^—is analysed. the time interval ½0; DTÞ. Secondly (t ¼ DT), the collabo- Secondly, the optimal switching time t is computed based ration is prolonged to t  9:74. Thirdly (t ¼ 2DT), the H H on the initial state x and T, cf. Step (2). This yields the switching time is shifted to t  10:27. Still, t ¼ 3DT  t optimal control strategy defined by (13), of which the first holds. Hence, the (measured) initial state x ^ is given by piece u j is applied. Then, the manufacturer and the x þ tx ¼ x þ 3DTx. Here, Step (2) of Algorithm (1) ½0;DTÞ 0 0 supplier meet again at time t þ DT to negotiate a new yields t  10:79, i.e., the collaboration stops within the contract. This initiates the process again, i.e.. the time frame ½t; t þ DTÞ. If the RHC scheme is further Time t 14 Page 8 of 12 Logist. Res. (2016) 9:14 applied, there occur collaboration intervals of shrinking 1. set t ¼ t in order to save negotiation costs, which length. would probably outweigh the achievable earning As already discussed in Sect. 5, if the contract is not growth. For the presented example, the supplier H H renewed, u ðtÞ is set to zero for t  t  9:21. In contrast development programme stops at 10.79 (still an increase of approximately 17.2 %) if the threshold is 1. to that, the RHC scheme prolongs the collaboration and, thus, increases the supply chain profit. To be more precise, 2. measure the current state x ^ ¼ xðtÞ and compute the optimal cost structure for contract periods of length T the profit generated by Algorithm 1 on ½0; T þ iDT, i 2f0; 1; 2; ...; T=DTg, by solving m1 m T=DTþi1 ðkþ1ÞDT m 2 2 mc Tx  ða  c  c x  Þþ 2bc ¼ 0 0 M 0 SD ða  c  c xðtÞ Þ  r M 0 c uðtÞdt SD 4b kDT k¼0 with respect to x . Then, set t ¼ t þðx  x ^Þ=x. In the considered example at time t ¼ 4DT, the measured H H is greater than its counterpart J ðu ; x Þþ V ðx ðTÞÞ T 0 iDT state is x ^ ¼ 10:79 while x   11:18. Hence, a collabo- consisting of the maximum of the original cost func- ration of length 0.39 time units is fixed. At all tion V ðx Þ¼ J ðu ; x Þ and a second (optimally oper- T 0 T 0 upcoming time instants, t ¼ t holds because the ated) contract on ½T; T þ iDT based on the reached cost optimal cost structure for contract periods of H H structure represented by x ðTÞ¼ x þ t x  x þ 0 0 length T ¼ 60 is already reached. 9:21x ¼ 10:21. In particular, this assertion holds in com- Clearly, the threshold should be chosen such that the profit parison with simply sticking to the cost structure based on increase outweighs the negotiation costs. t ðTÞ, i.e., Thus, Algorithm 1 allows both the manufacturer and the m 2 TþiDT H 2 ða  c  c xðt ðTÞÞ Þ  r M 0 supplier to prolong their supplier development programme J ðu ; x Þþ dt: T 0 4b without binding themselves for a time span longer than T and, thus, provides more flexibility. ð14Þ Remark 3 Algorithm 1 is a simplified version. Indeed, the While an increased switching time t may already time step DT may vary in time, e.g., longer time steps in increase the profitability within a supply chain during the the beginning (for example, DT ¼ t in the considered considered time span, the achieved cost reduction sus- setting), and shorter ones later on. For details on the so- tains. Hence, if the collaboration between the manufac- called time-varying control horizon, we refer to [15]. turer and the supplier lasts, the obtained effect is a sustainable one. In summary and with regard to the question how should In summary and referring to the question how does receding horizon control be arranged to optimize supply receding horizon control offer a straightforward method chain profit, two strategies are presented in order to make for dynamically extending the planning horizon, the find- the proposed receding horizon scheme, cf. Algorithm 1, ings show that dynamically extending contracts enhance applicable even if negotiation costs are taken into account. the supplier development process, because value genera- tion is facilitated while both the manufacturer and the supplier gain flexibility due to shorter contract periods. 6 Numerical results As seen in the previous section, applying the receding 5.2 Optimal point of collaboration horizon Algorithm 1 dynamically extends the collaboration within the supply chain and, thus, generates additional As observed in Fig. 3, the collaboration can stop within the profit within the supply chain. Next, we conduct a time interval ½t; t þ DTÞ meaning that the prerequi- numerical case study to obtain further managerial insights. site DT  t is no longer satisfied at time t. This leads to a HH To this end, we compare the outcome J of the pro- sequence of collaboration times of shrinking length. posed algorithm based on the second option presented in Summing up all of these intervals on the infinite horizon Sect. 5.2 and the supply chain profit resulting from the yields a total collaboration time of approximately 11.18 control time units. Hence, the total collaboration time is increased by 21.3 %. However, since the collaboration intervals are H x for t\t ðTÞ uðtÞ¼ ð15Þ becoming comparably short, implementing this strategy 0 for t  t ðTÞ may be impracticable. Here, we propose two remedies: If the new collaboration period at time t ¼ kDT, i.e., t  t, on the time interval ½0; 2T¼ ½0; 120. The control pol- is below a certain threshold value, icy (15) results from the basic optimal control problem 123 Logist. Res. (2016) 9:14 Page 9 of 12 14 considered on [0, 60] and, then, utilizing the achieved cost Second, we are interested in the interplay of the sup- plier’s learning rate m and receding horizon control. Thus, structure c ðtÞ¼ x þ t x on [60, 120] without further SC 0 investments in supplier development. The corresponding based on the parameters of Table 1, we perform a sensi- tivity analysis with respect to the parameter m with profit is given by (14). To fully understand the impact of receding horizon m 2f0:15; 0:14; 0:13; 0:12; 0:11; control on the supply chain profit in depth, we first vary the 0:1; 0:09; 0:08; 0:07; 0:06; 0:05g: following parameters of Table 1 Applying Algorithm 1 (T ¼ 60, DT ¼ 3), Fig. 5 shows a 2f192:5; 195; 197:5; 200; 202:5; 205; 207:5g; both the optimal switching time t (without receding b 2f0:007; 0:008; 0:009; 0:01; 0:011; 0:012; 0:013g; horizon control) compared to the optimal switching time c 2f70000; 80000; 90000; 100000; 110000; 120000; 130000g; SD HH t (with receding horizon control) in dependence of x 2f0:7; 0:8; 0:9; 1; 1:1; 1:2; 1:3g; m (left), and the profit growth with respect to the switching m 2f0:13; 0:12; 0:11; 0:1; 0:09; 0:08; 0:07g time for different learning rates (right). Again, the com- 5 putations are based on a simulation of 120 time units. Here, resulting in a total number of 7 = 16,807 instances. For we observe that the impact of receding horizon control each parameter combination, we then evaluate the respec- decreases for lower learning rates. tive profits. Hence, the results infer that especially firms in high- The depicted histogram in Fig. 4 shows the absolute learning industries, e.g., technology-based industries, ben- frequency with which a percentage of profit increase is efit most from applying the proposed receding horizon observed within our parameter set. The mean value scheme. is 3.36 % with a standard deviation of 1.06 %. In conclu- sion, receding horizon control significantly improves the profitability of the considered supply chain. 7 Conclusion In this paper, we investigated the impact of the contract period on supplier development. In particular, we showed that the supply chain partners’ incentives to commit rela- tionship-specific resources, i.e., to invest in cost-reduction efforts, critically depend on the length of the contract period. Given the fact that long-term contracts entail certain risks, we proposed a receding horizon control scheme to mitigate possible contractual hazards. In addition, we showed that dynamically extending contracts enhance the supplier development process, because value generation is 0 2 4 6 8 facilitated while both the manufacturer and the supplier Profit increase ratio (%) gain flexibility due to shorter contract periods. Further- Fig. 4 Profit increase ratio in percent more, we presented two strategies in order to make the Fig. 5 Optimal switching time x 10 H HH 5 t and t with respect to the t (m) parameter m (left) and earning t (m) growth with respect to the H 15 switching time t for different values of m (right) 0 5 10 15 20 −0.14 −0.12 −0.1 −0.08 −0.06 Parameter m Switching time t Number of instances Switching time Earning growth 14 Page 10 of 12 Logist. Res. (2016) 9:14 m1 m H H H H proposed receding horizon scheme, cf. Algorithm 1, fðt Þ :¼ðT  t Þzðt Þ ða  c  c zðt Þ Þ: M 0 applicable even if negotiation costs are taken into account. m2 0 H H Finally, we verified the reliability of the application by Then, the term f ðt Þ zðt Þ is a sum consisting of the H H performing Algorithm 1 for an extensive parameter set and positive summand zðt Þða  c  c zðt Þ Þ and M 0 demonstrated that receding horizon control leads to a sig- m m H H H ðT  t Þxð1  mÞða  c  c zðt Þ Þþ c mzðt Þ : M 0 0 nificant profit increase within the supply chain. Moreover, by means of a sensitivity analysis with respect to the Here, it was used that a  c  c  r [ 0 holds. Hence, M 0 learning rate, we showed that especially firms in high- we investigate the term learning industries benefit since supplier development m m H H programmes play a predominant role in order to optimize ð1  mÞða  c  c zðt Þ Þþ c mzðt Þ ð17Þ M 0 0 the cost structure of the supplier network. in order to determine the sign of the second summand using The study is based on a simple model to focus on the impact of dynamical decision-making in supplier devel- that ðT  t Þx [ 0 holds. To this end, the supply chain profit p :¼ a  c  c [ r [ 0 per unit plays a major opment. Clearly, a more elaborated model with less strin- M 0 gent assumptions like, e.g., a linear price distribution, role: (17) equals should be studied in the future. Moreover, the combination m m H H c ð1  mÞp=c þ mzðt Þ þð1  mÞðc  c zðt Þ Þ 0 0 0 0 of the proposed dynamic strategy with decentralized |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} approaches is of great importance and deserved a detailed analysis, see, e.g., the negotiation-based coordination because m\0 and t  0 hold. Positivity of the first sum- mechanism proposed in [36]. Another interesting direction mand is ensued from (10). Hence, (17) is positive and, for future research is to expand our study to a network thus, f is (strictly) decreasing. perspective, in which the supply chain consists of more In conclusion, the left-hand side of (16) is strictly than a single manufacturer and a single supplier, see, decreasing in t and strictly increasing in T. 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Towards dynamic contract extension in supplier development

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Copyright © 2016 by The Author(s)
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Engineering; Engineering Economics, Organization, Logistics, Marketing; Logistics; Industrial and Production Engineering; Simulation and Modeling; Operation Research/Decision Theory
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Logist. Res. (2016) 9:14 DOI 10.1007/s12159-016-0141-z ORIGINAL PAPER 1 2 3 2 • • • • Karl Worthmann Michael Proch Philipp Braun Jo¨rg Schlu¨chtermann Ju¨rgen Pannek Received: 4 January 2016 / Accepted: 13 July 2016 / Published online: 22 July 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We consider supplier development within a supply chain profit. Our findings are validated by a supply chain consisting of a single manufacturer and a numerical case study. single supplier. Because investments in supplier develop- ment are usually relationship-specific, safeguard mecha- Keywords Supply chain management  Supplier nisms against the hazards of partner opportunism have to development  Optimal control  Receding horizon be installed. Here, formal contracts are considered as the scheme  Dynamic systems primary measure to safeguard investments. However, for- mal contracts entail certain risks, e.g., a lack of flexibility, particular in an ambiguous environment. We propose a 1 Introduction receding horizon control scheme to mitigate possible con- tractual drawbacks while significantly enhancing the sup- Since manufacturing firms increasingly focus on their core plier development process and, thus, to increase the overall business activities, an efficient supply chain plays a major role in generating competitive advantages. However, sup- pliers too often lack the capability to perform adequately. In response, manufacturers across a wide range of indus- This article is part of a focus collection on ‘‘Dynamics in Logistics: tries are implementing supplier development programmes Digital Technologies and Related Management Methods’’. to improve supply chain performance [48]. According to & Ju¨rgen Pannek [22, p. 206], supplier development is defined as any effort pan@biba.uni-bremen.de by a buying firm to improve a supplier’s performance and/ Karl Worthmann or capabilities to meet the manufacturing firm’s short- and/ karl.worthmann@tu-ilmenau.de or long-term supply needs. Michael Proch In accordance with the relational view as proposed michael.proch@uni-bayreuth.de by [10], activities of supplier development, in which firms Philipp Braun convert general-purpose resources such as money, people philipp.braun@uni-bayreuth.de skills, or managerial knowledge into relationship-specific Jo¨rg Schlu¨chtermann resources, represent a rent-generating process. However, j.schluechtermann@uni-bayreuth.de relationship-specific resources are difficult or even impos- sible to redeploy outside the particular business relation- Institute for Mathematics, Technische Universita¨t Ilmenau, 98693 Ilmenau, Germany ship [54]. Thus, firms may see resources committed to supplier development as vulnerable to opportunistic expro- Faculty of Law, Business Administration and Economics, University of Bayreuth, 95440 Bayreuth, Germany priation [51]. Following this line of reasoning, supplier development activities with high levels of asset specificity Mathematical Institute, University of Bayreuth, 95440 Bayreuth, Germany should be safeguarded against the hazards of partner opportunism [27]. Here, contracts in terms of formalized, Dynamics in Logistics, BIBA, University of Bremen, legally binding agreements that explicitly specify the 28359 Bremen, Germany 123 14 Page 2 of 12 Logist. Res. (2016) 9:14 obligations of each firm, are usually viewed as the primary 2 Related literature means of safeguarding, particularly in a dynamically evolving environment [2, 7]. The drawback of long-term The topic of supplier development has received consider- contracts is, as the degree of uncertainty increases, both able attention from researchers in the past two decades. specifying ex ante all possible contingencies and verifying Previous research has provided good insights into the use ex post the performance of the business partner becomes of certain activities [47], the antecedents [22], critical increasingly difficult [54]. Therefore, firms might be reluc- success factors [27, 49], and the prevalence of supplier tant to sign long-term contracts, which potentially dimin- development in practice [24, 41]. ishes the firms’ propensity to invest in supplier development Supplier development has been applied in various fields activities and thus impedes the manufacturer’s initial strat- of application [44]. Within the automotive industry, Toy- egy to enhance supply chain performance [37]. ota initially began providing on-site assistance to help Given this background, the purpose of our research is to suppliers implement the Toyota Production System [39]. analyse the impact of dynamically extending a contract to Other manufacturers have followed this collaborative mitigate possible contractual hazards. In addition, we seek to approach to develop suppliers’ performance and/or capa- answer the following questions: How does the contract bilities, including Boeing, Chrysler, Daimler, Dell, Ford, period, i.e., planning horizon, affect firms’ willingness to General Motors, Honda, Nissan, Siemens, and Volkswagen commit relationship-specific resources to supplier develop- [34, 38]. Typically, manufacturing firms use a variety of ment? Does receding horizon control offer a straightforward supplier development activities, e.g., providing perfor- method for dynamically extending the planning horizon, mance feedback, training suppliers’ personnel, furnishing while simultaneously facilitating value generation within temporary on-site support to enhance further interaction, supplier development? Further, how should receding hori- providing equipment and tools, or even dedicating capital zon control be arranged to optimize supply chain profit? resources to suppliers [47, 50]. By answering these questions, the contribution of our Empirical studies support that supplier development is a paper is threefold. Firstly, we formulate a continuous-time key factor to attenuate inefficiencies within the supply optimal control problem characterizing the supplier devel- chain and, thus, strategically contributes to strengthen the opment investment decision. We conduct a detailed study, manufacturer’s competitiveness [28, 40]. Benefits resulting showing that the incentives for firms to participate in supplier from supplier development include, e.g., improvements in development critically depend on the contract period. Sec- cost efficiency, product quality and/or lead time [17, 25]. ondly, given the fact that long-term contracts entail certain However, [23] note that firms’ success in supplier devel- risks, e.g., a lack of flexibility, we utilize receding horizon opment varies. In particular, relationship-specific invest- control and show that the supplier development process can ments lead, in general, to a more satisfactory outcome. be enhanced by dynamically extending the contract, see [43] Further, [22] shows that the firms’ propensity to participate for the basic idea of prediction-based control. Based on this in supplier development activities is higher if a continua- result, a one-to-one map is derived linking the contract per- tion of the relationship is expected. Here, [49] adds that iod to the optimal level of supplier development (collabo- supplier development is more effective in mature as ration). The insight gained from these considerations allows opposed to initial phases of relationship life cycles. to either increase the supply chain efficiency or realize the According to [10], appropriate safeguard mechanisms same level of collaboration while being obliged to a shorter may influence both transaction costs and the willingness of contract period. Finally, we present a simple strategy slightly firms to commit relationship-specific resources to supplier modifying the proposed receding horizon control scheme in development, a condition that could be an important source order to avoid pathological behaviour of the supply chain. of competitive advantage. In the first case, firms achieve an This allows to realize the optimal level of collaboration while advantage by incurring lower transaction costs to realize a avoiding unnecessary transaction costs. The remainder of given level of supplier development specificity. In the this paper is structured as follows. Firstly, the related liter- second case, firms create relational rents by attaining a ature is briefly reviewed in Sect. 2. Then, in Sect. 3 the basic higher level of asset specificity [9, 46]. Following this line optimal control problem is described. In the subsequent of reasoning, the firms’ ability to align a considerable level Sect. 4, the dependence of the control policy on the contract of relationship-specific investments with an appropriate period is studied in detail. In Sect. 5, a receding horizon safeguard mechanism could enhance efficiency and effec- scheme is proposed and analysed before the effectiveness of tiveness of supplier development activities and thereby the developed methodology is demonstrated by means of a should be critical to the success of supplier development. numerical case study in Sect. 6 before conclusions are Scholars usually distinguish between two classes of drawn. governance mechanism: the first relies on third-party 123 Logist. Res. (2016) 9:14 Page 3 of 12 14 enforcement of agreements, e.g., legal contracts, whereas the coefficients a [ 0 and b [ 0 denote the prohibitive the second relies on self-enforcing agreements, e.g., rela- price and the price elasticity of the commodity, respec- tional norms, that make long-term gains from the ongoing tively. This market condition is comparable with an relationship exceed potential short-term payoffs from act- oligopolistic or monopolistic market structure, in which a ing opportunistically [8, 45]. Here, it has been suggested firm can increase market demand by lowering the sale that self-enforcing agreements are a less costly and more price. Similar approaches to specify the price distribution effective means of safeguarding relationship-specific curve have been proposed by [4, 20, 27]. investments in comparison with formal contracts [1, 35]. Despite the significant methodological and theoretical 3.1 Basic model contributions of these streams of research, empirical evi- dence shows that formal contracts are still viewed as the It is supposed that the decision-making process is struc- primary means of safeguarding against the hazards of tured such that M determines the quantity supplied to the partner opportunism, particular in an ambiguous environ- market obeying the paradigm of profit maximization. Note ment [2, 7]. However, contract research is moving away that we do not distinguish market demand from the pro- from a narrow focus on contract structure and its safe- duction quantity of the manufacturer because the market guarding function towards a broader focus that also price is endogenous to the quantity sold. Moreover, the highlights adaptation and coordination as shown in [42]. supplier produces the components to satisfy the demand d In [53] it is even suggested that contracts function as and thus does not decide on the production quantity. relationship management tools. Because the manufacturer’s goal is profit maximization, the Nevertheless, the application of formal decision-making production quantity d chosen by M is determined by models proposed for assisting firms in contract negotiations differentiating in order to adequately safeguard relationship-specific d ðpðdÞ c  cÞð1Þ M SC investments has received limited attention in the supplier development literature [3]. Without understanding the with respect to d and setting the resulting expression equal impact of the contract period on the firms’ incentives to to zero, i.e., commit relationship-specific resources to supplier devel- ð2Þ pðdÞ c  c  bd ¼ 0; M SC opment, its return will be negligible, perhaps even leading to the premature discontinuation of such collaborative cost- which yields the optimum production quantity d ¼ reduction efforts. ac c aþc þc M SC H M SC and the optimal sale price pðd Þ¼ . 2b 2 The trend to utilize mathematical models in general and Here, c and c denote the manufacturer’s unit produc- M SC control theory in particular in decision-making within tion costs and the supply costs per unit charged by S, supply chains is clearly visible [18] and [16]. Here, model respectively. We further assume that the supplier wants to predictive control (MPC), also termed receding (rolling) earn a fixed profit margin r. Thus, the supply costs c SC horizon control, plays a predominant role due to its ability consist of the supplier’s fixed profit margin r and the to deal with nonlinear constrained multi-input multi-output supplier’s unit production costs c , i.e., c ¼ r þ c .This S SC S systems on the one hand, see, e.g., [6, 14], and its inherent assumption is not completely new: Honda Motor Com- robustness on the other hand, see [31, 32, 57] for details. pany, e.g., first learns extensively about a suppliers cost Consequently, MPC is a well-established strategy to deal structure and then specifies a target price that combines with uncertainties in supply chains, see, e.g., [33, 52] and both the suppliers unit production cost and a percent [19]. In this paper, MPC is first used in supplier develop- margin [29]. Similar approaches to specify the supply costs ment to mitigate possible contractual hazards by dynamical have been proposed by [4, 21, 27]. Summing up, the supply extending the contract, see also our preliminary study [55]. chain profit is given by ða  c  c Þ a  c ðr þ c Þ M SC M S M S J ¼ J þ J ¼ þ r 3 Model description 4b 2b ða  c  c Þ  r M S We consider a particular supply chain consisting of a single ¼ : 4b manufacturer M and a single supplier S, in which M It is supposed that the manufacturer wants to decrease the assembles components from S and sells the final product to supplier’s unit production costs c by conducting supplier the market. We restrict ourselves to the linear price dis- S development projects to increase the market share if that tribution curve pðdÞ¼ a  bd, which establishes a con- increases the overall profit of the supply chain. To this end, nection between the production quantity d and the sale the sustainable effect of supplier development on the price p, in order to streamline the upcoming analysis. Here, 123 14 Page 4 of 12 Logist. Res. (2016) 9:14 Table 1 List of parameter 3.2 Solution of the optimal control problem Symbol Description Value Pontryagin’s maximum principle, see, e.g., [26], is used T Contract period 60 analogously to [20] to solve the optimal control problem a Prohibitive price 200 introduced in the preceding subsection. To formulate the b Price elasticity 0.01 necessary optimality conditions, we require the so-called c Variable cost per unit (M)70 Hamiltonian H, which is defined as c Variable cost per unit (S) 100 m 2 ða  c  c x Þ  r r Fixed profit margin (S)15 M 0 ð5Þ Hðx; u; kÞ :¼  c u þ ku: SD c Supplier development cost per unit 100,000 4b SD x Resource availability 1 From the necessary conditions, we obtain the system m Learning rate 0:1 dynamics H H H H x_ ðtÞ¼ H ðx ðtÞ; u ðtÞ; kðtÞÞ ¼ u ðtÞ; supplier’s unit production costs c is modelled by c ðxÞ¼ c x , where c [ 0 denotes the supplier’s unit S 0 0 the so-called adjoint k : ½0; T! R, which is characterized production cost at the outset, m\0 characterizes the sup- by plier’s learning rate, and x defines the cumulative number m1 m H H mc x ðtÞ ða  c  c x ðtÞ Þ 0 M 0 H H of realized supplier development projects. The latter is kðtÞ¼H ðx ðtÞ; u ðtÞ; kðtÞÞ ¼ ; 2b modelled as a time-dependent function x : ½0; T! R ð6Þ governed by the ordinary differential equation and the transversality condition _ ð3Þ xðtÞ :¼ xðtÞ¼ uðtÞ; xð0Þ¼ x ¼ 1; dt kðTÞ¼ 0: ð7Þ with u 2L ðR ; ½0; xÞ. Here, u(t) describes the number The solution u : ½0; TÞ! ½0; x of the optimal control of supplier development projects at time t; with capacity problem exhibits the structural property bound x [ 0 to reflect limited availability of resources in terms of time, manpower, or budget. Similar models of cost x if t\t reduction through learning have been proposed by u ðtÞ :¼ ð8Þ 0if t  t [4, 11, 20, 27, 56]. The costs of supplier development are integrated into the depending on the (optimal) switching time t 2½0; T, proposed model by a penalization term c uðtÞ, c  0. SD SD which is characterized by the equation Overall, this yields the supply chain’s profit function SC J : u 7! R m1 m H H mc ðx þ xt Þ ða  c  c ðx þ xt Þ Þ c 0 0 M 0 0 SD Z ¼ : m 2 T 2 ða  c  c xðtÞ Þ  r 2b ðt  TÞ M 0 J ðu; x Þ :¼  c uðtÞdt T 0 SD 4b ð9Þ ð4Þ In the following, (9) is called switching condition. Indeed, since the cost function is (strictly) convex and the system for a given time interval [0, T], which must be maximized subject to the control constraints 0  uðtÞ x, t 2½0; TÞ, dynamics are governed by a linear ordinary differential equation, it can be shown that this condition is necessary and the system dynamics (3). The contract period T is of particular interest since investments into the cost structure and sufficient for the considered problem, see [36] for a detailed derivation. We emphasize that the switching of the supply chain require their amortization during the runtime of the contractual agreement. A summary of the time t characterizes the optimal time of collaboration parameters is given in Table 1. since every investment in supplier development up to t results in an increased profit while expenditures spent after t do not amortize during the contract period and are, thus, not economically reasonable within the considered setting. The optimal value function V ðx Þ of the problem under T 0 Because supplier development is most often used as of the end of the growth stage as opposed to initial stages of a product’s life cycle, consideration reads we consider solely the learning that occurs through the cumulative SC V ðx Þ :¼ sup J ðu; x Þ T 0 0 number of realized supplier development projects without considering T u2L ð½0;TÞ;½0;xÞ further effects, e.g., total number of units produced [5, 30]. 123 Logist. Res. (2016) 9:14 Page 5 of 12 14 cooperate with a different supplier instead of adhere to the x 10 already existing business relation, see, e.g., [12] and [36] for the considered setting with multiple suppliers. Here, however, it is supposed that continuation of the collabo- ration is preferable since our focus is on the arrangement of the manufacturer/supplier cooperation. Hence, Option 1 corresponds to the scenario, in which supplier development cannot increase profitability within the supply chain and the cooperation with another supplier acting on the market is also not economically reasonable. Hence, we focus on the second case within this paper. Here, from the specific structure (8) of the optimal λ(t) SD control function we can conclude that all investments up to 0 10 20 30 40 50 60 H time t pay off during the contract period. Then, taking Time t into account the already reduced supply costs given by c ðtÞ¼ r þ c xðt Þ with SC 0 Fig. 1 The adjoint k : ½0; T! R computed based on the param- eters given in Table 1 Z m m H H H c xðt Þ ¼ c x þ u ðsÞdt ¼ c ð1 þ xt Þ ; 0 0 0 0 where the expression on the right-hand side is maximized further effort in terms of uðtÞ [ 0, t 2½t ; TÞ, does not lead subject to x_ðtÞ¼ uðtÞ, xð0Þ¼ x . V : R ! R maps the 0 T [ 0 to an increased profit. The latter holds true since cost-re- initial value x to the optimal value. The index T indicates duction efforts after t do not amortize within the the contract period and can be considered as a parameter— remaining time interval of at most length T  t and are, an interpretation, which is crucial for the upcoming thus, not economically reasonable. We show that a pro- analysis. longation of the contract period yields an augmentation of Evidently, investments (in the cost structure) pay off in the investments in supplier development, which corre- the long run: while all the effort is spent directly at the sponds to an increased switching time t . A proof of beginning of the collaboration, the resulting cost decreas- Lemma 1 is given in ‘‘Appendix 8’’. ing effect is exploited during the remainder of the contract period. Lemma 1 Suppose that the contract period T is chosen H H (long enough) such that t ¼ t ðTÞ [ 0 holds. In addition, Remark 1 At the switching time t , the marginal revenue let the condition of further investments in supplier development (given by the adjoint variable k) equals the marginal costs (given by ð1  mÞða  c  c Þþ c m  0 ð10Þ M 0 0 c ) as indicated in Fig. 1. This reasoning is expressed by SD hold. Then prolonging the contract period T, T [ T, im- the switching condition (9). H H plies a strictly larger switching time t ¼ t ðTÞ, H H i.e., t ðTÞ [ t ðTÞ. 4 Interplay of switching time and contract period Remark 2 The assumptions of Lemma 1 imply the inequality a  c  c  r [ 0 as a by-product because the M 0 If the desired contract between manufacturer M and sup- manufacturer cannot realize a profit per unit sold otherwise plier S ranges over the interval [0, T], two cases can be (prohibitive price is greater than the production cost per distinguished: unit at time t ¼ 0 from the manufacturer’s point of view). Hence, the seemingly technical Condition (10) links the 1. The (optimal) switching time is given by t ¼ 0 supplier’s production costs c with the difference of profit meaning that investments in supplier development do per unit a  c  c by the learning rate m. Note that the M 0 not pay off during the contract period. assumptions of Lemma 1 can be easily verified for a given 2. A switching time t [ 0 represents the scenario where dataset of parameters. investing into supplier development amortizes during Lemma 1 shows that investments in supplier develop- the contract period. ment are extended if the contract period is prolonged. After determining the outcome of a potential collaboration Hence, the collaboration continues after the previously over the interval [0, T], the overall market situation has to determined switching time t . As a result, the supplier’s be taken into account, e.g., does it make (more) sense to Adjoint variable λ(t) 14 Page 6 of 12 Logist. Res. (2016) 9:14 Fig. 2 Optimal switching x 10 H H 13 time t ¼ t ðTÞ in dependence of the length of the contract T ¼ 3.5 T þ i  DT (T ¼ 60, DT ¼ 3 and i ¼ 0; 1; .. .; 7) 2.5 1.5 60 65 70 75 80 0 5 10 15 20 Switching time t Contract period T SC H unit production costs are further decreased, the quantity i.e., J ð; xðDTÞÞ is considered. Since DT  t holds by offered is increased and the supply chain profit per time assumption, the new initial state xðDTÞ is given by unit grows. The argument that a longer contract period Z DT leads to larger switching times can also be validated ð11Þ xðDTÞ¼ xð0Þþ u ðsÞdt ¼ x þ DT  x numericallyasvisualizedinFig. 2. Here, we observe that the supply costs c ðtÞ¼ r þ c xðtÞ are further reduced SC 0 in view of Property (8). Hence, the profit on the new if both the manufacturer and the supplier agree on a contract period ½DT; T þ DT is determined by maximizing longer contract period. The relation between the contract m 2 T 2 ða  c  c x ~ðtÞ Þ  r H M 0 period T and the optimal switching time t ðTÞ is almost J ðu; xðDTÞÞ ¼  c uðtÞdt T SD 4b linear. In summary and according to the initial question how subject to uðtÞ2½0; x, t 2½0; TÞ and the differential does the contract period, i.e., planning horizon, affect equation (3) with initial condition x ~ð0Þ¼ xðDTÞ¼ firms’ willingness to commit relationship-specific resources x þ xDT. Here, we used the notation x ~ to distinguish the to supplier development, the findings show that the supply previously computed (state) trajectory xð; x Þ and its chain partners’ incentives to commit relationship-specific counterpart x ~ð; xðDTÞÞ depending on the new initial con- resources, i.e., to invest in cost-reduction efforts, critically dition xðDTÞ. Another option is to use the time invariance depend on the length of the contract period. of the linear differential equation x_ðtÞ¼ uðtÞ, which allows to rewrite the profit functional as TþDT m 2 5 Successive prolongation of the contract period ða  c  c xðtÞ Þ  r M 0 c uðtÞdt SD 4b DT The benefits of an increased switching time come along with initial value xðDTÞ given by (11) at initial time DT. with the inflexibility resulting from long-term contracts. In We point out that the resulting trajectory deviates from the this section, we propose a methodology for assisting supply previously computed one already before time T. In con- chain partners in contract negotiations to achieve the clusion, the implemented control strategy on ½0; T þ DTÞ is benefits of long-term contracts while committing them- given by selves only to agreements of a certain, prespecified (col- laboration) time period. To this end, it is assumed that the H SC u ðtÞ maximizing J ð; x Þ t 2½0; DTÞ manufacturer and the supplier are only content to make uðtÞ :¼ ; H SC u ðtÞ maximizing J ð; xðDTÞÞ t  DT contracts of length T. If the collaboration is successful for a ð12Þ certain amount of time ½0; DTÞ, DT  t , they might agree to renew the contract on the time interval ½DT; T þ DT. i.e., the first piece of the old policy concatenated with the Before we continue the discussion, let us briefly sketch newly negotiated strategy. This strategy yields an optimal the computation of the (optimal) control func- policy on the time span ½0; T þ DTÞ. Hence, the same tion u : ½DT; T þ DTÞ!½0; x. Here, the profit function overall supply chain profit is reached without the hazards has to be maximized based on the new (initial) state xðDTÞ, of being committed already at the beginning (time 0) as shown in the following corollary. Indeed, the slope of the curve is slightly increasing. SC J (u; x ) T +i·ΔT t (T ) Logist. Res. (2016) 9:14 Page 7 of 12 14 Corollary 1 Let the optimal switching time t deter- mined by Condition (9) be strictly greater than zero. Fur- thermore, let DT, DT\t , be given. Then, the control strategy defined in (12) and the corresponding supply chain profit on ½0; T þ DT equal their counterparts obtained by maximizing J ðu; x Þ with respect TþDT 0 to u : ½0; T þ DTÞ!½0; x Proof Since the profit J ðu; x Þ on the considered TþDT 0 time interval ½0; T þ DT with u from (12) is the sum of 0 3 6 9 12 15 m 2 DT 2 ða  c  c xðtÞ Þ  r M 0 Planed collaboration interval c x dt SD 4b Fig. 3 Application of Algorithm 1 to compute the optimal switching and times for T ¼ 60 and changing initial conditions x ^. The lengths of the collaboration intervals are decreasing m 2 TþDT 2 ða  c  c xðtÞ Þ  r M 0 þ  c uðtÞdt; SD 4b DT previously described steps are repeated, which is referred the dynamic programming principle yields the equality to as receding horizon principle. Note that since the underlying system dynamics are time invariant, the newly J ðu; x Þ¼ V ðx Þ; TþDT 0 TþDT 0 (measured) initial state x ^ represents all information which completes the proof. h required. In particular, no knowledge regarding the previ- ously applied control is needed to solve the adapted 5.1 Receding horizon control switching condition of Step (2) with respect to t . Figure 3 illustrates the outcome of Algorithm 1 with prediction The idea of an iterative prolongation of collaboration horizon T ¼ 60 (contract period) and control hori- contracts can be algorithmically formalized as receding zon DT ¼ 3 (time step) based on the parameters given in horizon control (RHC) also known as model predictive Table 1. control. Upon start, the manufacturer M and the supplier S agree Firstly (t ¼ 0), the original optimal control problem is H H on a collaboration for a given contract period of length solved resulting in t  9:21. Then, u  x is applied on T. Firstly, the status quo—represented by x ^—is analysed. the time interval ½0; DTÞ. Secondly (t ¼ DT), the collabo- Secondly, the optimal switching time t is computed based ration is prolonged to t  9:74. Thirdly (t ¼ 2DT), the H H on the initial state x and T, cf. Step (2). This yields the switching time is shifted to t  10:27. Still, t ¼ 3DT  t optimal control strategy defined by (13), of which the first holds. Hence, the (measured) initial state x ^ is given by piece u j is applied. Then, the manufacturer and the x þ tx ¼ x þ 3DTx. Here, Step (2) of Algorithm (1) ½0;DTÞ 0 0 supplier meet again at time t þ DT to negotiate a new yields t  10:79, i.e., the collaboration stops within the contract. This initiates the process again, i.e.. the time frame ½t; t þ DTÞ. If the RHC scheme is further Time t 14 Page 8 of 12 Logist. Res. (2016) 9:14 applied, there occur collaboration intervals of shrinking 1. set t ¼ t in order to save negotiation costs, which length. would probably outweigh the achievable earning As already discussed in Sect. 5, if the contract is not growth. For the presented example, the supplier H H renewed, u ðtÞ is set to zero for t  t  9:21. In contrast development programme stops at 10.79 (still an increase of approximately 17.2 %) if the threshold is 1. to that, the RHC scheme prolongs the collaboration and, thus, increases the supply chain profit. To be more precise, 2. measure the current state x ^ ¼ xðtÞ and compute the optimal cost structure for contract periods of length T the profit generated by Algorithm 1 on ½0; T þ iDT, i 2f0; 1; 2; ...; T=DTg, by solving m1 m T=DTþi1 ðkþ1ÞDT m 2 2 mc Tx  ða  c  c x  Þþ 2bc ¼ 0 0 M 0 SD ða  c  c xðtÞ Þ  r M 0 c uðtÞdt SD 4b kDT k¼0 with respect to x . Then, set t ¼ t þðx  x ^Þ=x. In the considered example at time t ¼ 4DT, the measured H H is greater than its counterpart J ðu ; x Þþ V ðx ðTÞÞ T 0 iDT state is x ^ ¼ 10:79 while x   11:18. Hence, a collabo- consisting of the maximum of the original cost func- ration of length 0.39 time units is fixed. At all tion V ðx Þ¼ J ðu ; x Þ and a second (optimally oper- T 0 T 0 upcoming time instants, t ¼ t holds because the ated) contract on ½T; T þ iDT based on the reached cost optimal cost structure for contract periods of H H structure represented by x ðTÞ¼ x þ t x  x þ 0 0 length T ¼ 60 is already reached. 9:21x ¼ 10:21. In particular, this assertion holds in com- Clearly, the threshold should be chosen such that the profit parison with simply sticking to the cost structure based on increase outweighs the negotiation costs. t ðTÞ, i.e., Thus, Algorithm 1 allows both the manufacturer and the m 2 TþiDT H 2 ða  c  c xðt ðTÞÞ Þ  r M 0 supplier to prolong their supplier development programme J ðu ; x Þþ dt: T 0 4b without binding themselves for a time span longer than T and, thus, provides more flexibility. ð14Þ Remark 3 Algorithm 1 is a simplified version. Indeed, the While an increased switching time t may already time step DT may vary in time, e.g., longer time steps in increase the profitability within a supply chain during the the beginning (for example, DT ¼ t in the considered considered time span, the achieved cost reduction sus- setting), and shorter ones later on. For details on the so- tains. Hence, if the collaboration between the manufac- called time-varying control horizon, we refer to [15]. turer and the supplier lasts, the obtained effect is a sustainable one. In summary and with regard to the question how should In summary and referring to the question how does receding horizon control be arranged to optimize supply receding horizon control offer a straightforward method chain profit, two strategies are presented in order to make for dynamically extending the planning horizon, the find- the proposed receding horizon scheme, cf. Algorithm 1, ings show that dynamically extending contracts enhance applicable even if negotiation costs are taken into account. the supplier development process, because value genera- tion is facilitated while both the manufacturer and the supplier gain flexibility due to shorter contract periods. 6 Numerical results As seen in the previous section, applying the receding 5.2 Optimal point of collaboration horizon Algorithm 1 dynamically extends the collaboration within the supply chain and, thus, generates additional As observed in Fig. 3, the collaboration can stop within the profit within the supply chain. Next, we conduct a time interval ½t; t þ DTÞ meaning that the prerequi- numerical case study to obtain further managerial insights. site DT  t is no longer satisfied at time t. This leads to a HH To this end, we compare the outcome J of the pro- sequence of collaboration times of shrinking length. posed algorithm based on the second option presented in Summing up all of these intervals on the infinite horizon Sect. 5.2 and the supply chain profit resulting from the yields a total collaboration time of approximately 11.18 control time units. Hence, the total collaboration time is increased by 21.3 %. However, since the collaboration intervals are H x for t\t ðTÞ uðtÞ¼ ð15Þ becoming comparably short, implementing this strategy 0 for t  t ðTÞ may be impracticable. Here, we propose two remedies: If the new collaboration period at time t ¼ kDT, i.e., t  t, on the time interval ½0; 2T¼ ½0; 120. The control pol- is below a certain threshold value, icy (15) results from the basic optimal control problem 123 Logist. Res. (2016) 9:14 Page 9 of 12 14 considered on [0, 60] and, then, utilizing the achieved cost Second, we are interested in the interplay of the sup- plier’s learning rate m and receding horizon control. Thus, structure c ðtÞ¼ x þ t x on [60, 120] without further SC 0 investments in supplier development. The corresponding based on the parameters of Table 1, we perform a sensi- tivity analysis with respect to the parameter m with profit is given by (14). To fully understand the impact of receding horizon m 2f0:15; 0:14; 0:13; 0:12; 0:11; control on the supply chain profit in depth, we first vary the 0:1; 0:09; 0:08; 0:07; 0:06; 0:05g: following parameters of Table 1 Applying Algorithm 1 (T ¼ 60, DT ¼ 3), Fig. 5 shows a 2f192:5; 195; 197:5; 200; 202:5; 205; 207:5g; both the optimal switching time t (without receding b 2f0:007; 0:008; 0:009; 0:01; 0:011; 0:012; 0:013g; horizon control) compared to the optimal switching time c 2f70000; 80000; 90000; 100000; 110000; 120000; 130000g; SD HH t (with receding horizon control) in dependence of x 2f0:7; 0:8; 0:9; 1; 1:1; 1:2; 1:3g; m (left), and the profit growth with respect to the switching m 2f0:13; 0:12; 0:11; 0:1; 0:09; 0:08; 0:07g time for different learning rates (right). Again, the com- 5 putations are based on a simulation of 120 time units. Here, resulting in a total number of 7 = 16,807 instances. For we observe that the impact of receding horizon control each parameter combination, we then evaluate the respec- decreases for lower learning rates. tive profits. Hence, the results infer that especially firms in high- The depicted histogram in Fig. 4 shows the absolute learning industries, e.g., technology-based industries, ben- frequency with which a percentage of profit increase is efit most from applying the proposed receding horizon observed within our parameter set. The mean value scheme. is 3.36 % with a standard deviation of 1.06 %. In conclu- sion, receding horizon control significantly improves the profitability of the considered supply chain. 7 Conclusion In this paper, we investigated the impact of the contract period on supplier development. In particular, we showed that the supply chain partners’ incentives to commit rela- tionship-specific resources, i.e., to invest in cost-reduction efforts, critically depend on the length of the contract period. Given the fact that long-term contracts entail certain risks, we proposed a receding horizon control scheme to mitigate possible contractual hazards. In addition, we showed that dynamically extending contracts enhance the supplier development process, because value generation is 0 2 4 6 8 facilitated while both the manufacturer and the supplier Profit increase ratio (%) gain flexibility due to shorter contract periods. Further- Fig. 4 Profit increase ratio in percent more, we presented two strategies in order to make the Fig. 5 Optimal switching time x 10 H HH 5 t and t with respect to the t (m) parameter m (left) and earning t (m) growth with respect to the H 15 switching time t for different values of m (right) 0 5 10 15 20 −0.14 −0.12 −0.1 −0.08 −0.06 Parameter m Switching time t Number of instances Switching time Earning growth 14 Page 10 of 12 Logist. Res. (2016) 9:14 m1 m H H H H proposed receding horizon scheme, cf. Algorithm 1, fðt Þ :¼ðT  t Þzðt Þ ða  c  c zðt Þ Þ: M 0 applicable even if negotiation costs are taken into account. m2 0 H H Finally, we verified the reliability of the application by Then, the term f ðt Þ zðt Þ is a sum consisting of the H H performing Algorithm 1 for an extensive parameter set and positive summand zðt Þða  c  c zðt Þ Þ and M 0 demonstrated that receding horizon control leads to a sig- m m H H H ðT  t Þxð1  mÞða  c  c zðt Þ Þþ c mzðt Þ : M 0 0 nificant profit increase within the supply chain. Moreover, by means of a sensitivity analysis with respect to the Here, it was used that a  c  c  r [ 0 holds. Hence, M 0 learning rate, we showed that especially firms in high- we investigate the term learning industries benefit since supplier development m m H H programmes play a predominant role in order to optimize ð1  mÞða  c  c zðt Þ Þþ c mzðt Þ ð17Þ M 0 0 the cost structure of the supplier network. in order to determine the sign of the second summand using The study is based on a simple model to focus on the impact of dynamical decision-making in supplier devel- that ðT  t Þx [ 0 holds. To this end, the supply chain profit p :¼ a  c  c [ r [ 0 per unit plays a major opment. Clearly, a more elaborated model with less strin- M 0 gent assumptions like, e.g., a linear price distribution, role: (17) equals should be studied in the future. Moreover, the combination m m H H c ð1  mÞp=c þ mzðt Þ þð1  mÞðc  c zðt Þ Þ 0 0 0 0 of the proposed dynamic strategy with decentralized |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} approaches is of great importance and deserved a detailed analysis, see, e.g., the negotiation-based coordination because m\0 and t  0 hold. Positivity of the first sum- mechanism proposed in [36]. Another interesting direction mand is ensued from (10). Hence, (17) is positive and, for future research is to expand our study to a network thus, f is (strictly) decreasing. perspective, in which the supply chain consists of more In conclusion, the left-hand side of (16) is strictly than a single manufacturer and a single supplier, see, decreasing in t and strictly increasing in T. 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Logistics Research – Springer Journals

Published: Jul 22, 2016

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