A Supplier Selection Model with Quality-Driven Demand and Capacitated Suppliers

Transcription

1 A Supplier Selection Model with Quality-Driven Demand and Capacitated Suppliers Abstract This paper considers an OEM who outsources to external suppliers when faced with capacity shortage. To explore how the external suppliers quality affects the OEM s outsourcing strategy as well as its downstream demand, a quality-driven demand function is formulated and imbedded into a finite-horizon model on supplier selection and order allocation decisions. We study a two-period model and find it is optimal for the OEM to select suppliers according to their per unit effective procurement cost in the second period, which is defined as the sum of purchase cost and quality cost, and identify the unimodality of the second period value function. n our numerical example, we find the OEM sometimes is better off by not meeting all the demand, even when the supply pool has enough capacity since the benefit of fulfilling all demand could be offset by the future demand decrease caused by products outsourced from low quality suppliers in the current period. Due to the complexity of the first period problem, we characterize the optimal policy under some special cases when certain conditions are met. We also study the multiple period problem using numerical examples to develop a Demand-Driven Policy and to offer some managerial insights on the outsourcing strategy. Keywords: supplier selection; quality-driven demand; Taguchi loss function. 1

2 1 ntroduction Firms in various industries have increasingly considered outsourcing as a strategic option to reduce cost and enhance core competences. Especially when facing supply shortage problem triggered by insufficient production capacity, the original equipment manufacturers (OEM) have conventionally coped with the issue by relying on outsourcing to external suppliers. n January 2002, BM outsourced the manufacturing of its NetVista desktop computers to Sanmina-SC as part of a broader strategy to reduce both fixed and variable costs (Delattre et al., 2003). This practice supported BM s strategy of keeping PCs as an important element of their e-business infrastructure offerings, while making its business even more cost-competitive in the marketplace. However, due to less control on the external supplier s quality, the goal of short term cost reduction had exposed the OEM to the risk of overall quality issues in the long run (Chen et al., 2015). Our research investigates the OEM s outsourcing decision in the presence of multiple external suppliers under a two-level supply chain setting. The OEM has a limited capacity and thus frequently encounters the unfulfilled demand from the retailer. Facing this situation, the OEM may consider either capacity expansion or outsourcing to external suppliers so as to alleviate supply shortage. n comparison with capacity expansion, outsourcing has the following major attractions: (1) lower upfront investment; (2) lowcost labor markets to gain more competitive advantages; and (3) flexibility to meet the changing business condition. Outsourcing, on the other hand, may bring risks to the OEM as external suppliers, whether domestic or foreign, get introduced to the existing supply chain. The quality of the products/parts outsourced from a selected supplier has a direct impact on the overall quality of the delivered products, which would significantly affect the demand of the retailer in the following periods. n this paper, we construct a two-period nonlinear programming model that incorporates the effect of suppliers quality to the OEM s future demand through a quality-driven demand function with the goal to understand how the quality level of the products outsourced would affect the OEM s outsourcing strategy. Our model is discussed un- 2

3 der two scenarios: n-house Priority and Non-Priority. Under the n-house Priority scenario, the OEM prioritizes using it own capacity first to fulfill demand while under Non-Priority scenario, the OEM regards itself as just one supplier among its supplier pool. Under each scenario, the OEM makes decision on both supplier selection and order splitting among those selected suppliers. We study a two-period model and show that in the last period it is optimal for the OEM to select suppliers based on a single parameter, the effective purchase cost during the second period, which is defined as the sum of purchase cost and quality related cost. We call this supplier selection approach Effective Procurement Cost Based Policy. We identify the unimodal structure of the second period value function and incorporate this structural result into the consideration of the first period problem. However, we see that the Effective Procurement Cost Based Policy no longer ensures profit maximization, and that the long-term effect of quality on demand plays a more significant role in the supplier selection procedure. Due to the complexity of the first period problem, we characterize the OEM s optimal first-period policy under two special cases when certain conditions are met: (1) when all suppliers quality levels are the same; and (2) when suppliers quality varies and the OEM should fulfill its demand when it has enough capacity. n the numerical study, we expand the planning horizon into four periods to discuss the OEM s capacity investment decision towards its supplier pool. We also develop a heuristic method to solve the multiple period problem, under which the OEM assigns a set of quality ratios to generate a parameter that weights the purchase cost and quality cost for each supplier. We compare this method with a rolling horizon method to generate more managerial insights into this research problem. The remainder of this paper is organized as follows: Section 2 provides a review of the existing literature on products quality issues in firms outsourcing decision making. Sections 3 introduces the quality-driven demand function and proposes the multiple period model for the OEM s supplier selection problem. n Section 4 we presents the structural results for the two-period models. The numerical study of the multiple-period model is included in Section 5 and the final section gives a conclusion as well as some 3

4 potential research extensions of this study. 2 Literature Review The OEM s outsourcing strategy has been studied extensively from the economic and supply chain perspective. Gunasekaran et al. (2015) present a review of the literature on outsourcing with particular reference to Performance Measures and Metrics (PMMs) used in arriving at outsourcing decisions and provide a list of specific tools and techniques for PMMs in outsourcing. We identify among those potential determinants of outsourcing that cost-saving and the impact of supplier-quality are the OEM s top concerns, which are jointly considered in this paper. Arya et al. (2008) study the make-or-buy decision in the presence of a monopolistic input supplier who also serves the firm s retail rival and show that the firm would buy an input for a price above its in-house production cost that limits the incentive the supplier would otherwise have to provide the input on particularly favorable terms to the retail rival. Grahovac et al. (2015) find that outsourcing can help firms avoid over-investment and specify what types of insourcing, outsourcing, and non-sourcing behaviors are likely to emerge in different parts of the parameter space, while suppliers quality are not considered in these work. Elmaghraby (2000) states in an overview of firms sourcing issues that quality is the most important factor that should be incorporated into the supplier selection process. Chen et al. (2015) study a cooperative quality investment (CQ) strategy and proposes a simple proportional investment sharing schedule in the outsourcing of a supply chain. Zhu and Zhang (2007) explore the roles of different parties in a supply chain in quality improvement and show that the buyer s involvement can have a significant impact on the profits of both parties and of the supply chain as a whole. Bae et al. (2010) investigate how the competitors quality strategy traits would affect an OEM s outsourcing strategy when it is facing both quality and cost pressures. They show that an OEM s outsourcing decisions are sensitive to market expectation and the quality performance of competitors at the same time. Different from these works, we focus on the OEM s outsourcing 4

5 strategy through the construction of a quality-driven demand and incorporate it into the OEM s joint supplier selection and order allocation problem with quality-driven demand and limited capacity of the OEM and the potential suppliers. Supplier selection has also been receiving considerable attention from researchers that study firm s outsourcing strategy. Various decision making techniques have been applied in supplier selection literature. Ho et al. (2010) review the literature of the multi-criteria decision-making (DM) approaches for supplier evaluation and selection based on journal articles published from 2000 to Chai et al. (2013) later provide a systematic review of literature published from 2008 to 2012 on the application of decision-making techniques for supplier selection. They identify those DM techniques from three perspectives: (1) Multi-criteria decision making techniques; (2) Mathematical programming (MP) techniques; and (3) Artificial intelligence techniques. According to this classification, our work is to be positioned in second category as we construct a nonlinear programming (NLP) model and apply dynamic programming techniques to study supplier selection problem. n the literature, NLP technique is applied in two directions by researchers to study supplier selection process. First is the simple utilization of NLP as a decision tool. Related literature includes Hsu et al. (2010) and Razmi et al. (2009). The second direction is to use Mixed nteger NLP formulations. Related literature includes Zhang and Ma (2009), Yeh and Chuang (2011), Rezaei and Davoodi (2012) and Chen and Li (2008). Besides NLP, other MP techniques applied to supplier selection problems include Data Envelopment Analysis, Linear Programming, Multiobjective Programming, Goal Programming and Stochastic Programming. Related literature applying these techniques can be found in Chai et al. (2013). Some literature apply MP techniques to jointly study supplier selection and order splitting problems. Ghodsypour and Brien (2001) consider supplier selection and ordersplitting problems jointly under a constant annual demand and fixed ordering costs. Qi (2007) studies the OEM s problem when it is facing a price-sensitive demand and capacitated suppliers. Ekici (2013) presents an improved model for supplier selection based on the work of Ghodsypour and Brien (2001) under capacity constraint and multiple criteria. 5

6 Feng et al. (2011) propose a multi-objective model for selecting a pool of suppliers for the provision of different service process/product elements. They apply collaborative utility concept between partner firms for supplier selection. While all of these work construct static models and do not consider the OEM s decisions from multiple period perspective, Razmi and Rafiei (2010) develop a multiple-period dynamic mixed integer NLP model to make order allocation decisions. Mafakheri et al. (2011) propose a two-stage multiple criteria dynamic programming approach combined with analytic hierarchy process method to solve supplier selection and order allocation problems. Ustun and Demirtas (2008) apply an integrated approach of analytic network process and multiobjective mixed integer linear programming to study these problems over multiple periods. None of these work, however, consider the effect of suppliers quality on the OEM s subsequent demand and outsourcing decision. Our work supplements the existing studies on supplier selection and order splitting by considering the interplays of supplier s product quality and OEM s capacity and outsourcing strategy. n our work, we apply Taguchi Loss Function to measure the overall product quality, which enables us to incorporate the quality factor into supplier selection problem. Some studies apply Taguchi Loss Function as a channel for OEMs supplier selection decisions. For example, Pi and Lao (2006) provide a method for quantifying the supplier s attributes to quality-loss using Taguchi loss function and transfers it into a variable for decision-making by an analytical hierarchy process (AHP). Teeravaraprug (2008) proposes a supplier selection model that applies Taguchi Loss Function to measure non-cash impacts. Ordoobadi (2009) proposes a procedure to help the decision maker to rank potential suppliers based on their aggregate loss score computed using Taguchi Loss Function. But none of these works applied Taguchi Loss Function to supplier selection and order splitting decision problem in the multiple-period dynamic scenario. 6

7 3 The Model We consider an OEM who operates under the mass production to manufacture standardized product and makes production and outsourcing decision for a multiple-period planning horizon. At the beginning of each period, it faces a deterministic demand from downstream retailers that depends on its overall product quality that customer experience during the last period. The OEM has a marginal production cost c and a lost-sale penalty cost g for each unit of unsatisfied demand. The OEM sells its products to retailers at a per unit wholesale price w and does not accept backorders, which can be justified by the presence of high fixed backlog cost. (Pekelman, 1974) The OEM has a supply pool of suppliers. Supplier i has a capacity of u i and charges for p i per unit. Each supplier s per unit quality loss E(L i ) that are quantified using the mean and variance of the product s defective rate. When mass production is assumed, based on Central Limit Theorem, defective rate d m follows a normal distribution with mean µ qm and variance σq 2 m when items are produced by the OEM and with mean µ qi and variance σq 2 i when they are outsourced to supplier i. We call p i + E(L i ) as the effective procurement cost of supplier i as it is the procurement price paid to a supplier plus associated quality cost. We assume the effective procurement cost is no higher than the wholesale price changed by the OEM, i.e., p i + E(L i ) w. The overall quality of the products in period t is captured by a weighted average loss of quality L t that is evaluated using Taguchi Loss Function, which measures quality loss by the square of deviation from the target value. L t is measured based on the OEM s own per unit quality loss E(L m ). We present the details on the generalization of the quality driven demand in Section 3.1. Our analysis first focuses on the n-house Priority Scenario where the external supplier capacity will be utilized only when the manufacturer s own capacity is used up, which is based on one of the motivations for outsourcing, capacity shortage. We present a further discussion of our model in Section 4.2 when the OEM is applying Non-Priority Scenario where it has no priority of using up its in-house capacity before 7

8 outsourcing to other suppliers. We define C t and y it as the OEM s in-house production and outsourcing quantity to external supplier i during period t. The level of y it should be chosen, after considering quality loss, within each supplier s own capacity level u i. The total outsourced quantity y it should not be greater than the total shortage. The OEM s problem is to select suppliers to outsource from and to decide the outsourcing quantity for each supplier in the pool. 3.1 Quality-Driven Demand and Measure of Quality We apply a price-and-quality-dependent demand function as presented from Xu (2009). The two most common forms of this type of demand function are defined in multiplicative or additive fashion. Since quality effect is of major interest in this study, we assume p to be exogenous and it is just used as an instrument to measure demand. Thus we define the quality-driven demand as a function of D t (L t ), which decreases with respect to the quality level L t. The overall quality of the product, in our paper, is measured by the average loss incurred due to imperfect quality. The expected value of the quality loss per unit, when the products are produced by the OEM, E(L m ), is shown in Equation (1) as in Teeravaraprug (2008): E(L m ) = r qm (µ 2 q m + σ 2 q m ). (1) The expected value of loss per unit when products are produced by each supplier i is E(L i ) is: E(L i ) = r qi (µ 2 q i + σq 2 i ) (2) The r qm and r qi in both Equations (1) and (2) are the coefficients of the quality loss functions for the OEM and the external supplier i, respectively. Both coefficients are the constants obtained by the corresponding costs of repair/replacement and the width of the specification limits. The loss function L t is a weighted average of the expected quality loss for the OEM and external suppliers evaluated at the beginning of each period t using Taguchi Loss Function. The L t is defined as follows: 8

9 C t 1 E(L m ) + y i,t 1 E(L i ) L t = (3) C t 1 + y i,t 1 The weighted average quality loss, as can be seen in the above equation, is obtained by multiplying the production quantity share for the OEM and external suppliers by expected quality loss per unit correspondingly. 3.2 The OEM s Supplier Selection and Order Allocation Problem Considering a finite planning horizon, the OEM s profit maximization problem involves both the selection of suppliers to outsource products to as well as the order allocation to each selected supplier during each period. ts demand during period t depends on the overall product quality at the beginning of that period, which is measured by the weighted average loss of quality L t. Since L t evolves in a nonlinear fashion as depicted in Equation (3), the OEM s problem is a nonlinear programming (NLP) problem: max {y it },...,;t=1,...,t T { (Ct + t=1 [D t (L t ) C t s.t. C t + L 1 = E(L m ); y it )w C t c y it ] + g y it D t (L t ), t = 1,..., T; y it p i y it E(L i ) } (4) 0 C t u m, t = 1,..., T; 0 y it u i, i = 1,...,, t = 1,..., T; To explore how product quality evolves and affects the OEM s decision for each period, we construct this NLP problem into a dynamic programming model and define the state variables in period t as {L t }. We define the value function V t (L t ), the OEM s maximized aggregate total profit from period t to the last period of the planning horizon, 9

10 in a recursive structure for the dynamic programming model in Equation (5). For t T: V t (L t ) = max {y it },2,..., {(C t + [D t (L t ) C t s.t. C t + y it )w C t c y it ] + g y it E(L i )} + αv t+1 (L t+1 ) L 1 = E(L m ); y it D t (L t ); y it p i (5) 0 C t u m ; 0 y it u i, i = 1,...,, t = 1,..., T; Define V T+1 = 0; The state transition follows L t = C t 1 E(L m )+ y i,t 1 E(L i ) C t 1 + y i,t 1, for t = 2, 3,..., T. The value function (5) consists of five parts: the revenue from selling both products that are manufactured in house and outsourced to the suppliers at a pre-specified wholesale price w; the cost of in-house production and outsourcing procurement; a penalty cost for unsatisfied demand; an additional cost for quality loss, e.g., the loss of goodwill; and the discounted next period value function V t+1 (L t+1 ) where parameter α represents the discount factor. The first constraint ensures the total quantity of products the OEM obtains during each period, either produce in-house or outsource from suppliers, would not exceed its demand level during that period. The next constraint specifies that the initial state L 1 be equal to the OEM s per unit quality loss. The third constraint is essentially the set of capacity constraints for each supplier in the supply pool. 10

11 4 Structural Results for the Two-Period Problem 4.1 n-house Priority Scenario n this section, we discuss the structure of a two-period problem to explore how product quality influences the OEM s outsourcing decisions under a n-house Priority Scenario. This scenario specifies that the OEM should use up its own capacity first, before turning to external suppliers, to fulfill the demand. This scenario is always preferable when the OEM has the capacity to produce the item more cost-effectively than, and with comparable quality as, external suppliers. Even when this is not the case, it may still be advisable to implement this policy, if we consider other factors, such as idle time costs, etc. When the demand exceeds the OEM s capacity, then the remaining unmet demand would be filled by external suppliers. Given that the OEM has capacity shortage at the beginning of the first period, the value function for a two-period planning horizon can be expressed as: V 1 (L 1 ) = max {C 1 (w c) [D 1 (L 1 ) C 1 ]g {y i1 },2,..., + s.t. C 1 + y i1 [w + g p i E(L i )]} + αv 2 (L 2 ) y i1 D 1 (L 1 ); C 1 = min{u m, D 1 (L 1 )}; (6) The state transition is given as: 0 y i1 u i, i = 1, 2,..., ; L 2 (y i1,..., y 1 ) = C 1 E(L m ) + y i1 E(L i ) (7) C 1 + y i1 We solve backwards from the second period value function V 2 (L 2 ) to derive the OEM s decision on each external supplier s share in the total outsourcing quantity. When D 2 (L 2 ) is no more than the capacity u m, it is not necessary for the OEM to make 11

12 a contract with any external supplier, and thus y i2 = 0, i = 1,...,. n the case when D 2 (L 2 ) > u m, the OEM s second-period value function can be simplified as: V 2 (L 2 ) = max {u m (w c) [D 2 (L 2 ) u m ]g {y i2 },2,..., + s.t. u m + y i2 [w + g p i E(L i )]} y i2 D 2 (L 2 ); 0 y i2 u i, i = 1, 2,..., ; (8) Theorem 1 When D 2 (L 2 ) u m, the OEM produces in-house only. When D 2 (L 2 ) > u m, the optimal supplier selection policy for the second period problem is to select the suppliers by the ascending order of their effective procurement cost, p i + E(L i ). Starting from the supplier with the lowest effective procurement cost, assign the outsourcing quantity up to the supplier s capacity respectively until the demand is fulfilled or the total capacity from all external suppliers are used up. We name this outsourcing policy as Effective Procurement Cost Based Policy. Algorithm 1 Effective Procurement Cost Based Policy Step 1: Re-index the suppliers such that p 1 + E(L 1 ) p 2 + E(L 2 )... p + E(L ); Step 2: Start with supplier 1, assigning y i1 up to min[u i, D 1 (L 1 ) u m ]; Step 3: Repeat step 2 with the next supplier until the demand is fulfilled or all external capacities are used up; Step 4: Assign the number of the last supplier who is selected in Step 3 to k, which is defined as the number of cooperative external suppliers chosen in the current period; Since V 2 (L 2 ) is a linear function of y i2 with coefficient w + g p i E(L i ) 0, it is straightforward to prove Theorem 1 which is omitted here. Based on the results stated in Theorem 1, we present the detailed solution algorithm for the second-period problem in Algorithm 1. We further show in Theorem 2 that V 2 (L 2 ) is a unimodal function. 12

13 Theorem 2 The second period value function V 2 (L 2 ) monotonically decreases with respect to L 2 in [L 0, + ) and monotonically increases in (0, L 0 ), where L 0 is the quality loss level that satisfies D 2 (L 0 ) = u m + u i. Proof of Theorem 2: First of all, define L as a threshold quality loss that satisfies D 2 (L ) = u m. Upon the acknowledgement of L 2, if L 2 L, it can be implied that D 2 is less than the OEM s capacity u m. Then the optimal value function in the second period is just equal to D 2 (L 2 )(w c), which is a decreasing function with respect to L 2. f L 2 < L, outsourcing becomes necessary. We already derived the optimal policy for single period scenario from Theorem 1. Based on that result, rewrite V 2 (L 2) with respect to D 2 (L 2 ) and the optimal number of suppliers that are selected by the OEM, k (L 2 ), given the state L 2 : V 2 (L 2 ) =C 2 (w c) [D 2 (L 2 ) C 2 ]g k 1 + u i [w + g p i E(L i )] k 1 + [D 2 (L 2 ) C 2 u i ][w + g p k E(L k )] (9) C 2 can be set equal to u m given the OEM is applying the n-house Priority scenario. Besides, both D 2 and k can be observed as implicit functions with respect to L 2. Since a decreasing initial overall quality loss of the product L 2 would result in higher demand D 2, it also implies that k (L 2 ) is non-increasing in L 2. Suppose L 2 drops to L 2, the problem can be discussed in the following three cases: (1) when the initial value of L 2 lies within (L 0, L ] and the drop of L 2 is not large enough to affect the value of k, then: V 2 (L 2) =u m (w c) [D 2 (L 2) u m ]g k 1 + u i [w + g p i E(L i )] + [D 2 (L 2) k 1 u m u i ][w + g p k E(L k )] (10) 13

14 (10) (9) =[D 2 (L 2) D 2 (L 2 )][w + g p k E(L k )] [D 2 (L 2) D 2 (L 2 )]g (11) =[D 2 (L 2) D 2 (L 2 )][w p k E(L k )] Since it has been assumed that p k + E(L k ) < w, in this case we can conclude that V 2 (L 2 ) > V 2 (L 2). k + 1: (2) when the initial value of L 2 lies within (L 0, L ] and the drop of L 2 makes k = This case only occurs when there is no extra capacity left from supplier k and the OEM must order from another supplier in face of the increased level of demand generated by the drop of L 2. To prove that V 2 (L 2 ) > V 2 (L 2) still holds in this case, we set D 2 (L 2 ) = u m + k u i. So, V 2 (L 2 ) =u m (w c) [D 2 (L 2 ) u m ]g + k u i [w + g p i E(L i )] (12) When L 2 drops to L 2 and makes k = k + 1: V 2 (L 2) =u m (w c) [D 2 (L 2) u m ]g + k u i [w + g p i E(L i )] + [D 2 (L 2) u m k u i ][w + g p k +1 E(L k +1 )] (13) (13) (12) =[D 2 (L 2) u m k [D 2 (L 2) D 2 (L 2 )]g u i ][w + g p k +1 E(L k +1 )] (14) =[D 2 (L 2) D 2 (L 2 )][w p k +1 E(L k +1 )] Since the effective procurement cost, (p k +1 + E(L k +1 )) of supplier (k + 1) is assumed to be less than the wholesale price w, then it is proved that V 2 (L 2 ) > V 2 (L 2). 14

15 (3) when the initial value of L 2 belongs to (0, L 0 ] and the value of k is already equal to the total number of suppliers in the pool: Any further drop of L 2 would lead to an excessive demand that can not be satisfied by outsourcing to external suppliers. f all the external capacity has been used up, excessive demand then incurs a penalty cost that needs to be considered. Suppose D 2 (L 2 ) = u m + u i and L 0 has been defined as the quality loss that satisfies D 2 (L 0 ) = u m + u i. f L 2 is further dropped to L 2 : V 2 (L 2 ) =u m (w c) [D 2 (L 2 ) u m ]g + =u m (w c) + u i [w + g p i E(L i )] u i ][w + p i E(L i )] (15) V 2 (L 2) =u m (w c) [D 2 (L 2) u m ]g + u i [w + g p i E(L i )] (16) t can be observed that (15) (16) = [D 2 (L 2 ) u m]g > 0. So now we have proved that V 2 (L 2 ) monotonically decreases in [L 0, + ) and monotonically increases in (0, L 0 ). The result shows that a higher overall product quality resulting from outsourcing decisions in the previous period generates more profits unless the quality-induced demand exceeds the total available capacity of the OEM and all suppliers. One implication of this result is that it does not necessarily pay off to outsource from a supplier, who offers superior quality but often charges a price premium, even though the supplier may help lift the overall quality level. When it comes to the OEM s decisions in the first period, the problem is a lot more complicated. t is challenging to derive the property of V 2 (L 2 ) with respect to the first period decisions {y 11,..., y 1 } due to unimodality of V 2 (L 2 ) and the nonlinearity of L 2 (y 11,..., y 1 ) in terms of each y i1. Specifically, the marginal profit that the OEM 15

16 makes by outsourcing one more unit in the first period to supplier i can be measured by w + g p i E(L i ). But it is hard to measure the outsourcing s marginal effect towards the second period s profit. Each time an additional unit is outsourced to supplier i during the first period, the updated value of L 2 may increase or decrease since both the numerator and denominator in Equation (7) increase at the same time and the direction of change in L 2 depends on which suppliers have been selected and the outsourcing quantity from each of them. As such, it is challenging to come up with a stopping criterion on a supplier selection algorithm. Furthermore, the structure of the optimal policy in a general model could be quite complex and unintuitive. For example, through the numerical study in Section 5.1, we find the OEM is better off not fulfilling all the demand during the first period even though capacity is ample enough. Therefore, we first identify some special cases under which supplier capacity slack and unmet demand never coexist. We consider a case where the product quality levels are the same among all the external suppliers. Define E(L i ) = L a, i. n this special case, applying Effective Procurement Cost Based Policy can still find the optimal solution for the second period while the ranking criteria specified in Step 1 is equivalent to ranking all the suppliers by their procurement costs p i. The overall quality loss of products at the beginning of the second period, L 2, can be expressed as a function of E(L m ), L a and the OEM s total outsourcing quantity to all selected suppliers, y i1 : L 2 = E(L m ) + ( 1 C 1 C 1 + y i1 ) [L a E(L m )] (17) n this special case, we show that the same supplier selection criterion based on per unit procurement cost can be optimally applied in both periods. Moreover, the OEM fulfills demand by using the capacity of his suppliers as much as possible. Proposition 1 Consider the special two-period case where the product quality levels are the same among all the external suppliers which is defined as L a. 16

17 (a) The optimal supplier selection criteria in both periods is to select from the supplier with lowest per unit procurement cost p i, which is equivalent to select based on per unit effective procurement cost p i + E(L i ). (b) Under the following two circumstances, the optimal solution in the first period is to select from the supplier with lowest procurement cost and order up to its capacity level until demand is fulfilled or all suppliers capacity are used up: (1) when E(L m ) L 0, L a E(L m ) and E(L m ) + [D 1(E(L m )) C 1 ] [L D 1 (E(L m )) a E(L m )] L 0 is satisfied; (2) when E(L m ) L 0, E(L m ) L a L 0 and E(L m ) + (1 C 1 is satisfied; C 1 + u i )[L a E(L m )] L 0 Proof of Proposition 1 Part (a): We re-index the suppliers by the ascending order of their effective procurement cost such that p 1 p 2... p. According to Equation (6), selecting from supplier 1 and assigning an order quantity up to his capacity u i until demand has been fulfilled would maximize y i1 [w + g p i E(L i )]. While for V 2 (L 2 ), no matter which suppliers the OEM selects to outsource, it is only affected by the total outsourcing quantity, y i1. t implies that starting from supplier 1 and only considering supplier 1 until his capacity is used up before moving to the next supplier would ensure the maximization of V 1 (L 1 ), which in other words, is the optimal supplier selection criteria for the OEM in the first period. Proof of Proposition 1 Part (b): t has been proved that selecting from the supplier with lowest procurement cost would maximize y i1 [w + g p i E(L i )] given E(L i ) = L a, i. Under the circumstance where E(L m ) L 0, it implies that the initial demand in the first period does not exceed all available capacity. When L a E(L m ), according to Equation (17), the increase of y i1 would cause L 2 to continuously decrease from y i1 = D 1 (E(L m )) C 1. As shown in Theorem (2), V 2 (L 2 ) monotonically decreases with respect E(L m ) to E(L m ) + [D 1(E(L m )) C 1 ] [L D 1 (E(L m )) a E(L m )] which is equal to L 2 evaluated at to L 2 in [L 0, + ) and monotonically increases in (0, L 0 ), where L 0 is the quality loss level that satisfies D 2 (L 0 ) = u m + u i. f E(L m ) + [D 1(E(L m )) C 1 ] [L D 1 (E(L m )) a E(L m )] L 0 is 17

18 satisfied, selecting from the supplier with lowest procurement cost and order up to its capacity level until demand is fulfilled would maximize V 2 (L 2 ) at the same time. Since both of the two parts of V 1 (L 1 ) are maximized, this is the optimal solution for the first period. For the second circumstance where the initial demand in the first period exceeds all available capacity, if suppliers quality level L a lies between E(L m ) and L 0, the increase of y i1 would cause L 2 to continuously increase from E(L m ) to E(L m ) + (1 C 1 y i1 = u i. Knowing V 2 (L 2 ) )[L a E(L m )] which is equal to L C evaluated at u i monotonically increases with respect to L 2 within (0, L 0 ), if E(L m ) + (1 C 1 C 1 + u i )[L a E(L m )] L 0 is satisfied, V 2 (L 2 ) is maximized at y i1 = u i. This implies that selecting from the supplier with lowest procurement cost and order up to its capacity until all external capacity is used up would maximize both of the two parts of V 1 (L 1 ) and is the optimal solution for the first period. Proposition 1 shows that in a special case when their quality levels are the same, it is optimal to select suppliers based on their procurement cost and exhaust a supplier s capacity once it is selected as long as there is unsatisfied demand. However, it is not clear if this holds in a general model with suppliers of different quality level. n Proposition 2, we partially characterize the optimal policy when the demand function is linear by showing that under certain conditions, the optimal quantity allocation strategy is to exhaust one supplier s capacity, once it has been selected, until the demand is satisfied. Proposition 2 Suppose the demand function is D t (L t ) = α βp γl t ; α, β, γ > 0 (18) Assume u m < D 1 (L 1 ) < u m + u i and all demand needs to be fulfilled. Re-index the suppliers such that p 1 + E(L 1 ) p 2 + E(L 2 )... p + E(L ) and define L 2 such that L 2 = u m E(L m )+ k 1 u i E(L i )+[D 1 (L 1 ) k 1 u i ]E(L k ), where k is the total number of suppliers the OEM D 1 (L 1 ) 18

19 outsources to fulfill D 1 (L 1 ) according to the (heuristic) effective procurement cost based policy. f L 2 > L 0 and either E(Li )+p i E(L j ) p j γ exceeds E(L j ) E(L i ) D 1 (L 1 ) [w p1 E(L 1 )] or is no greater than γ D 1 (L 1 ) [w p E(L )] is satisfied for any supplier i and j with E(L i ) < E(L j ), it is optimal to allocation the quantity up to its capacity to satisfy the demand once a supplier is selected before choosing another one. Proof of Proposition 2: Under the assumption that the OEM must use up all available capacity to fulfill the demand, L 2 = C 1 E(L m )+ y i1 E(L i ) min{d 1 (L 1 ),u m + u i } first period value function becomes:. Given u m < D 1 (L 1 ) < u m + u i, C 1 = u m and the V 1 (L 1 ) = s.t. max {y i1 },2,..., { y i1 = D 1 (L 1 ) u m ; y i1 [w p i E(L i )]} + V 2 (L 2 ) 0 y i1 u i, i = 1, 2,..., ; (19) where the state transition is given as: L 2 = u m E(L m )+ y i1 E(L i ). Suppose the OEM out- D 1 (L 1 ) sources to supplier i and j during the first period where E(L i ) < E(L j ) and the outsourcing quantity to each of them is less than their respective capacity level. Given the second period demand function is defined as D 2 (L 2 ) = α βp γl 2, we prove that the current order allocation policy cannot be optimal in the following two cases: (a) E(L i ) + p i E(L j ) + p j : Switching one unit from supplier j to i cuts down the cost in the first period by [E(L j ) + p j E(L i ) p i ] and raises the second period demand by γ D 1 (L 1 ) [E(Lj ) E(L i )] units. Knowing the lowest possible value of the second period quality level L 2 would not exceed L 0, it means that the additional demand in the second period can always be fulfilled by external capacity. Therefore, outsourcing to supplier i always outperforms outsourcing to supplier j under this case. (b) E(L i ) + p i > E(L j ) + p j : First we look at the marginal effect of switching one unit from supplier i to j. t cuts down the cost in the first period by [E(L j ) + p j 19

20 E(L i ) p i ] while causes the profit loss during the second period by γ[e(lj ) E(L i )] [w D 1 (L 1 ) p k E(L k )], where k is the total number of suppliers the OEM outsources to during the second period. With more units switching from supplier i to j during the first period, cost saving per unit of switching remains unchanged while the profit loss increases due to the drop of total number of suppliers to be outsourced in the second period. While the effect of switching one unit from supplier j to i is the opposite. t costs an additional amount of [E(L j ) + p j E(L i ) p i ] in the first period but raises the second period profit by γ[e(lj ) E(L i )] [w D 1 (L 1 p ) k E(L k )]. And this profit gain by per unit switching from supplier j to i diminishes due to the increased number of k. We re-index the suppliers in a way that p 1 + E(L 1 ) p 2 + E(L 2 )... p + E(L ). Therefore, if either E(L i )+p i E(L j ) p j E(L j ) E(L i ) > γ D 1 (L 1 ) [w p1 E(L 1 )] or E(Li )+p i E(L j ) p j E(L j ) E(L i ) γ D 1 (L 1 ) [w p E(L )] is satisfied, the switching between supplier i and j is to happen and will end until the switching supplier s quantity drops to zero or the other supplier s capacity is used up. Proposition 2 shows the optimal order allocation scheme once the suppliers are selected. Therefore, the OEM can enumerate all the possible supplier selections, evaluate each one based on our model and choose the one that would generate the highest total profit. 4.2 Non-Priority Scenario n this scenario, the OEM places no emphasis on using up the in-house capacity first before outsourcing demand to external suppliers. This policy would be more advantageous when external suppliers can also provide good quality items and when procurement cost is less than the manufacturer s cost. The problem can easily be adapted from the model in Section 4.1 by considering the OEM as a potential supplier. The value function for this case is the same as in (8) except that the first constraint becomes C 2 + y i2 D 2 (L 2 ). Based on the comparison between c and p i + E(L i ), i = 1,...,, the following proposition is presented as the solution mechanism for the second period problem under this extended setting. 20

21 Proposition 3 The optimal single period policy follows one of the following cases: (a) f c < p i + E(L i ), i, the OEM fills its demand by using its own capacity first and then outsource the rest to external suppliers. The order allocation policy among external suppliers follows Algorithm (1). (b) f there exist some supplier whose effective procurement cost is less than the OEM s production cost c, first rank all the suppliers by the ascending order of their effective procurement cost. Define supplier h as the one that p h + E(L h ) < c while p h+1 + E(L h+1 ) > c. Then the OEM fills its demand based on the following sequence until the entire demand is fulfilled or all capacity is used up: supplier 1 to supplier h OEM s own capacity supplier h+1 to the last supplier. Under this policy, the OEM itself performs as another supplier which has no priority regarding the sequence of filling demand. When the OEM s capacity is utilized to fill demand, it would not generate an additional quality loss while outsourcing to the other suppliers would. Therefore, the OEM can be regarded as supplier 0 with p 0 = c and E(L 0 ) = 0. Under this framework, it is straightforward that all the results presented from Theorem (2) still hold for this extension problem. 5 Numerical Study n this section, we consider an OEM that has an in-house production capacity equal to 50 units, with a supply pool of four external suppliers from which to outsource. External suppliers neither compete with, nor produce the same items as, the OEM unless requested. However, they do have the capacity to produce the items when the OEM provides the specification of the item. t is assumed that the price per unit and quality vary across the suppliers. The OEM s in-house production cost c is $1 per unit and it sells the products to the retailer at a price of $5 per unit. The retailer charges $8 for each unit of product it sells to the end customers. We consider both cases for an OEM whose penalty cost g is zero and non-zero. The former is the case where unsatisfied demand would not cause much profit loss since the OEM is the market leader and provides the products that cannot be substituted by competitors, whereas the latter is the case for 21

22 nonzero penalty cost. The penalty cost for unmet demand would increase as market competition intensifies. The OEM applies a price-and-quality-dependent demand function in the linear fashion as Equation (18) which has been defined in Proposition 2. All the initial quality evaluation for suppliers as well as the OEM are given in Table 1 and Table 2. The quality loss coefficients for both the OEM and suppliers (r qm and r qi ) are obtained by the proportion between the loss per unit and the tolerance range of the quality, and they are assumed to be the same. The effective procurement cost for suppliers is the sum of selling price p i and the average loss per unit for each supplier. The capacity u i for all suppliers is equally limited to 15. We conduct several numerical experiments to examine the model we proposed and to generate more managerial insights into the OEM s supplier selection decision. Entity r qm µ qm σ qm E(L m ) c OEM Table 1: Quality Evaluation of the OEM Supplier r qi µ qi σ qi E(L i ) p i Effective Procurement Cost Table 2: Quality Evaluation of External suppliers 22

23 5.1 Supplier Capacity Slack and Unmet Customer Demand n this section, we present two scenarios to illustrate how quality affects customer demand in the subsequent periods, using a two-period two-supplier (supplier 1 and 4) problem, where the OEM s in-house production capacity is equal to 50 units and each supplier has a capacity of 25 units. Both the OEM and suppliers quality evaluations are shown in Table 1 and Table 2. The first scenario shows that the OEM is better off not fulfilling the demand even if there is sufficient capacity, since the quality does not meet expectations, and there is a likelihood that the poor quality will decrease the future demands. And the second scenario shows the other case, in which the OEM can still increase the supply even when the quality is not satisfactory and there is quality loss. The value of the initial state L 1 is assigned the same as E(L m ), which indicates all the products sold are manufactured in house and the overall quality level of the products is reflected by the per unit quality loss of the OEM. L 1 would generate the 1st period demand that is equal to 100 units as we apply the value of (α, β, γ) to be (163, 3, 1000) using Equation (18), which is exactly equal to all the available capacity to the OEM. The objective is to search for an optimal set of outsourcing strategies for each period so as to maximize the total profit for the entire planning horizon. The second period outsourcing strategy follows the Effective Procurement Cost Based Policy presented from Theorem 1 upon observing the state L 2 which is dependent on the first period outsourcing strategy. The programs are set up using AMPL and are solved by KNTRO solver within the directory of Nonlinearly Constrained Optimization under Neos Solvers. We acknowledge that the optimization program cannot guarantee the optimal solutions while we utilize it for comparison purposes. The unimodal structure of V 2 (L 2 ) that has been presented in Theorem 2 implies that extreme values of L 2 may not be ideal for achieving the maximized total profits for the entire planning horizon. The solution shown in Table 3 is the first scenario and the OEM does not use up all the capacity (both in-house and external) to fulfill the first period demand. t suggests to outsource less to the low-quality supplier (supplier 1) and this outsourcing strategy generates the same value of the initial quality loss of the second period, L 2, as the value 23

24 of L 1. n comparison, Table 4 shows the second scenario in which the OEM uses up its own capacity, as well as the capacity of all the external suppliers, to meet the demand in the first period. t shows that the profit is smaller (by $28.96) than in the case shown in Table 3. Notice that the quality loss of the second period is higher than that of the initial value (L 2 > L 1 ). This example illustrates that the OEM is better off not meeting the customers demand in full when there is a likelihood that the external supplier s unsatisfactory quality will affect the next period s demand. Notice that in this example no penalty cost for unmet demand is assumed. Period t C t y 1t y 2t L t D t Total Profit: $ Table 3: Solution without Using up 1st Period Capacity Period t C t y 1t y 2t L t D t Total Profit: $ Table 4: Solution with Using up 1st Period Capacity To generalize our discussion, we now examine the same problem in which the penalty cost for unmet demand is non-zero due to contractual agreements and/or loss of goodwill. When the penalty cost is non-zero, the OEM should increase its supply to 24

25 fulfill the demand, even though the quality may not be perfect. Otherwise, the OEM s insufficient capacity will reduce its profit and eventually hurt its market share. Therefore, in cases where the penalty cost is non-zero, the decision whether or not to meet the demand in full depends on the size of the penalty cost. As an illustration, we revisit a previous example with various values of penalty cost per unit. As shown in Table 5, the optimal decision, when the penalty cost of 0.5, is not to meet the demand in full; in the face 100 unit demand (expected shortage is 50 units without outsourcing) in the first period, the OEM decides to outsource only 19 units instead of outsourcing the full 25 units, from the lower quality supplier 1, recognizing that lower quality units may affect the next period s demand. We observe that the total profit is still higher than the case of full supply, although lower than the case of no penalty cost by $ Also, notice that the value of quality loss in the second period is better than the case shown in Table 4. When the penalty cost is greater than 1, however, the optimal decision is to use up the OEM s in-house capacity, as well as all external capacities, to meet the demand in full. Period t C t y 1t y 2t L t D t Total Profit: $ Table 5: Solution with Non-Zero Penalty Cost (g=0.5) The implication is that there is a threshold penalty cost, and the optimal decision for the OEM is to meet the demand in full when the penalty is greater than the threshold, but not to fulfill the demand in full if the penalty cost is less than the threshold value. The OEM s decision regarding the optimal level of outsourcing has to be made in consideration of the two risks: one associated with quality (scenario 1), and the other associated with competition (scenario 2). 25

26 5.2 Supplier Capacity nvestment Decision n many supply chains with an established OEM whose resource flexibility is low and capital intensity is high, in face of capacity shortage, the OEM would rather make some investments to expand the suppliers capacity instead of its own as to guarantee itself a stable supply and avoid the supplier s bankruptcy (Qi et al., 2015). nspired by the result of the previous numerical example, in this Section 5.2, we present another example showing how the OEM s investments in each supplier s capacity would bring about different outsourcing strategies. n this example, all the four suppliers in the supply pool are available for the OEM. The suppliers quality evaluations, as well as corresponding procurement costs, are shown in Table 2. The planning horizon has been extended to 4 periods to see the effects of investment. t is assumed that the parameters of the demand function except the customer base remain the same as before. The customer base is now increased to 200 to show the effects of investment in supplier capacity increase. Table 6 demonstrates the solution for the OEM s outsourcing decision under a 4-period planning horizon when the set of (α, β, γ) is assigned as (200, 3, 1000) and the initial state L 1 is still the same as E(L m ). Given an initial demand of 137 units and each supplier s capacity equal to 15 units, the OEM can achieve a total profit of $ by executing the order allocation solution assigned to each supplier during each period as shown in Table 6. Suppose the OEM has a budget to invest in one of the suppliers capacities. To examine which supplier would be the best choice for capacity investment, we design four plans where an additional 20 unit s capacity has been added respectively to each of the four suppliers in the supplier pool from supplier 1 to supplier 4. Among those four suppliers, as shown in Table 2, suppliers 1 and 2 produce lower quality products and charge lower prices, while suppliers 3 and 4 produce higher quality products at higher prices. The total profit for each of the four investment plans has been computed and it is found that the total profit is most significantly improved when the capacity expansion is made to supplier 4 who produces the highest quality products. However, we also notice that suppliers 3 and 4 perform similarly in terms of total profits, although the quality of supplier 4 is higher (See the values of E(L i ) for suppliers 3 and 4). This similarity of 26

27 total profits obviously comes about because of the higher price of supplier 4. As such, it may seem reasonable for the OEM to choose the high quality supplier (Supplier 4) to help its capacity expansion. We need to investigate further the trade-off relationship between the quality and the price since the market demand is a function of both the quality and price, as shown in Equation (18). To examine how quality in relation to price impacts the capacity expansion decision, we run the model with five different arbitrary values of γ (e.g., γ = 800, 900, 1000, 1100, 1200) for the demand function. The result is summarized into Figure 1. As demonstrated, the best choice for the OEM s capacity expansion is supplier 3 in most cases, except the case of γ = 1, 000. t also shows that the differences in total profits between suppliers 3 and 4 are more significant when γ is small, and less significant when γ is large. This implies that the demand is relatively less (more) sensitive to the quality when γ is small (large). Period t C t y 1t y 2t y 3t y 4t L t D t Total Profit: $ Table 6: Local Optimum for the Four-supplier Four-period Problem Along with the exploration of the four-supplier four-period example, it has also been demonstrated from the optimal supplier selection strategy, shown in Table 6, that the OEM manifests much concern about quality investment during the early stages of the planning horizon, the OEM purchases more from the high-quality supplier, but then switches to the cost-saving strategy in later periods by purchasing more from the lower total-cost supplier instead. Observing this end-of-horizon effect, we propose a solution 27