Capacity Planning in a General Supply Chain with Multiple Contract Types

Transcription

1 Capacity Planning in a General Supply Chain with Multiple Contract Type Xin Huang and Stephen C. Grave M.I.T. 1 Abtract The ucceful commercialization of any new product depend to a degree on the ability of a firm to match it upply to market demand. In an emerging indutry where product have little imilarity with the product in exiting indutrie, it i very hard to predict demand pattern. In thi paper, we will develop a general mathematical model for providing deciion upport for the deign of upply chain for emerging indutrie. In particular, we will focu on how capacity invetment in a general upply chain can be made in the preence of demand uncertainty and different type of contract. We will develop an efficient and practical algorithm for finding the optimal capacity planning trategy in a multi-product and multi-tage upply chain model and tudy the propertie of the optimal trategie. Index Term Capacity planning, new product, upply chain deign, contract option. O I. INTRODUCTION NE of the key challenge of the commercialization proce of product in an emerging indutry i to deign an effective upply chain that can meet market demand with high quality product in a timely fahion at competitive price. Since there i little data on the commercial uptake of the product in thee indutrie, it i difficult to predict the demand pattern of the product. One example that motivate our reearch i the microfluidic device indutry. Reearcher have demontrated that micro-fluidic device will benefit many indutrie and reearch procee. Thee device can be ued by cancer reearch laboratorie and drug development companie to perform pecific biological analyi tet. Many companie believe that thee device have a bright future and are tarting to commercialize thee product. However, there i limited information on the commercial uptake of thee device by the pharmaceutical indutry in their reearch Manucript received November 8, Thi work wa upported by the Singapore MIT Alliance. Stephen C Grave i the Abraham J Siegel Profeor of Management Science at the Maachuett Intitute of Technology, Cambridge, MA USA ( grave@ mit.edu). Xin Huang i a doctoral tudent at the Electrical Engineering and Computer Science Department, Maachuett Intitute of Technology, Cambridge, MA USA ( xinhuang@ mit.edu). proce; thu, the micro-fluidic manufacturing firm mut plan their capacitie at a time when the demand pattern of thee device are currently unknown. Beide uncertainty of demand, the manufacturer alo face the difficultie of planning reource for multiple product at the ame time. Due to the wide range of application, the manufacturer need to produce a variety of generic or cutom-made micro-fluidic device to meet the requirement of their cutomer. Such variety in product add complexity to the manufacturer upply chain and require them to plan their reource in a general etting. Manufacturer, however, are looking for efficient and practical algorithm for olving capacity planning problem in a general etting. Since micro-fluidic device manufacturer are till in the early tage of deigning their upply chain, they have the privilege to incorporate different type of capacity contract without high adminitrative cot. Traditionally, a manufacturer etablihe a fixed-cot capacity contract with it upplier to buy a fixed amount of capacity. They need to pay the price whether they ue the capacity or not. In practice, the cot of capacity might have two component: a fixed cot and a variable cot. In an option contract, the manufacturer buy right to ue a fixed amount of capacity with an upfront fixed payment. If they decide to execute their right and ue thee capacitie, they need to pay an exercie price for each unit of capacity that they actually ue. In thi paper, we develop a mathematical model to tudy capacity planning in a multi-product and multi-tage upply chain with different type of capacity contract. We tudy the propertie of the optimal capacity planning trategie. We alo develop an efficient and practical algorithm to find the optimal capacity planning trategie. The ret of the paper i organized a follow. Section II tate the formulation of a multi-product multi-tage capacity planning problem. Section III outline and compare different algorithm for olving the capacity planning problem. Section IV tudie the propertie of the optimal capacity planning trategy. Section V dicue how to extend the ingle-period model to a multi-period etting. Finally, Section VI conclude the paper. Related Literature. There i a large amount of

2 literature tudying capacity planning under uncertainty and fixed-cot contract. Fine and Freund (1990) conider capacity invetment trategie in flexible reource, Barahona et al. (2005) examine capacity acquiition chedule in the context of emiconductor tool planning, Huang and Ahmed (2006) tudy the problem where the capacity deciion can be revied a more demand information i revealed, and Zhang, et al. (2004) look at the capacity expanion problem with pecial demand tructure. Thee work either have more retrictive aumption on upply chain tructure or demand ditribution or focu on ome particular indutrie uch a emiconductor. Van Meighem and Rudi (2002) propoe a newvendor network which i cloely related to the model that we ue. All of thee paper only conider a fixed-cot contract for determining the capacity level. The conideration of option contract in upply chain i a more recent reearch topic. Martinez-de-Albéniz and Simchi-Levi (2002) analyze the optimal option contract for a cae of ingle product and ingle upplier. Yazlali and Erhun (2006) conider option contract in a ingle product dual upply problem. Both of thee work take lead time into conideration. Even though we do not conider lead time, our model allow a more general etting. Another tream of literature that i related to our work i that for algorithm for tochatic linear programming. We refer reader to Kall and Mayer (2006) for a review. Our model for capacity planning problem in upply chain ytem can be viewed a a tochatic linear program. II. MODEL We conider a multi-product, multitage upply chain coniting of M product, J procee, and K reource. The production of each product require a certain amount (poibly zero) of each type of proce. For intance, we might have two proce type aembly proce and teting proce. A reource provide capacity for one or more procee. For intance, a reource might be an aembly line with the capability to aemble a ingle product type. A flexible reource might be an aembly line capable of aembling everal different product type. We might alo imagine a reource with capability to provide more than one type of proce; for intance, a reource might do both aembly and tet for a ingle product type. Without lo of generality, we aume that to produce one unit of product, it require one unit of each of it required procee; we alo aume that to get one unit of a proce, we need one unit of capacity from one of it reource. There are multiple option for procuring or reerving capacity for each reource. A firm can reerve capacity on a reource with a fixed-cot capacity contract; alternatively a firm can reerve capacity on a reource with an option contract where there i maller upfront fixed cot and then a variable cot for the ue of thi capacity. For intance, under a fixed-cot capacity contract, the price for one unit of capacity i 1 dollar. Under option contract, the firm might pay a fixed cot 30 cent upfront for one unit of the capacity. If the firm decide to ue the capacity that it ha reerved, it need to pay another 80 cent per unit. Given thee alternative, the firm want to find the type of reource and contract to ue o that the reulting upply chain can maximize the firm expected profit. We denote D A vector of random variable, with probability denity function f( D ), that repreent the demand of product. (Vector of ize M) d A realization of random demand D. (Vector of ize M) z Amount of product that are produced. (Vector of ize M) x Amount of reource k provided under a fixed-cot capacity contract that i ued to provide capacity to proce j. (Scalar) x The vector of x. (Vector of ize JK) y Amount of reource k provided under an option capacity contract that i ued to provide capacity to proce j. (Scalar) y The vector of y. (Vector of ize JK) A An J M matrix uch that 1, if product m require proce j; A( j, m) = 0, otherwie. B An J JK matrix uch that 1, if reource k can provide B( j,( j, k)) = capacity to proce j; 0, otherwie. H A K JK matrix uch that 1, if reource k can provide H( k,( j, k)) = capacity to proce j; 0, otherwie. C The amount of fixed-cot capacity that the firm ha reerved. (Vector of ize K) G The total amount of capacity, including fixed-cot and option capacity, that the firm ha reerved. (Vector of ize K) r Unit profit generated from filled product. (Vector of ize M) p Unit price of reource under fixed-cot contract. (Vector of ize K) q Unit upfront price of reource under option contract. (Vector of ize K) e Unit exercie price of reource under option contract. (Vector of ize K) We aume that any demand that cannot be filled i lot. We alo aume a two-tage equential deciion proce. In the firt tage, the firm determine the type and ize of the contract with it upplier or contract manufacturer.

3 In the econd tage, demand i realized and the firm allocate production capacity to meet demand. We now formulate the econd tage problem a a ingle period production planning problem with the objective to maximize the profit of the firm. We are given the demand realization d a well a the deciion on C, the amount of reource to reerve with fixed-cot contract and G, the total amount of reource to reerve. We have the following linear optimization problem: P ( C, G, d) = max π ( C, G, d) = r' z e' Hy 2 xyz,, t.. z d Az B( x + y) (1.1) Hx C Hy G C xyz,, 0. By olving thi optimization problem, we can find the profit maximizing production level for a given demand realization and the capacity planning deciion. The firm ultimately want to find the optimal capacity planning trategy under demand uncertainty: P1 = max ( C, G) = E[ P2( C, G, D)] p' C q'( G C) CG, (1.2) t.. C G Propoition 1: ( CG, ) i concave in both C and G. Propoition 1 guarantee the exitence of an optimal olution for problem (1.2) and alo the convergence of algorithm given in the following ection. III. SOLVING THE CAPACITY PLANNING PROBLEM In thi ection, we examine two alternative algorithm for olving the capacity planning problem (1.2). A. Sub-gradient Method Van Meighem and Rudi (2002) found the neceary and ufficient condition for a different but imilar capacity planning problem. In their model, the firm cannot reerve capacity through option. They propoe an algorithm for their problem: 1. Given capacity () i C 2, olve the LP (1.1) and find () i ( j) λ ( C, d ) the aociated dual variable numerically for each ample demand vector d ( j). () i ( j) Take the average of the λ( C, d ) over all j a an () i ( j) unbiaed etimate of E[ λ ( C, d )], and ue it to () compute an etimate of the ub-gradient ( C i ). () i 2. If ( C ) p i maller than ome tolerance level, then top. Otherwie, adjut capacity in the direction of the ub-gradient: ( i + 1) ( ) ( ) C = C i + ξ ( ( C i ) p), whereξ i ome tep-ize (or perform a line-earch), and iterate. The algorithm ue ub-gradient method. At each tep, it will need to olve S LP where S i the number of ample demand point that i ued to etimate the ub-gradient. The computational requirement at each tep can be very intenive depending upon the number of ample point. The algorithm can take a very long time converge, due to the following obervation: 1. The convergence rate i contrained by the bottleneck procee. To produce a product, the firm need to plan the capacity of all procee for the product at the ame time. If one of the procee i hort of capacity, the production i contrained by the bottleneck proce, which dictate the ubgradient. Conider the following example: The firm produce a ingle product that require two type of procee a and b. Reource 1 can provide fixed capacity to proce a at price 5 per unit and reource 2 can provide fixed capacity to proce 2 at price 4 per unit. The demand for the product follow a uniform ditribution between 100 and 120. The price for the product i 12 per unit. The optimal capacity trategy will be 100 C 1 = C for ome value of C 1 = C 2. Now, uppoe we tart with initial point C 1 = 10 and C 2 = 11. Since C 1 < C 2 < 100, C 1 = 12 5 = 7 and C 2 = 0 4 = -4. The ub-gradient algorithm will adjut the capacity a follow: C1, new C1, old 7 = + ξ C2, new C 2, old 4 We alo oberve that when C 2 < C 1 < 100, the ign of the ub-gradient i revered. Thu, depending upon how we et the tep ize, the ub-gradient algorithm can take a long time to converge a it will cycle back and forth between thee two ubgradient. 2. The convergence rate i contrained by the nonuniquene of the ub-gradient. In a typical capacity planning problem, the number of procee i larger than the number of product and the number of reource i larger than the number of procee. Therefore, for ome capacity planning trategie (C,G) and demand d, the olution of the dual problem of (1.1) i not unique. Therefore, the ubgradient at ome capacity trategie (C,G) i not unique. Following different ub-gradient will have very different convergence rate. 3. The convergence rate depend heavily on the tarting point. 4. The convergence rate depend heavily on the tep ize. 5. Lack of good termination criterion. Due to ampling error, the termination criteria, < ε, i hard to atify. B. Supporting Hyperplane Algorithm Let conider a new problem: 2 Since the firm cannot reerve option capacity, C = G.

4 min f 2. Simulate a demand realization d and let 1 t.. f + E[ P2 ( C, G, D)] p' C q'( G C) 0 (2.1) V = V d. Add linear cut C G f + ( C, G, V ) + (2.4) It can been hown that (C*,G*) olve problem (1.2) iff T [( CG, ) ( C, G)] (, ) 0 C G (C*, G*, f*) olve problem (2.1) with For k = 1.. 1, update all previou cut f * + E[ P2 ( C*, G*, D)] p' C* q'( G* C*) = 0. To olve k k k k k 1 problem (2.1), we can ue the upporting hyperplane f + ( C, G, V ) + P2 ( C, G, d ) + algorithm uggeted by Veinott (1967). Let C upper (C lower ) and G upper (G lower ) be the upper (lower) bound of the fixed ( k, k and total capacitie. Let f upper (f lower ) be the upper (lower) ) C G k k T + 1 [( CG, ) ( C, G)] bound of f. Let ( P T = {( C, G, f) : C [ Clower, Cupper ], G [ Glower, Gupper ], 2 k, P2 ) k C G + 1. f [ flower, fupper ], C G} (2.5) Let = 0, the algorithm conit of the following tep: Let the new et of contraint to be T. 1. Solve the linear program of minimizing f, ubject to 3. Solve the linear program of minimizing f, ubject to ( CG,, f) T, and let ( C, G, f ) be the optimal ( CG,, f) T, and let ( C, G, f ) be the optimal olution. If olution. If f + E[ P2 ( C, G, D)] p' C q'( G C ) ε f + E[ P2 ( C, G, V )] p' C q'( G C ) (2.2) ε (2.6) where ε i a mall poitive number choen by the where ε i a mall poitive number choen by the uer, top. Otherwie, go to tep 2. uer, top. Otherwie, et = + 1and go to tep Ue the imulation method given in the ub-gradient algorithm to calculate the ub-gradient C and IV. PROPERTIES OF OPTIMAL STRATEGY G. Add linear contraint In thi ection we will tudy the propertie of optimal f + ( C, G ) + (2.3) trategie. We firt look at the effect of change of T [( CG, ) ( C, G)] (, ) 0 demand. Let I be a J K matrix uch that C G 1 to the et T. Let the new et be T +. Set = + 1 1, if pk=min{ pn B( j,( j, n)) = 1}; I( j, k) = and go to tep 1. 0, otherwie. Geometrically, the upporting hyperplane method If I( j, k ) = 1, it mean that uing reource k i the cheapet approximate function ( CG, ) with hyperplane. At each way to provide capacity to proce j. WLOG, we aume that there i an unique k for each j uch that I( j, k ) = 1. tep, the algorithm ue all the ub-gradient that it ha calculated o far. Therefore, it overcome obervation 1 and 2 of the ub-gradient algorithm. By the nature of upporting hyperplane algorithm, it doe not require a tarting point or a tep ize. Finally, at each tep - f i an upper bound of ( C*, G*, D). Therefore, ε i an upper bound for ( C, G, D) ( C*, G*, D). Thi termination criterion i a better indicator of whether the olution i cloe enough to the optimum or not. The upporting hyperplane method alo uffer from the high cot of calculating the ub-gradient. Thi, however, can be improved by uing a tochatic update method uggeted by Higle and Sen (1991). At each iteration, the algorithm imulate one demand realization d. 1 LetV = { d,..., d } denote the et of demand realization that have been imulated o far. The upporting hyperplane algorithm with tochatic update i a follow: 0 1. Set = 1, V =, and 0 T = {( C, G, f) : C [ Clower, Cupper ], G [ Glower, Gupper ], f [ f, f ], C G} lower upper Propoition 2: Let ( C*, G *) be the optimal olution of capacity planning problem ( DABHrpqe,,,,,,, ). Let ˆD be another et of random demand that i different to D only in it firt moment. Let = E Dˆ E[ D]. Let C ˆ*, Gˆ * be the optimal olution of capacity planning ( ) problem ( ˆ,,,,,,, ) ˆ * * G G I' A DABHrpqe. Then ˆ * * C = C + I' A and = +. Thi propoition ugget that if the firt moment of the demand vector change, the firm doen t need to recalculate the optimal capacity planning trategy. The new optimal trategy can be obtained by uing the method uggeted in the propoition. The effect of unit profit and unit price on optimal capacitie are more complicated and le intuitive. For example, If unit profit for ome product increae, the optimal total capacitie for ome reource might decreae. When unit profit increae, one would

5 expect that the firm will reerve a leat a much capacity a before. Thi, however, might not alway be true. Let ( C*, G*) be the optimal capacity planning trategy for problem ( DABHrpqe.,,,,,,, ) Let aume that G* > C* ; a the unit profit r for ome product increae, the optimal fixed capacity for ome reource might alo increae. If G* > C*, C * indicate the optimal trade-off threhold between fixed capacity and option capacity. One might expect that thi threhold only depend on the price ratio between fixed capacity and option capacity a in the ingle product cae. However, for the cae of multi-product, it alo depend on the unit profit of the other product. To illutrate the effect of unit profit, we conider the following example which contain 5 product, 9 procee, and 9 reource. The tructure of the upply chain i given in Figure 1. The demand for each product follow a normal ditribution N (120, 10). We et r = [30, 50, 46, 41, 25], pk = 10 k, q = 8 k, and e = 3 k. We plot the k change of optimal trategy for one of the reource a unit profit increae in Figure 2. We can ee that both the optimal total capacity and the fixed capacity increae a unit profit increae. Alo, the ratio between the option capacity and the fixed capacity increae a the unit price increae. Thi mean that a unit price increae the firm will increae the amount of option capacity in the optimal trategy. Moreover, both curve have a concave tructure. Thi i becaue the utilization of an additional unit of capacity deceae a the total capacity and fixed capacity increae. Finally, we look at the effect of unit price. We et q+ e to be a contant and increae q. When q i mall, the firm pay le up-front cot to reerve capacity and a higher exercie price. When q i large, the firm will pay more to reerve and le to ue the capacity. If q+ e i a contant, the firm prefer to pay le up-front cot ince the penalty of over reervation i le. Thi intuition i confirmed by the plot given in Figure 3. When q i mall, the firm reerve more capacity in total and le fixed capacity and when q i large the firm reerve le capacity in total and more fixed capacity. V. MULTI-PERIOD CAPACITY PLANNING AND INTEGER CONSTRAINT In thi ection, we dicu how to extend the ingle period model to a multi-period etting and how to olve the problem if capacity only can be reerved in indiviible unit. Depending on the time length of the contract, there are different way to formulate a multi-period capacity planning problem. If the contract require a long term commitment, after the firm ign the contract to acquire capacity, the ame amount of capacity might need to be bought or reerved in each period until the end of the k planning horizon. On the other hand, if the contract are hort term, the firm can reerve different amount of capacity for different period. Huang, et al. (2006), Roundy et al. (2004), Barahona et al. (2005), and Martinez-de- Albéniz and Simchi-Levi (2002) conider long term contract while Yazlali and Erhun (2006) ue hort term contract. For both formulation, we can how that if demand from different period are independent, we can decompoe the multiple-period problem into a erie of ingle period problem. However, if demand from different period are not independent, a imple decompoition algorithm might not be applicable. We will addre thi problem in our future reearch. In practice, the capacity might only be procured or reerved in bulk unit. Thi require that the deciion variable, C and G, to be integer multiple of ome bae unit. Having integer deciion variable will increae the difficulty of olving the problem. Barahona et al. (2005) and Ahmed and Garcia (2003) have propoed ome approximation algorithm that can be ued to cope with thee difficultie. Their algorithm need to olve an LPrelaxation of the integer programming problem. Similar type of technique might be ued with the algorithm given in thi paper to olve integer capacity planning problem. VI. CONCLUSION In thi paper, we propoe a model to tudy capacity planning in a multi-product and multi-tage upply chain with multiple type of contract. The model i very general o that manufacturer can ue it to plan their reource and alo deign their upply chain tructure. We alo give a practical algorithm for olving the capacity planning problem. We believe that our work open the door to many future reearch topic. REFERENCES [1] J. A. Van Mieghem and N. Rudi, Newvendor Network: Inventory Management and Capacity Invetment with Dicretionary Activitie, Manufacturing & Service Operation Management, Vol. 4, No. 4, pp , [2] J. Higle and S. Sen, Stochatic Decompoition: An Algorithm for Two Stage Linear Program with Reource, Mathematic of Operation Reearch, vol. 16, No. 3, pp , [3] A. F. Veinott, The Supporting Hyperplane Method for Unimodal Programming, Operation Reearch, 18, pp , [4] Ö. Yazlali and F. Frhun, Managing Demand Uncertainty with Dual Supply Contract, Working paper, [5] V. Martine-de-Albeniz and D. Simchi-Levi, A Portfolio Approach to Procurement Contract, Production and Operation Management, vol. 14, iue 1, pp , [6] S. Ahmed and R. Garcia, Dynamic Capacity Acquiition and Aignment under Uncertainty, Annual of Operation Reearch, Vol. 124, No.1-4, pp [7] F. Barahona, S. Bermon, O. Günlük, and S. Hood, Robut Capacity Planning in Semiconductor Manufacturing, Technical Report RC22196, IBM Corporation, [8] K. Huang and S. Ahmed, The value of multi-tage tochatic programming in capacity planning under uncertainty, Working paper, [9] F. Zhang, R. Roundy, M. Çakanyildirim, and W. Huh, Optimal capacity expanion for multi-product, multi-machine manufacturing

6 ytem with tochatic demand, IIE Tranaction, Vol. 36, No. 1, pp , [10] C. H. Fine and M. Freund, Optimal Invetment in Product-Flexible Manufacturing Capacity, Management Science, Vol. 36, No. 4, pp , [11] P. Kall and J. Mayer, Stochatic Linear Programming: Propertie, Solution Method, and Application, Springer, Product Procee Reource Figure 1: A upply chain with 5 product, 9 procee, and 9 reource. Figure 2: Optimal Capacity v. Unit Profit Increment Figure 3: Optimal Capacity v. Up-front Reervation Price