Max-Flow Min-Cost Algorithm for A Supply Chain Network

Transcription

1 APDSI Full Paper (Jul, ) Ma-Flow Min-Cost Algorithm for A Suppl Chain Network Shu-Yi Chen ), Ching-Chin Chern ) ) Dept. of Information Management, National Taiwan Universit (d455@im.ntu.edu.tw) ) Dept. of Information Management, National Taiwan Universit (chern@chern.im.ntu.edu.tw) Abstract The management of suppl chain is getting more and more attention in a wired world demanding fast response and fleibilit with ecellent service qualit. There are, as there should be, man studies done to aid companies with better knowledge in operating their suppl chains. However, these studies are either too conceptual to be directl put into use, or too specific and comple to be applied to an individual case. The objective of this stud, hence, is to find the maimum flow with the minimum cost based on the given suppl chain network and product tree structure. Given the original suppl chain graph with its product tree graph, this stud constructed an algorithm that first transforms the original graph of the suppl chain into another form. During the transformation, all the information, such as the cost and capacit of each link, should be kept in the new form of the suppl chain. With the help of the minimum cost flows problems found in network flow models, this stud than appl the ma-flow min-cost algorithms to the transformed suppl chain. In the algorithm developed herein, the capacit and cost relating to each link in the suppl chain are taken into account to solve for the maimum possible output with the lowest cost possible in the suppl chain. Since this algorithm is general enough, hence it is suitable for a manager to use to assist his decision regarding maimum flows and minimum cost in a suppl chain. Although it ma onl be a first step toward a more complete model for suppl chain management problem, there are some findings worth discussing in the stud. An eample illustrating the algorithm is also presented and is followed b some managerial implications. A conclusion is also included at the end of this paper... Introduction In a fast changing world, it is more and more important for a compan to respond as quickl as possible so as not to fall behind its competitors or lose customers. To become responsive, however, a compan has to work not onl on its own performance but also on the management of the suppl chains of the products. Since a suppl chain is, as the name implies, "a network of facilities that performs the functions of procurement of material, transformation of material to intermediate and finished products, and distribution of finished products to customers" [9, p.85], the management of a suppl chain is b no means an eas job. For this reason, the management of suppl chains becomes an important issue for an business in the highl globall competitive world toda, as it does for an researcher interested in the business domains. There are quite a few studies focusing on the management of suppl chain. Generall speaking, studies can be grouped in two main categories. Studies in one of the two categories lend themselves to the clarifications of important concepts of suppl chain management [,, 8, ], while studies in the other categor serve as eamples with models, eperiences and problems encountered in real cases [, 6, 9]. Conceptual-oriented research helps a lot in correcting wrong ideas or pitfalls [8], but provides few help when a compan tries to improve its own suppl chain. On the contrar, case-oriented research can light up the wa for a compan with similar situation and read to build its own models, but if the cases were not properl generalized, the would be of little value. Regardless of what the virtues these two kinds of studies ma have, there is still a lack of a general mathematical model for an suppl chain to appl. The problem of lacking a general model is more critical when most of the midsized or small-sized businesses would probabl not have the budget to build their own mathematical models. Moreover, lacking a general model also makes it difficult for researchers to discuss and to discover certain problems in the management of suppl chain.

2 This stud tried to answer the above call with a general model of a suppl chain, with the help of maimum flow problem seen in network flow models []. With the help of the model, a manager can find the maimum flow in his suppl chain and makes the most use of the chain. The model is thus designed that with some modifications, it provides a fast and feasible, if not optimal, solution for an suppl chain. After this section, the paper presents the reviews of some important literatures in the following section. The formulation of the model is given in section, followed b an illustrative modeling eample in section 4 with managerial implications. A conclusion summarizing the findings and the possible research directions appears in the last section.. Literature Review Two categories are used to classif related studies, namel, conceptual-oriented and case-oriented studies. The differentiation line is drawn at how a stud presents and generalizes its result. A stud is said to be a conceptualoriented one if the stress of the stud is on the instructions of general concepts, while the stud can result from a real world case. On the contrar, a case-oriented stud focuses more on some special cases and the eperiences learnt. Ambiguit eists as it should, but the distinction is still clear. Lee and Billington [8], Billington [] and Fisher [] are eamples of conceptual-oriented studies. In this sense, the all introduced man great concepts in managing the suppl chain or the inventor in a suppl chain. However, the lack of detail description of how to build one's own model ma disappoint some managers. For eample, Lee and Billington [8] were quite descriptive with man illustrative eamples to clarif their point of view. In the article, fourteen often seen pitfalls were discussed, including the basic misunderstanding and misusage of suppl chain, the operational problems encountered, and some strategic issues. There were also some opportunities observed b them worth noticing. Fisher [] provided a good grid to identif the fitness of the product of a compan and its suppl chain. While it is ver helpful, a manager ma want to know more about how to find some details in his own situation. On the other hand, Arntzen et al. [], Lee and Billington [9] and Cohen and Mallik [6] serve as good eamples of case-oriented studies. Eperiences were illustrated and can be learnt for those companies who have the capabilities or are read to start building such models. However, medium to small companies ma not benefit much from these studies, since small companies wouldn't afford to build comple mathematical models. Besides, the formulations ma not appl to specific condition. There are still researches tring to provide more general et mathematical help. Chen and Chern [4, 5] have done some studies utilizing the network flow algorithms such as shortest path algorithm and maimum flow algorithm. In their studies, a general suppl chain graph, with the graph of the production tree of the suppl chain, is transformed into another form that is read to appl the shortest path or the maimum flow algorithm. With each algorithm, their algorithms are suitable for eas solution about the lowest cost production path or the maimum amount of production in the suppl chain. Hence each of their algorithms provides a simple et mathematical wa for suppl chain managers to make decisions regarding one of the above issues. However, their model works for either lowest cost or maimum production output onl, but it is usuall necessar to consider both objectives in a suppl chain decision. Therefore, it is possible to improve their studies b combining the two considerations. According to the above reviews, it is obvious that there is a need for a general but mathematical model for taking into account of cost and production amount. This stud, based on the results achieved b Chen and Chern [4, 5], tried to answer the call with the following algorithm that provides such a solution.. Model Formulation The algorithm of the maimum production output with the lowest cost consideration in a suppl chain works as follows. First the graph of the suppl chain is transformed to be read to appl an of the traditional minimum cost flow algorithms, such as those mentioned in []. After the maimum flow with the lowest cost path, i.e. the lowest cost wa to produce the final product, is found, the capacit of this path is also known. The most challenging part of this stud, hence, is the part of transforming the original suppl chain graph, which is not read for the traditional minimum cost flow algorithm, into the form that is read to directl appl the well-developed and efficient minimum cost flow algorithms. To appl the algorithm developed in this stud, some assumptions, terms and notations have to be defined. Most of them are borrowed from Chen and Chern [4, 5]. For the time being, there can be onl one kind of item shipped per arc. That is, although the items would be different on different arcs, it is thought of as the kind of single commodit problems. Moreover, a node in a suppl chain can onl have production function or distribution function or both. The definitions of production and distribution functions are given below along with other definitions and notations.

3 ( N A ) G, : A suppl chain network or a product tree can be represented b a directed graph A are the set of all nodes and all arcs in G, in which N and G, respectivel. Since the original suppl chain graph needs to be converted before appling the maimum flow algorithm, superscript is used to indicate the number of steps counted while converting the original graph, and subscript denotes what G stands for. For eample, G represents a suppl chain network after being converted once and G a product tree in its original configuration. B definition, is an integer and. (i, j): The directed arc from node i to node j in A. Production function: If a node performs some kind of processing work in a suppl chain, it is said to have the production function. B definition, a node with production function transforms components it receives into a different component or components in the product tree item set, N. Production node: If a node performs production function onl, and all components sent to this node are necessar in producing outward items, it is a production node. A production node can be further categorized as an internal or eternal production node. An internal production node not onl uses the components it receives as materials for production, but also takes the items it produces again as materials to further produce other items. (All items mentioned here could be found in the product tree.) In other words, in the product tree hierarch, an internal production node produces product items one or more levels higher than those product items it receives. On the other hand, an eternal production node produces items onl one level higher (in the product tree) than what it receives. That is, an eternal production node uses onl what it receives as materials to produce items without using what it produces as materials. A one-item production node is a production node that ships out eactl one kind of item. Distribution function: If a node ships out what it receives without processing them, it is said to have the distribution function. Distribution node: If a node performs distribution function onl, it is a distribution node. A node is further called a oneitem distribution node, if it is a distribution node and it ships out eactl one kind of item. CO i : The product component (item) produced/shipped b node i. M incost : The maimum flow path vector with minimum cost generated in G ( N, A ) from node s to t. f MinCost c MinCost is the cost. ( ) is used to denote the amount of the flow of this path, and ( ) U : The capacit of the path S in G ( N, A ). c : The cost incurred for i to process an item and/or deliver the item between (i, j) in ( N A ) ij G,. that it includes onl the added costs but not the material cost ecept for the raw material providers. n i,j : The number of CO i needed to produce one CO j. u, : The capacit limit of flow on (i, j) in G. The unit of u, is the number of CO i. i j i j c ij is thus defined Step. Before modeling a suppl chain, two etra virtual nodes and some arcs have to be added in the suppl chain. Node s and t represents the source and sink node respectivel. There should also be some arcs added to link s and t to other nodes. A node i without arcs linked into it, that is, a starting node without suppliers in a suppl chain, should add (s, i) with c si = and u s,i =. A node j without arcs linked to other node, that is, an ending node without demand nodes in a suppl chain, should add (j, t) with c jt = and u j,t =. G (, N A ) is formed from the original configuration of the suppl chain and these added nodes and arcs. With the above definitions, a suppl chain G (, N A ) can be mo dified and solved b the algorithm for the maimum flow with the lowest cost consideration. Step. Differentiate the production function from the one-item distribution function b splitting node i into sub- G N A nodes. That is, separate the one-item distribution function, if an, performed b an node in ( ), with sub-nodes managing these one-item distribution functions. Add and move necessar arcs and set a new capacit limitation for each arc, at the same time record all the information about cost and capacit. Create G N A after step. ( ), Make all production nodes become one-item eternal production nodes. That is to split all production nodes into one or more one-item eternal production nodes. Furthermore, a production node after the splitting should not receive the same component from different vendors. Add and move necessar arcs, and also G N A to denote the changes and calculate for the correct capacit limit for each arc. Create ( ), include new sub-nodes and arcs added or moved after step. For ever j in i_) to (j, i_k), add a constraint that u j = u k, i_ k j, i. N that (j, i) is split into (j,

4 Step. Combine the arcs carried different items into a production node in G to create G which is suitable for the maimum flow algorithm, so that these arcs and nodes form a new arc (Q, i) and a new node Q. In which Q is the set of these combined vendor nodes. Begin from the original starting nodes in N, i.e. nodes have no upstream arcs other than those to s, calculate for min { u n } u Q, i = j, i / j, i, in which j is an supplier node of i. And if u i, k > u, set Q, i u i, k = u. Also calculate for Q, i c = Q,i c. At last create j, i ( N A ) G for these changes., Step 4. Appl an of the minimum cost flow algorithms and search to find M incost. Step 5. Use M incost, n( MinCost ), (, G N A ), G ( N, A ) and G ( N, A ) to epand M incost and n( MinCost ) back into the path M incost and n( MinCost ) along with the minimum cost in G N A. ( ), 4. Eample and Managerial Implications This section provides a simple eample for all the steps in the previous section to show the feasibilit of the formulation. There are also some managerial implications after the eample. An eample suppl chain G (, N A ) of a compan is shown in Fig.. The CO : a / b notation above or at the right side of each arc (i, j) are used to denote the component and its quantit shipped from node i to j, in which CO is the component shipped, and a is the capacit u i,j of each (i, j), and b is the largest possible quantit of the final product made from these CO in the product tree. Furthermore, the number under or at the left side of each arc is the c ij of each (i, j). Moreover, the product tree ( N A ) G, of G is shown in Fig., in which the number after the slash is n i,j. j V : / 9 V CO : 6/5 5 CO : 55/ P : 8/8 D : 6/6 R V S CO : / : / 6 T CO V 4 : /4 4 CO P 5 : / : 4/4 D : 44/44 R V 5 Fig. ( N, A ) B carring out step to step, ( N A ), G of a compan or arcs being added or split. Since node P is split into P - and P -, G is created as shown in Fig.. Dotted circles and arcs are sub-nodes u should also split to P, P u and P, P P, P P, P u that u P, P + u P, P = u P, P. If it is assigned that u = P, P, then u = u P, P P, P. In this eample, u =, so u =, in which P P. The same logic can be applied on up, D and u P, D, and.,

5 CO CO / CO /5 CO / / Fig. ( N A ) G, of the suppl chain in Fig. 4 After step, some of the nodes are combined and ( N A ) G, is now shown in Fig. 4. B eecuting step 4 and search for the maimum flow with lowest cost, M incost is now found as M incost = { S, VV, P V4, P, D, R, T }, with f ( MinCost ) = 5 and c( MinCost ) = 6. Backtrack from M incost, M incost is shown in Fig. 5 with bold arcs. V V CO 5 CO P - CO P - D R S V V 4 CO : - /- CO : / P - : 6 T D : - /- R V 5 P - Fig. ( N, A ) G after step The solution found here can be used as a wa to find the lowest cost of manufacturing the final product from the suppl chain without performing comple optimization calculations. A manager can formulate the suppl chain with the algorithm established here, so that he can find the low cost choice to produce the products needed.

6 S V V P - V P D R T 6 P - V 4 P - D R P - V 5 5 P Figure 4 ( N, A ) G and M incost (in bold arc) after step 4 V 9 V CO 5 CO P D R S V CO 6 T V 4 P D R V 5 Figure 5 M incost in G ( N, A ) The ma flow-min cost path found in a network is not reall a path in its traditional meaning. It is a route which including necessar entities to produce the final product. Furthermore, the minimum cost is not necessaril the financial cost incurred, but it can represent time, actual distance or other relating factors from a source node to a sink node. Hence it is suitable to be used in a suppl chain to find the lowest cost, whatever it represents, production path with maimum output in the suppl chain. In a sense, it is like the critical path in the suppl chain, and a manager can arrange the allocation of resources/orders in his suppl chain according to the critical path. Besides, the model is itself general and eas to build without a large amount of mone or professional knowledge. The point is important because man companies, especiall in Asian regions, are small to middle size businesses. For this reason, it is not possible for them to hire or introduce comple modeling staff or technique to enhance their

7 management in suppl chains. The model developed here is helpful for them to adopt and modif into their own business considerations. Second, with the help of this algorithm, the model can be used to generate an initial result of the possible path to produce the products required in a suppl chain. It is especiall important when there is no time to develop comple models while the market are demanding fast response, or while the managers want to get an approimate et quantitative idea about the shortest production ccle or the lowest cost possible to produce a product in their suppl chains. At last, because the model is split in a product tree fashion to appl the shortest path algorithm, it can be seen that there seems to be different sections or cuts eisting in the suppl chain. The observation is furthered b eamine the different levels in a product tree that these levels have some relationships with the cuts in the suppl chain. These seemingl hierarchical levels do eist in the suppl chain networks. It is thus possible to find a wa to optimize in each cut to still approach some degree of optimal solution for the whole suppl chain. It is worth studing for the researchers or managers. B eamining their own suppl chain models, companies ma get some insight to better understand and manage their suppl chains. 5. Conclusion The algorithm developed in this stud is another step toward a general model of a suppl chain management problem. There are some points worth considering in the stud. First of all, the result ielded from the algorithm serves as an initial basis for a manager to make a decision about saving the most from his suppl chain so as to fulfill the demand of the market. Since the market is greatl fluctuant, sometimes a manager needs to make a quick decision. The algorithm developed herein is simple enough to assist the job. Second, such an algorithm is general enough to be put into practice with a few modifications. It is especiall suitable for those medium to small size companies, who ma not have enough resources to build their own specific models, et still need to evaluate their suppl chain based on a mathematical wa. Third, the algorithm, along with the formulation closel connected to the product tree, also serves as a ground for researchers to discuss the issue about the management of a suppl chain. More research should be done in this direction, and a more detailed et general enough model can be epected. At last, since the formulation honoring the single commodit ma flow-min cost problem, there are some future directions of research ling here. The most important one is the consideration of uncertaint. If uncertaint can be formulated into the model, it would be of great help. Multi-commodities maimum flow is another interesting topic. The suppl chains have become so comple that it's impossible for a single compan to handle. Therefore, closer cooperation and coordination are in urgent need. It is also true for the development of a better model for suppl chain management, no matter in practice or in academic research. A good model is surel helpful in this direction, and that's the reason wh this stud was set out. References [] Ahuja, Ravindra K., Thomas L. Magnanti, and James B. Orlin, Network Flows: Theor, Algorithms, and Applications, Prentice-Hall, 99. [] Arntzen, Bruce C., Gerald G. Brown, Terr P. Harrison, and Linda L. Trafton, Global Suppl Chain Management at Digital Equipment Corporation, Interface, Vol. 5, No., Januar-Februar 995, pp [] Billington, Core, Strategic Suppl Chain Management, OR/MS Toda, April 994, pp.-. [4] Chen, Shu-Yi, and Ching-Chin Chern, Shortest Path for A Suppl Chain Network, Proceedings of the 4th Asia Pacific Decision Sciences Institute Conference, Shanghai, China, June 9-, 999, pp [5], and, Maimum Flow for a Suppl Chain Network, Proceedings of the th Annual Meeting of the Decision Sciences Institute, New Orleans, Louisiana, USA, November -, 999, pp.4-6. [6] Cohen, Morris A. and Suman Mallik, Global Suppl Chains: Research and Applications, Working paper, Operations and Information Management Dept., Univ. of Penn., 996. [] Fisher, Marshall L., What Is the Right Suppl Chain for our Product? Harvard Business Review, March-April, 99, pp.5-6.

8 [8] Lee, Hau L., and Core Billington, Managing Suppl Chain Inventor: Pitfalls and Opportunities, Sloan Management Review, Spring 99, pp.65-. [9], and, Material Managing in Decentralized Suppl Chains, Operations Research, Vol. 4, No. 5, September-October 99, pp [] Lee, Hau L., V. Padmanabhan, and Seungjin Whang, The Bullwhip Effect in Suppl Chains, Sloan Management Review, Spring 99, pp.9-.