Multiperiod Planning of Enterprise-wide Supply Chains Using an

Multiperiod Planning of Enterprise-wide Supply Chains Using an Operation. Policy. Jun-hyung Ryu*,† and Efstratios N. Pistikopoulos‡. Department of...
0 downloads 0 Views 526KB Size
Download PDF
8058

Ind. Eng. Chem. Res. 2007, 46, 8058-8065

Multiperiod Planning of Enterprise-wide Supply Chains Using an Operation Policy Jun-hyung Ryu*,† and Efstratios N. Pistikopoulos‡ Department of Chemical Engineering, POSTECH, Pohang, Korea 790-784, and Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, London SW7 2BY, United Kingdom

A new modeling framework for multiperiod supply chain planning problems is proposed in this paper. The problem is transformed into a set of two optimization problems. At first, demands in distinctive geographical locations over multiple time periods are aggregated and distributed to multiple entities using an operation policy that was employed by Ryu and Pistikopoulos (Ind. Eng. Chem. Res. 2005, 44, 2174). Individual singlesite multiperiod planning problems are then constructed based on their allocated demands. These optimization problems are mathematically formulated as mixed-integer linear programming (MILP) problems and numerically demonstrated with two examples. From the results of the examples, it is seen that the key issue in the multiperiod supply chain planning problem is the distribution of multiple demands to multiple entities. 1. Introduction An enterprise-wide supply chain consists of multiple industrial entities such as suppliers, plants, warehouses, and retailers that are often in distinct geographical places. Recently, there has been a great industrial and academic interest in how to design and operate them because of their economic impact under severe competition.1-3 The most important feature of the enterprise-wide supply chains is multiplicity. The presence of multiplicity can be explained in two ways. First, a supply chain consists of multiple entities that may do the same role, e.g., a product may be produced at more than one plant, at a plant in east Asia or in South America, simultaneously. That is to say, the total amount of products ordered by all demands may be met by one plant or even by all participating plants. This is a typical supply chain problem whether one plant or multiple plants are assigned for manufacturing a specific product. Second, a supply chain consists of multiple connections of “supplier” and “demand”: an entity may play the role of a supplier and a demand at the same time. For instance, a warehouse plays the role of a demand when it gets products from plants or other warehouses. On the other hand, it is a supplier when it delivers products to retailers or customers. In practice, products are delivered through multiple connections from the initial raw material suppliers to the final customer, and a supply chain itself is also often connected with other supply chains, e.g., raw materials is also the outcome of material producer supply chains. Because of the effect of multiplicity, a large number of alternative paths exist in a supply chain from the start to the end, depending on which entities are gone through. Since the demand of one market and the demand of the other market may be met by one manufacturing plant (or warehouse) or the demand of a market can be met by more than two plants (or warehouses), the operation of an entity affects and is affected by the operations of other entities. Since an operation of an entity may be affected by other entities, the decisions of which plants make which products for which market in which quantity * To whom correspondence should be addressed. E-mail: jun2002@ postech.ac.kr. Tel.: 82-54-279-5956. Fax: 82-54-279-5528. † POSTECH. ‡ Imperial College.

should be computed globally with a view to maximizing the objectives of the entire supply chain. This poses a new challenge surpassing the scopes of the previous research, which mainly focused on maximizing individual performances relatively, neglecting the overall integration of the entire enterprise-wide supply chains. There may be many types of supply chain problems, e.g., design, planning, control, etc. In this paper, we are particularly interested in supply chain planning, because they occur frequently in practice, e.g., companies want to analyze what will be the benefit for the next 6 months when current facilities are expanded over 20%, or how much their profit will increase if they make a contract to use a new third-party warehouse to prepare the high season, or whether a couple of companies are merged, etc. An enterprise-wide supply chain planning problem is, thus, concerned with deciding the capacities of the supply chain entities and their connection paths over the aggregated time horizons. That is to say, we have to decide which entity makes which product for which demand over which time periods. In order to address the supply chain planning problem, this paper is motivated by the one by Ryu and Pistikopoulos,4 which presented a concept of an operation policy as a guidance in the design of an enterprise-wide supply chain. As an illustration, they proposed three operating policies, which are, namely, coordination, cooperation, and competition policies. They are distinguished by placing importance on the distinctive function of supply chains: coordination policy is a one-product-oneplant policy. Products are grouped by their type regardless of their geological locations of demands. Entire amounts of a product in all demands are made at a specific plant or plants. In cooperation policy, all demands of multiple products in a market are satisfied at one plant. It is a one-market-one-plant policy: markets are grouped and their entire demands are met by a specific plant. That is to say, a plant produces all products of a market, or grouped markets. This is a market-oriented production policy because the market can select a plant as a supplier. In competition policy, each plant may make any product for any market. The demand of a market can be satisfied, either fully or in part, by any plant, so long as the corresponding configuration meets objectives. A competition policy is different from others in that demands of a market can be divided. It is possible for the model using a competition policy to give the

10.1021/ie070508b CCC: $37.00 © 2007 American Chemical Society Published on Web 10/12/2007

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8059

2.1. Problem Statement. We aim to construct a decisionmaking model for an enterprise that consists of multiple plants for which the index is denoted by l, multiple markets denoted by d, and multiple products denoted by i over multiple time periods denoted by t. The model consists of two optimization problems, which are demand-allocation and single-process planning of individual entities. 2.2. Demand-Allocation Models. The objective of the demand allocation denoted by RProfit is to obtain a configuration of suppliers satisfying multiple demands over multiple time periods. Figure 1. Illustrations of process networks according to three suggested operating policies: (a) coordination, (b) cooperation, (c) competition (Ryu and Pistikopoulos4).

same result as one using coordination or cooperation policy. In short, an operation policy is a strategy for the production of multiple plants to satisfy the demands of multiple markets. When the key concern of a supply chain is given on plant performance, coordination policy would be appropriate. When logistics is the major concern in a supply chain, cooperation policy is used, and a competition policy is used on competition. It is possible to assume that the supply chain under competition policy is the one without any operation policy but maximizing cost. Refer to Figure 1 for the graphical illustration of supply chain design using operation policy.4 The rest of this paper is constructed as follows: At first, demands in distinctive geographical locations over multiple time periods are aggregated and distributed to multiple entities using an operation policy that was employed by Ryu and Pistikopoulos.4 On the basis of the assigned demands, single-site multiperiod planning problems are, thus, constructed for individual entities. This two-stage optimization framework is motivated since we assume that the complexity in the supply chain planning problem is mainly due to the existences of multiple combinations between multiple suppliers and multiple demands. Since an entity may send any amount of products to any entity over multiple time periods, a large number of combinatorial alternatives should be reviewed with the additional decisionmaking issue of their timing. Instead of addressing these issues simultaneously, we handle the key decision of the demand allocation first and the corresponding production planning later. The results of numerical examples are presented with some remarks. 2. Multiperiod Planning Model of an Enterprise-wide Supply Chain Planning models are generally constructed with an aim to compute the optimal operation levels or configurations that maximize the objective, often profit, subject to various constraints such as (i) production capacity with or without capacity expansion, (ii) demand, (iii) availability, (iv) inventory requirements, and (v) material balance. The production configuration takes the form of selecting particular unit or processing types in response to varying external conditions, which are, in most cases, demands over multiple time periods (Sahinidis et al.5 and Ierapetritou and Pistikopoulos6). The major focus of the planning model is the discrete decision in the form of selecting which unit or process type or when to expand the process capacity in what amount, etc. As a result, the multiperiod planning models are mathematically formulated into mixed-integer linear programming (MILP) problems. See the Nomenclature section for variables used to present the ideas in the mathematical models.

RProfit )

∑ ∑ ∑ scd,i,tDMd,i,t - ∑ ∑ ∑ pcl,i,tXl,i,t l∈L i∈N t∈T ∑ ∑ ∑ ∑ dcl,d,i,txxl,d,i,t (1) l∈L i∈N d∈D t∈T

d∈D i∈N t∈T

where DMd,i,t is the demand of product i at market d in time t, Xl,i,t is the production quantity of product i at plant l in time t, and xxl,i,d,t is the specific production quantity for market d at time t. The first term on the right-hand side represents sales revenue of the enterprise, and the other two terms represent production and delivery costs, respectively. The constraints for the above objective function can be formulated according to the distinctive operation policy of the supply chain. As an illustration, this paper is going to consider three operation policies that were proposed by Ryu and Pistikopoulos.4 First, the demand-allocation problem according to coordination policy can be formulated as follows:

∑ ∑ ∑ scd,i,tDMd,i,t ∑ ∑ ∑ pcl,i,tSCRl,i,t l∈L i∈N t∈T ∑ ∑ ∑ ∑ dcl,i,d,tSCRl,i,t l∈L i∈N d∈D t∈T s.t. ∑ Ycrl,i,t ) 1, ∀i, t l∈L ∑ DMd,i,tYcrl,i,t ) SCRl,i,t d∈D

Max RProfitDCR )

d∈D i∈N t∈T

∀i, t

(2)

where Ycrd,i,t is the binary variable that is 1 when plant l makes the sum of product i wanted by all markets at time t and is 0 otherwise. A demand-allocation problem according to cooperation policy can be formulated as follows:

∑ ∑ ∑ scd,i,tDMd,i,t ∑ ∑ ∑ pcl,i,tSCOPl,i,t l∈L i∈N t∈T ∑ ∑ ∑ ∑ dcl,i,d,tSCOPl,i,d,t l∈L i∈N d∈D t∈T s.t. ∑ Ycopl,d,t ) 1, ∀d, t linL ∑ DMd,i,tYcopl,d,t ) SCOPl,i,t, d∈D

Max RProfitDCOP )

d∈D i∈N t∈T

∀l, i, t (3)

where Ycopl,d,t is the binary variable that is 1 when plant l makes all demands of a market d at time t and is 0 otherwise and SCOPl,i,t is the production quantity according to cooperation policy. Finally, a demand-allocation problem according to competition policy is as follows:

8060

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007

∑ ∑ ∑ scd,i,tDMd,i,t ∑ ∑ ∑ pcl,i,tSCMPl,i,t - ∑ ∑ ∑ ∑ dcl,i,d,tXl,d,i,t l∈L i∈N t∈T l∈L i∈N d∈D t∈T

Max RProfitDCMP )

d∈D i∈N t∈T

s.t.

Ycmpl,i,d,t e NOPL, ∑ l∈L

∀i, d, t

∑ Ycmpl,i,d,t e NOMR,

∀l, i, t

CPl,t ) CPl,t-1 + CEl,t (4)

d∈D

∑ Xl,i,d,t e SCMPl,i,t,

policy is selected as an operation policy, SCRl,i,t is used for Sl,i,t; when a cooperation policy is used, SCOPl,i,t is used for Sl,i,t; and SCMPl,i,t is Sl,i,t under a competition policy. The current capacity of plant l, CPl,t can be maintained as the capacity of the previous stage or expanded as much as CEl,t:

The expansion quantity, CEl,t should be between the lower and upper bounds,

∀l, i, t

Yl,tCELl,t e CEl,t e Yl,tCEUl,t

d∈D

U L e SCMPl,i,d,t e SCMPl,i,d,t SCMPl,i,d,t

where Ycmpl,i,d,t is the binary variable that is 1 when plant l makes product i for market d at time t and is 0 otherwise. NOPL denotes the number of total plants available, while NOMR is the number of total markets. SCMPl,i,t is the production quantity L U and SCMPl,i,t are the according to competition policy. SCMPl,i,t lower and upper bound production quantities for plant l, respectively. By solving one of the above three allocation problems (eqs 2, 3, and 4), multiple demands of multiple markets over multiple time periods are distributed to multiple plants. Then, in the next subsection, the individual plants operation will be planned optimally based on its results. 2.3. Single-Process Planning of Individual Entities. It is determined which plant make(s) which products in which quantity at which time by computing the solution of the demandallocation optimizations. The next is to arrange how individual plants are operated by deciding which capacity of a plant is expanded or whether to increase the inventory levels of products to satisfy the requested demands of individual plants. Therefore, the objective of the multiplant planning level is to minimize the operating cost, COSTOP, which is the sum of process operating costs and inventory holding costs of all plants,

COSTOP )

(5)

where CPl,i is the capacity of plant l at time t, yl,t is the binary variable that is 1 when the capacity is expanded and is 0 otherwise, and ecl,t and fl,t are the cost coefficient for capacity operation and the fixed discrete expansion cost coefficient, respectively. On the other hand, the inventory cost denotes the total inventory holding cost from the start to the end of planning horizon except the final-stage inventory, which does not cause cost yet, T-1

(6)

where Il,i,t is the inventory level and ihl,i,t is the inventory holding cost coefficient. The material balances of product i at plant l at time t are formulated with the following set of the inequality equations,

bl,i,tCPl,i + Il,i,t g Sl,i,t + Il,i,t+1

where it is decided whether to expand the capacity or not by introducing the binary variable, yl,t. The inventory is limited by the lower bound, which can be 0 or a safety stock value and upper bound: L U Il,i,t e Il,i,t e Il,i,t

(10)

The total inventory of plant l at time t should be limited as follows,

cil,i,tIl,i,t e TotalIl,t ∑ i∈N

(11)

where cil,i,t is the inventory cost coefficient. Finally, the planning problem of multiple plants in an EPN is formulated mathematically into the following mixed-integer linear programming problem:

(PLAN) Min COSTOP )

∑ ∑ ecl,tCPl,t + ∑ ∑ fl,tYl,t + l∈L t∈T l∈L t∈T T-1

∑ ∑ ∑ ihl,i,tIl,i,t l∈L i∈N t)t1 subject to

CPl,t ) CPl,t-1 + CEl,t

T-1

∑ ∑ ∑ ihl,i,tIl,i,t ) ∑ ∑ ∑ ihl,i,tIl,i,t - ihl,i,TIl,i,T l∈L i∈N t)t1 l∈L i∈N t∈T

(9)

bl,i,tCPl,i + Il,i,t g Sl,i,t + Il,i,t+1

∑ ∑ ecl,tCPl,t + ∑ ∑ fl,tYexpl,t + l∈L t∈T l∈L t∈T ∑ ∑ ∑ ihl,i,tIl,i,t l∈L i∈N t)t1

(8)

(7)

where bl,i,t is the performance coefficient for product i at plant l at time t and Sl,i,t is the production quantity decided previously in the demand-allocation optimization. When a coordination

cil,i,tIl,i,t e TotalIl,t ∑ i∈N L U Yl,tCEl,t e CEl,t e Yl,tCEl,t L U Il,i,t e Il,i,t e Il,i,t

The overall proposed framework is illustrated in Figure 2. 2.4. Examples. 2.4.1. Example 1. Consider a planning problem of a manufacturing enterprise involving 3 plants and 3 markets for 4 products over 10 time periods. The application of the presented framework yields the results summarized in Tables 1 and 2 and graphically illustrated in Figures 3 and 4. The problem is solved using CPLEX in SUN ultra10 workstation. As can be seen in Figure 3, we can see that it is necessary to expand the capacity in plants 1 and 2 using any operation policy to meet all demands. Another point from the figure is that the capacities by coordination and cooperation policies are much higher than the one by competition policy. This result shows that competition policy is more capacity-efficient than the others. From Figure 4, how supply chains are planned by the effect of their operation policy is illustrated. Under competition policy, all plants are operated consistently throughout the time periods.

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8061

of $46.62M. As can be seen in Table 5, most of the computation resources are consumed in the demand-allocation part. From this result, we can see that the key issue in the multiperiod supply chain planning problem is the distribution of multiple demands to multiple entities. 3. Discussion

Figure 2. Proposed total framework of the enterprise-wide process networks planning. Table 1. Result of Example 1 coordination

cooperation

competition

32.736M 4.157M 4.862M 7.843M 16.863M 16.107M

32.005M 2.198M 3.740M 5.537M 11.475M 19.409M

34.772M 2.048M 2.462M 3.370M 12.390M 22.382M

sum of revenues ($) O.C.a of plant 1 ($) O.C. of plant 2 ($) O.C. of plant 3 ($) sum of O.C. ($) sum of profits ($) a

O.C. ) operating cost.

Table 2. Capacity Expansion Result of Example 1 capacity expansion time bucket for each operation policy plant

coordination

cooperation

competition

plant 1 plant 2 plant 3

t3, t5, t8, t9 t2, t4, t6, t8 t2

t3, t5, t9 t5, t6, t8 t2

t3, t9 t2, t6

Regarding the coordination policy, production items are fixed at each plant. In practice, we may more expect the improvement of production efficiencies because they specialize the fixed items. On the other hand, the cooperation planning forces a very unbalanced operation over the total time period: some time periods produce a very small amount, while some produce a very large amount. This contrasting operation would not be welcomed practice. It would cause frequent setup costs under such a case. More discussion is done about this in a later section of this paper. 2.4.2. Example 2. Another similar type of supply chain planning problem with a bigger size involving 5 plants, 10 markets, and 10 products is solved. The problem is also solved using CPLEX in SUN Ultrasparc workstation. The results are summarized in Tables 3 and 4. The optimal solutions of the proposed planning problems according to the coordination and cooperation policies are computed using CPLEX in a reasonable time as in Table 5. However, the solution according to the competition policy is the result calculated after 20 000 s with the best possible solution

There are a few issues worthwhile to mention on the proposed framework. The first is if the operation policy is really necessary for the multiperiod planning of supply chains. When individual entities focus their own issues assuming other conditions fixed, how to address the interaction between participating entities remains unanswered. The key contribution of supply chain management (SCM) is to disclose such unnoticed potential of the interactions between supply chain entities and reflect it in their design and operation planning. Since the supply chain planning problem addresses which entity makes which product by how much for whom over which time, the resulting problem scope is, thus, very wide, and its degree of freedom is very high. Because of a need for a guidance of operation as to which one to choose first or later, we introduce an operation policy to address the problem. It can be different depending on industry specifications, and the operation policy plays the role of guidance by shifting emphasis on manufacturing, logistics, or cost, etc. Therefore, we think that operation policy is indispensable in supply chain operation. With this regard, supply chain without using operation policy is, in fact, the model using the competition policy. The difference between the two models (cooperation and coordination) and the competition model is that manufacturing or transportation efficiencies are more preferred in the two models. Regarding the validity of operation policy, a simple LP model for supply chain planning without any operation policy can be formulated and solved to decide the optimal activities of supply chains. Because of fewer constraints, this model can return a higher profit than the results from the proposed model. Then the question is why the managers need to use the operating policies to get lower optimal profit, instead of following the optimal decisions from the simply planning problem without it. From years of industrial experience, we have seen supply chains are not designed by only one factor of short-term cost perspective. There are many issues that cannot be measured quantitatively besides the direct economic criteria because of various reasons such as industry specific, legislation issue, strategic preparation, etc. The better-looking optimal solution from the fewer constraints may not actually be implemented in practice. The worse-looking solution based on the operation policy model would be taken for granted by the practitioners. As indicated in the previous work by the authors on enterprisewide design (Ryu and Pistikopoulos),4 in the design problem where setup is considered as a single campaign, the results by competition and coordination policies are similar. That is to say, when we consider every possible industry-specific condition, the production in each plant has relatively simple configurations. Therefore, we think we can obtain relatively inexpensively the supply chain planning model accepted by practitioners when we use another operation policy model instead of computing long times using a competition model. The second is on the planning horizon for the proposed model. Capacity expansion is usually considered in the strategic level for long-term planning. However, one can point that the discussed operation policies lie more on the operational level, which is better suited for short-term planning. Since the operation policy represents the guidance of manufacturing

8062

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007

Figure 3. Graphical illustration of the result: capacity of plants according to different operating policies.

in supply chains, we think that it affects the supply chain throughout their entire time horizon: it affects the long-term as well as short-term operation at the same time. Then the next issue is how to address its role on the corresponding long- and short-term decision-making problems. Capacity expansion and inventory would be the major leverage in the long-term planning problem, and backlog and inventory are the major leverage in the short-term planning. Since the proposed model addresses the supply chain problem in the relatively midterm or longterm horizon, the proposed approach reflects the capacity expansion and inventory without involving the backlog. It is

assumed that demands are distributed to individual entities as much as each can afford to handle. The entities should operate as much amount as they are assigned. If not, their capacities are expanded to meet them. In the very short-term operation planning such as scheduling level, backlog is the essential issue that should be considered, but the single-site planning is concerned with the relatively longer-term horizon, where daily based backlog is neglected. The third is about the validity of the framework. In the proposed framework, inventory costs are not considered in the optimization of demand allocation. It is thought that the

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8063

Figure 4. Graphical illustration of the result: production at plants according to different operating policies (coordination: a1, a2, a3; cooperation: b1, b2, b3; and competition: c1, c2, c3). Table 3. Result of Example 2

Table 4. Capacity Expansion Result of Example 2 operation policy cooperation

competition

plant

coordination

154.92M 17.73M 13.15M 17.59M 14.04M 19.68M 82.20M 72.72M

113.26M 0 0 11.50M 0 9.98M 21.48M 91.78M

94.85M 10.40M 7.18M 11.10M 7.49M 12.06M 48.22M 46.63M

plant 1 plant 2 plant 3 plant 4 plant 5

t2, t6, t7, t8, t9 t2, t3, t7 t3, t7, t8, t9, t10 t2, t3, t4 t4, t5, t10

RProfit ($) O.C.a of plant 1 ($) O.C. of plant 2 ($) O.C. of plant 3 ($) O.C. of plant 4 ($) O.C. of plant 5 ($) sum of O.C. of ($) sum of profits ($) a

capacity expansion time bucket for each operation policy

coordination

O.C. ) operating cost.

inventory plays a limited role in the demand allocation. Once the demands are allocated to individual plants, their inventories are considered in the multiperiod planning to satisfy the previously allocated demand. In addition to the inventory cost, this paper assumed that the objective in this paper is to maximize the overall profit instead of the intermediate revenue. Regarding Example 1, the tabulated results in Table 1 show that the profit according to the cooperation policy ($19.409M) is bigger than

cooperation

competition

t3, t4, t9, t10

t2 t2, t3, t7 t7, t8, t10 t3, t4 t3

t7

the one according to the coordination policy ($16.107M), even though the revenue profit by the cooperation policy ($32.005M) is less than the one by the coordination policy ($32.736M). This result is justified because the objective in this paper is to maximize the overall profit instead of the intermediate revenue. In terms of modeling, the major difference between the design model of an enterprise-wide supply chain and the model of demand allocation is if the resource limit constraint of the following form is included or not:

∑i Rl,iXl,i e resourcel

∀l

(12)

8064

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007

Parameters

Table 5. Computational Statistics of Example 2 calculation time (s)

coordination

cooperation

allocation model planning model plant 1 plant 2 plant 3 plant 4 plant 5 computation time

0.120

0.30

0.060 0.070 0.080 0.060 0.080 0.350

0.060 0.090 0.450

competition g20 000 0.040 0.070 0.060 0.080 0.070 g20 000

In the design model, the approximated resource is an important factor. However this paper assumes that planning considers the operation of the already designed processes and can expand its capacities or resources without consideration of the above resource limit constraints in the demand-allocation optimization stage. The fourth is how to select the operation policy over multiple time periods. In the proposed problem, a fixed operation policy is used for the demand allocation at each period. However, different operating policies can be used for individual time periods. For example, cooperation policy is used for 1 month and coordination policy is used for the next 1 month, etc. It can be said that introducing the distinctive operating policies for separate time periods can allow the processes to respond more efficiently to varying demands. However, it would not be desirable because the frequent change of operation policy causes frequent setup costs at plants. It would be a good topic for further research to review how the result according to different operating policies is effected against the change of parameters. 4. Conclusion A multiperiod planning problem of supply chains has to deal with macro issues of overall supply chain activities as well as micro issues of individual plant operations simultaneously. In order to address the problem, this paper has proposed a twostage optimization framework using a concept of supply chain operation policy. As an illustration, three policies by Ryu and Pistikopoulos4 were employed, which are coordination, cooperation, and competition. There may be different approaches for the problem, but the above issues would still pose a challenge, which is escalated in its impact because of the presence of multiplicity in supply chains. Complexity in a supply chain can be in various forms with regard to individual industry’s practices, for example, numerous delivery routes in logistics, complex manufacturing processes in semiconductor manufacturing, etc. The proposed framework allows us to respond to such various practices by shifting the emphasis of supply chains using different operation policies. The proposed work can be further expanded by inventing more operating policies suitable to various industries. Acknowledgment J. Ryu gratefully acknowledges the partial financial support from the BK (Brain Korea) 21. Nomenclature Indices i ) product 1...N l ) plant 1...L d ) market demand 1...D t ) time period t 1...T

DMd,i,t ) demand of product i at market d in time t (ton) scd,i,t ) per unit sale price of product i at market d in time t ($) pcl,i,t ) per unit production cost of product i at plant l in time t ($) dcl,i,d,t ) per unit delivery cost of product i from l to d in time t ($) ecl,t ) cost coefficient for capacity operation at plant l in time t ($) fl,t ) fixed cost coefficient for expansion cost at plant l in time t ($) bl,i,t ) per unit performance coefficient for material balance cil,i,t ) per unit inventory cost of product i at plant l in time t ($) ihl,i,t ) per unit inventory cost of product i at plant l in time t ($) L CEl,t ) lower bound of capacity expansion at plant l in time t (ton) U CEl,t ) upper bound of capacity expansion at plant l in time t (ton) L Il,i,t ) lower bound of inventory level of product i at plant l in time t (ton) U Il,i,t ) upper bound of inventory level of product i at plant l in time t (ton) NOPL ) number of available plants for an enterprise NOMR ) number of markets for an enterprise Discrete Variables Yexpl,i ) binary variable; if capacity of plant l is expanded in time t, then 1, else 0 Ycrl,i,t ) binary variable; if product i is made at plant l in time t according to coordination policy, then 1, else 0 Ycopl,d,t ) binary variable; if plant l produces products for market d in time t according to cooperation policy, then 1, else 0 Ycmpl,i,t ) binary variable; if product i is made at plant l in time t according to competition policy, then 1, else 0 Continuous Variables RProfit ) revenue of an enterprise ($) COSTOP ) sum of operational costs of all plants including production, inventory, and capacity expansion ($) xxl,i,d,t ) production quantity of product i at plant l for demand d in time t (ton) Sl,i,t ) production quantity of product i at plant l in time t (ton) SCRl,i,t ) production quantity of product i at plant l in time t according to coordination policy (ton) SCOPl,i,t ) production quantity of product i at plant l in time t according to cooperation policy (ton) SCMPl,i,t ) production quantity of product i at plant l in time t according to competition policy (ton) CPl,t ) capacity at plant l in time t (ton) CEl,t ) expanded capacity at plant l in time t (ton) Il,i,t ) inventory level of product i at plant l in time t (ton) Literature Cited (1) Varma, V. A.; Reklaitis, G. V.; Blau, G. E.; Pekny, J. F. Enterprisewide modeling & optimizationsAn overview of emerging research challenges and opportunities. Comput. Chem. Eng. 2007, 5-6, 692. (2) Shah, N. Pharmaceutical supply chains:key issues and strategies for Optimization. Comput. Chem. Eng. 2004, 6-7, 929.

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8065 (3) Grossmann, I. Enterprise-wide optimization: A new frontier in Process Systems Engineering. AIChE J. 2005, 51, 1846. (4) Ryu, J.; Pistikopoulos, E. N. Design and Operation of an Enterprisewide Process Network using Operation Policies. 1. Design. Ind. Eng. Chem. Res. 2005, 44, 7, 2174. (5) Sahinidis, N. V.; Grossmann, I. E.; Fornari, R.; Chathathi, M. Optimization model for long range planning in the chemical industry. Comput. Chem. Eng. 1989, 13, 1049.

(6) Ierapetritou, M. G.; Pistikopoulos, E. N. Novel Optimization Approach of Stochastic Planning Models. Ind. Eng. Chem. Res. 1994, 33, 1930.

ReceiVed for reView April 10, 2007 ReVised manuscript receiVed August 17, 2007 Accepted August 17, 2007 IE070508B