A strategic model for supply chain design with logical constraints: formulation and solution

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Computers & Operations Research 30 (2003) 2135 – 2155

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A strategic model for supply chain design with logical constraints: formulation and solution Hong Yan∗ , Zhenxin Yu, T.C. Edwin Cheng Department of Management, The Hong Kong Polytechnic University, Kowloon, Hong Kong Received 1 March 2002; received in revised form 1 September 2002

Abstract This paper proposes a strategic production–distribution model for supply chain design with consideration of bills of materials (BOM). Logical constraints are used to represent BOM and the associated relationships among the main entities of a supply chain such as suppliers, producers, and distribution centers. We show how these relationships are formulated as logical constraints in a mixed integer programming (MIP) model, thus capturing the role of BOM in the selection of suppliers in the strategic design of a supply chain. A test problem is presented to illustrate the e5ectiveness of the formulation and solution strategy. The results and their managerial implications are discussed. Scope and purpose Supply chain design is to provide an optimal platform for e6cient and e5ective supply chain management. The problem is often an important and strategic operations management problem in supply chain management. This paper shows how the mixed integer programming modeling techniques can be applied to supply chain design problem, where some complicated relations, such as bills of materials, are involved. We discuss how to solve such a complicated model e6ciently. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Supply chain; Bill of materials (BOM); Logical constraint; Mixed integer programming (MIP) model

1. Introduction The formulation of strategic production–distribution models for supply chain design has been a popular research topic in the 8eld of Supply Chain Management for two decades. Most of these ∗

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0305-0548/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 2 ) 0 0 1 2 7 - 2

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formulations are in the form of mixed integer programming (MIP) models. The problem commonly arises in the following scenario: a number of production producers supply a collection of distribution centers (DCs) with multiple products, which, in turn, supply customers with speci8ed demand quantities of the di5erent products. The challenge is to determine the number, location, capacity, and type of production producers and DCs to use so as to minimize the total cost, or to maximize the after-tax pro8t, of the supply chain. For the entire chain, it is also necessary to select the set of suppliers, hence to determine the number of supply contracts. A diversity of mathematical programming models dealing with di5erent issues of production and distribution can be found in the literature. This paper attempts to present strategic analysis models of the production–distribution system with consideration of the bills of materials (BOM). In their pioneering paper, Geo5rion and Graves [1] described a multi-commodity single-period production–distribution problem and solved it by Benders Decomposition. This is probably the 8rst paper that presents a comprehensive MIP model for the strategic design of supply chains. Another main contribution of this paper was the special method proposed for solving the problem under study. Subsequently, Geo5rion et al. [2] presented a status report on research in strategic distribution system planning based on decomposition techniques. But neither paper considered the supplier factors. Cohen and Lee [3] developed a comprehensive modeling framework for linking material management activities throughout the material production–distribution supply chain. The framework consists of four stochastic sub-models. The optimal solution for each sub-model is solved individually under some assumptions. However, it would be extremely di6cult to 8nd the optimal solution if all sub-models are integrated. A heuristic procedure was designed and demonstrated to be able to yield quality approximate optimal policies. A dynamic, nonlinear MIP model presented by Cohen et al. [4] considered the operation of a network of suppliers, producers and markets. The main contribution of this model was the explicit inclusion of supplier supply contracts in the model. Bills of materials were mentioned, but they were simply treated as material requirement balance constraints in the model. Cohen and Lee [5] presented a single-period multi-commodity model that analyzed resource deployment decisions for an international 8rm. This model was a simpli8ed version of the model presented by Cohen et al. [4]. It considered producer production capacity, material requirements at each producer (major components and sub-assemblies based on BOM), balance constraints at producers and DCs, demand limits, feasible Jow constraints, and capacity of suppliers. BOM were treated as a source of production data for the development of consistency constraints that balanced the material Jows among suppliers, producers and DCs. Arntzen et al. [6] considered BOM as a collection of rooted arborescence, on which each vertex represents a product and the fabrication facility. The contribution of this paper was the inclusion of BOM constraints and duty considerations in an international supply chain. But, it is not clear how the suppliers and DCs were considered in the formulation. In a recent review, Vidal and Goetschalckx [7] highlighted that there was a lack of models with the inclusion of BOM constraints. They argued that BOM should be considered as constraints in a strategic production–distribution model while designing a complete supply chain, but they conceded that it was di6cult to formulate such constraints in a mathematical model. Most past e5orts considered the coordination between supply and production–distribution activities separately. As a result, BOM just acts as some consistency constraints and many of the complex supply chain inter-relationships are neglected, despite that some researchers such as Arntzen et al. [6] and Cohen and Lee [5] suggested BOM be exploited to coordinate the behavior of suppliers with the production and distribution activities. The set of suppliers was seldom included under

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the assumption that they were less involved in the coordination with production and distribution. In practice, it is necessary to consider suppliers in the comprehensive design of a supply chain. BOM provides key information for coordinating the activities between material procurement and production planning. The introduction of BOM as constraints in an MIP-based supply chain design models will be a reasonable way to consider the selection of suppliers, especially for assembly systems, where di5erent parts making up a 8nished product may come from di5erent locations or even countries. There the challenge is to 8nd a method to e5ectively include BOM in the strategic supply chain design model. The early researchers often focus their attention on formulating the material balance constraints between suppliers and producers. However, this cannot truthfully reJect the close relationships between suppliers and manufacturers. Recently, with new developments in logic and integer programming, logical rules and propositions play an increasingly important role in mathematical programming models. Such simple logical constraints as “if A is true, then either B or C must be true” have been explicitly considered in mathematical programming problems (see Yan and Hooker [8]). Introducing logical constraints in an MIP model not only achieve substantial savings in solution time (see Hooker et al. [9] and Raman and Grossmann [10]), but also enable adequate descriptions of relevant relationships among the entities involved in the model. ReJecting intimate material supply relationships with suppliers, the bottom level of BOM can be described as a list of speci8ed quantities of materials or components supplied by di5erent suppliers. According to the types and capacity limits of suppliers and producers, each candidate supplier can provide a subset of materials (or components) in a 8nite amount and each proposed producer can produce a subset of products with throughput limit. To ful8ll the customer demand, we can incorporate BOM into the logical rules, which express the logical relationships between the suppliers and producers. An example of these logical rules is presented as “if product i will be produced in at least s of the proposed producers, then at least t of the candidate suppliers must be selected”. This formulation of BOM makes it possible to capture the role of suppliers in the strategic supply chain design model. We show how BOM logical rules are formulated as constraints in our proposed MIP model to assist in the selection of suppliers, and how linear representation of logical constraints is developed. A numerical example is solved by LINDO to illustrate the e5ectiveness of including BOM constraints for supplier selection. The rest of the paper is structured as follows. In Section 2, we formulate the model which is a multi-commodity, multi-echelon, single-period MIP model for the design of a supply chain. BOM logical constraints are analyzed in Section 3. Section 4 illustrates representation of logical constraints in linear programming formulation. A test problem is solved and the results are discussed in Section 5. Conclusions are presented in Section 6. 2. Formulation The MIP model developed in this section aims to select suppliers from a candidate set of material (or component) suppliers, as well as to locate a given number of production producers, and DCs, subject to producer and DC capacity restrictions. We assume that the customer zone locations and their speci8c demand estimates for multiple products are given in advance. The potential producer and DC locations as well as their capacities

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are also known. For each open producer and DC, a decision must be made on the total units of products that need to be transported from the open producer to the open DC, and the total units of products that need to be distributed from the open DC based on a given service level. In addition, we assume that production of one unit of a product requires one unit of producer capacity, regardless of which product it is. The similar assumption is adopted to suppliers and DCs. We then present a multi-commodity, multi-echelon, and single-period MIP model with the objective of minimizing the total cost, with consideration of BOM logical constraints for supplier selection. 2.1. Parameter notations and de/nitions Before formulating the model, we introduce the basic parameter notations and de8nitions. In this study, we use the following indices: j ∈ J , a set of candidate suppliers; k ∈ K, a set of potential producers; l ∈ L, a set of possible distribution centers; n ∈ N , a set of customer zones; m ∈ M , a set of materials (or components) needed for production, and, i ∈ I , a set of products. We de8ne the problem parameters and decision variables as follows: I. Parameters: FOj gk fl OCmjk PCik TCikl DCiln CVmj CPk LCik ; UCik Wl Din

8xed cost to open and operate supplier j 8xed cost to open and operate producer k 8xed cost to open and operate DC l unit cost of material (or component) m ordered from producer k to supplier jS unit production cost of product i in producer k unit transportation cost of product i from producer k to DC l unit distribution cost of product i from DC l to customer zone n capacity limit of material (or component) m of supplier j capacity limit of producer k lower and upper production capacity limits of product i in producer k throughput limit of DC l demand of product i in customer zone n

II. Decision variables: Gmjk Hiln Fikl x˜j y˜ k z˜l

total units of material (or components) m purchased from supplier j for producer k total units of product i distributed from DC l to customer zone n total units of product i shipped from producer k to DC l a 0 –1 variable indicating whether supplier j is selected (x˜j = 1) or not (x˜j = 0) a 0 –1 variable indicating whether producer k is open (y˜ k = 1) or not (y˜ k = 0) a 0 –1 variable indicating whether DC l is open (z˜l =1) or not (z˜l =0)

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xmj

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a 0 –1 variable indicating whether a contract for material (or component) m is entered into with supplier j (xmj = 1) or not (xmj = 0) a 0 –1 variable indicating whether any amount of product i is produced in producer k (yik = 1) or not (yik = 0) a 0 –1 variable indicating whether any amount of product i is delivered to DC l (zil = 1) or not (zil = 0)

yik zil

The problem is to select a subset of suppliers according to the BOM for supplying some materials (or components) required, and to choose a subset of producers and DCs to open for producing and distributing some products to satisfy the demand requirements at the customer zones in order to minimize the overall supply chain management cost. 2.2. The objective function The total cost of the supply chain includes purchasing cost, production cost, transportation and distribution cost, and 8xed costs such as the 8xed ordering cost, the 8xed cost to open and operate a producer, and the 8xed cost to open and operate a DC. Therefore, the objective function to be minimized is given by:



Min Z = OCmjk Gmjk + PCik Fikl + TCikl Fikl + DCiln Hiln m; j; k

+




FOj x˜j +

j


k

i; k;l

gk y˜ k +




i; k;l

fl z˜l :

i;l; n

(1)

l

The 8rst term in the objective function is the total purchasing cost of materials and components from all suppliers. The second term is the total production cost in all producers. The third and fourth terms are the total transportation cost (from producers to DCs) and distribution cost (from DCs to customer zones). The remaining three terms are the various 8xed costs including the 8xed ordering cost, the 8xed cost to open and operate producers, and the 8xed cost to open and operate DCs. 2.3. The constraints For an MIP supply chain design model, there are many generic constraints to be considered, including the balance constraints of materials (or components) and products, the capacity limit constraints, the throughput limit constraints, and the service level constraints (for example, see Geo5rion and Graves [1]). They are discussed under constraints I–V below. We also introduce some new constraints concerning the logical rules involving BOM and ensuring logical consistency, as discussed under constraints VI and VII below. I. Material requirements: Rmi : Units of material (or component) m required to produce one unit of product i according to the product BOM.

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The materials (or components) balance constraint is

Fikl Rmi 6 Gmjk for all k; m:

(2)

j

i;l

II. Supplier’s capacity limits: The capacity limits, for supplier j, can be formulated as
Gmjk 6 CVmj xmj for all m:

(3)

k

III. Production capacity limits of producers: There are lower and upper production capacity limits of product i in producer k. So, for each producer k, the production capacity limits constraints are
LCik yik 6 Fikl 6 UCik yik for all i: (4) l

IV. The throughput limit constraints: For DC l, the throughput limit is
Hiln 6 Wl zi; l for all i:

(5)

n

For producer k, the throughput limit is
Fikl 6 CPk : i;l

(Note: In each candidate producer site, we must have V. Service level constraints: #: Percentage of customer demands met.

(6)  i

UCik = CPk :)

Under the circumstances of deterministic customer demands, the customer service level can be de8ned as a percentage of the total demand met. We can formulate the service level constraints as
# Din for all i; n; Hiln ¿ (7) 100 l

where


k

Fikl ¿




Hiln

for all i; l:

(8)

n

VI. BOM logical constraints: Taking into consideration the types and capacity limits of suppliers and producers, we 8nd that the logical relationships between the candidate suppliers and proposed producers can be connected by BOM. Each candidate supplier can provide a subset of materials (or components) in a 8nite amount and each proposed producer can produce a subset of products with throughput limit, while generally the bottom level of BOM can be described as a list of speci8ed quantities of materials or components needed by the set of products. Considering the customer demands, we can incorporate BOM into such logical rules as “if at least p of the proposed producers are open, then at least q of the candidate suppliers should be chosen” and “if product i will be produced in at least s of the proposed producers, then at least t of the candidate suppliers should be selected to supply the material m according to

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the BOM”. These two types of logical rules are represented in the following forms, known as cardinality rules (see McKinnon and Williams [11]): (y˜ 1 ∨ y˜ 2 ∨ · · · ∨ y˜ K )p ⇒ (x˜1 ∨ x˜2 ∨ · · · ∨ x˜J )q

(9)

and (yi1 ∨ yi2 ∨ · · · ∨ yiK )s ⇒ (xm1 ∨ xm2 ∨ · · · ∨ xmJ )t :

(10)

Note that we assume the same priority to all the suppliers for selection process in this model. To deal with the problem of supplier selection with di5erent priorities, it simply adds another set of logical rules or by directly assigning weights to suppliers. The research scope of this paper does not cover this issue. The number of open producers is subject to the producer type and the throughput limit of each producer. These logical rules will be discussed in more detail and their formulation illustrated in Section 3. VII. Logical consistency constraints: Another important set of constraints represents the fact that if a product is to be produced, then the corresponding type and amount of parts or materials must be supplied. This set of constraints is called logical consistency constraint. For example, for supplier m, the variables xmj should be consistent with x˜j in the MIP model. The logical rule representing this consistency relationship is “if at least one xmj is true (xmj = 1), then x˜j is true”. Such logical consistency also exists for the binary variables related to producers and DCs. We list the relevant logical consistency constraints in the following: (1) Whether a supplier is selected is subject to if any amount of materials (or components) is purchased from it. The material purchasing indicators xmj should be consistent with the supplier selection indicators x˜j . So, we have (x1j ∨ x2j ∨ · · · ∨ xMj ) ⇒ x˜j :

(11)

It means if any of the indicators xmj is true (some amount of material m is ordered from supplier j), then x˜j is true (supplier j is selected). (2) Whether a producer should be open is subject to if any product is produced in that producer. This can be formulated into logical consistency constraints as follows: (y1k ∨ y2k ∨ · · · ∨ yIk ) ⇒ y˜ k :

(12)

(3) The same logic can be applied to DCs. If any product is distributed from a DC, then it is open. Thus, (z1l ∨ z2l ∨ · · · ∨ zIl ) ⇒ z˜l :

(13)

The logical rules (11), (12) and (13) above are also cardinality rules. 3. BOM constraints As a basic input to manufacturing resources planning, BOM plays a pivotal role in production planning and scheduling. In supply chain management, BOM also plays a fundamental role in the chain design. However, there does not exist a formal and consistent way to represent BOM constraints

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in strategic supply chain design models (see Vidal and Goetschalckx [7]), although some researchers (such as Arntzen et al. [6] and Cohen and Lee [5]) proposed that BOM information be exploited to coordinate the behavior of suppliers with the production and distribution activities of a manufacturer. In fact, many important interactions and their logical consistency among the entities of a supply chain can be captured via expressing them as BOM logical constraints. Based on advances in logical integer programming (Williams and Brailsford [12], Yan and Hooker [8]), we propose to capture BOM information as logical constraints and introduce them in a strategic supply chain design model. The introduction of BOM as constraints in MIP models will enable the inclusion of supplier selection in the strategic design of supply chains. In a supply chain design model, the limits of the supplier’s capacity are commonly expressed as constraints for supplier selection. Generally, BOM can be described as a hierarchical product structure that speci8es the quantity and lead time of each item, ingredient, or material needed to assemble, mix, or produce the end product. The bottom level of BOM consists a set of materials (or components) provided by suppliers. Considering other related important constraints such as supplier’s type, producer’s type, and producer’s throughput limits, we identify relevant logical relationships between suppliers and production producers. Such relationships are expressed as cardinality rules which say that “if at least p of the proposed producers are open, then at least q of the candidate suppliers should be chosen”, or “if product i will be produced in at least s di5erent producers, there are at least t suppliers to be chosen for supplying the materials (or components) of product i”. In the following, we discuss these two types of logical rules, respectively. A. Consider all products with all suppliers and producers involved: For a multi-commodity supply chain model as discussed in Section 2, a set of products is to be produced to meet the demands of customer zones. From the BOM of these products, we obtain information about a set of materials (or components) supplied by suppliers. Each supplier can only supply a subset of these materials (or components) within its capacity limit. Meanwhile, each producer only produces a subset of products demanded within its throughput limit. Considering the demands in customer zones, we can state the logical rule “if at least p of the proposed producers are open, then at least q of the candidate suppliers should be chosen”, and formulate it as (y˜ 1 ∨ y˜ 2 ∨ · · · ∨ y˜ K )p ⇒ (x˜1 ∨ x˜2 ∨ · · · ∨ x˜J )q (see constraint VI). In the model presented in Section 2, we use x˜j as the 0 –1 indicator for candidate suppliers and y˜ k as the 0 –1 indicator for potential producers. B. Consider one product with certain suppliers and producers involved: Each product has its own set of materials (or components) as input for production. According to producer type, certain types of products can be produced in each producer. And, at the same time, each supplier can only supply certain materials (or components). Therefore, for a particular product, it can be produced only by a subset of proposed producers, and its materials (or components) can only be supplied by a subset of candidate suppliers. We can formulate this constraint as a logical rule if product i will be produced in at least s di5erent producers, there are at least t suppliers to be chosen for supplying the materials (or components) of product i” represented by (yi1 ∨ yi2 ∨ · · · ∨ yiK )s ⇒ (xm1 ∨ xm2 ∨ · · · ∨ xmJ )t (see constraint VI). We see, from the above, that applying logical rules to formulate BOM constraints is a natural way to take BOM into account in a supply chain design model. The advantage of such logical rules is

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that they precisely reJect the inherent logical relationships among products, suppliers and producers. In practice, there could be other factors, such as the quantitative amount of materials, that need to be represented in BOM. In such a case, we can say that it would not be di6cult to add more linear equations or to modify the above rules to represent the quantitative relationship. In addition, some related factors can also be stated as logical rules formulated via BOM constraints such as producer type, supplier’s capacity, throughput limit and customer demand. Therefore, the above descriptions give basic rules for BOM formulation, and reveal the fundamental structure of BOM. To illustrate how to formulate BOM as logical constraints, an example is given in Appendix A.1. The BOM data of this example are from the illustration problem in Section 5. 4. Linear representation of logical constraints The supply chain design model we presented in Section 2 is an MIP model which, excluding the constraints VI and VII, can be solved by Benders Decomposition (Geo5rion and Graves [1]) or factorization methods (Brown and Olson [13]). In general, a logic programming problem belongs to a wider group of problem called constraint logic programming. A number of systems, such as Prolog, Prolog III, ILOG solver and CHIP, have been developed to solve the problem. Alternatively, the problem can be translated into a integer linear programming problem by incorporating the sets of logical constraints VI and VII into the model directly involving 0 –1 variables. For the practical problems with a relatively small amount of such logic rules, it is often easier to solve the problem with all linear constraints. In this section, we brieJy introduce some theoretical background about logical constraints. One conventional method for representing logical relations in an MIP model involves two stages. A logical rule is 8rst rewritten as a conjunction of logical “clauses”, i.e., in the conjunctive normal form (CNF) (Williams and Brailsford [12]). A clause is a disjunction of literals, such as x1 ∨ @x2 ∨ x3 ; where @ means logic “not”, and ∨ means logic “or” . Each clause is then written as an inequality in 0 –1 variables, which for this example is x1 + (1 − x2 ) + x3 ¿ 1; where x is interpreted as true when x = 1 and false when x = 0. Generally speaking, the speci8c logical relationships among the units in the process can be explicitly converted into the CNF form: [ ∨i∈P1 (xi ) ∨i∈PT1 (@xi )] ∧ [ ∨i∈P2 (xi ) ∨i∈PT2 (@xi )] ∧ · · · ∧ [ ∨i∈Ps (xi ) ∨i∈PTs (@xi )]; where Pj is the subset of positive literals and PT j is the subset of negative literals in the jth clause (for details, see example, Hooker and Chandru [14]). Each conjunction in the CNF can be written as an inequality in 0 –1 variables like the example above. A logical relationship is thus converted into the CNF, and then the CNF is transformed as linear inequalities in binary variables. However, the linear inequalities so obtained do not usually give a “tight” representation of the original logic rule. In their work, Yan and Hooker [8] discussed the cardinality logic rules, and provided a convex hull representation via a simple recursive procedure.

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In this paper, two types of logical constraints are formulated. One includes the logical constraints from the BOM requirements (constraint VI), while the other includes some logical consistency constraints related to some 0 –1 variables for each entity of the supply chain (constraint VII). Each of these logical constraints can thus be transformed into the corresponding convex hull representation, which can then be embedded in the MIP model. In the following, we will brieJy exhibit these two types of logical constraints. 4.1. Logical consistency constraints We discuss the logical consistency constraints 8rst, because the logical consistency constraints in the model follow a special form of the cardinality rules. The form of logical consistency constraints in Section 2 is (x1 ∨ x2 ∨ · · · ∨ x n ) ⇒ y. This can be converted into the CNF: (@x1 ∨ y) ∧ (@x2 ∨ y) ∧ · · · ∧ (@x n ∨ y): And, they can be represented by the linear inequalities: − xi + y ¿ 0;

xi ; y ∈ {0; 1};

i = 1; 2; : : : ; n:

(14)

This representation adds n linear inequalities in the MIP model. According to the result by Yan and Hooker [8], the logical rule (x1 ∨ x2 ∨ · · · ∨ x n ) ⇒ y can be alternatively represented by − (x1 + x2 + · · · + x n ) + ny ¿ 0:

(15)

Theorem 1 in Appendix A.2 shows that inequality (15) is equivalent to all the equalities (14) in the sense that they have the same satis8ability set. In the work of Yan and Hooker [8], (15) is called a “main facet” of the convex hull representation. In this simple case, it is more compact than (14). Thus, the logical consistency constraints (11)–(13) (x1j ∨ x2j ∨ · · · ∨ xMj )

⇒

x˜j ;

(y1k ∨ y2k ∨ · · · ∨ yIk )

⇒

y˜k ;

(z1l ∨ z2l ∨ · · · ∨ zIl )

⇒

z˜l ;

can be represented as linear inequalities, respectively, −(x1j + x2j + · · · + xMj ) + M x˜j

¿

0;

−(y1k + y2k + · · · + yIk ) + I y˜k

¿

0;

−(z1l + z2l + · · · + zIl ) + I z˜l

¿

0:

4.2. BOM logical constraints According to special considerations of all products and one product in Section 3, the logical constraints (in Section 2) derived from BOM are expressed in two di5erent logical rules (y˜ 1 ∨ y˜ 2 ∨ · · · ∨ y˜ K )p ⇒ (x˜1 ∨ x˜2 ∨ · · · ∨ x˜J )q and (yi1 ∨ yi2 ∨ · · · ∨ yiK )s ⇒ (xm1 ∨ xm2 ∨ · · · ∨ xmJ )t :

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Both of them are cardinality rules, and in the same form. Thus, we only consider the former type of BOM logical constraints above: (y1 ∨ y2 ∨ · · · ∨ yK )p ⇒ (x1 ∨ x2 ∨ · · · ∨ xJ )q : The convex hull of this rule is given by Theorem 1 of Yan and Hooker [8]. The convex hull representation has the following inequality as the main facet: − qey + (1 + K − p)ex ¿ q(1 − p);

(16)

where e is a vector of 1s with corresponding dimension. In general, the convex hull representation consists of many linear inequalities. In order to illustrate the convex hull description of this rule, we consider a simple example which states that “at least two suppliers will be chosen from three candidate suppliers, and at least two producers will be open among three potential producer sites”. This logical rule can be written as (y1 ∨ y2 ∨ y3 )2 ⇒ (x1 ∨ x2 ∨ x3 )2 ;

(17)

where xi is a 0 –1 indicator for suppliers and yj is a 0 –1 indicator for producers. It needs to point out that the linear representation of logical rules could make the problem look di6cult for computation, due to the large number of linear inequalities involved. However, for many practical problems, the size of LP formulation would still be within a controllable range. Yan and Hooker [8] discussed this in detail. Next sub-section further discuss the way to reduce the size of the LP formulation without hurting the optimality of the solution. 4.3. Reduction of linear inequalities The convex hull of a cardinality rule provides a tight representation for that rule. However, such a tight representation often involves a large number (it is exponential in the number of variables) of linear inequalities. When several cardinality rules are considered in an MIP model simultaneously, it would sometimes render the computation impractical. As shown earlier in this section, for logical consistency constraints, only the main facet is needed to represent the rule since it is equivalent to the CNF form in terms of integer feasibility. For the general BOM constraints, we may need to use some heuristics to speed up the e6ciency of computation. Yan [15] discussed the computational strategies for this problem. A set of symbolic constraints, which are the logic rules separated from the MIP model, are used as cutting rules to prune further branching on solutions whenever these solutions potentially violate the rules. In his work, the following three rules are tested and compared: (1) logic rules are represented by the linear inequalities which fully describe the convex hull, (2) logic rules are separated from the model and used as symbolic constraints, and (3) logic rules are used as symbolic constraints, but the main facet of each logic rule is added into the linear constraints to tighten the feasible region of the LP relaxation. It is shown that the third strategy is preferable to speed up computation. In this study, we propose a new strategy where the “sense of management” is used. Consider the example given above. The convex hull description of logical rules (17) contains 16 linear inequalities. If some speci8c information about suppliers is provided, e.g. one of the three suppliers is considered an “important supplier” to the manufacturer, we can remove some of the facet representing inequalities provided that they are not the main facet inequality. An “important

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supplier” may imply that the supplier is designated to supply some essential material (or component), or has some strategic relationship with the manufacturer. The important supplier should have a high priority to be chosen as a supplier, although it is possible that it may still be excluded due to high ordering cost or other related restrictions. The common way to deal with this kind of information is to assign a weight to the important supplier arti8cially in the model. A more natural way to give priority to an important supplier in the model is to remove some inequalities de8ning the logical constraints. In this example, the inequality − y 1 − y2 + x 2 + x 3 ¿ − 1

(18)

says that “if y1 and y2 are true, then at least one of x2 and x3 must be true”. Thus, withdrawing this constraint implies that the opportunity for suppliers 2 and 3 to be selected is reduced. Therefore, higher priority is shifted to supplier 1. Inequality (18) is not the main facet de8ning inequality. It is easy to show that to remove such inequalities will not change the optimal solution (see Yan and Hooker [8]). Clearly, this is another way to balance the problem size and the tightness of representation. Now, assume that supplier 1 is an important supplier in the above example. We can remove the following inequalities from the original linear descriptions of the logical constraints, since x1 is not involved −(y1 + y2 + y3 ) + 2(x2 + x3 ); −y1 − y2 + x2 + x3 ;

¿ − 1;

−y1 − y3 + x2 + x3 ;

¿ − 1;

−y2 − y3 + x2 + x3 ;

¿ − 1:

¿ − 1;

This heuristic leads to a great reduction in the number of inequalities of the linear representation. Generally, we suppose that there are K potential producers and J candidate suppliers. According to BOM considerations, we have such a logical rule as “if at least s producers are open, there are at least t suppliers to be chosen for supplying materials (or components)”. This logical rule can be formulated as (y1 ∨ y2 ∨ · · · ∨ yK )s ⇒ (x1 ∨ x2 · · · ∨ xJ )t :

(19)

The number of linear inequalities generated from the BOM logical rule (19) can be expressed as a number function F(K; J; s; t) = (CKK + CKK −1 + CKK −2 + · · · + CKs )(CJJ + CJJ −1 + CJJ −2 + · · · + CJt ):

(20)

If there are n (n ¡ s) suppliers which have good relationships with the manufacturer, for example they have some priority to be chosen as suppliers, we should exclude the logical cuts of these n suppliers. Then, the number function F(K; J; s; t) will become −n J − n− 1 n + · · · + CJt − Fn (K; J; s; t) = (CKK + CKK −1 + · · · + CKs )(CJJ− n + CJ − n −n ):

(21)

So, the reduction in the number of inequalities is F(K; J; s; t) − Fn (K; J; s; t). To illustrate how e6cient this reduction is, we consider a supply chain in which there are four potential producers and six candidate suppliers. The logical rule based on BOM is “if at least two producers is open,

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then at least two suppliers should be chosen”. From Eq. (20), we see that the total number of linear inequalities from the logical rule is F(4; 6; 2; 2) = (C66 + C65 + C64 + C63 + C62 )(C44 + C43 + C42 ) = 627: If one supplier has priority to be chosen, then the number of linear inequalities for such a case from Eq. (21) is F1 (4; 6; 2; 2) = (C55 + C54 + C53 + C52 + C51 )(C44 + C43 + C42 ) = 341; after removing some inequalities—those that include logical cuts of the important supplier. The resulting number of linear inequalities is F − F1 = 627 − 341 = 286, amounting to a reduction of 46 percent. Therefore, to exclude some inequalities of logical rule for the important suppliers is not only a natural way to give priority to these suppliers in the model, but also enables a great reduction in the solution time of the MIP model. Having the logical constraints expressed as integer programming constraints, the model we formulate in Section 2 becomes a tractable production–distribution MIP model. As discussed above, this paper aims at introducing a natural representation for building the BOM into a mathematical programming model. For at least the practical problem without too complicated relationship or too large amount of entities, the model can be translated into a linear integer programming problem and solved within reasonable time. The general computational issues are discussed in (Yan and Hooker [9]). Therefore, in this paper, we do not conduct the simulated computational test on the computational time against the problem size. Rather, through a real example, we show the formulation process and how the representation techniques.

5. An illustration problem This section presents a small-scale supply chain design problem adapted from a real-life situation. The purpose is neither to show any advantage of the modeling process by comparing with other MIP models, nor to exhibit the e6ciency of problem solving by benchmarking the computation time to other algorithms. Indeed, we aim to illustrate the e5ectiveness and convenience of the formulation by introducing the logic constraints into the model and the solution strategy, and to reveal insight of the problem structure through the modeling process. We discuss the results obtained, on issues of the viability, validity, and sensibility of the proposed model. The potential design of a supply chain being considered by an international computer company in Southeast Asia is illustrated in Fig. 1, which includes four suppliers, three producers, three DCs, and four customer zones. Based on the company’s historical marketing data, we acquire some basic information about costs and demands for use as input to our general model. However, full and exact commercial data are modi8ed numerically to preserve business privacy. We adopt some assumptions for the information in order to generate adequate data needed to formulate the model. The unit ordering costs are generated from 20 to 100 according to di5erent components. The unit transportation and distribution costs are in the range between 10 and 20 percent of the unit product prices. The 8xed costs related to suppliers, producers, and DCs are generated over a range of values to provide

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Vendors

Plants

Distribution centers

Customer zones

V1

C1

P1 D1

V2

C2

P2 V3

D2

C3

D3

C4

P3 V4

Fig. 1. Entities of the proposed supply chain.

Vendors

Plants

Distribution centers Customer zones C1

V2 P1 D2

V3

C3

P3 V4

C2

D3

C4

Fig. 2. Result of the supply chain design.

a realistic scenario for the supply chain design problem. The BOM of the two products and the demands of customer zones are discussed in the example of Appendix A.1. The objective of this supply chain design problem is to choose supply chain entities in order to minimize the system cost while satisfying all customer demand. Therefore, the decision variables include all binary (0 –1) variables which represent the selection of supply chain entities and the connection between them. Some integer variables which represent the unit of product Jow in the network are also included. Based on our general model, we formulate the problem as an MIP model which is composed of 80 variables (including 26 binary variables) and 260 rows of constraints. The representation of the BOM logical constraints and logical consistency constraints follows the formulation presented for the example in Appendix A.1. We code the procedure of linear representation of logical rules and the MIP problem is solved by LINDO. The resulting supply chain network is shown in Fig. 2. The solid lines connecting entities represent the relative decision variables of connecting the two entities are 1. The actual supply chain design problem of the company is shown by the arrow line connections in Fig. 3. To observe the sensitivity of the model to di5erent operating conditions, we 8rst increase the unit ordering cost, unit transportation cost, and unit distribution cost, and then, increase the 8xed ordering cost and the 8xed costs of producers and DCs in order to detect their e5ects on the total cost. The unit production cost is not discussed here since it is small relative to others. The computational

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Legends: Vendors Plants DCs Customers

2

2 3 1

2

1

3 3

1

4

3 4

2 1

Fig. 3. Illustration of a supply chain design problem. Table 1 Change in total cost against increasing unit ordering, transportation, and distribution costs Percentage change (%)

10 20 30 40 50

Increasing unit ordering cost

Increasing unit transportation cost

Increasing unit distribution cost

Minimum total cost

Change in total cost (%)

Minimum total cost

Change in total cost (%)

Minimum total cost

Change in total cost (%)

99810.0 105260.0 110710.0 114580.0 121610.0

5.776 11.552 17.327 21.429 28.879

94820.0 95280.0 95740.0 96000.0 96660.0

0.487 0.975 1.462 1.738 2.437

94762.0 95140.0 95490.0 95840.0 96160.0

0.426 0.827 1.198 1.568 1.908

results and discussion are given as follows: (1) Increase unit ordering, transportation, and distribution costs by a step increment of 10 percent. The resulting minimum total cost and percentage change in total cost are shown in Table 1. (2) Increase 8xed ordering cost, 8xed cost of each producer, and 8xed cost of each DC by a step increment of 10 percent. The resulting minimum total cost and percentage change in total cost are given in Table 2. A comparison of the e5ect on the total cost by increasing the three unit costs and increasing the three 8xed costs is presented in Fig. 4. It can be seen that the total cost is relatively sensitive to

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Table 2 Change in total cost against increasing the three 8xed costs Percentage change (%)

10 20 30 40 50

Increasing 8xed order cost

Increasing 8xed cost of plants

Increasing 8xed cost of DCs

Minimum total cost

Change in total cost (%)

Minimum total cost

Change in total cost (%)

Minimum total cost

Change in total cost (%)

94434.0 94508.0 94582.0 94656.0 94730.0

0.078 0.157 0.235 0.314 0.392

94690.0 95020.0 95350.0 95680.0 96010.0

0.350 0.699 1.049 1.399 1.749

94480.0 94600.0 94720.0 94840.0 94950.0

0.127 0.254 0.382 0.509 0.625

changes in unit ordering cost. Increasing unit ordering cost by 10 percent will cause a more than 5 percent increase in total cost. Changes in any of the other 8ve costs only have minor e5ects on total cost—a 10 percent increase in any of the 8ve costs will cause a no more than 0.5 percent increase in total cost. In Fig. 4, we also see that all unit costs (unit ordering cost, unit transportation cost, and unit distribution cost) have more inJuences on total cost than 8xed costs (8xed order cost, producers’ 8xed cost, and DCs’ 8xed cost). Therefore, ordering cost is an important cost component in the company’s cost structure, and the main element of the ordering cost is related to supplier selection. So we can draw the conclusion that supplier selection is a controlling factor for supply chain design. From a practical perspective, the results obtained from our model are also seen to be reasonable. Generally, logistics costs (including transportation and distribution costs) are very high in Asian companies which account for about 15 –20 percent of total cost, particularly for consumer products. But, for equipment or industrial products, these costs are very low, about 3 percent of total cost. Ordering cost is a main component in the cost structure for any company in Asia. A manager of the computer company, from which we acquired the data for our test problem, admitted that, in this company, logistics costs accounted for no more than 10 percent of total manufacturing cost and increasing logistics costs by 10 percent will roughly cause about a 1 percent increase in total cost. From the solution of this numerical example, we see that formulating BOM as logical constraints provides a new and viable approach to incorporate BOM constraints in supply chain design models. It also enables the explicit inclusion of supplier selection in the strategic design of supply chains. 6. Conclusions Although there is a wealth of literature and research on modeling of strategic supply chain design, there is an apparent lack of theoretical consideration of bills of materials (BOM) constraints. In this paper, we formulate a strategic supply chain design model which includes BOM in the form of logical constraints. We express the relationships among products, suppliers, and producers as logical constraints via BOM considerations. These logical constraints include two types: BOM logical

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35

30

Percentage change in total cost (%)

25

20

15

10

5

0 10

20

30

40

50

Percentage change in unit or fixed costs (%) unit ordering cost

unit transportation cost

unit distribution cost

fixed order cost

plants' fixed cost

DCs' fixed cost

Fig. 4. E5ect of di5erent unit costs on the total cost.

constraints and logical consistency constraints. Through these two types of logical constraints, we can capture the role of BOM in the selection of suppliers in an mixed integer programming (MIP)-based model for the strategic design of supply chains. In order to make the MIP model with logical constraints to be tractable, we extensively discuss the representation of logical constraints. We present logical constraints as linear inequalities based

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H. Yan et al. / Computers & Operations Research 30 (2003) 2135 – 2155 Product A: PC

Product B: Server

PC Motherboard (a1)

Server Motherboard (b1)

SDRAM (a2)

RDRAM (b2)

PC CPU Chip (a3)

Server CPU Chip (b3)

Fig. 5. The BOM of products A and B.

on the conjunctive normal form (CNF) representation of logical rules. But the di6culty with CNF representation is not how to represent individual logical rules. The large number of clauses in the CNF will result in a great number of inequalities embedded in the original model. We show how to achieve simpli8ed representation of the logical consistency constraints and obtain an e6cient set of inequalities with the help of additional information about suppliers of the supply chain. We formulate a test problem and solve it to demonstrate that the model is valid and viable as a tool for realistic design of strategic supply chains. The solution also reveals that making use of logical constraints not only provides a new and e5ective means to incorporate BOM constraints, but also enables an explicit consideration of supplier selection in supply chain design. Acknowledgements The authors wish to thank the editor and two anonymous referees for their valuable comments and suggestions. This research is partially supported by The Hong Kong Polytechnic University under grant number G-YD50. Appendix A. A.1. An example of formulating BOM logical constraints Consider a simpli8ed supply chain operated by an international computer company located in Southeast Asia. The supply chain includes four candidate suppliers, three proposed producers, three potential DCs, and four customer zones (please see the solution of this example in Section 5). The total demands for two products of the four customer zones are: A (personal computer) 300 units and B (server) 250 units. The BOM of the two products are illustrated in Fig. 5. Table 3 give the supplier type and suppliers’ capacity limits. Information about the producer type and producers’ throughput limits is given in Table 4.

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Table 3 Supplier type and their capacity limits for the example Vendors

Components supplied (capacity limits)

V1 V2 V3 V4

a2 (300); b2 (250) a1 (300); a3 (300); b1 (250); b3 (250) a2 (300); a3 (300); b2 (250); b3 (250) a1 (300); b1 (250)

Table 4 Producer type and their throughput limits for the example Plants

Plant type (throughput limits)

P1 P2 P3

Product A (300 units) Products A and B (600 units) Product B (300 units)

From the data in Tables 3 and 4, together with demands of customer zones, we can formulate the following logical constraints based on BOM considerations: • For both products A and B, at least one producer (P2 ) should be open in order to satisfy customer demands. And, according to the BOM, at least two suppliers should be selected for supplying the components needed. This can be expressed as a logical rule “if at least 1 of the proposed producers is open, then at least 2 of the candidate suppliers should be chosen”, formulated as (y˜ 1 ∨ y˜ 2 ∨ y˜ 3 ) ⇒ (x˜1 ∨ x˜2 ∨ x˜3 ∨ x˜4 )2 :

(A.1)

• For each product A or B, if it is produced in a particular producer, the necessary components should be ordered from the corresponding suppliers. We use yA; P1 to indicate if product A is produced in producer P1 and xa1 ;V2 to indicate if component a1 is ordered from supplier V2 . Based on the BOM of product A and supplier type information given in Table 1, we have the following logical constraints (yA; P1 ∨ yA; P2 )

⇒

(xa1 ;V2 ∨ xa1 ;V4 );

(yA; P1 ∨ yA; P2 )

⇒

(xa2 ;V1 ∨ xa2 ;V3 );

(yA; P1 ∨ yA; P2 )

⇒

(xa3 ;V2 ∨ xa3 ;V3 ):

Similar logical constraints can be formulated for product B. A.2. Proof of Theorem 1 Theorem 1. If xi ; y ∈ {0; 1}; i = 1; 2; : : : ; n, then the inequality −(x1 + x2 + · · · + x n ) + ny ¿ 0;

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is equivalent to the inequalities −xi + y ¿ 0;

i = 1; 2; : : : ; n:

Proof. If −xi + y ¿ 0, for all i = 1; 2; : : : ; n, then −(x1 + x2 + · · · + x n ) + ny ¿ 0: On the other hand, if (−x1 + y) + (−x2 + y) + · · · + (−x n + y) ¿ 0; then there is at least one i, such that −xi + y ¿ 0: We consider two cases: (i) if y = 1, then for any i ∈ {1; 2; : : : ; n}; xi ∈ {0; 1}, we must have −xi + y ¿ 0; (ii) if y = 0, then from (16), we have xi = 0 for all i; therefore, −xi + y ¿ 0 for i = 2; : : : ; n. So, we have −(x1 + x2 + · · · + x n ) + ny ¿ 0 equivalent to −xi + y ¿ 0;

xi ; y ∈ {0; 1};

i = 1; 2; : : : ; n:

References [1] Geo5rion AM, Graves GW. Multicommodity distribution system design by Benders decomposition. Management Science 1974;20:822–44. [2] Geo5rion AM, Graves GW, Lee SJ. Strategic distribution system planning: a status report. In: Hax AC, editor. Studies in operations management. Amsterdam: North-Holland, 1978. p. 179–204. [3] Cohen MA, Lee HL. Strategic analysis of integrated production–distribution systems: models and methods. Operations Research 1988;36:216–28. [4] Cohen MA, Fisher M, Jaikumar R. International manufacturing and distribution networks: a normative model framework. In: Ferdows K, editor. Managing international manufacturing. Amsterdam: North-Holland, 1989. p. 67–93. [5] Cohen MA, Lee HL. Resource deployment analysis of global manufacturing and distribution networks. Journal of Manufacturing and Operations Management 1989;2:81–104. [6] Arntzen BC, Brown GG, Harrison TP, Trafton LL. Global supply chain management at Digital Equipment Corporation. Interfaces 1995;25:69–93. [7] Vidal CJ, Goetschalckx M. Strategic production–distribution models: a critical review with emphasis on global supply chain models. European Journal of Operational Research 1997;98:1–18. [8] Yan H, Hooker JN. Tight representation of logical constraints as cardinality rules. Mathematical Programming 1999;85:363–77. [9] Hooker JN, Yan H, Grossman IE, Raman R. Logic cuts for processing networks with 8xed charges. Computers and Operations Research 1994;21:265–79. [10] Raman R, Grossmann IE. Symbolic integration of logic in MILP branch and bound methods for the synthesis of process networks. Annals of Operations Research 1993;42:169–91. [11] McKinnon KIM, Williams HP. Constructing integer programming models by the predicate calculus. Annals of Operations Research 1989;21:227–46.

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[12] Williams HP, Brailsford SC. Computational logic and integer programming. In: Beasley JE, editor. Advances in linear and integer programming. Oxford: Oxford Science Publications. [13] Brown GG, Olson MP. Dynamic factorization in large-scale optimization. Mathematical Programming 1994;64: 17–51. [14] Hooker JN, Chandru V. Optimization methods for logical inference. New York: Wiley Inter-Science, 1999. [15] Yan H. An ILP model with logical constraints for mail sorting facility selection. Journal of Operational Research Society 1998;49:273–7. Hong Yan is an associate professor of Department of Management, associate head and associate professor of Department of Shipping and Transport Logistics, The Hong Kong Polytechnic University. He is also the deputy program leader of the Doctor of Business Administration program of the university. He obtained his Ph.D. from Carnegie Mellon University. In addition, he is also a guest professor and academic advisor for several universities in China mainland. Zhenxin Yu has obtained a Ph.D. in Department of Management from The Hong Kong Polytechnic University. He is pursuing his second Ph.D. in Olin School of Business at Washington University in St. Louis. His research interest is supply chain management, e-business information sharing and distribution channel coordination. T.C. Edwin Cheng is Chair Professor of Management at The Hong Kong Polytechnic University. He obtained a bachelor’s, master’s and doctoral degree from the Universities of Hong Kong, Birmingham and Cambridge, respectively. His expertise is in operations management; in particular, quality management, business process re-engineering and supply chain management. He has published widely in these areas. Professor Cheng regularly provides management training and executive development to public and private organizations.