European Journal of Operational Research 163 (2005) 668–687 www.elsevier.com/locate/dsw

Negotiation-based collaborative planning between supply chains partners Gregor Dudek *, Hartmut Stadtler Department of Production and Supply Chain Management, Darmstadt University of Technology, Hochschulstrasse 1, Darmstadt 64283, Germany Available online 26 February 2004

Abstract It is often proposed that operations planning in supply chains can be organized in terms of a hierarchical planning system. However, the hierarchical approach assumes a single, centralized planning task for synchronizing operations across the supply chain. As central coordination can usually be realized only for isolated parts of an overall supply chain, the question arises whether there are alternative ways of coordination. In this paper we propose a non-hierarchical, negotiation-based scheme which can be used to synchronize plans between two independent supply chain partners linked by material flows. Assuming that plans are generated based upon mathematical programming models, we show how modified versions of these models can be utilized for evaluating material orders or supplies proposed by the supply chain partner and for generating counter-proposals. Resulting is an iterative, negotiation-like process which establishes and subsequently improves a consistent overall plan. Computational tests suggest that the scheme comes close to optimal results as obtained by central coordination.
2004 Elsevier B.V. All rights reserved. Keywords: Supply chain planning; Collaborative planning; Mixed- integer programming

1. Introduction Supply chain management (SCM) deals with the management of the multiple relationships across the supply chain, i.e. the network of organizations involved in creating final customer products and services (Christopher, 1998). As such, SCM embraces various business processes which are of relevance for servicing customers (e.g. order fulfillment, customer service management, product development) and explicitly accounts for the structure of the supply chain (SC) (Cooper et al., 1997). Planning and control of operations, i.e. production, storage, and distribution processes, across the SC clearly forms a key aspect of SCM. Rohde et al. (2000) identified the various planning tasks of interest and

*

Corresponding author. E-mail addresses: [email protected], [email protected] (G. Dudek).

0377-2217/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.01.014

G. Dudek, H. Stadtler / European Journal of Operational Research 163 (2005) 668–687

procurement

long-

production

distribution

669

sales

Strategic Network Planning

term mid-

Master Planning

term

short-

Demand Material Requirements Planning

term

Production Planning

Distribution Planning

Scheduling

Transport Planning

Planning

Demand Fulfilment

Fig. 1. Supply chain planning matrix (taken from Rohde et al., 2000).

showed how they can be organized in terms of a hierarchical planning structure, the supply chain planning matrix (Fig. 1). At the operational planning level the task of master planning (MP) plays a crucial role. It serves to balance supply with demand over the planning horizon and to synchronize operations across the SC (Rohde and Wagner, 2002). In order to achieve this purpose, a single centralized planning task is proposed for the entire SC as indicated in Fig. 1. The implementation of a centralized MP however requires a high degree of integration among participating organizational units. In practice, centralized MP can therefore only be realized for relatively small, isolated parts of the SC, such as for entities belonging to a single company. Thus, the question arises of how to link and coordinate planning between these isolated parts of the SC. This paper contributes to the question in laying out a negotiation-like coordination scheme for two parties, a buyer and a supplier, which establishes and subsequently improves a consistent overall plan. The coordination process of autonomous, yet inter-connected MP activities is referred to as collaborative planning (CP) in the following. The integrated parts of the SC for which centralized MP is realized are called planning domains (in analogy to Kilger and Reuter, 2002). The resulting situation is depicted in Fig. 2. It shows the facility networks pertaining to each planning domain and the corresponding planning processes. Within each planning domain, centralized MP takes place and coordinates subsequent, shortterm planning activities. In absence of any supply chain integration each MP task is accomplished with an isolated view of the corresponding domain and based on local demand forecasts. However, since operations

Buyer Domain

Supplier Domain Collaborative Planning

supplies

final demand

external demand

Fig. 2. Collaborative planning between adjacent planning domains.

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at the distinct domains actually interrelate due to supplies required at the buyer domain, uncoordinated planning results in sub-optimization and inefficiencies such as unnecessary inventory buffers or frequent plan adjustments. Therefore, to improve supply chain performance, domain-specific MP tasks can be linked by CP. The scheme developed in the following assumes that mathematical programming models are used for domainspecific MP. The idea is to pass order proposals (generated by the buyer) and supply proposals (generated by the supplier) as well as associated cost effects between the parties in an iterative manner. A proposal received from the partner is analyzed for its consequences on local planning, and a counter-proposal is generated by introducing partial modifications. Resulting is a negotiation-like process which subsequently improves supply chain wide costs without centralized decision making and with limited exchange of information between the partners. Specifically, only the respective order/supply proposals and associated effects on local cost are exchanged. The paper is organized as follows. A literature review is presented in the next section. Thereafter, we further detail the decision situation outlined above and introduce a quantitative modeling framework. The CP coordination scheme is described in Section 4, followed by computational results which demonstrate its performance in Section 5. A summary and some final remarks conclude the paper.

2. Literature review Relevant literature originates from several fields of contemporary research on SCM. First, there is a large and growing stream of literature on SC coordination by contracts. An often studied situation within this context is the news vendor problem where a retailer faces random demand and has to buy a certain quantity from the manufacturer prior to the realization of demand (e.g. Silver et al., 1998; Lariviere and Porteus, 2001). Another class of papers is concerned with individual and SC wide operating costs when a manufacturer and retailer face lot-sizing decisions for replenishments. Monahan (1984) shows how the manufacturer can induce the retailer to order globally optimal quantities by offering a quantity discount. The analysis is generalized by Lee and Rosenblatt (1986) and enhanced by additional features e.g. by Weng (1995) who adds price sensitivity of demand. Lee and Whang (1999) devise an incentive scheme for optimal control of a SC resembling a serial multiechelon inventory system managed by independent entities. Fransoo et al. (2001) demonstrate how exchange of demand information and definition of service level constraints can be used to improve coordination of a divergent SC with one supplier and several retailers. Reviews of articles on supply contract design can be found in Tsay et al. (1998) and Cachon (in press). A second area of research deals with multi-agent systems for the coordination of SCs. Agent orientation is a paradigm of constructing software solutions such that autonomous entities (agents), which are able to interact with each other, are responsible for sub-tasks of an overall problem (Yu, 2001). Fox et al. (2000) describe a system for modeling SCs with functional agents responsible for e.g. order acquisition, logistics, transport, or scheduling. Similarly, Swaminathan et al. (1998) develop a software library which contains functional agents such as plants, suppliers, etc. and control agents for inventory management, transportation flows or demand planning. In operating the SC, functional agents utilize control agents to support their decision making. An overview of various articles on agent-based solutions for production planning and control as well as SCM is given by Grolik et al. (2001). Finally, a field of research is concerned with mathematical programming models for SC planning at both, the strategic level of network design and the operational planning level. At the operational planning level, Ereng€ uc et al. (1999) present sub-models for production, distribution, and inventory planning in SCs. € Ozdamar and Yazgac (1999) develop an aggregate and a detailed planning model for a production–distribution system. Z€ apfel and Wasner (2000) propose a planning model for the SC of a steel manufacturer

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671

which covers procurement decisions besides manufacturing and distribution. Reviews of mathematical programming planning models within the SC context are given in Bhatnagar et al. (1993), Thomas and Griffin (1996) and Ereng€ uc et al. (1999). The models mentioned above assume central decision making and hence are not applicable to coordinating planning between independent partners. A few contributions however combine mathematical programming approaches with distributed decision making. Here, planning models are used for planning within separate domains. In order to account for the inter-relations between domains, coordination mechanisms are proposed. The simplest coordination scheme is discussed by Bhatnagar et al. (1993) and is referred to as ‘‘upstream planning’’ in the following. It consists of determining plans level by level or hierarchically. Beginning with the most downstream domain, the plan is prepared and defines (among others) supply requirements for its suppliers. These requirements are passed to suppliers, and the procedure continues in upstream direction. The scheme is easy to implement and results in an improvement vs. completely isolated planning at each domain based on local information only. However, it ignores effects of local decisions on upstream partners and thus yields sub-optimal plans compared to central coordination (Bhatnagar et al., 1993). The degree of sub-optimization is studied through computational tests by Simpson and Ereng€ uc (2001) for a three-tier SC. In comparing upstream to central planning results they observe an average gap in total SC cost of 14.1%. As upstream planning can be easily implemented, but results in sub-optimization of the SC as a whole, some authors propose extensions to upstream planning intended to improve the quality of resulting plans. € ur (1999) consider a SC with a distribution and a production stage and plans by a Barbarosoglu and Ozg€ mechanism which rests on upstream planning. However, they extend the scheme by a heuristic method which modifies distribution decisions, if capacity shortages occur at the production stage. Zimmer (2001) proposes a coordination scheme for a buyer and a supplier which, too, grounds on upstream planning. She extends the basic scheme by anticipation at the buyer domain. That is, a simplified model of the supplier decision situation is included in the buyerÕs planning model. For the scenario studied, significantly improved results vs. mere upstream planning are obtained. Finally, a differing approach is presented by Ertogral and Wu (2000). They develop a coordination mechanism based on Lagrangean relaxation of a total SC model. Balance constraints linking adjacent stages are relaxed by introducing Lagrangean multipliers. In this way the total SC model decomposes into domain-specific sub-models. A central agent iteratively sets target values for supply quantities and multipliers and evaluates resulting domain-specific plans for convergence. This work draws from and combines aspects of all three research domains: contract design, agent-based solutions, and mathematical programming. The coordination mechanism for aligning individual planning domains is based on a mathematical programming approach. The coordination mechanisms reviewed above suffer from shortcomings such as a poor quality of solutions (upstream planning), extensive information needs about partners (anticipation approach), or being tailored to specific problem structures € ur, 1999; Ertogral and Wu, 2000). Therefore, a novel approach is and models (Barbarosoglu and Ozg€ presented in the following. As collaborative planning affects the cost outcomes of individual partners, contract terms need to be adapted based on the planning result in order to achieve mutual benefits. Finally, the coordination scheme can be considered and implemented as a multi-agent system for SC planning.

3. Decision situation and modeling In this section we outline the model assumed being used for intra-domain master planning. Also, links to adjacent planning domains will be explicitly considered by additional constraints, so that the model can be employed to collaborative planning.

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As mentioned above, master planning serves to synchronize operations and material flows across the SC at a medium term such that market demands are met at minimum cost. Planning decisions may concern purchasing, production, transportation, and inventory holding as the major types of operations within SCs (Ereng€ uc et al., 1999). Also, availability of resources is planned for as far as it can be adjusted at a medium term (working time). The planning interval is divided into several periods. The objective is to minimize the total cost of fulfilling given, deterministic customer demand. In order to create feasible plans, capacity constraints and the multi-level structure of operations (consumption of intermediate and raw components) are considered. The majority of these relationships can be modeled by linear equations and inequalities. Some decisions may however demand binary ‘‘go’’/‘‘no go’’ variables. In fact these decisions often are the ones of particular importance e.g. due to their impact on costs. Discrete decisions can, among others, regard lotsizing in production and transportation, availability of resources, or activation of supply quantity discounts. The inclusion of binary variables converts the otherwise linear model into a mixed-integer program. In modeling the above decision situation we adopt a framework as presented in Ereng€ uc et al. (1999). Setup decisions for all operations are included in the formulation resulting in a multi-level capacitated lotsizing problem (e.g. Stadtler, 2003). Although in practical applications binary decisions are usually required only for a subset of the operations considered, this is not reflected in the model for ease of exposition. Also, setup times are neglected. Model 1 MP (master planning model) Indices t planning period 2 T j operation 2 J m resource 2 M Index sets T set of planning periods J set of operations M set of resources Sj set of direct successor operations of j Data cvj chj cfj com Dj;t Cm;t Bj am;j rj;k

unit cost of operation j unit holding cost of operation j fix setup cost of operation j unit cost of overtime (capacity expansion) at resource m (external) demand for operation j in period t capacity at resource m in period t large constant unit requirement of resource m by operation j unit requirement of operation j by successor operation k

Variables c total cost xj;t output level at operation j in period t ij;t inventory level of operation j in period t yj;t setup variable of operation j in period t om;t overtime at resource m in period t

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673

Formulation min c s:t: c ¼

XX t2T

ðcvj xj;t þ chj ij;t þ cfj yj;t Þ þ

j2J

ij;t
1 þ xj;t ¼ Dj;t þ

X

XX

ð3:1Þ ð3:2Þ

com om;t ;

t2T m2M

rj;k xk;t þ ij;t

8j 2 J ; t;

ð3:3Þ

k2Sj

X am;j xj;t 6 Cm;t þ om;t

8m 2 M; t;

ð3:4Þ

j2J

xj;t 6 Bj yj;t 8j 2 J ; t; xj;t P 0; ij;t P 0 8j 2 J ; t; om;t P 0 8m 2 M; t; yj;t 2 f0; 1g 8j 2 J ; t:

ð3:5Þ ð3:6Þ ð3:7Þ ð3:8Þ

The model plans output ðxj;t Þ and inventory levels ðij;t Þ for all operations considered as well as expansions of resource capacity ðom;t Þ. Operations represent production, transport or other value-adding activities. The objective function minimizes the value of variable c which due to constraint (3.2) represents the total cost incurred from operation levels, lot-sizing, inventory holding, and capacity expansion. Constraints (3.3) capture the flow balance between output, inventory and consumption by external demand or successor operations. Constraints (3.4) represent capacity restrictions, while lot-sizing relationships are expressed in (3.5). Constraints (3.6) through (3.7) specify domains of variable values. Assigning total cost to the extra variable c is not strictly required at this point, but will prove handy in later sections. The above model depicts the isolated perspective of a single planning domain facing only external (market) demand. Accounting for collaborative planning entails extensions in order to explicitly model links to SC partners. Those links are composed of supply or order proposals received from the collaboration partner. They are introduced in turn for buyer and supplier (only additional data and variables are listed). Extension 1: Buyer domain Index sets JSð 6 J Þ set of supplied items (operations) Data XSj;t

proposed supply quantity of j in period t

Variables xsj;t supply quantity of j in period t isj;t supply inventory of j in period t Formulation s:t: c ¼

XX t2T

ðcvj xj;t þ chj ij;t þ cfj yj;t Þ þ

j2J

isj;t
1 þ xsj;t ¼

X

XX t2T m2M

rj;k xk;t þ isj;t

8j 2 JS; t;

com om;t þ

XX t2T

chj isj;t ;

ð3:9Þ

j2JS

ð3:10Þ

k2Sj

xsj;t ¼ XSj;t 8j 2 JS; t; isj;t P 0 8j 2 JS; t:

ð3:11Þ ð3:12Þ

Constraints (3.9) through (3.12) can be used in two ways. Assuming that the supplier has announced supply proposals XSj;t they can be added to Model 1 as presented above in order to find the optimal intra-domain

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plan given supply quantities XSj;t ((3.9) replaces the original cost function of (3.2)). Supply quantities XSj;t are input to the model due to constraints (3.11). Balance equation (3.10) link supplies to their consumption, thereby restricting internal operations by the availability of supply items. However, inventory holding of supplied quantities is permitted, in order not to fully dictate internal operations by supply quantities. Consequently, the cost function in (3.9) is enhanced by inventory holding costs of supply items. Even though these costs may in fact be covered by the supplier, their inclusion is important for ensuring a proper trade-off between supply inventory holding and remaining operations. Alternatively, when no supply proposals are known, constraints (3.11) can be skipped. In this case supply quantity variables xsj;t take any desired values, and no inventory holding in the form of isj;t occurs. In this mode, constraints (3.10) reveal order quantities (values of xsj;t ) that have to be requested from the supplier in order to realize the resulting plan. The situation is similar at the supplier domain. Extension 2: Supplier domain Index sets JOð J Þ set of ordered items (operations) Data XOj;t

proposed order quantity of j in period t

Variables xoj;t order quantity of j in period t Formulation s:t: ij;t
1 þ xj;t ¼ Dj;t þ xoj;t þ

X

rj;k xk;t þ ij;t

8j 2 J ; t;

ð3:13Þ

k2Sj

xoj;t ¼ XOj;t

8j 2 JO; t:

ð3:14Þ

Assuming that the buyer has announced order proposals XOj;t constraints (3.13) and (3.14) can be used to incorporate the order quantities into intra-domain planning and generate the optimal plan given the proposed pattern of orders. Constraints (3.13) replace the original flow balance equation (3.3), so that two sources of demand are considered: external demand Dj;t and orders by the buyer. If no order quantities are announced by the buyer, (3.14) can be skipped, and the extended model can be used to generate a proposal of supply quantities (values of xoj;t ). However, for doing so lower and upper bounds on xoj;t must be specified at minimum, otherwise values of xoj;t would be zero. This aspect is discussed in more detail in the following section. 4. Collaborative planning scheme In this section we present the negotiation-based scheme for collaborative planning. Thereby, the planning model presented above is utilized throughout every stage of the process. The following section gives an overview of the coordination process in question. Thereafter, the process steps carried out by buyer and supplier in each iteration are described in more detail. Section 4.3 is concerned with the total process flow and financial implications resulting from the negotiation scheme. 4.1. Coordination scheme overview As described in the literature review, coordination between planning domains is achieved in the simplest way by upstream planning. The scheme developed in the following uses upstream planning as a starting

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675

Table 1 Negotiation process example Data exchange

Period item.

1

2

3

4

BfiS

1 2 3

168 77 247

230 239 347

363 239 548

397 375 650

SfiB

1 2 3

122 239 247

363 239 299

397 239 548

BfiS

1 2 3

95 77 247

363 239 347

SfiB

1 2 3

95 77 347

397 404 442

Cost B

Cost S

Cost total

98,667

129,574

228,241

397 431 650

102,727

120,122

222,849

363 239 548

426 404 650

100,679

120,459

221,038

397 404 442

397 404 869

105,451

106,228

211,679

point. The simple, hierarchical coordination process is however extended in a way which establishes a negotiation-like process among the partners. The resulting scheme assumes an equal role for both partners. As a metaphor we can think of supply chains planners holding a meeting to manually coordinate their respective plans. Each planner will analyze the consequences of partner plans and actively propose modifications which improve his situation. Eventually, the planners will commit to a compromise solution. A similar logic applies to the scheme presented below. In a first step, any given partner order/supply proposal is analyzed. Second, as partner requirements usually cause a deviation from the locally optimal plan, targeted modifications to the order/supply pattern are generated and proposed to the partner. The partner then carries out the same process of analyzing the modified order/supply pattern and introducing new modifications to it. It should however be clear that modifications can only be made to a limited extent. Otherwise, the parties would re-generate their original order/supply pattern as their preferred situation. Therefore, the underlying idea is to only allow for the most effective modifications, i.e. those offering the largest marginal reduction of local cost. These modifications yield the largest local cost improvement per unit change, thereby giving the greatest chance for a total SC cost improvement. The negotiation process is visualized by an example in Table 1 which contains the first four periods of an order/supply pattern (the example is taken from one the computational tests presented below with a planning horizon of 12 periods). Note that quantities represent cumulated orders/supplies from period 1 through t which immediately indicate excess or short supply up to a given period. The top contains the initial order pattern requested by the buyer based on his locally optimal plan. If the supplier fully covers the orders, he faces costs of 129,574 monetary units (MU), and total supply chain costs sum up to 228,241 MU. Based on this outcome, the supplier proposes a modified supply pattern as shown in the second section of Table 1 (excess supplies (vs. the initial orders) are printed bold, short supplies italic and bold). The modifications result in a cost decrease for the supplier of about 9500 MU. Once the modifications are communicated to the buyer, he evaluates the new proposals finding a cost increase of about 4000 MU. In summation, the new proposal however creates net savings of 5500 MU compared to the initial situation. The buyer proposes additional modifications (Section 3) by partly returning to its initial orders and partly introducing new changes. The modifications decrease local cost by 2100 MU, but in turn increase cost at the supplier by about 300 MU. In total, additional net savings of 1800 MU can be realized. According to the iterative nature of the process the supplier suggests further modifications at this point, thereby once more generating significant additional savings.

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In order to implement the above process, the collaboration partners need to exchange the respective order/supply patterns as well as effects on local cost incurred by the modifications. From a contractual perspective, we assume that a fixed unit-price contract exists between buyer and supplier initially. Since the negotiation scheme does not affect the total volume provided to the buyer, the payment received by the supplier remains unchanged and is not regarded during the negotiation. However, since the cost outcomes per partner are affected by the scheme, an adaptation of the contract terms needs to take place after the negotiations in order to ensure that both partners realize benefits. Also, the negotiation process is presumed having a short duration compared to the length of a planning period, such that effects of time-passing or changes to underlying data can be neglected. With this overview in mind, we can now consider the outlined scheme in more detail. The following section presents the distinct process steps carried out by the collaboration partners in each iteration. 4.2. Iterative planning steps As we have seen above each partner repeatedly evaluates received order/supply proposals and generates compromises in reply. An intermediate step required for the compromise generation is to determine a locally best outcome that can be reached by modifying the given order/supply pattern. These basic planning activities hence represent the distinct process steps carried out in each iteration and are described in detail subsequently. 4.2.1. Evaluating a proposed order/supply pattern In order to determine the cost resulting from the current order/supply proposal, the planning model as described in Section 3 can be directly applied. From the buyerÕs perspective the situation is as follows: The buyer receives supply proposals XSj;t for all supplied items j 2 JS from the supplier which are input to the model through constraints (3.11). Solving the model hence reveals the optimal local plan given the supply proposals received from the supplier. The supplier is confronted with an analogous situation: He receives order proposals XOj;t for items j 2 JO from the buyer. The order proposals are input to his model through constraints (3.14) as explained above such that its solution yields the optimal local plan given the order proposals. The optimal solution to the model at either domain (buyer and supplier) provides the basis for evaluating a given partner proposal. The major insight naturally comes from resulting cost c . c is in every case greater (or equal) than the cost resulting, if a deviation from the partner proposal is permitted to some degree. Therefore, we reference it as CBmax ¼ c ;

CSmax ¼ c

ð4:1Þ

for the buyer and supplier, respectively. 4.2.2. Determining the best outcome The purpose of this second step is to find all modifications to the received order/supply pattern which improve the local cost situation. Therefore in contrast to step one, deviations from the received order/ supply proposal are permitted and explored in their cost effect. Resulting is the most preferred order/supply pattern that can be devised from the one just received from the planning partner. An important issue here concerns the degree to which modifications can be introduced. As we will see shortly, some limits to the degree of change should be specified. For the moment, we assume that those limits exist and are known at this stage of the process. Again the model as presented above can be used to determine the preferred order/supply pattern after introducing a few extensions. The resulting, modified model for the buyer is given below.

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Model 2 CP-B (Buyer) Data cum;min XSj;t minimum cumulated supply quantity of j, in periods 1 through t cum;min XSj;t maximum cumulated supply quantity of j, in periods 1 through t e arbitrarily small number ð 1Þ Variables þ
dj;t =dj;t supply shift to next/previous period of j in t Formulation min c þ e

XX

þ
ðdj;t þ dj;t Þ

ð4:2Þ

j2JS t2T

s:t: ð3:3Þ–ð3:8Þ; ð3:9Þ; ð3:10Þ; ð3:12Þ; þ
þ
xsj;t þ dj;t þ dj;t ¼ XSj;t þ dj;t
1 þ dj;tþ1 t X cum;min xsj;s P XSj;t 8j 2 JS; t; s¼1 t X

cum;max xsj;s 6 XSj;t

8j 2 JS; t:

8j 2 JS; t;

ð4:3Þ ð4:4Þ ð4:5Þ

s¼1 cum;min cum;max þ
, XSj;t and variables dj;t =dj;t . The data items represent the Model 2 contains new input data XSj;t discussed modification limits, expressed as cumulated minimum/maximum supply quantities. The variables capture modifications vs. the original supply proposals. All constraints of the original model are still in place except for (3.11) which are replaced by constraints (4.3). Hence, supply variables xsj;t are no longer strictly linked to proposed supply quantities. Instead, according to the LHS of (4.3) supply quantities can þ
be shifted to the next ðdj;t Þ or previous ðdj;t Þ period. Likewise, the RHS contains proposed supply quantities plus shifts from the previous and next period. The degree to which shifts are introduced is limited by constraints (4.4) and (4.5). They guarantee that the resulting supply pattern stays within the specified limits. The objective is still to minimize total cost in the first place. The second term of the objective function makes sure that ineffective shifts, which do not improve resulting cost, are avoided. The corresponding model for the supplier is largely equivalent. The only difference stems from the fact that modifications are introduced vs. order quantities. Hence, constraints (4.14) are extended in analogy to (4.3). Accordingly, shift limits as expressed in (4.4) and (4.5) apply to received order quantities. Solving Model 2 reveals the locally best, minimum cost solution which can be reached within the specified shift limits. Since the solution obtained at this process step incurs the lowest possible cost, the resulting cost c will be referred to as

CBmin ¼ c ;

CSmin ¼ c :

ð4:6Þ

Also, it contains more modifications as will be present in the compromise solution. To capture the amount of changes present, we introduce an item-specific difference measure X þ;
; Dmax ¼ ðdj;t þ dj;t Þ 8j 2 JS; JO; ð4:7Þ j t2T þ;
; =dj;t which measures the total amount of modifications for each supplied or ordered item, respectively (dj;t þ
refer to the values of variables dj;t =dj;t: at the modelÕs optimal solution). The solution obtained at this stage depends on the allowed deviation from the original order/supply pattern. The question in setting the maximum deviation limits, i.e. XOcum;min =XOcum;max for the supplier, is to which extent the proposed order/supply pattern can be reasonably modified without yielding unacceptable

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results for the collaboration partner. Defining precise, acceptable limits is clearly difficult, if not impossible. However, a simple rule is proposed in the following which draws from a lot-sizing heuristic developed by Simpson and Ereng€ uc (1998). The idea is to define the maximum deviation (for the supplier) as a shift of the entire order quantity XOj;t to the previous (or the next) order period, that is the nearest preceding (or following) period with an order greater than zero. Since the partner requests a non-zero quantity in neighboring order periods, he has a general need of the item. Hence, there is a realistic chance that he can make use of a surplus supply with small adjustments to his current plan. Resulting minimum and maximum cumulated quantities then are computed as cum;max XOj;t ¼ XOj;nextt ;

cum;min XOj;t ¼




XOcum j;prevt XOcum j;prevt
1

ð4:8Þ  if XOj;t > 0 ; else

ð4:9Þ

where prevt =nextt represent the previous and next order periods relative to a given period t. The maximum cumulated amount which can be supplied up to a period t corresponds to the cumulative order up to the next order period. The minimum amount equals to the cumulated order up to the previous order period, if period t is an order period. Otherwise ðXOj;t ¼ 0Þ the order quantity of prevt can be shifted beyond t and the 1 minimum amount to be supplied through to t correspond to XOcum j;prevt
1 . The computation of limits for the buyer is equivalent, but based on the supply proposals XSj;t . 4.2.3. Generating a compromise proposal The minimum cost solution obtained above contains all modifications which yield a decrease of local cost. However, among those modifications some incur significant marginal savings while others only have a minor impact on cost. Since we can assume that modifications tend to increase cost for the collaboration partner, only (locally) most advantageous modifications should be included in a compromise proposal. The underlying assumption here is that the most effective modifications offer the greatest chance to improve the overall cost outcome, since they create the greatest local cost savings per unit change. In order to identify the most effective modifications and their cost effects, the model above can be used, once more in an extended version. Principally, the objective of this process step is to maximize the cost savings (vs. C max ) per unit deviation from the order/supply pattern. Mathematically this can be expressed as max

ðC max
cÞ=d;

ð4:10Þ

where d corresponds to a measure of the total deviation. As both c and d depend on the modifications in place, the term in (4.10) is non-linear. Methods for dealing with non-linear objective functions in mathematical programming models have been developed known as ‘‘successive linear programming algorithms’’ (see e.g. Zhang et al., 1985). The approach taken there is to linearize the objective function around a given point and then to repeatedly solve the LP and update the linearization. As this method is fairly complex and requires many iterations. The basic idea is to define two (conflicting) goals as (1) maximize the cost savings and (2) minimize the amount of modifications. Each goal by itself is computed linearly (numerator and denominator in (4.10)). In order to pursue both goals simultaneously a goal programming approach can be applied (see e.g. Tamiz et al., 1998). Solving the corresponding two-objective goal program does not guarantee maximization of (4.10). However, it produces a compromise close to the maximum, as it contains only modifications which have a higher impact on cost savings than on the deviation measure.

1 If the possibility of order shifts causing backorders or lost sales at the buyer shall be avoided, the supplier can be additionally notified of latest, minimum supply requirements based on a bill-of-material explosion of final demand forecasts at the buyer.

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Implementing this approach requires to define the total deviation measure d. The item-specific maximum deviations Dmax observed by solving Model 2 can be utilized at this point. Given Dmax an item-specific j j percentage deviation can be easily computed. The total deviation then follows by averaging item-specific values. The resulting definition is X wj X þ
d¼ ðdj;t þ dj;t Þ: ð4:11Þ Dmax j j t2T In case all item quantities are measured in identical units (or can be converted to such), a reasonable weight assignment P is to use an itemÕs maximum deviation compared to the total deviation across all items (wj ¼ Dmax = k Dmax j k ). Resulting target values for goal programming are zero for the deviation (d  ¼ 0) and minimum cost for cost (c ¼ C min ). As both objectives are of different magnitude, an appropriate normalization has to be put in place. Based on the definition in (4.11) d takes values between 0 and 1. Since the total deviation serves as an estimator of the ‘‘harm’’ caused to the collaboration partner, d is scaled by an estimate DP of the partnerÕs cost increase that would follow from suggesting the minimum cost solution C min as counterproposal. In that way an anticipation of partner cost increases is included in the goal programming model. It is realized by a linear function which assumes proportionality between total deviation d and the partnerÕs cost increase above his current outcome. The resulting goal programming model formulation for the buyer is presented in Model 3. Model 3 CP-C (Buyer) Data C max C min DP Dmax j wj

maximum cost (Model 1 solution) minimum cost (Model 2 solution) estimated partner cost increase associated with C min -solution maximum deviation in supply units of j weight of operation (item) j in total deviation calculation

Variables D deviation from minimum cost d percentage distance in supply pattern Formulation min D þ DPd

ð4:12Þ

s:t: ð3:3Þ–ð3:8Þ; ð3:9Þ; ð3:10Þ; ð3:12Þ; þ
þ
xsj;t þ dj;t þ dj;t ¼ XSj;t þ dj;t
1 þ dj;tþ1 t X cum;min xsj;s P XSj;t 8j 2 JS; t; s¼1 t X s¼1

d¼

cum;max xsj;s 6 XSj;t

8j 2 JS; t;

X wj X þ
ðdj;t þ dj;t Þ; max D j t2T j2JS

c
D ¼ C min :

8j 2 JS; t;

ð4:13Þ ð4:14Þ ð4:15Þ ð4:16Þ ð4:17Þ

Except for the objective function and the additional constraints (4.16) and (4.17), the model is equivalent to Model 2. (4.16) and (4.17) capture the off-sets from target goal values. Since the target value for dðd  Þ is

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zero, (4.16) simply represent formula (4.11). D captures the cost increase above C min . The objective function contains the sum of both off-sets, where d is scaled by the estimated cost increase for the collaboration partner as explained above. Once again, the corresponding supplier model is identical, except that modifications vs. order values (XOj;t ) instead of supplies are considered. A final point here concerns the questions of how to determine the parameter DP . The exact way to obtain it is by sending the order pattern which correspond to the C min solution to the partner for evaluation. However, to avoid such a direct inquiry, the partnerÕs likely cost increase is estimated from the effect of the previous compromise proposal. Given that the previous proposal caused a cost increase DP act to the partner and was based on a total deviation of d act , the estimate is obtained as 2 DP ¼ DP act =d act :

ð4:18Þ

The savings realized with the compromise are DC ¼ C max
c ;

ð4:19Þ

where c is the cost associated with the solution to Model 3. 4.3. Total process flow and implications on contract terms We have seen above that, based on a given solution, each new compromise order/supply pattern yields savings to its initiator but (typically) a cost increase to the other partner. Therefore, the compromise pattern and associated cost effects have to be exchanged between the partners in order to determine the total cost effect corresponding to a proposal. Assuming team behavior, i.e. truthful communication of cost effects, this can be accomplished such that the initiator transmits the compromise pattern and associated local savings to his partner. Given this information, the partner determines the total cost effect of the proposal as the difference between reported savings DC and his local cost increase vs. the previous iteration. If the process continues, the partner generates a new compromise proposal in return yielding a new order/supply pattern and local savings DC. He then transmits the respective data items for further inspection to his partner. Apparently, it seems reasonable to continue this process as long as additional net savings occur. However, a degradation of total cost can be tolerated, too, because temporary degradations may give way for additional improvements in subsequent iterations. Therefore, a stochastic acceptance function can be applied to degraded solutions as known for example from simulated annealing (Van Laarhoven and Aarts, 1987; Johnson et al., 1989; Pesch and Voss, 1995). That is, denoting DC Total as the total cost effect (net savings) of a new solution, we define an acceptance function of the form
 1 if DC Total > 0 Total p¼ DC ð4:20Þ lnðN Þ Cumulated DC else e and draw a random number r from a uniform distribution over the [0,1] interval. If r is smaller than p, the new solution gets accepted and the process continues. Otherwise, the process is terminated. In (4.20) N represents the number of degraded solutions accepted so far in the process. Hence, the more degraded solutions have been accepted, the smaller the chances to accept yet another one. DC Cumulated refers to the cumulated savings generated so far and takes the role of a normalizing factor.

2 Since no previous DP act is known in the first iteration, an initial value DP ¼ C max
C min is used which assumes that the cost increase accruing to the partner is equivalent to the savings generated locally.

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Once the process is terminated by denial of a degraded solution, the partners can determine the best solution and the associated supply pattern by inspecting cumulated net savings of the respective outcomes, i.e. by adding net savings per iteration up to each outcome. Since the negotiation process starts with the buyerÕs locally optimal plan, it is clear that any other outcome bears cumulated savings to the supplier but cost increases to the buyer, as also observable from Table 1. Therefore, the question of contractual consequences becomes apparent in order to offer an incentive to the buyer. This can be achieved by adding a savings sharing component to the basic fixed price contract. It can be specified as a ‘‘quantity commitment with bonus reward’’ (Anupindi and Bassok, 1998) which suggests that the buyer is rewarded a bonus if he complies to the order quantities of the final negotiation outcome. A win-win situation is established, if the bonus comprises a compensation of the buyerÕs cost increase vs. his initial solution plus a pre-negotiated share of net savings. The process flow just described rests on the assumption of cooperative team behavior of the partners. This is not overly optimistic in well-established, long-term SC partnerships which typically rely on trustful cooperation (Lambert et al., 1996) and may be supported by agreed-to cost controlling and auditing processes. However, the negotiation process can also be applied in loose relationships where self-interested, opportunistic behavior may occur. In such a setting, each partner tries to maximize his share of the net savings generated during the negotiation process regardless of the effect on his partner. Therefore, the initiator of a compromise proposal is reluctant to announce true savings accruing to him, while the other partner will usually report an exaggerated cost increase in order to receive a high compensation. Now, to counteract these cheating opportunities the ‘‘rules of the game’’ can be changed such that the initiator is no longer expected to announce the actual savings accruing to him. Instead he can make an offer of what he is willing to ‘‘pay’’ for the otherÕs acceptance of the new compromise. The partner can then analyze the compromise pattern for its cost increase and either accept the new proposal (and payment offer) or deny it (typically, when the offer is smaller than his cost increase). From a total SC perspective this procedure can yield identical results as the team approach. However, the savings accruing to each partner are affected depending on the offers submitted during the process. Also, this mode of negotiation bears the risk that proposals which actually improve total SC costs are discarded, if the initiatorÕs offer is insufficient to cover the cost increase faced by his partner. Once the negotiations end, accepted payment offers can be cumulated over all iterations to form a single bonus rendered to the buyer as in the team approach discussed above.

5. Computational results Computational tests have been conducted and are discussed in the following in order to explore the performance of the CP scheme. Thereby, the team approach with correct announcement of cost effects underlies all computations. An automated version of the collaboration process was implemented in MS Visual Basic. The optimization models were solved using the ILOG CPLEX 7.0 standard mathematical programming solver. The structure of input parameters considered here is taken from Tempelmeier and Destroff (1996). Resource capacities are constrained; they can however be expanded up to 20% of given base capacity per period incurring overtime costs. The planning horizon covers 12 periods for all problems. Five classes of test structures are considered as shown in Table 2. The major difference lies in the structure of operations and in the number of resources per planning domain. Small test classes S1 to S3 contain six or seven items, while 10 items are present in classes M1 and M2. Class S1 includes a single resource per planning partner and a two-level bill-of-material. Three levels and two resources at one of the partners are regarded in the remainder (classes S2 to M2).

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Table 2 Problem structure by test class Small

S1

S2

S

B

S3

S

S

B

1

5

4

2

6

6

4

2

5

3

7

1

M1

3

5

3

M2

S

8

1 2

6

4

Medium

B

S

B

5

9

6

10

7

1 2 3 4

8

B

1

7

9 10

2 5 6

3 4

Six demand series were randomly generated for each test class based on a constant, weakly seasonal and strongly seasonal average demand and a coefficient of variation of 0.1 or 0.2. Also, three cost structures were considered for each class based on the average ratio between holding and setup costs at buyer and supplier (constant, high at buyer/low at supplier, and low at buyer/high at supplier). Finally, seven capacity utilization profiles were regarded as shown in Table 3. In profiles 3 and 4 available capacity varies over time, the numbers in brackets show the corresponding periods. Available resource capacity, which is input to the planning model (constraints (3.4)), is calculated from the average capacity need derived from the final demand series and the utilization factor as given in Table 2. Resulting are 126 (6  3  7) test instances for each class which gives a total of 630 test instances. Two benchmark solutions are used in the following for evaluating the performance of the negotiation scheme. First, pure upstream planning results are considered. They represent an upper bound on resulting total cost, as upstream planning forms the starting point of the negotiation. Second, as a lower bound the best solution to a single, global planning model containing both buyer and supplier domain, (central planing) is considered. 3 An overview of the test results is given in Table 4. Concerning upstream planning, the number of capacity infeasible test problems, i.e. instances where given capacity and overtime budget are insufficient to cover buyer orders, is listed first. In these situations the supplier is not able to fulfill all orders on time. Secondly, the percentage gaps of upstream to corresponding central planning solutions, i.e. ðC res;UP
C central Þ=C central

3

Solutions with proven optimality to the central planning model are available in 528 out of 630 test problems based on a computational time limit of 600 seconds. for classes S1 to S3 and 1200 s. for classes M1 and M2. In remaining cases, the best, yet not optimality-proven, integer solution is used.

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683

Table 3 Capacity utilization profiles Profile

Buyer

Supplier

1 2 3 4 5 6 7

90% 70% 90% (1–3,10–12), 70% (4–9) 70% (1–3,10–12), 90% (4–9) 90% 70% 50%

90% 70% 70% (1–3,10–12), 90% (4–9) 90% (1–3,10–12), 70% (4–9) 70% 90% 50%

Table 4 Test results overview Class

Upstream planning

Negotiations

Iterations

Gap closure

Cap. inf.

Gap to central

#

Av. (%)

Std. dev. (%)

Gap to central Av. (%)

Std. dev. (%)

Av. (%)

Std. dev. (%)

Av. (%)

Total

64

22.4

40.9

1.6

1.9

69.8

37.9

4.6

S1 S2 S3 M1 M2

25 4 22 6 7

25.6 33.9 18.3 18.8 15.3

32.6 61.7 32.7 38.7 21.7

2.4 1.5 0.9 1.5 1.6

2.4 1.8 1.2 1.7 1.8

71.4 61.7 71.3 69.1 76.1

35.2 39.9 38.0 35.1 38.9

5.3 3.7 4.4 4.7 4.3

are given for the remaining test instances. Gaps to central planning are also shown for negotiation outcomes as a major indicator of the solution quality. Please note that capacity infeasible solutions do not occur in negotiation results, indicating that capacity shortages present in initial upstream outcomes are successfully corrected during the negotiations in all instances. Furthermore, a percentage ‘‘gap closure’’ is shown of the form ðC res;UP
C res

NEGO

Þ=ðC res

UP


C central Þ:

It serves as a measure of the improvement realized with the negotiation scheme and is captured only in test instances without capacity overrun in upstream planning. Finally, the number of iterations is presented. For all criteria average and standard deviation are listed. The top row shows total results over all test instances. As can be seen, capacity overruns occur in upstream planning in 64 out of 630 instances. Upstream solutions of remaining test problems deviate on average by 22.4% from central planning. Thereby, the standard deviation of 40.9% indicates that individual outcomes vary strongly in solution quality. In contrast, the results obtained with the negotiation scheme deviate on average by a mere 1.6% from central planning. Also, a standard deviation of 1.9% implies that the majority of results falls into the vicinity of central planning. Inspecting the gap closure reveals that, on average, 69.8% of the gap between upstream and central planning are closed. Here, the standard deviation is however higher in magnitude suggesting that individual outcomes are distributed over a large interval (potentially between 0% and 100%). The average number of iterations comes to 4.6. That is, around nine order/supply patterns (one per iteration and partner) are exchanged on average before the process is terminated.

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Upstream & Negotiation Gaps vs. Central percentage test instances [%]

100%

cumulated

80% 60%

negotiation upstream

40% 20% 0%