- Email: [email protected]
Supply chain coordination with uncertain just-in-time delivery
Int. J. Production Economics 77 (2002) 1–15
Supply chain coordination with uncertain just-in-time delivery Kirstin Zimmer* Department of Business Administration and Operations Research, University of Mannheim, D-68131 Mannheim, Germany Received 24 January 2001; accepted 31 May 2001
Abstract Coordinating producer and supplier is one of the main issues of supply chain management. The paper investigates this problem by means of a single-period order and delivery planning model within a Just-in-Time setting. The aim of the paper is to find a coordination mechanism that allows the system to perform just as well as a centralized one. Since the acceptance of a coordination mechanism depends on the associated expected costs for each party of the supply chain, the paper focuses not only on the overall performance but also on the allocation of the cost. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Supply chain management; Coordination; Distributed decision making
1. Introduction Recent years have seen a growing globalization of markets and the concentration of companies on their core competencies. As a result, increasingly legally and economically independent companies are involved in the value added process as part of an entire supply chain. Since this requires increased cooperation between the companies as well as the coordination of logistical decisions, the interest in the field of supply chain management has grown recently among both academics and practitioners. Supply chain management may be considered to be an extension of traditional logistics. Whereas logistics investigates the flow of information, materials, capital and manpower in the internal *Tel.: +49-621-181-1655; fax: +49-621-181-1653. E-mail address: [email protected] (K. Zimmer).
supply chain owned by a single firm, supply chain management deals with the coordination of logistic processes within the external supply chain. The main goal of both traditional logistics and supply chain management is ‘‘to deliver superior customer value at less cost to the supply chain as a whole’’ (see [1, p. 18]). But unlike traditional logistics, supply chain management involves the coordination of independently managed companies who seek to maximize their own profits. Although overall performance of the supply chain depends on the companies joint performance, the operational goals may conflict and result in inefficiencies for the entire chain. Therefore, one of the main issues of supply chain management is to find suitable mechanisms to coordinate the logistical processes that are controlled by various independent companies in order to achieve overall minimal cost. In addition, the mechanism must guarantee that in an existing supply chain partner-
0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 1 ) 0 0 2 0 7 - 9
2
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
ship each individual company will achieve at least the same profit as in the initial situation before applying the mechanism. The closer the parties of a supply chain are linked together the more important the coordination of the entire supply chain becomes. For this reason, we will consider a supply chain with Justin-Time delivery. When Just-in-Time purchasing is implemented, the production of the finished products largely depends on the on-time delivery of the components, since buffer inventories are typically reduced. While both producer and supplier focus on eliminating inefficiencies, shortage situations are usually costlier for the producer than for the supplier. As a result the supplier makes less of an effort to avoid shortage situations than would be optimal for the entire supply chain. The aim of our paper is to find a coordination mechanism that ensures that the decentralized system performs as well as a centralized one while neither the producer nor the supplier are worse off, when compared to the initial situation. The literature on supply chain management can be roughly split into two classes. The first considers the supply chain from the point of view of one decision maker and determines those decisions that minimize overall total cost. In these approaches the supply chain is considered as a fully vertically integrated firm where all information is common knowledge and the material flow is controlled by a single decision maker (see e.g. [2–5]). The second class considers the supply chain as a network (or chain) of antagonistically behaving companies and analyzes the implications of contract forms (for an overview of this literature see [6]). The underlying models describe the supply chain on a highly aggregated level and often consider only two decision makers: A buyer and a supplier. One of the first studies in this area is that of Monahan [7] who investigates the effect of quantity discounts on the order quantity of a buyer and the profit of the supplier. He shows that a supplier is able to increase his profit by offering quantity discounts. Banerjee [8,9] and Corbett/de Groote [10] extend his work and show that the underlying contract is also optimal from the point of view of the entire
supply chain. In the last couple of years the interest in the field of supply chain contracting has grown, thus leading to a considerable amount of research, especially in the field of quantity discounts and quantity commitments (see [11– 14]). However, since the literature in many cases addresses the problem in an agency-theoretic framework there is often no attempt to analyze whether the contract allows the decentralized system to perform just as well as a centralized one (see e.g. [15]). The literature on Just-in-Time related issues can be split into two classes in a similar way. Let us consider the production process of components and the production (or assembly) of the final products as two independent units of a supply chain. Then, the first class describes the supply chain from the point of view of one decision maker and determines the production decisions simultaneously, where the production yield of the components is either deterministic (see e.g. [16]) or random (see e.g. [17–19]). The second class which is most related to our work takes the decentralization of decisions into account and analyzes the effects of penalty-costs and side-payments between suppliers. Gurnani [20], Gerchak/Wang [21] and Gurnani/Gerchak [22] consider a supply chain consisting of one producer and two suppliers, where the suppliers may, due to production yield losses, deliver only a random fraction of the order quantity. Gurnani [20] shows that side-payments between the suppliers have a direct impact on the optimal policy of the suppliers and can be used to increase their efforts. However, he neither analyzes whether the side-payments allow the decentralized system to perform just as well as a centralized one nor he determines the optimal values of the side-payments. In contrast, Gurnani/Gerchak [22] calculate the optimal penalty costs and analyze the cases in which the implementation of these penalty costs achieves the same performance as a centralized system. However, a drawback of all these approaches is that they lead to only one specific cost allocation between producer and supplier. That means that although the overall cost of the supply chain is minimized, a situation may occur in which the cost of one decision
3
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
maker decreases while the cost of another decision maker increases. Our paper differs from the existing literature in that we not only attempt to find contract parameters that allow the decentralized system to perform just as well as a centralized one, but we investigate more general coordination mechanisms which in addition give us higher flexibility in allocating cost. Furthermore, we not only consider the effect of incentives but also the impact of order decisions on the efforts of the supplier. The paper is organized as follows. In the next section we present the basic assumptions of the model and discuss the cost functions of the producer and the supplier. Section 3 considers two extreme cases of the supply link. This worst case/best case analysis illustrates the difference between decentralized and centralized systems and is used in the following sections as a benchmark to evaluate the coordination mechanisms. In Section 4, which is the main section, we develop two different coordination mechanisms and show that if one of these mechanisms is implemented the decentralized system performs as well as a centralized one. Finally, we summarize the results in Section 5 and conclude with directions for further research.
process the order of the considered producer as being limited to a portion aC (with aA½0; 1). At the time the producer places his order, the realized value of a is not known to the supplier (because he is still waiting for further orders) but only its probability density function f ðaÞ: This leads to supply uncertainty in the considered supply chain. To cover the demand of the producer the supplier can build up extra capacity D: In contrast to the normal capacity the extra capacity is built up exclusively for the considered supplier and therefore is deterministic. As depicted in Fig. 1, the delivery quantity d depends on the available capacity aC; the extra capacity D and the order quantity q: Assuming that the capacity consumption rate equals 1, the delivery quantity is restricted by the following capacity constraint: dpaC þ D:
ð1Þ
Since the supplier does not deliver more than what is ordered the delivery quantity d equals the minimum of the order quantity q and the maximum deliverable quantity aC þ D based on the available capacity aC: d ¼ minfaC þ D; qg:
ð2Þ
2. Problem We consider a supply chain with one supplier and one producer in a one-period setting. The producer faces a known demand D for a finished product which is assembled using a special component or primary product that he orders in quantity q from the supplier. Without loss of generality, we assume that the producer requires one unit of the component to produce one unit of the finished product. That is, to meet the demand for the finished product the producer requires D units of the component in total. The normal (long-term) capacity of the supplier is denoted by C: Since a supply chain usually is embedded into a complex supply network, the supplier might not have just one but several producers. In capturing this influence of the network, we describe the supplier’s capability to
Producer
D
q
d (q, ,∆)
Supplier
d ≤ ⋅C + ∆ Fig. 1. Supply chain.
4
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
Demand for the finished product is known D Order quantity q
Available capacity of the supplier is realized
1
0
Delivery quantity d
t=1
Producer:
Supplier:
Delivery Date
Decision about order quantity
Decision about excess capacity Fig. 2. Chronological description of order- and delivery-planning.
Fig. 2 shows the course of events within the supply chain: Based on the given external demand D for the finished product, the producer determines the order quantity q at time t0 : After the order is made and before the available capacity aC becomes known, the supplier has to fix the extra capacity (at time t1 ) in order to meet the demand of the producer. The delivery quantity depends on the extra capacity as well as on the available normal capacity aC of the supplier and might not match the ordered amount at the delivery date.
amount. The second is the cost of holding excess components, after demand has been satisfied. The penalty cost per unit per period of the final product is denoted by p while the holding (recycling) cost per unit per period of the component is denoted by g: Now, for a given value of the delivery quantity d; the cost for the producer is given by
2.1. The producer’s problem
CP p D d g
The producer faces the problem of determining the optimal order quantity. We first describe the actual total cost for the producer at the delivery date and then determine the expected total cost depending on the random portion a of the supplier. In addition to the procurement cost pd; there are two types of costs which arise to the producer when the delivery quantity differs from the demand for the finished item. The first is the penalty cost the producer is charged if he cannot cover the external demand, since the delivered amount is less than the required
C P ¼ p½D d þ þ g½d Dþ þ pd:
ð3Þ
Notations
p
total cost for the producer shortage cost per unit for the finished product demand for the finished product delivery quantity holding (recycling) cost of the primary product per unit purchase cost of the primary product per unit.
When the producer has to place his order q only the distribution function FðaÞ and the density function f ðaÞ are known to the system. Taking into account Eq. (2), the expected total cost for the producer, which depends on his order quantity q as well as on the capacity decision D of the
5
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
supplier, is given by EðC P Þ ¼ p
Z
2.2. The supplier’s problem
minfxðq;DÞ;yðDÞg
ðD ðaC þ DÞÞ f ðaÞ da
0
þ pðD qÞ þ g
1
f ðaÞ da1fD>qg
xðq;DÞ
Z
xðq;DÞ
ðaC þ D DÞ f ðaÞ da yðDÞ
þ gðq DÞ þ p
Z
Z
Z
1
f ðaÞ da1fDpqg xðq;DÞ
xðq;DÞ
ðaC þ DÞ f ðaÞ da
0
þ pq
Z
1
f ðaÞ da
ð4Þ
xðq;DÞ
where yðDÞ :¼ ðD DÞ=C; xðq; DÞ :¼ ðq DÞ=C and 1fD>qg is an Indicator Function defined as: ( 1 if D > q; 1fD>qg ¼ 0 otherwise: In the cost function described above, the first two terms represent the expected customer penalty cost, the second two terms show the expected inventory cost and the last two terms describe the expected procurement cost. The limits xðq; DÞ and yðDÞ of the integrals result from the capacity constraint (1) and the case distinctions of the cost function (3). For instance, to integrate a over the range of ½yðDÞ; xðq; DÞ means to consider the case that the maximum deliverable quantity aC þ D exceeds the required quantity D but is below the ordered quantity q; or in mathematical terms: D D q D pap C C 3DpaC þ Dpq:
yðDÞpapxðq; DÞ 3
The supplier’s choice of extra capacity is a tradeoff between lost profits and the cost of capacity overshoot. The cost for building up extra capacity is fixed at w per unit. To avoid trivial problems, we assume that the revenue of one unit of the component exceeds the cost of procuring this unit with extra capacity, that is p > w: Without loss of generality, we further assume that the supplier must store unsold components at a value h per unit. If the supplier does not use all of the available capacity to produce components which must be stored, this cost may correspond to the cost for overshooting capacity, for instance to store unused raw materials. Now, for a given value of the order quantity q of the producer and in view of Eq. (2), the profit for the supplier is given by CqS ¼ p minfaC þ D; qg h½aC þ D qþ wD: ð5Þ Notations CqS p C a D q h w
profit for the supplier price for the component normal capacity random fraction of the normal capacity extra capacity order quantity cost for overcapacity (holding cost) cost to increase the capacity by one unit
With the knowledge of the density function f ðaÞ the expected profit for the supplier is then calculated by EðCS Þ ¼ p
Z
ðaC þ DÞf ðaÞ da þ pq
0
h Note that the producer is able to determine the optimal order quantity q only if the distribution function of a as well as the capacity decision of the supplier are common knowledge. We come back to this point later in Section 3 when we discuss different kinds of the supply link.
xðq;DÞ
Z
Z
1
f ðaÞ da
xðq;DÞ 1
ðaC þ D qÞf ðaÞ da wD:
ð6Þ
xðq;DÞ
The expected profit comprises the expected revenue, the expected holding cost and the actual cost of building up extra capacity. Since the supplier knows the density function of the random variable a when determining the
6
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
amount of extra capacity, he will choose that quantity that achieves maximal expected profit, viz. Dopt ¼ arg max EðC S ðDÞÞ: DA½0;q
3. Worst case/best case analysis After having described both problems separately, we will now show how the models can be linked in different ways. We consider first a worst case situation in which producer and supplier optimize their objective functions independently and apart from the order quantity they do not exchange any further information. This situation represents the worst case which unfortunately can be found often in real-life situations. Then, we compare the worst case with the (theoretical) best case where there is one decision maker choosing the optimal decision that yields minimal expected total cost for the entire supply chain. On the one hand, the worst case/best case analysis will show whether it is necessary to coordinate the supply chain under consideration. On the other hand, we will use the optimal decisions and their associated expected total costs later in Section 4 as benchmarks to assess the coordination mechanism. 3.1. Worst case: No information exchange As seen in Section 2.1 the producer is able to determine an optimal order quantity that achieves its minimal expected cost only if all information of the supplier are common knowledge, so that he can anticipate the optimal capacity decision of the supplier in order to calculate his own expected total cost. Assuming that in the worst case the producer has no idea of the suppliers problem, he will precisely order that quantity that he requires to meet the external demand D; that is qworst opt ¼ D: Based on this order quantity q the supplier determines the optimal extra capacity Dworst opt which incurs maximal expected profit for him, i.e. S Dworst opt ¼ arg max EðC ðDÞÞ: DA½0;D
The setting can be characterized as a newsvendor problem (for an explanation see [23]) and the can be calculated solving optimal solution Dworst opt the following equation: ! q Dworst p w opt ; ð7Þ 1 F ¼ pþh C where FðaÞ denotes a continuous probability distribution of a: Note that the left term of Eq. (7) represents the probability that the available capacity, i.e. the realized capacity aC plus the extra capacity D of the supplier is sufficient to meet the order. Mathematically, ! ! q Dworst q Dworst opt opt 1 F ¼ P aX C C ¼ PðaC þ Dworst opt XqÞ: The supplier will choose that value of extra capacity which ensures that the probability of meeting the order equals the value of the fraction ðp wÞ=ðp þ hÞ: For later considerations, it is important to note that the value of the fraction increases in p: We now assume the special case that the random variable a of the supplier follows a uniform distribution over the range ½a; a% : Thus the density % function f ðaÞ is given by 8 < 1 if apapa% ; % f ðaÞ ¼ a% a : % 0 otherwise; which means that the realized normal capacity of the supplier is at least aC and at most a% C: Taking into account that the % producer orders precisely that quantity that he requires to meet the external demand, then the left term of Eq. (7) results in ! Z a% D Dworst 1 opt 1 F da ¼ worst a a C % ðD Dopt Þ=C % Dworst %C opt D þ a : Cða% aÞ % Substituting from above into Eq. (7) and after some simplification we obtain the optimal capacity decision of the supplier in a worst case ¼
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
scenario, i.e. Dworst opt ¼ D C
wða% aÞ þ a% h þ pa % %: hþp
ð8Þ
In the following we compare the optimal decision of the supplier in the worst case scenario with the optimal capacity decision in a best case scenario in order to analyze the impact of the worst case decision on the overall total cost of the supply chain. 3.2. Best case: Joint optimal solution From the point of view of supply chain management, it would be optimal to determine an extra capacity Djoint opt which incurs minimal expected total cost for the entire supply chain, using the total costs for both the producer and the supplier. Depending on D and a the actual total cost of the supply chain is given by C joint ¼ p½D ðaC þ DÞþ þ minfg; hg½aC þ D Dþ þ wD:
ð9Þ
The cost function consists of shortage cost for not meeting the external demand, holding cost, and cost for building up extra capacity. Note that in the cases where there is more capacity available than required to satisfy external demand ðaC þ D > DÞ; that party of the supply chain will bear the cost which has the lesser holding cost ðminfg; hgÞ: The expected total cost is then given by EðC joint Þ Z ¼p
ðD DÞ=C
ðD ðaC þ DÞÞf ðaÞ da 0
þ minfg; hg
Z
1
ðaC þ D DÞf ðaÞ da ðD DÞ=C
þ wD:
ð10Þ
Since in the best case the external demand D is common knowledge to both the producer and the supplier, it is not necessary to specify an amount of the order quantity. Therefore, the only quantity which must be determined is the extra capacity Djoint opt at the least expected total cost, i.e. joint Djoint Þ: opt ¼ arg opt EðC DA½0;D
7
As in the worst case, the setting can be characterized as a newsvendor problem. Therefore, the joint optimal solution Djoint opt can be calculated by solving the following equation: ! D Djoint p w opt : ð11Þ 1 F ¼ p þ minfg; hg C In order to analyze the differences between the worst case and best case decisions, we compare Eq. (11) with Eq. (7). Keeping in mind that the term in the left-hand side of Eqs. (7) and (11) represents the probability of meeting demand, it is straightforward to verify that in the best case scenario more extra capacity will be built up than in the worst case scenario. Since, on the one hand, the shortage cost for the finished product p is usually higher than the purchase cost p of the primary product and on the other hand the holding cost in the best case might be less than in the worst case, the fraction in the right-hand side of Eq. (11) is greater than the fraction of Eq. (7). This means that in the worst case the demand will be satisfied with smaller probability than in the best case. From that it can be deduced that the supplier will build up less extra capacity in the worst case than would be optimal for the entire supply chain. For the special case that the random variable a of the supplier follows a uniform distribution over the range ½a; a% ; the joint optimal decision can be determined %to be wða% aÞ þ a% minfg; hg þ pa Djoint ð12Þ % %: opt ¼ D C minfg; hg þ p A comparison with Eq. (8) shows that only for the unusual case in which the shortage cost p equals the purchase cost p and hpg the worst case and the best case decision correspond. For the more general case pop the supplier builds up less extra capacity in the worst case than in the best case, which leads to a higher total cost for the entire supply chain.
4. Coordination mechanisms The worst-case/best-case analysis shows that in a decentralized system without coordination the
8
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
expected total cost of the entire supply chain is usually higher than in a centralized one. In the following, we will determine appropriate coordination mechanisms which allow the decentralized system to achieve the same performance (the same minimal overall expected total cost) as a centralized supply chain. At the same time, we try to design the coordination mechanisms to be as flexible as possible to allow different allocations of the associated total cost. Recall that the aim of our investigation is to minimize the expected total cost of the entire supply chain without increasing the initial expected cost or decreasing the initial expected profit of any party of the supply chain. The easiest way to achieve flexible cost allocation is to introduce an incentive function Y: A negative incentive in the form of a penalty cost for the supplier decreases the cost of the producer and the profit of the supplier, while a positive incentive, which might be a bonus for the supplier, increases the profit of the supplier and the cost of the producer. So, there is a direct impact of Y on the actual total cost of producer and supplier and therefore on the allocation of the total cost of the supply chain, which can be seen from the following equation: C P ðd; YÞ ¼ p½D dþ þ g½d Dþ þ pd þ Y; C S ðq; YÞ ¼ pd h½aC þ D qþ wD þ Y:
4.1. Penalty cost In the following, we consider a variable penalty cost which the supplier has to pay to the producer whenever he does not meet the order. Let K denote the penalty cost the supplier has to pay for every unit which is ordered but not delivered. Then the negative incentive function Y yields Y ¼ K½q dþ ¼ K½q ðaC þ DÞþ : That means that the supplier must pay a penalty cost whenever the delivery quantity, which depends on his available capacity (see Eq. (2)), is less than the order quantity q: The expected profit of the supplier after introducing the variable penalty costs is then given by EðCS ðKÞÞ Z xðq;DÞ ¼p ðaC þ DÞ f ðaÞ da 0
þ pq h
Z
Z
K
1
f ðaÞ da
xðq;DÞ 1
ðaC þ D qÞ f ðaÞ da wD
xðq;DÞ Z xðq;DÞ
ðq ðaC þ DÞÞ f ðaÞ da
0
¼ pq ðp þ KÞ h
Z
Z
xðq;DÞ
ðq ðaC þ DÞÞ f ðaÞ da 0
1
ðaC þ D qÞ f ðaÞ da wD:
ð13Þ
xðq;DÞ
The two objective functions above show that the incentive function Y has only an indirect impact on the total cost of the supply chain, since in ‘‘summing up’’ the functions ðC P C S Þ the expenditures ðpd þ YÞ of the producer and the receipts ðpd þ YÞ of the supplier cancel each other out. Hence, to achieve minimal expected total cost, the incentive function must rather have a direct impact on the capacity decision of the supplier. We will show in the next two subsections that, in finding an optimal combination of order decision and (positive or negative) incentive, a coordination mechanism can be provided enabling the supplier to choose the overall optimal decision while achieving a variety of different cost allocations.
Again, as in the worst case and the best case scenario the setting can be characterized as a newsvendor problem, where in contrast to the worst case situation (see Eq. (7)) the purchase price p is replaced by the expression p þ K: Then, the optimal decision DK opt depending on the penalty cost K can be calculated by solving the following equation: ! q DK pþK w opt : ð14Þ 1 F ¼ pþK þh C By taking a closer look at the equation above, we can see the steering effect of the penalty cost. As mentioned in Section 3.1 the fraction on the right side increases in p: Since, in contrast to the worst
9
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
case scenario (see Eq. (7)), we add the penalty cost K to the price p; the probability of meeting the demand is higher than in the worst case, which means that the supplier builds up more extra capacity. In summarizing, the penalty cost does not only compensate the shortage cost of the producer but moreover have a direct impact on the capacity decision of the supplier and consequently on the expected total cost of the producer as well as on the overall expected total cost of the supply chain. Considering the special case of a being uniformly distributed over the range ½a; a% ; the optimal % extra capacity can be calculated by solving Eq. (14) resulting in DK opt ðq; KÞ ¼ q C
wða% aÞ þ a% h þ ðp þ KÞa % %: pþK þh
ð15Þ
Note that the optimal capacity decision of the supplier, which minimizes his expected total cost, depends on both the penalty cost K and the order quantity q: Since the extra capacity is linearly proportional to the order quantity the steering effect of the penalty cost decreases with increasing cost values. Fig. 3 illustrates the typical behavior of the extra capacity for a given value q ¼ 100 of the order quantity. The underlying parameter values correspond to the values of an example which will be discussed in the appendix. The horizontal axis
represents the penalty cost, the vertical axis represents the supplier’s choice of extra capacity. The figure shows that the slope of the function decreases as the penalty cost K increases. There are two different ways for the producer to influence the optimal capacity decision of the supplier: The order quantity and the penalty cost. We call the order quantity decision the ‘executionoriented decision’ and the decision about the penalty cost the ‘control-oriented decision’. Since we presume a team-like behavior, we assume that the producer chooses the penalty cost K and the order quantity q such that the overall expected total cost of the entire supply chain is minimized. To put it differently, we assume that the producer directs the supplier to choose the overall optimal decision Djoint opt : Let us consider once more the optimal decision in the best case scenario, which can be calculated by solving Eq. (11): ! D Djoint p w opt 1¼F : þ p þ minfg; hg C On the other hand, in the coordination case, described above, the supplier chooses a capacity value so that Eq. (14) holds: ! q DK pþK w opt 1¼F : þ pþK þh C To ensure that the supplier chooses this value of extra capacity decision that incurs overall minimal expected total cost, the producer must fix the order quantity q and penalty cost K such that the following equation holds: joint DK opt ¼ Dopt
q Djoint opt 3F C ¼
Fig. 3. Extra capacity behavior depending on the underlying penalty cost.
!
D Djoint opt F C
p w pþK w : p þ minfg; hg p þ K þ h
!
ð16Þ
The term in the left-hand side describes the probability that the maximum deliverable quantity of the supply product depending on the available capacity of the supplier will be greater than the required quantity D and less than the ordered quantity q: The equation above shows that the
10
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
producer can attain the coordination of the entire supply chain through different combinations of order quantity q and penalty cost K: For instance, consider the case that the producer orders precisely that quantity that he requires ðq ¼ DÞ and assume that the cost for overcapacity h is less than the inventory cost g of the producer. Then he set fix the penalty cost K equal to p p (shortage cost minus purchase cost of the supply product) to reach overall minimal expected total cost. For the special case that a follows a uniform distribution the function which represents the relation between optimal order quantity and optimal penalty cost can easily be set up. For this we assume that the optimal decisions given by Eqs. (12) and (15) are equal and resolve the resulting equation in q: Then we get the optimal order quantity as a function of the penalty cost K: qopt ðKÞ ¼ D þ C C
wða% aÞ þ a% h þ ðp þ KÞa % % pþK þh
Whereas the penalty cost punishes the supplier for not meeting the order, bonus payments are an incentive to provide on-time delivery and lower the cost of the supplier. Unlike the penalty cost which has been assumed to be linearly proportional to the number of units short, we now consider fixed bonus payments which are only paid to the supplier when order and delivery quantity coincide. Therefore the incentive function Y can be written as ( A for d ¼ q; Y¼ 0 otherwise; or in a mathematical term, using an indicator function, as Y ¼ A1ðaCþDÞXq : Then, the expected profit of the supplier is given by Z xðq;DÞ ðaC þ DÞf ðaÞ da EðC S ðAÞÞ ¼ p 0
þ pq
wða% aÞ þ a% minfg; hg þ pa % % minfg; hg þ p
¼ Djoint opt þ C
4.2. Bonus
h
Z
Z
1
xðq;DÞ
wða% aÞ þ a% h þ ðp þ KÞa % %: pþK þh
wD þ A
ðaC þ D qÞf ðaÞ da Z
1
f ðaÞ da:
ð18Þ
xðq;DÞ
ð17Þ Each combination of penalty cost and order quantity that satisfies the above equation ensures that the supplier chooses the overall optimal capacity decision that leads to minimal expected total cost for the entire supply chain. On the other hand the function shows that, the lower the penalty cost, the higher the required order quantity. That means that the cost allocation of the overall total cost of the supply chain depends on the combination of q and K: So far, we have found a coordination mechanism that gains an overall optimal performance of the entire supply chain and offers different ways to allocate the overall total cost of the supply chain between the supplier and the producer. To extend the range of possible cost allocations we consider in the next subsection bonus payments which lower the cost of the supplier.
f ðaÞ da
xðq;DÞ 1
Unlike the previous cases the setting is no more transferable into a newsvendor problem due to the special structure of the incentive function. Nevertheless, we can obtain the optimal decision of the supplier in case a is uniformly distributed. Substituting the density function in (18) and using the first order condition, we obtain after some algebraic simplification the optimal policy for the supplier as a function of order quantity q and bonus A as DA opt ðq; AÞ ¼ q
Cðwða% aÞ þ a% h þ paÞ A : % % pþh
ð19Þ
The equation shows that the supplier builds up proportionately as much extra capacity as the bonus. As in the case of the penalty cost there are two ways to influence the supplier: The order quantity (execution-oriented decision) and the bonus (control-oriented decision). Again, we
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
assume that the producer selects those values of order quantity and bonus which provide the least expected total cost for the entire supply chain. Then, the optimal combination of order quantity and bonus can be calculated by equating the optimal decision of the supplier (19) with the overall optimal decision (11). After some conversions we get the optimal order quantity of the producer as a function of the bonus Cðwða% aÞ þ a% h þ paÞ A % % pþh wða% aÞ þ a% minfg; hg þ pa C % % minfg; hg þ p Cðwða% aÞ þ a% h þ paÞ A : ð20Þ ¼ Djoint % % opt þ pþh
qopt ðAÞ ¼ D þ
Note that each combination of bonus and order quantity which satisfies the above equation ensures that the supplier builds up as much extra capacity as is optimal from the point of view of the entire supply chain. The function also shows that the lower the value of the bonus, the more quantity units must be ordered and vice versa. However, unlike the case where a penalty cost is charged, the bonus payment and the order quantity have the same effect on the cost allocation. Both a higher bonus value and a higher order quantity result in higher total cost for the producer and a higher profit for the supplier.
11
as the centralized one. This means, that the resulting expected total cost of the entire supply chain is always as low as in the best case, whereas, unlike the case of a central planner, the involved parties do not lose their decision making authorities. Furthermore, the approach provides a flexibility of cost allocation between the two parties of the supply chain. Since the paper primarily focused on finding general analytical results, the supply chain was described on a highly aggregated level and the results are based on assumptions which might not completely hold in practice. But it can be shown (see [24]) that the coordination mechanism is also applicable to a realistic situation where the parties of the supply chain have some private information. Another important extension to consider is the coordination of more than two parties in a supply chain. In particular it would be worth pursuing whether the concept of combining an ‘execution-oriented decision’ with a ‘controloriented decision’ leads to similar flexible cost allocations for these multiple decisions, as observed here.
Acknowledgements The author thanks Prof. Christoph Schneeweiss for his continuous support and Prof. Jay Sankaran as well as the anonymous referees for their helpful comments and suggested improvements.
5. Conclusions The paper considered a supply chain, consisting of one producer and one supplier, in a Just-inTime environment where the supply of the component is uncertain due to an uncertain availability of the capacity of the supplier. First we compared the worst case, where no information exchange between the two parties takes place, with the best case, where all decisions are chosen simultaneously by a central planner. Then we developed a coordination mechanism which combined an ‘execution-oriented decision’ with a ‘control-oriented decision’. We showed that the coordination mechanism ensures the decentralized system to perform as well
Appendix A. Example In order to get a better understanding of the underlying coordination problem and to illustrate the flexibility of cost allocation using the described coordination mechanism we consider a numerical example. (For a more general and deeper discussion of the behavior of the coordination mechanism, and particularly the range over which the total cost of the supply chain can be allocated, see [23].) The parameters are specified as follows: The external demand of the final product is D ¼ 100: The price charged by the supplier to the producer
12
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
for each unit of the supply product is p ¼ $10: The producer must store any unsold supply product at a value of g ¼ $3 per unit and pays a shortage cost for the final product, which is p ¼ $60 per unit. The normal capacity of the supplier is C ¼ 100 and a is uniformly distributed over the range ½0:7; 1: To increase the capacity by one unit costs the supplier w ¼ $3 and the holding cost is h ¼ $1 per unit. Thus, the supplier can meet the order (in case the order coincides with the external demand) without building up extra capacity only in case the realization of a is 1. A.1. Worst and best case Let us first consider the worst and the best case. As mentioned in Section 3.1 we assume that in the worst case scenario the producer orders precisely that quantity that he requires, i.e. q ¼ 100: After that, the supplier chooses that amount of extra capacity that provides his most expected profit, that is Dworst opt ¼ 19:09 (see Eq. (8)). As a result of both decisions the expected total cost for the entire supply chain is EðCjoint ðq; Dworst opt ÞÞ ¼ $182:35; in which the expected cost of the producer is EðC P Þ ¼ 1099:17; whereas the expected profit of the supplier is EðC S Þ ¼ 916:82: On the other hand, the optimal capacity decision in the best case scenario is Djoint opt ¼ 28:03 (see Eq. (12)) which achieves the minimal expected total cost of E
ðC joint ðDjoint opt ÞÞ ¼ $101:06 for the entire supply chain. That means that in our numerical example the expected total cost of the entire supply chain in the best case is more than 55% less than in the worst case. Even if this concrete value depends on the specification of the parameters it gives a hint as to what cost saving potential may exist in a supply chain. The results are summarized in Table 1. Next, we analyze the performance of the coordination mechanism, which combines the
Table 1 Worst case compared with best case
Worst case Best case
qopt
Dopt
EðCP Þ
EðCS Þ
EðCjoint Þ
100 F
19.09 28.03
1099.17 F
916.82 F
182.35 101.06
order quantity as an execution-oriented decision with a penalty cost and a bonus payment, respectively, as a control-oriented decision. A.2. Penalty cost In case of penalty cost the optimal order quantity as a function of the penalty cost (see Eq. (15)) is given by 100ð8:9 þ 0:7KÞ : qopt ðKÞ ¼ 28:03 þ 11 þ K Table 2 shows the mutual dependency of order quantity and penalty cost for some specific discrete values. We observe that by increasing the penalty cost the expected total cost of the producer and the expected profit of the supplier decrease. Recall that the optimal decision of the supplier as well as the overall expected total cost of the entire supply chain correspond to those of the best case. Furthermore, the table shows that in charging a penalty cost the most expected profit of the supplier is EðCS Þ ¼ 979:12: A.3. Bonus To increase the expected profit of the supplier and thus to extend the range of possible cost allocations, we now consider the bonus case. From Eq. (20) we get the bonus as a function of the order quantity, that is AðqÞ ¼ 890 þ 11ð28:03 qÞ: Table 3 shows the dependency between bonus and order quantity. It can be observed that the higher the order quantity, the lower the bonus has to be. By increasing the order quantity (decreasing the bonus, respectively) the expected total cost of the
Table 2 Optimal order quantity in dependency of the penalty cost K
qopt ðKÞ
EðCS Þ
EðCP Þ
EðCjoint Þ
0 1 2 5 10
108.93 108.03 107.26 105.53 103.74
979.12 971.22 964.29 948.71 932.64
1080.18 1072.28 1065.35 1094.77 1033.70
101.06 101.06 101.06 101.06 101.06
13
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
producer and the expected profit of the supplier decrease. A.4. Cost allocation The overall range in which the expected total cost of the supply chain can be allocated by using the coordination mechanism is shown in Fig. 4. In contrast to the Tables 2 and 3 the figure does not depict the expected profit of supplier and the expected total cost of the producer in all, but shows only those cost components which concern the supply uncertainty directly, i.e. the purchase cost of $1000; which the producer would pay to the
Table 3 Optimal bonus in dependency of the order quantity qopt
Aðqopt Þ
EðCS Þ
EðCP Þ
EðCjoint Þ
100 101 102 103 108
98.33 87.33 76.33 65.33 10.33
994.04 990.95 988.22 985.86 979.55
1095.10 1092.01 1089.28 1086.92 1080.61
101.06 101.06 101.06 101.06 101.06
supplier in case of certainty, is not included. Recall that in adding up the expected total cost of the producer and the negative expected profit of the supplier the purchase costs cancel each other out anyway. The horizontal axis shows the different combinations of order quantity and incentive. The vertical axis represents the associated expected total cost for the supplier, the producer and the entire supply chain. Let us first have a closer look at the horizontal axis. The center of the axis describes the case when the producer neither offers a bonus nor charges a penalty cost but influences the supplier only by a maximal order quantity q ¼ qmax : This situation corresponds to the case described in the first line of Table 2. Starting in the center and moving to the right means that the value of the penalty cost increases (up to Kmax ) while the order quantity decreases (down to q ¼ D). On the other hand moving left from the center represents an increasing bonus value (up to Amax ) and a decreasing order quantity (down to q ¼ D). Thus, the figure is divided into two parts: the left part describing the bonus case and the right part describing the penalty case.
200 180
Expected total cost (worst case) 160
Expected costs
140
Expected total cost (best case)
120 100 80
Expected cost of the producer 60
Expected cost of the supplier
40
Expected cost of the supplier
20 0
A max (q =D )
A = K= 0 (q =q max)
Fig. 4. Allocation of the total cost in dependency of bonus and penalty cost.
K max (q =D )
14
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
Let us now have a look at the cost behavior. First, since the expected total cost of the entire supply chain using the coordination mechanism equals the expected total cost of the best case, the gap between the worst and the best case shows the cost savings which can be achieved through the considered coordination mechanism. Note that in an existing supply chain partnership the expected total cost will always be situated between the best and the worst case. Furthermore, in combining the order quantity with an incentive we get a wide range of different cost allocations. The figure shows that in case of the maximal bonus value Amax the producer must bear almost the whole expected total cost of the supply chain, while in case of the maximal penalty cost Kmax the supplier has to bear the expected total cost alone. Note that if we had determined only an optimal incentive to coordinate the supply chain, which is common in the literature, we would have achieved the same performance for the entire supply chain but would have allocated the total cost only in one possible manner. Assuming the producer orders precisely what he requires, he has to offer a bonus payment at a value of Amax (the point farthest to the left in Fig. 4) or charge a penalty cost at the value of Kmax per unit (the point farthest to the right) in order to minimize the overall expected total cost of the entire supply chain. But compared to an initial situation, which might be better than the worst case represented in Table 1, either the producer or the supplier might make losses. Thus, in practice the parties of the supply chain would probably not agree to any of these two incentive schemes and the implementation of such a coordination mechanism would fail. In summarizing, the example shows on the one hand which substantial cost savings might be achieved in practice by implementing the developed coordination mechanism. This result is of particular interest from the general point of view of supply chain management, since the cost savings refer to the overall cost of the entire supply chain. On the other hand the example demonstrates that the coordination mechanism is flexible enough to enable different allocations of these overall cost thus allowing both parties in an existing supply chain partnership to make a profit.
This aspect is of importance for the implementation of such a mechanism and is mostly ignored in literature.
References [1] M. Christopher, Logistics and Supply Chain Management: Strategies for Reducing Cost and Improving Service, Financial Times Pitman, London, 1998. [2] B.C. Arntzen, G.G. Brown, T.P. Harrison, L.L. Trafton, Global supply chain management at digital equipment corporation, Interfaces 25 (1) (1995) 69–93. [3] R.L. Breitman, J.M. Lucas, PLANETS: A modeling system for business planning, Interfaces 17 (1) (1987) 94–106. [4] G.G. Brown, G.W. Graves, M.D. Honczarenko, Design and operation of a multicommodity production/distribution system using primal goal decomposition, Management Science 33 (11) (1987) 1469–1480. [5] M.A. Cohen, S. Moon, An integrated plant loading model with economies of scale and scope, European Journal of Operational Research 50 (1991) 266–279. [6] R. Anupindi, Y. Bassok, Supply contracts with quantity commitments and stochastic demand, in: S. Tayur, R. Ganeshan, M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1999, pp. 198–232. [7] J.P. Monahan, A quantity discount pricing model to increase vendor profits, Management Science 30 (6) (1984) 720–726. [8] A. Banerjee, A joint economic-lot-size model for purchaser and vendor, Decision Science 17 (1986) 292–311. [9] A. Banerjee, Notes: On ‘‘A quantitative discount pricing model to increase vendor profits’’, Management Science 32 (11) (1986) 1513–1517. [10] Ch.J. Corbett, X. de Groote, Integrated supply-chain lot sizing under asymmetric information, Working Paper, The Anderson School at UCLA, 1997. [11] R. Aderohunmu, A. Mobolurin, N. Bryson, Joint vendor– buyer policy in JIT manufacturing, Journal of the Operational Research Society 46 (3) (1995) 375–385. [12] M.A. Lariviere, Supply chain contracting and coordination with stochastic demand, in: S. Tayur, R. Ganeshan, M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1999, pp. 233–268. [13] P.H. Ritchken, Ch.S. Tapiero, Contingent claims contracting for purchasing decisions in inventory management, Operations Research 34 (6) (1986) 864–870. [14] D. Barnes-Schuster, Y. Bassok, R. Anupindi, Coordination and flexibility in supply contracts with options, Working Paper, Graduate School of Business, University of Chicago, 1998. [15] Ch.J. Corbett, C.S. Tang, Designing supply contracts: Contract type and information asymmetry, in: S. Tayur, R.
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
[16]
[17]
[18] [19]
[20]
Ganeshan, M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1999, pp. 269–297. R. Kolisch, Coordination of assembly and fabrication for make-to-order production, Pre-prints of the 10th International Working Seminar on Production Economics 3 (1998) 333-338. Igls/Innsbruck. Y. Gerchak, Y. Wang, C.A. Yano, Lot sizing in assembly systems with random component yields, IEE Transactions 26 (2) (1994) 19–24. C.A. Yano, H.L. Lee, Lot sizing with random yields: A review, Operations Research 43 (2) (1995) 311–334. H. Gurnani, R. Akella, J. Lehoczky, Supply management in assembly systems with random yield and random demand, Working Paper, The Hong Kong University of Science and Technology, 1998. H. Gurnani, Supply management in assembly systems with uncertain supply and inter-supplier penalties, Working
[21]
[22]
[23]
[24]
15
Paper, The Hong Kong University of Science and Technology, 1998. Y. Gerchak, Y. Wang, Coordination in decentralized assembly systems with random demand, Working Paper, Department of Management Sciences, University of Waterloo, 1999. H. Gurnani, Y. Gerchak, Coordination in decentralized assembly systems with uncertain component yields, Department of Management Sciences, University of Waterloo, 1999. K. Zimmer, Hierarchische Coordination im Supply Chain Management, Ph.D. Thesis, Department of Operations Research, University of Mannheim, 2000: Ch. Schneeweiss, K. Zimmer, Hierarchical coordination mechanisms within the supply chain, Working Paper, Department of Operations Research, University of Mannheim, 2000:
Supply chain coordination with uncertain just-in-time delivery Kirstin Zimmer* Department of Business Administration and Operations Research, University of Mannheim, D-68131 Mannheim, Germany Received 24 January 2001; accepted 31 May 2001
Abstract Coordinating producer and supplier is one of the main issues of supply chain management. The paper investigates this problem by means of a single-period order and delivery planning model within a Just-in-Time setting. The aim of the paper is to find a coordination mechanism that allows the system to perform just as well as a centralized one. Since the acceptance of a coordination mechanism depends on the associated expected costs for each party of the supply chain, the paper focuses not only on the overall performance but also on the allocation of the cost. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Supply chain management; Coordination; Distributed decision making
1. Introduction Recent years have seen a growing globalization of markets and the concentration of companies on their core competencies. As a result, increasingly legally and economically independent companies are involved in the value added process as part of an entire supply chain. Since this requires increased cooperation between the companies as well as the coordination of logistical decisions, the interest in the field of supply chain management has grown recently among both academics and practitioners. Supply chain management may be considered to be an extension of traditional logistics. Whereas logistics investigates the flow of information, materials, capital and manpower in the internal *Tel.: +49-621-181-1655; fax: +49-621-181-1653. E-mail address: [email protected] (K. Zimmer).
supply chain owned by a single firm, supply chain management deals with the coordination of logistic processes within the external supply chain. The main goal of both traditional logistics and supply chain management is ‘‘to deliver superior customer value at less cost to the supply chain as a whole’’ (see [1, p. 18]). But unlike traditional logistics, supply chain management involves the coordination of independently managed companies who seek to maximize their own profits. Although overall performance of the supply chain depends on the companies joint performance, the operational goals may conflict and result in inefficiencies for the entire chain. Therefore, one of the main issues of supply chain management is to find suitable mechanisms to coordinate the logistical processes that are controlled by various independent companies in order to achieve overall minimal cost. In addition, the mechanism must guarantee that in an existing supply chain partner-
0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 1 ) 0 0 2 0 7 - 9
2
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
ship each individual company will achieve at least the same profit as in the initial situation before applying the mechanism. The closer the parties of a supply chain are linked together the more important the coordination of the entire supply chain becomes. For this reason, we will consider a supply chain with Justin-Time delivery. When Just-in-Time purchasing is implemented, the production of the finished products largely depends on the on-time delivery of the components, since buffer inventories are typically reduced. While both producer and supplier focus on eliminating inefficiencies, shortage situations are usually costlier for the producer than for the supplier. As a result the supplier makes less of an effort to avoid shortage situations than would be optimal for the entire supply chain. The aim of our paper is to find a coordination mechanism that ensures that the decentralized system performs as well as a centralized one while neither the producer nor the supplier are worse off, when compared to the initial situation. The literature on supply chain management can be roughly split into two classes. The first considers the supply chain from the point of view of one decision maker and determines those decisions that minimize overall total cost. In these approaches the supply chain is considered as a fully vertically integrated firm where all information is common knowledge and the material flow is controlled by a single decision maker (see e.g. [2–5]). The second class considers the supply chain as a network (or chain) of antagonistically behaving companies and analyzes the implications of contract forms (for an overview of this literature see [6]). The underlying models describe the supply chain on a highly aggregated level and often consider only two decision makers: A buyer and a supplier. One of the first studies in this area is that of Monahan [7] who investigates the effect of quantity discounts on the order quantity of a buyer and the profit of the supplier. He shows that a supplier is able to increase his profit by offering quantity discounts. Banerjee [8,9] and Corbett/de Groote [10] extend his work and show that the underlying contract is also optimal from the point of view of the entire
supply chain. In the last couple of years the interest in the field of supply chain contracting has grown, thus leading to a considerable amount of research, especially in the field of quantity discounts and quantity commitments (see [11– 14]). However, since the literature in many cases addresses the problem in an agency-theoretic framework there is often no attempt to analyze whether the contract allows the decentralized system to perform just as well as a centralized one (see e.g. [15]). The literature on Just-in-Time related issues can be split into two classes in a similar way. Let us consider the production process of components and the production (or assembly) of the final products as two independent units of a supply chain. Then, the first class describes the supply chain from the point of view of one decision maker and determines the production decisions simultaneously, where the production yield of the components is either deterministic (see e.g. [16]) or random (see e.g. [17–19]). The second class which is most related to our work takes the decentralization of decisions into account and analyzes the effects of penalty-costs and side-payments between suppliers. Gurnani [20], Gerchak/Wang [21] and Gurnani/Gerchak [22] consider a supply chain consisting of one producer and two suppliers, where the suppliers may, due to production yield losses, deliver only a random fraction of the order quantity. Gurnani [20] shows that side-payments between the suppliers have a direct impact on the optimal policy of the suppliers and can be used to increase their efforts. However, he neither analyzes whether the side-payments allow the decentralized system to perform just as well as a centralized one nor he determines the optimal values of the side-payments. In contrast, Gurnani/Gerchak [22] calculate the optimal penalty costs and analyze the cases in which the implementation of these penalty costs achieves the same performance as a centralized system. However, a drawback of all these approaches is that they lead to only one specific cost allocation between producer and supplier. That means that although the overall cost of the supply chain is minimized, a situation may occur in which the cost of one decision
3
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
maker decreases while the cost of another decision maker increases. Our paper differs from the existing literature in that we not only attempt to find contract parameters that allow the decentralized system to perform just as well as a centralized one, but we investigate more general coordination mechanisms which in addition give us higher flexibility in allocating cost. Furthermore, we not only consider the effect of incentives but also the impact of order decisions on the efforts of the supplier. The paper is organized as follows. In the next section we present the basic assumptions of the model and discuss the cost functions of the producer and the supplier. Section 3 considers two extreme cases of the supply link. This worst case/best case analysis illustrates the difference between decentralized and centralized systems and is used in the following sections as a benchmark to evaluate the coordination mechanisms. In Section 4, which is the main section, we develop two different coordination mechanisms and show that if one of these mechanisms is implemented the decentralized system performs as well as a centralized one. Finally, we summarize the results in Section 5 and conclude with directions for further research.
process the order of the considered producer as being limited to a portion aC (with aA½0; 1). At the time the producer places his order, the realized value of a is not known to the supplier (because he is still waiting for further orders) but only its probability density function f ðaÞ: This leads to supply uncertainty in the considered supply chain. To cover the demand of the producer the supplier can build up extra capacity D: In contrast to the normal capacity the extra capacity is built up exclusively for the considered supplier and therefore is deterministic. As depicted in Fig. 1, the delivery quantity d depends on the available capacity aC; the extra capacity D and the order quantity q: Assuming that the capacity consumption rate equals 1, the delivery quantity is restricted by the following capacity constraint: dpaC þ D:
ð1Þ
Since the supplier does not deliver more than what is ordered the delivery quantity d equals the minimum of the order quantity q and the maximum deliverable quantity aC þ D based on the available capacity aC: d ¼ minfaC þ D; qg:
ð2Þ
2. Problem We consider a supply chain with one supplier and one producer in a one-period setting. The producer faces a known demand D for a finished product which is assembled using a special component or primary product that he orders in quantity q from the supplier. Without loss of generality, we assume that the producer requires one unit of the component to produce one unit of the finished product. That is, to meet the demand for the finished product the producer requires D units of the component in total. The normal (long-term) capacity of the supplier is denoted by C: Since a supply chain usually is embedded into a complex supply network, the supplier might not have just one but several producers. In capturing this influence of the network, we describe the supplier’s capability to
Producer
D
q
d (q, ,∆)
Supplier
d ≤ ⋅C + ∆ Fig. 1. Supply chain.
4
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
Demand for the finished product is known D Order quantity q
Available capacity of the supplier is realized
1
0
Delivery quantity d
t=1
Producer:
Supplier:
Delivery Date
Decision about order quantity
Decision about excess capacity Fig. 2. Chronological description of order- and delivery-planning.
Fig. 2 shows the course of events within the supply chain: Based on the given external demand D for the finished product, the producer determines the order quantity q at time t0 : After the order is made and before the available capacity aC becomes known, the supplier has to fix the extra capacity (at time t1 ) in order to meet the demand of the producer. The delivery quantity depends on the extra capacity as well as on the available normal capacity aC of the supplier and might not match the ordered amount at the delivery date.
amount. The second is the cost of holding excess components, after demand has been satisfied. The penalty cost per unit per period of the final product is denoted by p while the holding (recycling) cost per unit per period of the component is denoted by g: Now, for a given value of the delivery quantity d; the cost for the producer is given by
2.1. The producer’s problem
CP p D d g
The producer faces the problem of determining the optimal order quantity. We first describe the actual total cost for the producer at the delivery date and then determine the expected total cost depending on the random portion a of the supplier. In addition to the procurement cost pd; there are two types of costs which arise to the producer when the delivery quantity differs from the demand for the finished item. The first is the penalty cost the producer is charged if he cannot cover the external demand, since the delivered amount is less than the required
C P ¼ p½D d þ þ g½d Dþ þ pd:
ð3Þ
Notations
p
total cost for the producer shortage cost per unit for the finished product demand for the finished product delivery quantity holding (recycling) cost of the primary product per unit purchase cost of the primary product per unit.
When the producer has to place his order q only the distribution function FðaÞ and the density function f ðaÞ are known to the system. Taking into account Eq. (2), the expected total cost for the producer, which depends on his order quantity q as well as on the capacity decision D of the
5
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
supplier, is given by EðC P Þ ¼ p
Z
2.2. The supplier’s problem
minfxðq;DÞ;yðDÞg
ðD ðaC þ DÞÞ f ðaÞ da
0
þ pðD qÞ þ g
1
f ðaÞ da1fD>qg
xðq;DÞ
Z
xðq;DÞ
ðaC þ D DÞ f ðaÞ da yðDÞ
þ gðq DÞ þ p
Z
Z
Z
1
f ðaÞ da1fDpqg xðq;DÞ
xðq;DÞ
ðaC þ DÞ f ðaÞ da
0
þ pq
Z
1
f ðaÞ da
ð4Þ
xðq;DÞ
where yðDÞ :¼ ðD DÞ=C; xðq; DÞ :¼ ðq DÞ=C and 1fD>qg is an Indicator Function defined as: ( 1 if D > q; 1fD>qg ¼ 0 otherwise: In the cost function described above, the first two terms represent the expected customer penalty cost, the second two terms show the expected inventory cost and the last two terms describe the expected procurement cost. The limits xðq; DÞ and yðDÞ of the integrals result from the capacity constraint (1) and the case distinctions of the cost function (3). For instance, to integrate a over the range of ½yðDÞ; xðq; DÞ means to consider the case that the maximum deliverable quantity aC þ D exceeds the required quantity D but is below the ordered quantity q; or in mathematical terms: D D q D pap C C 3DpaC þ Dpq:
yðDÞpapxðq; DÞ 3
The supplier’s choice of extra capacity is a tradeoff between lost profits and the cost of capacity overshoot. The cost for building up extra capacity is fixed at w per unit. To avoid trivial problems, we assume that the revenue of one unit of the component exceeds the cost of procuring this unit with extra capacity, that is p > w: Without loss of generality, we further assume that the supplier must store unsold components at a value h per unit. If the supplier does not use all of the available capacity to produce components which must be stored, this cost may correspond to the cost for overshooting capacity, for instance to store unused raw materials. Now, for a given value of the order quantity q of the producer and in view of Eq. (2), the profit for the supplier is given by CqS ¼ p minfaC þ D; qg h½aC þ D qþ wD: ð5Þ Notations CqS p C a D q h w
profit for the supplier price for the component normal capacity random fraction of the normal capacity extra capacity order quantity cost for overcapacity (holding cost) cost to increase the capacity by one unit
With the knowledge of the density function f ðaÞ the expected profit for the supplier is then calculated by EðCS Þ ¼ p
Z
ðaC þ DÞf ðaÞ da þ pq
0
h Note that the producer is able to determine the optimal order quantity q only if the distribution function of a as well as the capacity decision of the supplier are common knowledge. We come back to this point later in Section 3 when we discuss different kinds of the supply link.
xðq;DÞ
Z
Z
1
f ðaÞ da
xðq;DÞ 1
ðaC þ D qÞf ðaÞ da wD:
ð6Þ
xðq;DÞ
The expected profit comprises the expected revenue, the expected holding cost and the actual cost of building up extra capacity. Since the supplier knows the density function of the random variable a when determining the
6
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
amount of extra capacity, he will choose that quantity that achieves maximal expected profit, viz. Dopt ¼ arg max EðC S ðDÞÞ: DA½0;q
3. Worst case/best case analysis After having described both problems separately, we will now show how the models can be linked in different ways. We consider first a worst case situation in which producer and supplier optimize their objective functions independently and apart from the order quantity they do not exchange any further information. This situation represents the worst case which unfortunately can be found often in real-life situations. Then, we compare the worst case with the (theoretical) best case where there is one decision maker choosing the optimal decision that yields minimal expected total cost for the entire supply chain. On the one hand, the worst case/best case analysis will show whether it is necessary to coordinate the supply chain under consideration. On the other hand, we will use the optimal decisions and their associated expected total costs later in Section 4 as benchmarks to assess the coordination mechanism. 3.1. Worst case: No information exchange As seen in Section 2.1 the producer is able to determine an optimal order quantity that achieves its minimal expected cost only if all information of the supplier are common knowledge, so that he can anticipate the optimal capacity decision of the supplier in order to calculate his own expected total cost. Assuming that in the worst case the producer has no idea of the suppliers problem, he will precisely order that quantity that he requires to meet the external demand D; that is qworst opt ¼ D: Based on this order quantity q the supplier determines the optimal extra capacity Dworst opt which incurs maximal expected profit for him, i.e. S Dworst opt ¼ arg max EðC ðDÞÞ: DA½0;D
The setting can be characterized as a newsvendor problem (for an explanation see [23]) and the can be calculated solving optimal solution Dworst opt the following equation: ! q Dworst p w opt ; ð7Þ 1 F ¼ pþh C where FðaÞ denotes a continuous probability distribution of a: Note that the left term of Eq. (7) represents the probability that the available capacity, i.e. the realized capacity aC plus the extra capacity D of the supplier is sufficient to meet the order. Mathematically, ! ! q Dworst q Dworst opt opt 1 F ¼ P aX C C ¼ PðaC þ Dworst opt XqÞ: The supplier will choose that value of extra capacity which ensures that the probability of meeting the order equals the value of the fraction ðp wÞ=ðp þ hÞ: For later considerations, it is important to note that the value of the fraction increases in p: We now assume the special case that the random variable a of the supplier follows a uniform distribution over the range ½a; a% : Thus the density % function f ðaÞ is given by 8 < 1 if apapa% ; % f ðaÞ ¼ a% a : % 0 otherwise; which means that the realized normal capacity of the supplier is at least aC and at most a% C: Taking into account that the % producer orders precisely that quantity that he requires to meet the external demand, then the left term of Eq. (7) results in ! Z a% D Dworst 1 opt 1 F da ¼ worst a a C % ðD Dopt Þ=C % Dworst %C opt D þ a : Cða% aÞ % Substituting from above into Eq. (7) and after some simplification we obtain the optimal capacity decision of the supplier in a worst case ¼
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
scenario, i.e. Dworst opt ¼ D C
wða% aÞ þ a% h þ pa % %: hþp
ð8Þ
In the following we compare the optimal decision of the supplier in the worst case scenario with the optimal capacity decision in a best case scenario in order to analyze the impact of the worst case decision on the overall total cost of the supply chain. 3.2. Best case: Joint optimal solution From the point of view of supply chain management, it would be optimal to determine an extra capacity Djoint opt which incurs minimal expected total cost for the entire supply chain, using the total costs for both the producer and the supplier. Depending on D and a the actual total cost of the supply chain is given by C joint ¼ p½D ðaC þ DÞþ þ minfg; hg½aC þ D Dþ þ wD:
ð9Þ
The cost function consists of shortage cost for not meeting the external demand, holding cost, and cost for building up extra capacity. Note that in the cases where there is more capacity available than required to satisfy external demand ðaC þ D > DÞ; that party of the supply chain will bear the cost which has the lesser holding cost ðminfg; hgÞ: The expected total cost is then given by EðC joint Þ Z ¼p
ðD DÞ=C
ðD ðaC þ DÞÞf ðaÞ da 0
þ minfg; hg
Z
1
ðaC þ D DÞf ðaÞ da ðD DÞ=C
þ wD:
ð10Þ
Since in the best case the external demand D is common knowledge to both the producer and the supplier, it is not necessary to specify an amount of the order quantity. Therefore, the only quantity which must be determined is the extra capacity Djoint opt at the least expected total cost, i.e. joint Djoint Þ: opt ¼ arg opt EðC DA½0;D
7
As in the worst case, the setting can be characterized as a newsvendor problem. Therefore, the joint optimal solution Djoint opt can be calculated by solving the following equation: ! D Djoint p w opt : ð11Þ 1 F ¼ p þ minfg; hg C In order to analyze the differences between the worst case and best case decisions, we compare Eq. (11) with Eq. (7). Keeping in mind that the term in the left-hand side of Eqs. (7) and (11) represents the probability of meeting demand, it is straightforward to verify that in the best case scenario more extra capacity will be built up than in the worst case scenario. Since, on the one hand, the shortage cost for the finished product p is usually higher than the purchase cost p of the primary product and on the other hand the holding cost in the best case might be less than in the worst case, the fraction in the right-hand side of Eq. (11) is greater than the fraction of Eq. (7). This means that in the worst case the demand will be satisfied with smaller probability than in the best case. From that it can be deduced that the supplier will build up less extra capacity in the worst case than would be optimal for the entire supply chain. For the special case that the random variable a of the supplier follows a uniform distribution over the range ½a; a% ; the joint optimal decision can be determined %to be wða% aÞ þ a% minfg; hg þ pa Djoint ð12Þ % %: opt ¼ D C minfg; hg þ p A comparison with Eq. (8) shows that only for the unusual case in which the shortage cost p equals the purchase cost p and hpg the worst case and the best case decision correspond. For the more general case pop the supplier builds up less extra capacity in the worst case than in the best case, which leads to a higher total cost for the entire supply chain.
4. Coordination mechanisms The worst-case/best-case analysis shows that in a decentralized system without coordination the
8
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
expected total cost of the entire supply chain is usually higher than in a centralized one. In the following, we will determine appropriate coordination mechanisms which allow the decentralized system to achieve the same performance (the same minimal overall expected total cost) as a centralized supply chain. At the same time, we try to design the coordination mechanisms to be as flexible as possible to allow different allocations of the associated total cost. Recall that the aim of our investigation is to minimize the expected total cost of the entire supply chain without increasing the initial expected cost or decreasing the initial expected profit of any party of the supply chain. The easiest way to achieve flexible cost allocation is to introduce an incentive function Y: A negative incentive in the form of a penalty cost for the supplier decreases the cost of the producer and the profit of the supplier, while a positive incentive, which might be a bonus for the supplier, increases the profit of the supplier and the cost of the producer. So, there is a direct impact of Y on the actual total cost of producer and supplier and therefore on the allocation of the total cost of the supply chain, which can be seen from the following equation: C P ðd; YÞ ¼ p½D dþ þ g½d Dþ þ pd þ Y; C S ðq; YÞ ¼ pd h½aC þ D qþ wD þ Y:
4.1. Penalty cost In the following, we consider a variable penalty cost which the supplier has to pay to the producer whenever he does not meet the order. Let K denote the penalty cost the supplier has to pay for every unit which is ordered but not delivered. Then the negative incentive function Y yields Y ¼ K½q dþ ¼ K½q ðaC þ DÞþ : That means that the supplier must pay a penalty cost whenever the delivery quantity, which depends on his available capacity (see Eq. (2)), is less than the order quantity q: The expected profit of the supplier after introducing the variable penalty costs is then given by EðCS ðKÞÞ Z xðq;DÞ ¼p ðaC þ DÞ f ðaÞ da 0
þ pq h
Z
Z
K
1
f ðaÞ da
xðq;DÞ 1
ðaC þ D qÞ f ðaÞ da wD
xðq;DÞ Z xðq;DÞ
ðq ðaC þ DÞÞ f ðaÞ da
0
¼ pq ðp þ KÞ h
Z
Z
xðq;DÞ
ðq ðaC þ DÞÞ f ðaÞ da 0
1
ðaC þ D qÞ f ðaÞ da wD:
ð13Þ
xðq;DÞ
The two objective functions above show that the incentive function Y has only an indirect impact on the total cost of the supply chain, since in ‘‘summing up’’ the functions ðC P C S Þ the expenditures ðpd þ YÞ of the producer and the receipts ðpd þ YÞ of the supplier cancel each other out. Hence, to achieve minimal expected total cost, the incentive function must rather have a direct impact on the capacity decision of the supplier. We will show in the next two subsections that, in finding an optimal combination of order decision and (positive or negative) incentive, a coordination mechanism can be provided enabling the supplier to choose the overall optimal decision while achieving a variety of different cost allocations.
Again, as in the worst case and the best case scenario the setting can be characterized as a newsvendor problem, where in contrast to the worst case situation (see Eq. (7)) the purchase price p is replaced by the expression p þ K: Then, the optimal decision DK opt depending on the penalty cost K can be calculated by solving the following equation: ! q DK pþK w opt : ð14Þ 1 F ¼ pþK þh C By taking a closer look at the equation above, we can see the steering effect of the penalty cost. As mentioned in Section 3.1 the fraction on the right side increases in p: Since, in contrast to the worst
9
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
case scenario (see Eq. (7)), we add the penalty cost K to the price p; the probability of meeting the demand is higher than in the worst case, which means that the supplier builds up more extra capacity. In summarizing, the penalty cost does not only compensate the shortage cost of the producer but moreover have a direct impact on the capacity decision of the supplier and consequently on the expected total cost of the producer as well as on the overall expected total cost of the supply chain. Considering the special case of a being uniformly distributed over the range ½a; a% ; the optimal % extra capacity can be calculated by solving Eq. (14) resulting in DK opt ðq; KÞ ¼ q C
wða% aÞ þ a% h þ ðp þ KÞa % %: pþK þh
ð15Þ
Note that the optimal capacity decision of the supplier, which minimizes his expected total cost, depends on both the penalty cost K and the order quantity q: Since the extra capacity is linearly proportional to the order quantity the steering effect of the penalty cost decreases with increasing cost values. Fig. 3 illustrates the typical behavior of the extra capacity for a given value q ¼ 100 of the order quantity. The underlying parameter values correspond to the values of an example which will be discussed in the appendix. The horizontal axis
represents the penalty cost, the vertical axis represents the supplier’s choice of extra capacity. The figure shows that the slope of the function decreases as the penalty cost K increases. There are two different ways for the producer to influence the optimal capacity decision of the supplier: The order quantity and the penalty cost. We call the order quantity decision the ‘executionoriented decision’ and the decision about the penalty cost the ‘control-oriented decision’. Since we presume a team-like behavior, we assume that the producer chooses the penalty cost K and the order quantity q such that the overall expected total cost of the entire supply chain is minimized. To put it differently, we assume that the producer directs the supplier to choose the overall optimal decision Djoint opt : Let us consider once more the optimal decision in the best case scenario, which can be calculated by solving Eq. (11): ! D Djoint p w opt 1¼F : þ p þ minfg; hg C On the other hand, in the coordination case, described above, the supplier chooses a capacity value so that Eq. (14) holds: ! q DK pþK w opt 1¼F : þ pþK þh C To ensure that the supplier chooses this value of extra capacity decision that incurs overall minimal expected total cost, the producer must fix the order quantity q and penalty cost K such that the following equation holds: joint DK opt ¼ Dopt
q Djoint opt 3F C ¼
Fig. 3. Extra capacity behavior depending on the underlying penalty cost.
!
D Djoint opt F C
p w pþK w : p þ minfg; hg p þ K þ h
!
ð16Þ
The term in the left-hand side describes the probability that the maximum deliverable quantity of the supply product depending on the available capacity of the supplier will be greater than the required quantity D and less than the ordered quantity q: The equation above shows that the
10
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
producer can attain the coordination of the entire supply chain through different combinations of order quantity q and penalty cost K: For instance, consider the case that the producer orders precisely that quantity that he requires ðq ¼ DÞ and assume that the cost for overcapacity h is less than the inventory cost g of the producer. Then he set fix the penalty cost K equal to p p (shortage cost minus purchase cost of the supply product) to reach overall minimal expected total cost. For the special case that a follows a uniform distribution the function which represents the relation between optimal order quantity and optimal penalty cost can easily be set up. For this we assume that the optimal decisions given by Eqs. (12) and (15) are equal and resolve the resulting equation in q: Then we get the optimal order quantity as a function of the penalty cost K: qopt ðKÞ ¼ D þ C C
wða% aÞ þ a% h þ ðp þ KÞa % % pþK þh
Whereas the penalty cost punishes the supplier for not meeting the order, bonus payments are an incentive to provide on-time delivery and lower the cost of the supplier. Unlike the penalty cost which has been assumed to be linearly proportional to the number of units short, we now consider fixed bonus payments which are only paid to the supplier when order and delivery quantity coincide. Therefore the incentive function Y can be written as ( A for d ¼ q; Y¼ 0 otherwise; or in a mathematical term, using an indicator function, as Y ¼ A1ðaCþDÞXq : Then, the expected profit of the supplier is given by Z xðq;DÞ ðaC þ DÞf ðaÞ da EðC S ðAÞÞ ¼ p 0
þ pq
wða% aÞ þ a% minfg; hg þ pa % % minfg; hg þ p
¼ Djoint opt þ C
4.2. Bonus
h
Z
Z
1
xðq;DÞ
wða% aÞ þ a% h þ ðp þ KÞa % %: pþK þh
wD þ A
ðaC þ D qÞf ðaÞ da Z
1
f ðaÞ da:
ð18Þ
xðq;DÞ
ð17Þ Each combination of penalty cost and order quantity that satisfies the above equation ensures that the supplier chooses the overall optimal capacity decision that leads to minimal expected total cost for the entire supply chain. On the other hand the function shows that, the lower the penalty cost, the higher the required order quantity. That means that the cost allocation of the overall total cost of the supply chain depends on the combination of q and K: So far, we have found a coordination mechanism that gains an overall optimal performance of the entire supply chain and offers different ways to allocate the overall total cost of the supply chain between the supplier and the producer. To extend the range of possible cost allocations we consider in the next subsection bonus payments which lower the cost of the supplier.
f ðaÞ da
xðq;DÞ 1
Unlike the previous cases the setting is no more transferable into a newsvendor problem due to the special structure of the incentive function. Nevertheless, we can obtain the optimal decision of the supplier in case a is uniformly distributed. Substituting the density function in (18) and using the first order condition, we obtain after some algebraic simplification the optimal policy for the supplier as a function of order quantity q and bonus A as DA opt ðq; AÞ ¼ q
Cðwða% aÞ þ a% h þ paÞ A : % % pþh
ð19Þ
The equation shows that the supplier builds up proportionately as much extra capacity as the bonus. As in the case of the penalty cost there are two ways to influence the supplier: The order quantity (execution-oriented decision) and the bonus (control-oriented decision). Again, we
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
assume that the producer selects those values of order quantity and bonus which provide the least expected total cost for the entire supply chain. Then, the optimal combination of order quantity and bonus can be calculated by equating the optimal decision of the supplier (19) with the overall optimal decision (11). After some conversions we get the optimal order quantity of the producer as a function of the bonus Cðwða% aÞ þ a% h þ paÞ A % % pþh wða% aÞ þ a% minfg; hg þ pa C % % minfg; hg þ p Cðwða% aÞ þ a% h þ paÞ A : ð20Þ ¼ Djoint % % opt þ pþh
qopt ðAÞ ¼ D þ
Note that each combination of bonus and order quantity which satisfies the above equation ensures that the supplier builds up as much extra capacity as is optimal from the point of view of the entire supply chain. The function also shows that the lower the value of the bonus, the more quantity units must be ordered and vice versa. However, unlike the case where a penalty cost is charged, the bonus payment and the order quantity have the same effect on the cost allocation. Both a higher bonus value and a higher order quantity result in higher total cost for the producer and a higher profit for the supplier.
11
as the centralized one. This means, that the resulting expected total cost of the entire supply chain is always as low as in the best case, whereas, unlike the case of a central planner, the involved parties do not lose their decision making authorities. Furthermore, the approach provides a flexibility of cost allocation between the two parties of the supply chain. Since the paper primarily focused on finding general analytical results, the supply chain was described on a highly aggregated level and the results are based on assumptions which might not completely hold in practice. But it can be shown (see [24]) that the coordination mechanism is also applicable to a realistic situation where the parties of the supply chain have some private information. Another important extension to consider is the coordination of more than two parties in a supply chain. In particular it would be worth pursuing whether the concept of combining an ‘execution-oriented decision’ with a ‘controloriented decision’ leads to similar flexible cost allocations for these multiple decisions, as observed here.
Acknowledgements The author thanks Prof. Christoph Schneeweiss for his continuous support and Prof. Jay Sankaran as well as the anonymous referees for their helpful comments and suggested improvements.
5. Conclusions The paper considered a supply chain, consisting of one producer and one supplier, in a Just-inTime environment where the supply of the component is uncertain due to an uncertain availability of the capacity of the supplier. First we compared the worst case, where no information exchange between the two parties takes place, with the best case, where all decisions are chosen simultaneously by a central planner. Then we developed a coordination mechanism which combined an ‘execution-oriented decision’ with a ‘control-oriented decision’. We showed that the coordination mechanism ensures the decentralized system to perform as well
Appendix A. Example In order to get a better understanding of the underlying coordination problem and to illustrate the flexibility of cost allocation using the described coordination mechanism we consider a numerical example. (For a more general and deeper discussion of the behavior of the coordination mechanism, and particularly the range over which the total cost of the supply chain can be allocated, see [23].) The parameters are specified as follows: The external demand of the final product is D ¼ 100: The price charged by the supplier to the producer
12
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
for each unit of the supply product is p ¼ $10: The producer must store any unsold supply product at a value of g ¼ $3 per unit and pays a shortage cost for the final product, which is p ¼ $60 per unit. The normal capacity of the supplier is C ¼ 100 and a is uniformly distributed over the range ½0:7; 1: To increase the capacity by one unit costs the supplier w ¼ $3 and the holding cost is h ¼ $1 per unit. Thus, the supplier can meet the order (in case the order coincides with the external demand) without building up extra capacity only in case the realization of a is 1. A.1. Worst and best case Let us first consider the worst and the best case. As mentioned in Section 3.1 we assume that in the worst case scenario the producer orders precisely that quantity that he requires, i.e. q ¼ 100: After that, the supplier chooses that amount of extra capacity that provides his most expected profit, that is Dworst opt ¼ 19:09 (see Eq. (8)). As a result of both decisions the expected total cost for the entire supply chain is EðCjoint ðq; Dworst opt ÞÞ ¼ $182:35; in which the expected cost of the producer is EðC P Þ ¼ 1099:17; whereas the expected profit of the supplier is EðC S Þ ¼ 916:82: On the other hand, the optimal capacity decision in the best case scenario is Djoint opt ¼ 28:03 (see Eq. (12)) which achieves the minimal expected total cost of E
ðC joint ðDjoint opt ÞÞ ¼ $101:06 for the entire supply chain. That means that in our numerical example the expected total cost of the entire supply chain in the best case is more than 55% less than in the worst case. Even if this concrete value depends on the specification of the parameters it gives a hint as to what cost saving potential may exist in a supply chain. The results are summarized in Table 1. Next, we analyze the performance of the coordination mechanism, which combines the
Table 1 Worst case compared with best case
Worst case Best case
qopt
Dopt
EðCP Þ
EðCS Þ
EðCjoint Þ
100 F
19.09 28.03
1099.17 F
916.82 F
182.35 101.06
order quantity as an execution-oriented decision with a penalty cost and a bonus payment, respectively, as a control-oriented decision. A.2. Penalty cost In case of penalty cost the optimal order quantity as a function of the penalty cost (see Eq. (15)) is given by 100ð8:9 þ 0:7KÞ : qopt ðKÞ ¼ 28:03 þ 11 þ K Table 2 shows the mutual dependency of order quantity and penalty cost for some specific discrete values. We observe that by increasing the penalty cost the expected total cost of the producer and the expected profit of the supplier decrease. Recall that the optimal decision of the supplier as well as the overall expected total cost of the entire supply chain correspond to those of the best case. Furthermore, the table shows that in charging a penalty cost the most expected profit of the supplier is EðCS Þ ¼ 979:12: A.3. Bonus To increase the expected profit of the supplier and thus to extend the range of possible cost allocations, we now consider the bonus case. From Eq. (20) we get the bonus as a function of the order quantity, that is AðqÞ ¼ 890 þ 11ð28:03 qÞ: Table 3 shows the dependency between bonus and order quantity. It can be observed that the higher the order quantity, the lower the bonus has to be. By increasing the order quantity (decreasing the bonus, respectively) the expected total cost of the
Table 2 Optimal order quantity in dependency of the penalty cost K
qopt ðKÞ
EðCS Þ
EðCP Þ
EðCjoint Þ
0 1 2 5 10
108.93 108.03 107.26 105.53 103.74
979.12 971.22 964.29 948.71 932.64
1080.18 1072.28 1065.35 1094.77 1033.70
101.06 101.06 101.06 101.06 101.06
13
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
producer and the expected profit of the supplier decrease. A.4. Cost allocation The overall range in which the expected total cost of the supply chain can be allocated by using the coordination mechanism is shown in Fig. 4. In contrast to the Tables 2 and 3 the figure does not depict the expected profit of supplier and the expected total cost of the producer in all, but shows only those cost components which concern the supply uncertainty directly, i.e. the purchase cost of $1000; which the producer would pay to the
Table 3 Optimal bonus in dependency of the order quantity qopt
Aðqopt Þ
EðCS Þ
EðCP Þ
EðCjoint Þ
100 101 102 103 108
98.33 87.33 76.33 65.33 10.33
994.04 990.95 988.22 985.86 979.55
1095.10 1092.01 1089.28 1086.92 1080.61
101.06 101.06 101.06 101.06 101.06
supplier in case of certainty, is not included. Recall that in adding up the expected total cost of the producer and the negative expected profit of the supplier the purchase costs cancel each other out anyway. The horizontal axis shows the different combinations of order quantity and incentive. The vertical axis represents the associated expected total cost for the supplier, the producer and the entire supply chain. Let us first have a closer look at the horizontal axis. The center of the axis describes the case when the producer neither offers a bonus nor charges a penalty cost but influences the supplier only by a maximal order quantity q ¼ qmax : This situation corresponds to the case described in the first line of Table 2. Starting in the center and moving to the right means that the value of the penalty cost increases (up to Kmax ) while the order quantity decreases (down to q ¼ D). On the other hand moving left from the center represents an increasing bonus value (up to Amax ) and a decreasing order quantity (down to q ¼ D). Thus, the figure is divided into two parts: the left part describing the bonus case and the right part describing the penalty case.
200 180
Expected total cost (worst case) 160
Expected costs
140
Expected total cost (best case)
120 100 80
Expected cost of the producer 60
Expected cost of the supplier
40
Expected cost of the supplier
20 0
A max (q =D )
A = K= 0 (q =q max)
Fig. 4. Allocation of the total cost in dependency of bonus and penalty cost.
K max (q =D )
14
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
Let us now have a look at the cost behavior. First, since the expected total cost of the entire supply chain using the coordination mechanism equals the expected total cost of the best case, the gap between the worst and the best case shows the cost savings which can be achieved through the considered coordination mechanism. Note that in an existing supply chain partnership the expected total cost will always be situated between the best and the worst case. Furthermore, in combining the order quantity with an incentive we get a wide range of different cost allocations. The figure shows that in case of the maximal bonus value Amax the producer must bear almost the whole expected total cost of the supply chain, while in case of the maximal penalty cost Kmax the supplier has to bear the expected total cost alone. Note that if we had determined only an optimal incentive to coordinate the supply chain, which is common in the literature, we would have achieved the same performance for the entire supply chain but would have allocated the total cost only in one possible manner. Assuming the producer orders precisely what he requires, he has to offer a bonus payment at a value of Amax (the point farthest to the left in Fig. 4) or charge a penalty cost at the value of Kmax per unit (the point farthest to the right) in order to minimize the overall expected total cost of the entire supply chain. But compared to an initial situation, which might be better than the worst case represented in Table 1, either the producer or the supplier might make losses. Thus, in practice the parties of the supply chain would probably not agree to any of these two incentive schemes and the implementation of such a coordination mechanism would fail. In summarizing, the example shows on the one hand which substantial cost savings might be achieved in practice by implementing the developed coordination mechanism. This result is of particular interest from the general point of view of supply chain management, since the cost savings refer to the overall cost of the entire supply chain. On the other hand the example demonstrates that the coordination mechanism is flexible enough to enable different allocations of these overall cost thus allowing both parties in an existing supply chain partnership to make a profit.
This aspect is of importance for the implementation of such a mechanism and is mostly ignored in literature.
References [1] M. Christopher, Logistics and Supply Chain Management: Strategies for Reducing Cost and Improving Service, Financial Times Pitman, London, 1998. [2] B.C. Arntzen, G.G. Brown, T.P. Harrison, L.L. Trafton, Global supply chain management at digital equipment corporation, Interfaces 25 (1) (1995) 69–93. [3] R.L. Breitman, J.M. Lucas, PLANETS: A modeling system for business planning, Interfaces 17 (1) (1987) 94–106. [4] G.G. Brown, G.W. Graves, M.D. Honczarenko, Design and operation of a multicommodity production/distribution system using primal goal decomposition, Management Science 33 (11) (1987) 1469–1480. [5] M.A. Cohen, S. Moon, An integrated plant loading model with economies of scale and scope, European Journal of Operational Research 50 (1991) 266–279. [6] R. Anupindi, Y. Bassok, Supply contracts with quantity commitments and stochastic demand, in: S. Tayur, R. Ganeshan, M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1999, pp. 198–232. [7] J.P. Monahan, A quantity discount pricing model to increase vendor profits, Management Science 30 (6) (1984) 720–726. [8] A. Banerjee, A joint economic-lot-size model for purchaser and vendor, Decision Science 17 (1986) 292–311. [9] A. Banerjee, Notes: On ‘‘A quantitative discount pricing model to increase vendor profits’’, Management Science 32 (11) (1986) 1513–1517. [10] Ch.J. Corbett, X. de Groote, Integrated supply-chain lot sizing under asymmetric information, Working Paper, The Anderson School at UCLA, 1997. [11] R. Aderohunmu, A. Mobolurin, N. Bryson, Joint vendor– buyer policy in JIT manufacturing, Journal of the Operational Research Society 46 (3) (1995) 375–385. [12] M.A. Lariviere, Supply chain contracting and coordination with stochastic demand, in: S. Tayur, R. Ganeshan, M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1999, pp. 233–268. [13] P.H. Ritchken, Ch.S. Tapiero, Contingent claims contracting for purchasing decisions in inventory management, Operations Research 34 (6) (1986) 864–870. [14] D. Barnes-Schuster, Y. Bassok, R. Anupindi, Coordination and flexibility in supply contracts with options, Working Paper, Graduate School of Business, University of Chicago, 1998. [15] Ch.J. Corbett, C.S. Tang, Designing supply contracts: Contract type and information asymmetry, in: S. Tayur, R.
K. Zimmer / Int. J. Production Economics 77 (2002) 1–15
[16]
[17]
[18] [19]
[20]
Ganeshan, M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1999, pp. 269–297. R. Kolisch, Coordination of assembly and fabrication for make-to-order production, Pre-prints of the 10th International Working Seminar on Production Economics 3 (1998) 333-338. Igls/Innsbruck. Y. Gerchak, Y. Wang, C.A. Yano, Lot sizing in assembly systems with random component yields, IEE Transactions 26 (2) (1994) 19–24. C.A. Yano, H.L. Lee, Lot sizing with random yields: A review, Operations Research 43 (2) (1995) 311–334. H. Gurnani, R. Akella, J. Lehoczky, Supply management in assembly systems with random yield and random demand, Working Paper, The Hong Kong University of Science and Technology, 1998. H. Gurnani, Supply management in assembly systems with uncertain supply and inter-supplier penalties, Working
[21]
[22]
[23]
[24]
15
Paper, The Hong Kong University of Science and Technology, 1998. Y. Gerchak, Y. Wang, Coordination in decentralized assembly systems with random demand, Working Paper, Department of Management Sciences, University of Waterloo, 1999. H. Gurnani, Y. Gerchak, Coordination in decentralized assembly systems with uncertain component yields, Department of Management Sciences, University of Waterloo, 1999. K. Zimmer, Hierarchische Coordination im Supply Chain Management, Ph.D. Thesis, Department of Operations Research, University of Mannheim, 2000: Ch. Schneeweiss, K. Zimmer, Hierarchical coordination mechanisms within the supply chain, Working Paper, Department of Operations Research, University of Mannheim, 2000: