Open Access is an initiative that aims to make scientific research freely available to all. To date our community has made over 100 million downloads. It’s based on principles of collaboration, unobstructed discovery, and, most importantly, scientific progression. As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access publisher that levels the playing field for scientists across the world. How? By making research easy to access, and puts the academic needs of the researchers before the business interests of publishers.
We are a community of more than 103,000 authors and editors from 3,291 institutions spanning 160 countries, including Nobel Prize winners and some of the world’s most-cited researchers. Publishing on IntechOpen allows authors to earn citations and find new collaborators, meaning more people see your work not only from your own field of study, but from other related fields too.
To purchase hard copies of this book, please contact the representative in India:
CBS Publishers & Distributors Pvt. Ltd.
www.cbspd.com
|
customercare@cbspd.com
According to the Council of Supply Chain Management Professionals (September 2007), we have the following description for supply chain management:
“supply chain management encompasses the planning and management of all activities involved in sourcing and procurement, conversion, and all logistics management activities. Importantly, it also includes coordination and collaboration with channel partners, which can be suppliers, intermediaries, third-party service providers, and customers. In essence, supply chain management integrates supply and demand management within and across companies.”From this description, it is obviously true that a supply chain in general has multiple channel members (usually called stages) and the coordination and collaboration among these members is a crucial task in supply chain management.
In the literature, various policies for supply chain optimization and channel coordination have been proposed. Among them, setting a supply chain contract between individual parties has received much attention in recent years Tsay et al. 1999, Cachon 2003). Contracts such as buy-back contract, revenue sharing contract, quantity flexibility contract and rebates contract are all known forms of contract which can help to achieve channel coordination in a supply chain. However, in the majority of the literature works, the channels' and supply chain’s objectives are either maximizing the expected profit or minimizing the expected cost. There is no discussion on the level of risk associated with these contracts. As a result, the contract parameters under which coordination is achieved may be viewed as unrealistic by decision makers. In light of this, we conduct in this paper a mean-variance analysis on some popular forms of supply chain contracts such as buy-back contract. By including a constraint on profit uncertainty, we illustrate how decision makers can make a scientifically sound and tailored decision with respect to their degrees of risk aversion. Managerial implications are discussed.
The organization of the rest of this chapter is as follows: We briefly review some related literature in Section 2, the discussion of the supply chain’s structure is presented in Section 3. The mean-variance analyses on the buy-back contract and wholesale pricing profit sharing contract are conducted in Sections 4 and 5, respectively. We conclude with some discussions on managerial implications in Section 6.
For a notational purpose, we use the following notation in many places throughout this chapter: P = profit, EP = expected profit, SP = standard deviation of profit, MV = mean-variance. The subscripts “M, R, SC” represent “Manufacturer, Retailer, Supply Chain”, respectively.
Pioneered by Nobel laureate Harry Markowitz in the 1950s, the mean-variance formulation has become a fundamental theory for risk management in finance (Markowitz 1959). In decision sciences, the mean-variance approach and the von Neumann-Morgenstern utility approach (called utility function approach in short) are two well established methodologies for studying decision making problems with risk concerns. The utility function approach is more precise but its application is limited owing to the difficulty in getting a closed form expression of the utility function for every individual decision maker in practice. The mean-variance approach, as what Van Mieghem (2003) mentioned, aims at providing an implementable, useful but approximate solution. It is true that a utility function in general cannot be expressed fully in terms of mean and variance only. However, it is shown in Van Mieghem (2003) that maximizing a utility function with a constant coefficient of risk aversion is equivalent to maximizing a mean-variance performance measure (also see Luenberger 1998, Choi et al. 2008 for some supplementary discussions). There are also evidences in the literature which demonstrate that the mean-variance approach yields a solution which is close to the optimal solution under the utility function approach (see Levy & Markowitz 1979, Kroll et al. 1984, and Van Mieghem 2003). Moreover, some meaningful and applicable objectives, such as the safety first objective (Roy 1952), can be formulated under the mean-variance framework. Despite all kinds of arguments on the mean-variance approach, it is adopted as the performance measure in this chapter because it’s “applicable, intuitive and implementable”. In addition, more analytical results can be generated under this approach. On the other hand, even though the mean-variance and utility function approaches are well-established in finance, their applications in supply chain management are not yet fully revealed. In fact, most research works on this important topic appear only in recent years. We review some of them as follows.
First, in Lau (1980), instead of maximizing the expected profit, the author derives an optimal order quantity which maximizes an objective function of the expected profit and standard deviation of profit for the classic newsvendor problem. Next, Eechhoudt et al. (1995) study the classic newsvendor problem with risk averse newsvendor via a utility function approach and obtain some interesting findings on the optimal stocking quantity. Later on, Lau and Lau (1999) directly extend the work of Pasternack (1985) and study a single-manufacturer single-retailer supply chain model under which both the retailer and manufacturer seek to maximize a linear objective function of the expected profit and variance of profit. Choi et al. (2008) analyze via a mean-variance approach the supply chains under returns policy in both decentralized and centralized settings. Implications for setting returns contracts for achieving channel coordination with risk considerations are discussed. Some other recent research works which analyse the risk issues in supply chain management include a qualitative discussion on proactive supply management and its close relationship with risk management (Smeltzer & Siferd 1998), a quantitative analysis of the role of intermediaries in supply chains to reduce financial risk (Agrawal & Seshadri 2000), a mean-variance analysis of single echelon inventory problems (Chen & Federgruen 2000), a study of the risk-free perishable item returns policy with a risk neutral retailer in a two-echelon supply chain (Webster & Weng 2000), an investigation of the use of capacity options in managing risk from demand uncertainty (Tan 2002), an analysis of the use of commitment-option for supply chain contract setting with forecast updates (Buzacott et al. 2003), a study on contracting scheme with risk preferences considerations (Bassok & Nagarajan 2004), a mean-variance analysis for the newsvendor problem with and without the opportunity cost of stock out (Choi et al. 2007a), and a study on channel coordination in supply chains under mean-variance objectives (Choi et al. 2007b)
Consider a two-echelon supply chain with one manufacturer and one retailer. The retailer sells a fashionable product and faces an uncertain market demand. The manufacturer bears a unit product cost of c and sells the product to the retailer with a unit wholesale price w. For the retailer the unit product’s selling price is r. At the end of the selling season, there is a salvage market in which any product leftover can be salvaged at a unit price v. Let the market demand faced by the retailer be x with a probability density function f(x), and a corresponding cumulative distribution function F(x). We assume that there is a one-to-one mapping between F( ) and its argument. We consider the following sequence of action: The manufacturer will first announce the wholesale price and other parameters (with respect to different kinds of contracts) to the retailer, the retailer will react by placing an order with a quantity q. We assume that the manufacturer can always fulfil the required order quantity placed by the retailer. For a notational purpose, define:
ξ(q)=2q∫0qF(x)dx−2∫0qxF(x)dx−(∫0qF(x)dx)2E1
Table 1 below gives the profit, expected profit, standard deviation of profit of the simple supply chain described above. Observe that the manufacturer is risk free and can always make a positive profit when the wholesale price is larger than the production cost under this simple supply chain.
Supply Chain
Retailer
Manufacturer
P
(r−c)q−(r−v)(q−x)+
(r−w)q−(r−v)(q−x)+
(w−c)q
EP
(r−c)q−(r−v)∫0qF(x)dx
(r−w)q−(r−v)∫0qF(x)dx
(w−c)q
SP
(r−v)ξ(q)
(r−v)ξ(q)
0
Table 1.
Profit, Expected Profit, and Standard Deviation of Profit of the Simple Supply Chain without Additional Contracts
We now consider two kinds of contracts, the buy-back contract and the wholesale-pricing profit-sharing contract, in the following.
3.1. Buy-back contract
Under the buy-back contract, by the end of the selling season, the retailer can return the unsold products to the manufacturer for a partial refund with a unit buy-back price b, wherev≤bw. The returned products have a unit value of v to the manufacturer. We can derive the profit, expected profit, and standard deviation of profit under the buy-back contract for the supply chain, the retailer, and the manufacturer respectively as shown in Table 2 (see Choi et al. 2008 for the details of derivations).
Supply Chain
Retailer
Manufacturer
P
(r−c)q−(r−v)(q−x)+
(r−w)q−(r−b)(q−x)+
(w−c)q−(b−v)(q−x)+
EP
(r−c)q−(r−v)∫0qF(x)dx
(r−w)q−(r−b)∫0qF(x)dx
(w−c)q−(b−v)∫0qF(x)dx
SP
(r−v)ξ(q)
(r−b)ξ(q)
(b−v)ξ(q)
Table 2.
Profit, Expected Profit, and Standard Deviation of Profit under the Buy-back Contract
Notice that the supply chain’s expected profit and standard deviation of profit are not affected by the presence of the buy-back contract.
3.2. Wholesale pricing and profit sharing contract
Under the wholesale pricing and profit sharing contract, the manufacturer controls the wholesale price w, where w can be set to be c, i.e., the manufacturer is supplying at cost and makes zero profit from the direct supply. On the other hand, the manufacturer will share the retailer’s profit with a proportion of(1−α), where0α1. To be specific, we can derive the following the profit, expected profit and standard deviation of profit under the wholesale pricing and profit sharing contract for the supply chain, the retailer, and the manufacturer, respectively:
Supply Chain
Retailer
Manufacturer
P
(r−c)q−(r−v)(q−x)+
α[(r−w)q−(r−v)(q−x)+]
(w−c)q+(1−α)⋅ [(r−w)q−(r−v)(q−x)+]
EP
(r−c)q−(r−v)∫0qF(x)dx
α[(r−w)q−(r−v)∫0qF(x)dx]
(w−c)q+(1−α)⋅ [(r−w)q−(r−v)∫0qF(x)dx]
SP
(r−v)ξ(q)
α(r−v)ξ(q)
(1−α)(r−v)ξ(q)
Table 3.
Profit, Expected Profit, and Standard Deviation of Profit under the Wholesale Pricing and Profit Sharing Contract
Remarks and findings:
i. Please notice that under both buy-back contract and the wholesale pricing and profit sharing contract, the expected profit functions of both the retailer and supply chain are concave in q,and their standard deviation of profit functions are increasing in q (see Choi et al. 2007a for more details).
ii. A direct observation from the expected profit and standard deviation of profit expressions for the manufacturer in Tables 1, 2 and 3 indicates that the manufacturer is basically risk free under the simple supply chain without additional contracts. However, under both the buy-back contract and wholesale pricing and profit sharing contract, the manufacturer needs to bear a higher risk. As a result, depending on the degree of risk aversion of the manufacturer, exercising one of these contracts is not always beneficial because the risk level for the manufacturer is higher.
iii. From Tables 1, 2 and 3, we can see that the sum of retailer’s SP and manufacturer’s SP equals the supply chain’s SP. The same applies for the expected profit EP. As a result, a change of the contract parameter, of either the buy-back contract and the wholesale pricing and profit sharing contract, can lead to a reallocation of benefit (expected profit) and risk (standard deviation of profit) between the manufacturer and the retailer. Bargaining power hence plays a crucial role especially for the wholesale pricing and profit sharing contract.
We now consider the above proposed supply chain in which the manufacturer acts as a supply chain coordinator. Here, instead of maximizing the supply chain’s expected profit, the manufacturer adopts the following MV objective for the supply chain:
(P1)
maxqEPSC(q)s.t.SPSC(q)≤kSCE2
The objective of (P1) is to maximize the supply chain’s expected profit subject to a constraint on the supply chain’s standard deviation of profit, where kSC is a positive constant. Represent by qSC,EP*=F−1[(r−c)/(r−v)] the product quantity which maximizesEPSC(q). The efficient frontier for (P1) can be constructed withq∈[0,qSC,EP*], and [0,qSC,EP*] is the efficient region. In (P1), a smaller kSC implies that the manufacturer (who is the decision maker) is more conservative and risk averse. We thus call kSC the supply chain’s risk aversion threshold. Notice that when kSC∈ [0,SPSC(qSC,EP*)], a smaller value of kSC would lead to a smaller optimal quantity for (P1) because in this region: EPSC(q)is increasing and concave,SPSC(q) is increasing, and the constraint SPSC(q)≤kSC is active. When kSCSPSC(qSC,EP*), the SP constraint becomes “inactive” as the optimal solution is alwaysqSC,EP*. Represent the optimal solution of (P1) byq*. It is easy to show that q* exists and can be uniquely determined (see Choi et al. 2007a for the details). Similar to the model setting in (P1), the retailer’s decision making problem is modelled as follows,
(P2)
maxqEPR(q)s.t.SPR(q)≤kRE3
In (P2), the retailer tries to maximize his expected profit with the corresponding standard deviation of profit under control, i.e., SPR(q)≤kR, where kRis a positive constant and it is the retailer’s risk aversion threshold. When the manufacturer has specified the details on the wholesale price and other contract parameters, the retailer will determine an order quantity qR* which optimizes (P2). Observe that there exists a unique qR,MV* (see Choi et al. 2007a for the details).
In general, q*and qR,MV* are different. In this chapter, we consider the best product quantity for the supply chain in the mean-variance domain asq*. As a consequence, the manufacturer who acts as the supply chain coordinator can consider using some incentive alignment schemes to try to entice the retailer to order in a quantity which is equal toq*. We will now explore how the buy-back contract and the wholesale pricing and profit sharing contract can help to achieve this kind of coordination in a mean-variance domain. We separate the analysis into two parts in the next two sections.
5. Coordination by the buy-back contract in the mean-variance domain
Under the presence of the buy-back contract, we rewrite (P2) into (P2(b)) as follows,
(P2(b))
maxqEPR[q;b]s.t.SPR[q;b]≤kRE4
where EPR[q;b]=(r−w)q−(r−b)∫0qF(x)dxSPR[q;b] =(r−b)ξ(q) (see Table 2), and b is the buy-back price offered by the manufacturer. Denote the optimal order quantity for (P2(b)) byqR,BB*(b). Following the approach in Choi et al. (2008), for any given b, we define the following:
qR,2*(b)=argq{SPR(q|b)−kR=0}E5
qR,1*(b)=F−1[(r−w)/(r−b)]E6
Notice that qR,1*(b) is the order quantity which maximizes the retailer’s expected profit with a given b. The following procedure, Procedure 1, provides the steps to identify the buy-back price which can achieve coordination (bSC,MV*):
Procedure 1
Step 1. Compute q* by solving (P1).
Step 2. Determine a parameter b1* which makes qR,1*(b) = q* as follows:
• IfSPR(qR,1*|b1*)≤kR, thenqR,BB*(b1*)=qR,1*(b1*). Thus, setting b=b1* would yieldqR,BB*(b1*)=qR,1*(b)=q*. Set bSC,MV*=b1* and stop.• IfSPR(qR,1*|b1*)kR, then qR,BB*(b1*)=qR,2*(b1*)=. However, setting b=b1* would not yield qR,BB*(b1*)=q* since setting b=b1* can only achieveqR,1*(b)=q*, but here qR,BB*(b1*)=qR,2*(b)=. Go to Step 5.
Step 5. Check for the feasibility ofbSC,MV*=b2* (after Step 4):
• IfSPR(qR,1*|b2*)kR, then qR,BB*(b2*)=qR,2*(b2*)=. Thus, setting b=b2* would yieldqR,BB*(b)=qR,2*(b)=qSC,MV*. Set bSC,MV*=b2* and stop.• IfSPR(qR,1*|b2*)≤kR, thenqR,BB*(b2*)=qR,1*(b2*). In this case, setting b=b2* can only achieve qR,2*(b)=q* (but not qR,1*(b2*)=q* which impliesqR,BB*(b2*)=q*). Thus, we are not able to achieveqR,BB*(b2*)=q*. In this situation, setting both bSC,MV*=b1* and bSC,MV*=b2* cannot achieve coordination in the MV domain.
Procedure 1 gives us the detailed steps for identifying the buy-back price which can achieve coordination in a mean-variance domain. Since the buy-back price is bounded between v and w, i.e.v≤bw, a checking on the computed value of bSC,MV* with respect to this bound is a required feasibility test.
6. Coordination by the wholesale pricing and profit sharing contract in the mean-variance domain
With the wholesale pricing and profit sharing contract, we rewrite (P2) into (P2(w,α)) as follows,
(P2(w,α))
maxqEPR[q;w,α]s.t.SPR[q;w,α]≤kRE10
whereEPR[q;w,α]=α[(r−w)q−(r−v)∫0qF(x)dx],SPR[q;w,α]=α(r−v)ξ(q), (see Table 3), αis the proportion of profit that the retailer takes and w is wholesale price offered by the manufacturer to the retailer. Represent the optimal quantity which maximizes (P2(w,α)) byqR,WP*(w,α). Similar to the idea in Section 4, we define the following:
qR,2*(w,α)=argq{SPR(q|w,α)−kR=0}E11
qR,1*(w)=F−1[(r−w)/(r−v)]E12
Notice that qR,1*(w) is the order quantity which maximizes the retailer’s expected profit with a given w and it is independent ofα. Suppose that α is initially set to be αo(where0αo1) upon the negotiation between the retailer and the manufacturer. The following procedure gives the steps to identify the wholesale price and/or the necessary adjustment in α in order to achieve coordination in the mean-variance domain:
Procedure 2
Step 1. Compute q* by solving (P1).
Step 2. Determine a parameter w* which makes qR,1*(w) = q* as follows:
Step 4. Check for the feasibility of setting the wholesale price w = w* with α=αo
• IfSPR(qR,1*(w=w*)|αo)≤kR, then setting w=w* with α=αo can already makeqR,WP*(w,α)=q*. Thus, we can set the wholesale price w=w* withα=αo, and stop; otherwise, go to Step 5.
Step 5. Check for the feasibility of setting another value ofα.
• IfSPR(qR,1*(w=w*)|αo)kR, then:
• Option 1: The manufacturer can try to negotiate with the retailer and set a value of α=α1 (where0α11) with whichSPR(qR,1*(w=w*)|α1)≤kR
• Option 2: The manufacturer can check and see ifα*1. Ifα*1, then the manufacturer can propose to the retailer by setting a value of α=α* (where0α*1) which can makeqR,WP*(w,α)=q*
Procedure 2 provides to us some guidelines for determining the contract parameters of the wholesale pricing and profit sharing contract which can help to achieve coordination in the mean-variance domain.
In this chapter, we have conducted a mean-variance analysis for supply chains under a buy-back contract and a wholesale pricing and profit sharing contract. We characterize in the supply chain the return and the risk by the expected profit and the standard deviation of profit, respectively. We focus our discussions on the centralized supply chains. From the structural properties of the supply chain, we find that the buy-back price and the wholesale price are simply internal money transfers between the retailer and the manufacturer. A change of these prices will lead to a change of the profit and risk sharing between the retailer and the manufacturer. We illustrate how a buy-back contract and a wholesale pricing and profit sharing contract can coordinate a supply chain in a mean-variance domain. Efficient procedures are proposed. The necessary and sufficient conditions for the optimal contract parameters to be found in its feasible region can then be determined. Observe that channel coordination in the mean-variance domain is not always achievable. This finding is important because when we ignore the risk aversions of the individual supply chain members (as what most papers in the literature assume), channel coordination can always be achieved by setting a buy-back contract and a wholesale pricing and profit sharing contract. However, in the real-world, different supply chain members have different degrees of risk aversion, and hence a realistic contract should be set with respect to the risk aversions of these individual decision makers. Moreover, intuitively, when the risk aversions between the supply chain coordinator and the retailer are too far away, channel coordination may not be achievable and this point can be revealed by using our analytical models. From the studies in this chapter, we can see that the mean-variance model can provide a systematic framework for studying channel coordination issues in stochastic supply chain models with risk and profit considerations. This framework can be further extended and used to study a large variety of supply chain contracts.
This work is partially supported by the RGC Competitive Earmarked Research Grant PolyU5146/05E, and the internal fundings provided by the Hong Kong Polytechnic University. The author would like to dedicate this piece of work to Bryan Choi.
References
1.ChoiT. M.LiD.YanH.2007a Mean-Variance Analysis of Newsvendor Problem. To appear in IEEE Transactions on Systems, Man, and Cybernetics: Part A.
2.ChoiT. M.LiD.YanH.2008 Mean-variance analysis of a single supplier and retailer supply chain under a returns policy. European Journal of Operational Research, 184356376 .
3.ChoiT. M.LiD.YanH.ChiuC. H.2007b Channel coordination in supply chains with agents having mean-variance objectives. Forthcoming in Omega, available online in ScienceDirect.com, doi : 10.1016/j.omega.2006.12.003.
4.AgrawalV.SeshadriS.2000 Risk intermediation in supply chains. IIE Transactions, 32819831 .
5.BassokY.NagarajanM.2004 Contracting under risk preferences. Working paper, University of Southern California.
6.BuzacottJ.YanH.ZhangH.2003 Risk analysis of commitment-option contracts with forecast updates. Working paper, York University.
7.CachonG. P.2003 Supply chain coordination with contracts. Working paper, University of Pennsylvania,
8.ChenF.FedergruenA.2000 Mean-variance analysis of basic inventory models. Working paper, Columbia University.
9.EeckhoudtL.GollierC.SchlesingerH.1995 The risk averse (prudent) newsboy. Management Science, 41786794 .
10.KrollY.LevyH.MarkowitzH. M.1984 Mean-variance versus direct utility maximization,” Journal of Finance, 394761 .
11.LauH. S.1980 The newsboy problem under alternative optimization objectives. Journal of the Operational Research Society, 31525535 .
12.LauH. S.LauA. H. L.1999 Manufacturer’s pricing strategy and returns policy for a single-period commodity. European Journal of Operational Research, 116291304 .
13.LevyH.MarkowitzH. M.1979 Approximated expected utility by a function of mean and variance. American Economics Review, 69308317 .
14.LuenbergerD. G.1998Investment Science. Oxford University Press.
15.MarkowitzH. M.1959Portfolio Selection: Efficient Diversification of Investment. New York: John Wiley & Sons.
16.PasternackB. A.1985 Optimal pricing and returns policies for perishable commodities. Marketing Science, 4166176 .
17.RoyA. D.1952 Safety first and the holding of assets. Econometrica, 20431449 .
18.SmeltzerL. R.SiferdS. P.1998 Proactive supply management: The management of risk. International Journal of Purchasing & Materials Management, Winter, 3845 .
19.TanB.2002 Managing manufacturing risks by using capacity options. Journal of the Operational Research Society, 53232242 .
20.TsayA. A.NahmiasS.AgrawalN.1999 Modelling supply chain contracts: a review. In: Quantitative Models for Supply Chain Management, Tayur S et al. (Eds), Kluwer Academic Publishers, 299336 .
21.Van MieghemJ. A.2003 Capacity management, investment, and hedging: Review and recent developments. Manufacturing and Service Operations Management, 5269301 .
22.WebsterS.WengZ. K.2000 A risk-free perishable item returns policy. Manufacturing and Service Operations Management, 2100106 .