Keywords: Inventory; Manufacturing; Retailing; Game theory
1. Introduct ion
Establishing the size of its inventories is a major decision of the retail firm. Maintaining large invento- ries is costly, but it has the advantage of minimizing occasions where a customer cannot find a product on the shelf. A careful retailer will set inventory levels to balance the costs of ordering and holding inven- tory with the costs of stockouts. The costs of stock- outs vary from firm to firm, depending on the nature of demand and on the management policies of the firm. If stockouts lead to backorders, the firm will
* Corresponding author.
pay administrative costs to keep track of and to fulfil backorders. If the disappointed customer chooses to buy a less-preferred substitute, the retailer will lose good will and possibly future sales. For many goods, stockouts will lead to immediate lost sales for the retailer as customers search for the product at an- other location. In the long run, both backorders and immediate lost sales may lead to reduced demand as customers learn to avoid a store with an inadequate service level.
The inventory decision of the retailer is conceptu- ally straightforward, although there are a number of practical difficulties surrounding the measurement of stockout costs. The problem of setting appropriate inventory levels becomes more complex when the interests of the manufacturer are taken into account.
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A manufacturer that is dependent on retail sales for its revenues would like to see the retailer maintain a positive inventory of its product at all times, since no sales then are lost to competing products. If a single entity owns both manufacturing and retail facilities the inventory problem is solved by taking into ac- count the costs of stockouts on both facilities. If separate entities own the facilities the problem be- comes complex, because the objectives of the two entities cannot generally be optimized simultane- ously. In this paper we provide a solution to the problem of how a manufacturer can provide incen- tives to a separately-owned retailer to raise its ser- vice level above the level it would choose on its o w n .
The nature of the interaction between manufac- turer and retailer depends on several features of the industry. Perhaps the most important factor is the nature of demand. To take an extreme case, the manufacturer will have no interest at all in the retailer's service level if all demand not met immedi- ately is backordered. If all demand is backordered, and backordering has no effect (through lost good will) on future sales, the manufacturer will eventu- ally sell all the product for which there is demand. Thus it will have no incentive to raise retail service levels. However, this is not the usual case. If, at the retailer's chosen service level, there is a substantial amount of lost sales or lost good will the manufac- turer will want to consider ways to influence the retailer to improve service. The value of doing so will depend on, among other factors, the ratio of backorders to lost sales.
Several other factors may have a bearing on the nature of the manufacturer-retailer interaction. First, the inventory policy used by the retailer will obvi- ously influence the level of inventories and the ser- vice provided, and may alter the incentives available to the manufacturer to improve service. Second, the manufacturer itself can use a variety of means to influence the retailer, including quantity discounts, expedited delivery programs, side payments, and so on. Finally, the distribution of power between the two players, which itself may depend on the nature of the product being sold as well as its relative importance to both manufacturer and retailer, will influence the nature of the solution to the service problem.
In this paper we model the interaction between manufacturer and retailer as a leader-follower game (Von Stackelberg, 1952; Gardner, 1995). We assume that the retailer, who markets thousands of products, acts as the follower to the manufacturer, who as leader takes into account the reaction of the retailer to its own actions. We also assume that a portion of retail demand is sensitive to the service level main- tained by the retailer, so that sales increase with the service level. Our approach shows how the manufac- turer can create an incentive for the retailer to in- crease its inventories and raise its service level above the level that is optimal for it alone. We also investi- gate how total profits and the individual profits of the two firms change under this incentive scheme.
We do not take into account explicitly in our model the costs of administering a service-level agreement between manufacturer and retailer. Such costs may be significant, but are highly specific to the firms involved. Our analysis focuses on the potential benefits of certain types of coordination within a supply chain, thus providing a reference point against which to measure the costs in a specific application. Clearly, unless the benefits appear sub- stantial to both parties, neither will have an incentive to investigate the costs involved.
Our work is an application of game theory, in which we tackle directly the problem posed by sepa- rate ownership of the two firms and the resulting conflict over the gains from cooperation. We utilize a plausible model of service-sensitive demand, one which is relatively free of ad hoc assumptions. Finally, our approach has the strength that it pro- poses a credible solution to the problem of manufac- turer-retailer cooperation when profits are sensitive to retail service levels.
The paper is organized as follows. In the Section 2 we review the related literature, particularly in the areas of service-sensitive demand and manufac- turer-retailer coordination. Then in Section 3 we present the formulation of our model. In Section 4 we illustrate through numerical examples how the model determines the optimal incentive from manu- facturer to retailer, and how joint and individual profits are affected. We also present extensive sensi- tivity analyses. Implementation issues are addressed in Section 5. The paper concludes with a summary of results and suggestions for future research.
R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998)437-450 439
2. Literature review
In this section we review the literature in two areas. The first is studies of the inventory decision when demand is sensitive to inventory levels or to service levels. The second involves the workings of various coordination mechanisms between manufac- turer and retailer (or buyer and seller). The latter area has come to be called the integrated inventory prob- lem.
The standard treatment of the inventory decision assumes that demand is either deterministic or a stationary stochastic process, and is not influenced by the decisions of the retailer. There have been a few attempts to model the sensitivity of demand to the inventory behavior of the retailer. One line of research begins with Baker and Urban (1988), who have develop algorithms for determining optimal inventory policies for a single firm when its demand is proportional to a fractional power of the current inventory level. In this formulation, demand (and sales) decreases as inventories decrease, so there is an incentive to hold higher inventories than in the case with exogenous demand. Also, under these assumptions there is no demand when inventories reach zero so the analysis can ignore lost sales.
In a related paper, Mandel and Phaujdar (1989) developed inventory policies for the case of deterio- rating goods with a demand rate that is linearly dependent on current inventories. Unlike Baker and Urban they allowed for shortages. Datta and Pal (1990) extended the work of Baker and Urban to the case where the demand rate declines with inventories to a certain level and then becomes constant. Finally, Urban (1990) has corrected the analysis of Datta and Pal by recognizing that inventories do not necessar- ily fall to zero at the end of the cycle under the optimal policy.
In a separate line of research an attempt has been made to explicitly model the loss of goodwill that arises from stockouts. Schwartz (1966, 1970) was the first to attempt to go beyond the usual assump- tion of a fixed penalty cost for stockouts. In his work the effect of a stockout is felt in the future buying behavior of the customer. He coins the term per- turbed demand for this phenomenon, since stockouts perturb the underlying demand for the product. In the later paper he formalized the concept in a model
where the ratio of demand with stockouts to demand without stockouts depends on two parameters, one measuring the proportion of stockouts to total de- mand and the other the rate at which customers forgive the seller for the stockout (and return to previous buying behavior). The analysis focuses on the steady-state, where customers have adapted their buying behavior to the stockout proportion chosen by the retailer.
Hill (1976) has generalized this work by including ordering and holding costs and shows that the opti- mal lot size reorder point policy is either to order the EOQ and allow no stockouts, or to order an amount equal to available capacity. Caine and Plaut (1976) developed analogous results for the continuous re- view, deterministic demand case. They then studied a stochastic demand, periodic review model in which demand is exponentially distributed with a mean value determined by the proportion of stockouts. For future reference, we note that in this model the standard deviation of demand increases at the same rate as the mean, since the coefficient of variation (cv) of the exponential distribution is always one.
Robinson (1988) further generalized this work by analyzing a periodic review model with non-sta- tionary demand in which both the mean and variance of demand change over time in response to the service level. He assumes a linear relationship be- tween mean demand and the numbers of satisfied and dissatisfied customers in the previous time pe- riod. In this formulation, if service improves both the mean and variance of demand increase. The general form of this model does not yield a stationary policy. Special cases that do result in stationary solutions include the constant variance and the constant cv cases. Robinson shows that the optimal policy is bounded by these special cases.
Ernst and Cohen (1992) have utilized a demand formulation similar to Robinson's in an analysis of stock control policies in a two-level distribution sys- tem where either entity in the system may be domi- nant. They assumed that the increase in demand is proportional to the increase in service level. Since their analysis focuses on mean values they do not explicitly model the behavior of the variance of demand with changes in service level.
Ernst and Powell (1995) developed this model further in order to analyze optimal order-up-to inven-
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tory policies when both the mean and variance of demand are sensitive to service levels. Their basic model for the mean and standard deviation of de- mand as a function of the service level takes the form:
B,(SL) = (1 + ol(SL - SLo)),u, o,
= ( 1 +/32, (SL - .
Here the parameter a measures the increase in mean demand that results from a one percentage point increase in the service level, /3 measures the relative variability of the additional demand that is created by an increase in service level, and /-~o and o" o are the mean and standard deviation of demand at the initial service level SL o.
One of the interesting features of this model, which is derived from first principles of consumer behavior, is that there is no combination of parame- ters for which the cv is constant, which is a common assumption in existing models in the literature. When the variability parameter/3 is 1.0, so that additional demand has the same variability as the original demand, the cv of demand declines (slowly) with an increasing service level. When additional demand is twice as variable as original demand the cv first increases and then decreases with the service level; when additional demand has no variability (/3 = 0) the cv declines sharply with the service level. Ernst and Powell discovered, however, that their model with /3 = 1.0 behaves in essentially the same manner as a constant-cv model, at least for low to moderate values of a. In fact, both the optimal service level and the resulting profit are relatively insensitive to /3 while being quite sensitive to ct.
We turn now to a review of studies of the inte- grated inventory problem. The approach taken by Monahan (1984) in studying the seller's choice of an optimal discount schedule is typical of this literature. In this analysis the seller wishes to entice the buyer to increase his existing order size by a certain frac- tion. Monahan assumes the seller must ensure that inventory costs to the buyer do not rise. This solution to the problem of how to model the strategic interac- tion of buyer and seller can be seen as an extreme form of leader-follower model in which the follower satisfices at a preexisting cost level.
Another approach is taken by Chakravarty and
Martin (1988) who developed a joint buyer-seller discount pricing model for periodic review inventory policies. Here the issue of how the parties share the gains from joint optimization is dealt with by assum- ing the existence of a "predetermined shared-sav- ings agreement." Rubin and Carter (1990), in a related article, showed that cooperation between buyer and seller is always superior from the stand- point of cost reduction. However, they do not pro- vide a model of how the cost savings are likely to be shared.
Another recent contribution is due to Joglekar and Tharthare (1990), who examine the determination of lot sizes in a two-level supply system with determin- istic demand. They propose a policy in which the seller requires the buyer to pay the seller's order costs. The seller then lowers its price by the ratio of order cost to the buyer's EOQ. Now, when the buyer optimizes, its order quantity is larger. Joglekar and Tharthare show that this policy has lower system- wide costs than the joint economic lot size approach of Monahan. However, the approach can only be seen as an ad hoc solution to the general buyer-seller coordination problem, in that it does not result from explicit optimizing behavior on the part of the agents involved.
Finally, Ernst (1990) has used the Nash equilib- rium concept in a study of buyer-seller coordination in which the gains are due to improved service and reduced lost sales. Ernst assumes that an increase in service level will increase total profits to the system, and that the seller has a predetermined service level toward which it aims. In line with the Nash concept, he also assumes that neither buyer nor seller has greater bargaining power. The emphasis of the study is on the sequential bargaining process which deter- mines whether the seller achieves its objective of higher service.
Summarizing this literature, we observe that a number of studies have been carried out that deter- mine jointly (or individually) optimal solutions to the integrated inventory problem. Some of these deal with the question of how the parties are likely to share the gains from cooperation by imposing arbi- trary constraints of one sort or another, such as predetermined sharing agreements. However, the lit- erature is weak on analyses of how optimizing be- havior and the relative power of the players will
R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998) 437-450 441
affect the outcome. Here one would expect game theory to be useful, but we find only one application of game theory to this topic.
Our work can be seen as an application of game theory to the integrated inventory problem, in which we tackle directly the problem posed by separate ownership of the two firms and the resulting conflict over the gains from cooperation. We utilize a plausi- ble model of service-sensitive demand, one which is relatively free of ad hoc assumptions. Finally, our approach has the strength that it proposes a credible solution to the problem of manufacturer-retailer co- operation when profits are sensitive to retail service levels.
3. Model formulation
3.1. Assumptions
Our model deals with the simplest two-echelon distribution system, in which a manufacturer supplies a single retailer, and consumer demand is met only at the retail location. Demand comes from two distinct sources. One group of customers (Loyals) will back- order any time they cannot buy from stock, and their future demand for the product is not affected by the frequency with which they must backorder. The remaining customers (Nonloyals), by contrast, are sensitive to the service level provided by the retailer; if they find the retailer out of stock they will go without and will not backorder.
The retailer uses a periodic review, base stock inventory policy, in which it sets an order-up-to level that covers both mean demand and a safety stock to cover demand variability. The safety stock is deter- mined by trading off the costs of holding inventory, of backorders and lost sales.
Finally, the manufacturer has full knowledge of the characteristics of consumer demand and the in- ventory policy of the retailer. It also has available to it a financial incentive mechanism through which it can reward the retailer for increasing its service level. The manufacturer's incentive to use this mech- anism is to capture some percentage of the sales revenue that would be lost to other manufacturers as a consequence of the retailer choosing an initial service level below 100%.
3.2. Consumer demand
We pointed out in the introduction that if con- sumers always backorder and there are no long-run effects on demand due to loss of good will, the manufacturer will have no incentive to change the retail service level. This is because all units de- manded will eventually be sold, even if backordered. In order to provide a simple but plausible model for demand sensitivity to service levels we assume con- sumer demand arises from two distinct groups: Loy- als, who always backorder, and Nonioyals, who never do. It is the latter group whose effective demand will increase if the retailer holds more inventory and thereby improves its service level.
For demand from the Nonloyals we adopt the service-sensitive demand model of Ernst and Powell (1995). The mean of total demand is the sum of mean demand from the Loyals and mean demand from the Nonloyals, the latter being a function of the service level. The variance of total demand, simi- larly, is the sum of the variances of demand from each group. Thus we have
/XT(SL ) = /x L +/XNL
= / z L + [1 + o~(SL - SLo) ] /.tNL(SLo),
( l )
= +
= o "2 + [1 + fl2a(SL - SLo)] o'r~L(SLo).
(2)
SL o represents a pre-existing service level, at which demand from the Nonloyal group has mean /,tNL(SL 0) and variance tyN2L(SLo ), against which the effects of raising service level can be measured.
3.3. Retailer inventory decision
The retailer in our model acts as a follower to the manufacturer, which means that it takes as given a set side payment for service when it determines an optimal service level SL*. The resulting order-up-to level Z will balance the revenues (which themselves depend on SL both through sales to consumers and through the side payment from the manufacturer) with the costs of holding inventory and the costs of
442 R. Ernst, S.(7,. Powell / European Journal of Operational Research 104 (1998) 437-450
backorders for Loyals. Note that Z is itself a func- tion of SL; the higher the service level desired the higher will be the order-up-to level.
Let PR denote the retail unit profit margin, C B and C n the unit backorder and holding costs per cycle, respectively, and W the marginal side pay- ment value, i Further, let the ratio of backorders to total sales be b(SL), and f ( - ) the demand density. Finally, SL 0 is the optimal service level chosen by the retailer prior to the introduction of the side payment incentive scheme. Then we can write the retailer's objective as
max PR (SL)
[ /zT(SL ) - (1 - b (SL) ) X
X - Z ( S L ) ) f ( fz(sL)( X x ) d x ]
- b(SL)C,jz(sL)( x - Z ( S L ) ) f ( x ) d x
- Cn f0Z(SL) (Z (SL) - x ) f ( x ) dx
+ W(SL - SLo), (3)
where the first term represents the retailer's revenue net of lost sales, the second and third terms represent backorder and holding costs respectively, and the last term represents the manufacturer's side payment. The result of this optimization on the part of the retailer is the reaction function SL* (W). Note that as the follower the retailer does not take into account the effect changes in SL might have on the manufac- turer's choice of W, but takes W as a fixed parame- ter.
3.4. Manufacturer determination of side payment
The manufacturer's problem is to choose the opti- mal side payment rate W, given a wholesale profit
i We assume for simplicity in our basic model a linear side- payment scheme: i.e., the manufacturer offers the retailer a fixed dollar amount for every percentage point the retailer increases its service level above the pre-existing level. In Section 4.2 we will present results on alternative side-payment arrangements.
margin PM. This problem is complicated by the effects that changes in W have on the choice of service level SL made by the retailer. As the leader in our model, the manufacturer will take the reaction of the retailer into account when it chooses W. If we write the retailer's optimal service level as SL* (W), we can write the manufacturer's objective function as
max PM {W}
[ p~T(SL* ( W ) ) - (1 - b(SL* ( W ) ) ) X
×
- W[SL" ( W ) - SL0]. (4)
3.5. Solution procedure
The solution to this leader-follower game is a pair (W*, SL* (W*)) that simultaneously optimizes (3) and (4). Although both the manufacturer and retailer face what are essentially Newsvendor prob- lems, no closed-form solution is available to the joint optimization problem resulting from a Von Stackel- berg game model. Therefore, we use numerical methods to develop solutions and to conduct sensi- tivity analysis. The solution process was carried out in Mathematica (Wolfram, 1991; model available from authors). In what follows we assume that de- mand follows a normal distribution; similar methods could be used with any demand distribution.
Our solution procedure involves first determining SL 0, the optimal retail service level with no side payment, and then determining W * (SL*), the opti- mal side-payment rate for the manufacturer (when the retailer optimally chooses SL* in response).
In the first step, we determine the optimal service level (and profit) for the retailer in the absence of any side payment from the manufacturer. This is done using a simple grid search over a range of values for SL from, say, 0.70 to 1.00. Since the Newsvendor problem is concave (Nahmias, 1989) we can be assured of finding the true optimum within the accuracy of our grid. Given the complex dependence of the retailer's objective function on
R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998) 437-450 443
SL, we carry out this search in stages. For each value of SL we first calculate /xr(SL) and t r r (SL) using Eq. (1) and Eq. (2). Then we determine the ratio of backorders to total sales:
b ( S L ) = / X L / / X r ( S L ) .
Next we solve for the required safety stock k(SL) that will give the appropriate orderTup-to-level Z, using the implicit relationship (see Silver and Peter- son, 1985):
C.(kl T(k) S L ( k ) = b ( k ) 1 t x r ( k )
+( l -b (k) ) re(k) "
(51
In Eq. (5),
G , ( k ) = ~ ( u - k l g ( u ) d u
and g(u ) is the standard normal density. The order-up-to level is then Z ( S L ) = , t t r ( S L ) +
k(SL)t r r (SL) . Finally, we can evaluate the retai ler 's objective function (3) for the given value of SL. This procedure gives us the optimal service level SL 0 for the retailer before the manufacturer attempts to alter the retai ler 's behavior through use of a side payment.
The second step in solving this problem is to determine the manufacturer ' s optimal side payment W *. This requires a nested loop: an outer loop in which we search for W *, and an inner loop in which we search for the optimal retail service level S L ( W ) for each value of W. Within the inner loop, that is for a fixed W, we optimize retailer profit using the same sequence of calculations we used to determine SL 0. This procedure mimics the Von Stackelberg game model, in which the leader finds an optimal choice by taking into account the optimizing re- sponse on the part of the fol lower to the leader ' s choices. This procedure is again guaranteed to find the true opt imum because the Newsvendor problem is concave.
We stress that the starting point for all our calcu- lations, and for the numerical results that will be presented next, is the assumption that the retailer is at a prof i t -maximizing posit ion (SL 0) before the
Table 1 Base case parameters
PM = 20 a = 0,I0 PR = 10 /3 = 1.0 C a = 8 /x r = 100 C n = 80 o" r = 30 b= 0.5
imposition of the manufacturer 's side payment incen- tive.
4. Numerical examples
4.1. Base case
In this section we use a variety of numerical examples to illustrate the operation of our model and its implications for manufacturer and retailer behav- ior. We begin with a base case defined by the parameters defined in Table 1.2
We assume at some arbitrary initial time that half the population backorders and that mean demand is 100 units per period, with a standard deviation o f 30 units. Since the population is split evenly between Loyals and Nonloyals, each orders 50 units with a standard deviation of 3 0 - 2 t - i/2). The retailer sets
its initial (non-optimal) service level at 0.6. If the service level is increased, mean demand will increase by 0.1 units per percentage point increase (ct = 0.10), and the variabili ty of this incremental demand will be the same as in the initial population (/3 = 1.0).
The first step in our analysis is to determine the optimal service level for the retailer before the man- ufacturer institutes its side payment incentive scheme. A search over the retai ler 's service level reveals that
2 These parameters are chosen primarily to ensure the manufac- turer has an incentive to use the side-payment scheme. For example, holding costs are assumed to be high so that the retailer will choose a relatively low service level; if the retailer's service level is initially high there is little incentive for the manufacturer to increase it. Likewise, the manufacturer's profit margin is higher than the retailer's to give the manufacturer a significant incentive to improve service. It is an interesting empirical question, al- though beyond the scope of this paper, to ask what types of firms have this cost structure and can therefore consider the policy described here.
444 R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998) 437-450
the optimal service level is 0.720, which is achieved with a safety factor (k 0) of - 0 . 9 7 0 standard devia- tions of demand. (Note that the safety factor is negative in this case because of the high relative cost of holding inventory.) Revenues from Loyals are 500, while revenues from Nonloyals are 727 (the higher revenues from Nonloyals are due to the in- crease in the service level, which induces higher Nonloyal demand), Retail profits are 730, while profits of the manufacturer are 2454.
Next we determine the optimal side payment from the manufacturer. We assume that the retailer oper- ates initially at its profit maximizing service level of 72%. Searching over the side payment rate W (while optimizing Z for each W) reveals that the manufac- turer optimizes its profits at W * = 8, which induces an increase in retail service level to 0.734. Retail profits increase to 832, an increase of 14%; manufac- turer profits increase to 2483, an increase of 1%. The total increase in profit is 131, of which the manufac- turer receives 22%. A detailed comparison of manu- facturer and retailer profits and service before and after the side payment is given in Table 2.
In Fig. 1 we show in more d&ail how the manu- facturer's choice of side payment rate influences profits and service levels. Fig. l(a) shows how the service level chosen by the retailer varies for side payment rates up to $20 per percentage point. In this case the service level chosen by the retailer in the absence of a side payment is 72%. As the manufac- turer increases the side payment rate the retailer can afford to increase safety stock and thereby raise the service level. Over the range of side payment rates shown here the relationship between service level and side payment is nearly linea r, with a hint of diminishing returns. Now the costs to the retailer of
Table 2 Base case results
Without side Optimal side Difference payment payment
Manufacturer Profit ($) 2454 2483 Side payment rate ($) 8
29
(a) 0.755
0.75
0.745
0.74
0.735
0.73
0.725
0 .72 ,
0.715
0.71
0.705
0.7
Manufac turer '= Incentive
(b) 1050.00
1000.00 •
9 5 0 0 0 .
900.00 -
850.00 -
8¢0.00 -
750.00 -
700.00 -
650.00 -
600.00 -
Manu fac tu re r ' s Incentive
(C) 2485.oo.
2480.00 •
2475.00 •
2470.00 . a=
2465.00
i 2460.00.
2455.00 ;
2450.00
2445.00 •
2440.00 -
2435.00 o ~ ® ~ ~
Manufacturer's Incentive
(d) 350ooo
3450.00
3400.00
3350.00
_ 3300.00
L 325000
~ 3 L ~ . ~
3100.00
3050.00
3000.00
Manufacturer's Incentive
Fig. 1. (a) Service level; (b) Retailer's profit; (c) Manufacturer's profit; (d) Total profit.
increasing the service level themselves increase with the service level because larger and larger incre- ments of safety stock are needed to achieve a given increase. It stands to reason, then, that as the manu- facturer raises the side payment rate the retailer will be able to afford a smaller increase in the service level.
Fig. l(b) shows how retailer profits vary with the side payment rate. We would expect retailer profits to increase with the side payment, since the retailer optimizes for any level of side payment. What is particularly interesting in the figure is that retailer profits increase at an increasing rate. This is a result of two factors: the nature of the side payment mech- anism, and the amount of safety stock needed to achieve a given service level. We pointed out above
R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998) 437-450 445
that successive increments in the side payment rate bring about decreasing increments in the service level. But an increase in the side payment rate (paid per unit of increase above a base level) increases retailer profits not only on the resulting increase in service level, but also on all previous increases. For example, assume the retailer initially has a service level of 90% and the manufacturer has offered a $5 side payment which brings the service level to 95%. If the manufacturer raises the rate to $6, retailer profits increase for two reasons: the retailer can afford to raise the service level once again, and the side payment rate on the earlier 5 percentage point increase has itself increased.
We see in Fig. l(c) that manufacturer profits initially increase with the side payment, but eventu- ally decrease as the side payment rate increases. This is because successive increments in the side payment rate bring about progressively smaller increases in service levels and manufacturer profits. And increas- ing the side payment factor increases the total trans- fer from manufacturer to retailer at any final level of service.
Finally, Fig. l(d) shows total profit for the manu- facturer and retailer together as a function of the side payment rate. Interestingly, total profits increase steadily over this range of side payments. This is a consequence of the particular parameters used in this illustration, since the successive increments in re- tailer's profits exceed the corresponding decrements in manufacturer's profits. In other cases, in particular when a is low, total profits are concave in the side payment.
The results of this single numerical example do not, of course, generalize for all parameter values. To illustrate this point, we show how the relative
gains to manufacturer and retailer vary as a function of one parameter, the manufacturer's margin PM- Table 3 summarizes the relative changes in profit due to use of the side-payment scheme for manufac- turer and retailer for values of PM from 10 to 100.
Whereas the manufacturer gains 22% of the total gain in profit from use of the side-payment scheme in the base case (PM-----20), it gains 32% when the margin is $50 and 55% when it is $100. Thus the relative shares of each party in the gains are sensitive to the underlying parameters.
4.2. Sensitivity analysis
In this section we report the results of a series of tests of our model under different assumptions as to the make-up of market demand. The purpose of these tests is to understand how the interaction between manufacturer and retailer varies with changes in the parameters of the market. Five key factors in our model are studied: the increase in mean demand due to an increase in service (c t), the standard deviation in the new demand induced by an increase in service (f l) , the initial standard deviation of demand (try), the proportion of backorders (b), and the specifica- tion of the side-payment scheme. Table 4 summa- rizes the results for the first four factors, both for the initial optimal retailer service level and for the situa- tion after the manufacturer offers an optimal side payment. In order to interpret these results properly, note that changes in the underlying parameters nec- essarily change the initial optimal service level for the retailer. For example, the higher the parameter a the higher will be the service level chosen by the retailer before the imposition of the manufacturer's side payment.
Table 3 Sensitivity of base case to manufacturer's margin
PM Additional Additional Manufacturer retail profit manufacturer percent of
4.2.1. Sensitivity to a We begin with the sensitivity to c~, the increase in
mean demand as a function of the retailer's service level. In the base case, a = 0.1, the optimal service level is 0.720 and retailer profits are 730. As we vary a from a low level of 0.01 to a high level of 0.2 we see that the retailer's service level varies within a narrow range, from 0.68 to 0.72. Retailer revenues vary dramatically, from a low of 839 to a high of 1596. However, these increases in revenues are sub-
446 R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998) 437-450
stantially offset by increases in ho ld ing costs, which
rise f rom 145 to 562. Backorder costs, on the o ther
hand, remain essential ly constant s ince the popula-
tion o f Loya l s is also constant. The net result is that
retai ler profi ts vary f rom a low o f 545 to a high o f
893. Manufac ture r profits also range wide ly ove r this
range o f parameters , f rom a low o f 1678 to a high o f
3192.
The opt imal side payment for the base case is 8.
O v e r this range o f values o f a we find that the side
payment ranges f rom a low o f 4 to a high o f 15. The
resul t ing service level ranges f rom 0.706 to 0.734,
whi le retai ler profits range f rom 582 to 1083 and
manufac ture r profi ts f rom 1689 to 3262. In general ,
at low levels o f a we find that the opt imal manufac-
turer side payment is low, retai ler profi ts increase
modes t ly (7%), and manufac ture r profi ts increase by
a negl ig ib le amount . Fo r high values o f ct, in which
case the Nonloya l populat ion is h ighly sensi t ive to
service levels , the manufac tu re r ' s side payment is
quite high, retai ler profits increase substantially
(21%), but manufac ture r profi ts cont inue to show
only a modes t increase (2%).
4.2.2. Sens i t i v i t y to
In the base case we have assumed that the popula-
tion induced to purchase because o f an increase in
the service level has the same variabil i ty in demand
as the original populat ion ( f l = 1.0). In this set o f
tests we vary this parameter f rom a low value o f 0.0
to a high value o f 2.0.
Table 4 shows that the initial opt imal service and
profi t levels are highly sensi t ive to this parameter.
The service level varies f rom 0.915 to 0.594, whi le
retail profi ts vary f rom 1957 to 515 and manufac-
turer profi ts f rom 4772 to 1487. In general, low
levels o f fl lead to high service levels and high
profits, whi le high levels lead to low levels o f all
three factors. W h e n fl is low, the result o f increasing
the service level is to bring in more demand (at a
Table 4 Summary of sensitivity results
SL o k o Ret~lerprofit Manufactu~rprofit W ° SL * k * Retailer profit Manufacturerprofit
Base case 0.720 -0.970 730 2454 8 0.734 - 0.889 832 2483
R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998) 437-450 447
rate given by a ) without any additional variability. Thus no additional safety stock is required, so there is no additional holding cost to offset the increase in revenues.
The optimal side payment rates for these five cases range from 0 to 40. At low levels of /3 the retailer is already setting a high service level, and there is no incentive for the manufacturer to raise the level. At high levels of /3, the reward to the manu- facturer of inducing an increase in service levels is high, so a high rate is chosen, and a significant increase in manufacturer profits results. For example, when /3 = 2.0, the manufacturer offers a side pay- ment of 40, the retailer raises the service level slightly and retailer profits increase by a negligible amount, but manufacturer profits increase by a fairly significant amount (49, or 3.3%).
4.2.3. Sensitivi~ to trr In the base case, the demand variability in the
underlying population was assumed to be 30 (rela- tive to mean demand of 100). We test the sensitivity of our model to this parameter by varying it from a low of 10 to a high of 50. The results in Table 4 show that both the initial and final solutions are sensitive to this parameter. At a low level of variabil- ity, the retailer initially sets a high service level and enjoys high profits due to high revenues with low holding costs. The manufacturer also enjoys high profits. As the variability increases, the optimal ser- vice level falls and so too do profits to both retailer and manufacturer.
When we examine the results of the manufac- turer's choice of side payment, we see that low levels of variability lead to no side payment, because service is already high, but high levels of variability lead to high side payments. In the case where tr r = 50, the optimal incentive is 40, which increases retailer profits by 73% and manufacturer profits by 14%.
4.2.4. Sensitivity to b Next, we analyze the sensitivity of the model to
the initial distribution of the population between Loyals, who backorder, and Nonloyals, who do not backorder but whose demand is sensitive to the service level. In our base case this parameter was set at 0.5, splitting the population evenly. In Table 4 we report on the cases where b = 0.25 and 0.75 as well.
Before examining these results, we should explain how the parameter b is used to allocate initial mean demand and demand variability. We define b to represent the proportion of initial demand made up of Loyals. Thus for any value of b we have
/x L = 100. b,
/XNL = 100- (1 -- b).
Also, in order that total demand variability maintains a standard deviation of 0- T, we apportion demand variability according to
O" L = O" T . b l / 2 '
~rNL = o 'T" (1 - - b) v2. The initial situation, in which the retailer chooses
an optimal service level without a side payment, proves to be quite sensitive to the parameter b. The higher the proportion of Loyals, the lower the retailer sets the service level and the lower the resulting profit levels to both retailer and manufacturer. This may appear counterintuitive, but it is Nonloyals who are the target of the manufacturer: the higher the proportion of Nonloyals the larger the population that can be induced to buy with higher service levels and side payments.
4.2.5. Sensitivity to the form of side-payment The side-payment scheme we have studied, in
which the manufacturer offers to pay the retailer a fixed dollar amount for each percentage point in- crease in the service level, is only one of many possible arrangements. Our formulation has the ad- vantage of simplicity in what is already a complex model. However, it is interesting to consider how the equilibrium between manufacturer and retailer might change if a different side-payment arrangement were in place.
To investigate this question we considered alter- native side-payment formulations in which the man- ufacturer pays W for some power of the increase in service level. Specifically, we determined the opti- mal level of W when the side-payment formula was W(SL - SLo )1/2 and W(SL - SL0 )2. The results are summarized in Table 5.
These results show that retail and manufacturer profits can be sensitive to the side-payment arrange- ment adopted. It would appear from these limited
448 R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998) 437-450
Table 5 Sensitivity to side-payment scheme
Incentive W ' SL ' Reta i l Manufacturer profit profit
cases that both player's profits increase with the power of the side-payment scheme, but that the optimal side-payment rate W * first increases and then decreases. While these results are interesting, the larger question of determining not just the opti- mal W but the optimal side-payment function itself is considerably more difficult and beyond our scope here.
5. Implementation issues
The primary purpose of this paper is to investigate the implications of a joint arrangement between manufacturer and retailer over service levels. Our purpose has not been to determine which firms in reality have the characteristics necessary for such an arrangement, nor to discuss how such an arrange- ment could be implemented. However, a brief dis- cussion of important implementation issues is war- ranted.
In our model we have assumed away all costs related to implementing the side-payment arrange- ment. As discussed earlier, this is because these costs are likely to depend on factors specific to the firms involved, and not to depend on the choices we study here, which involve side-payment and service levels. However, the direct costs of implementation could be substantial for both manufacturer and retailer, as could the costs of monitoring compliance. We be- lieve that our analysis of the benefits from this form of cooperation will help motivate firms to investigate the practicality of implementing such a scheme, in- cluding a determination of costs. We should point out, however, that point-of-sale systems have drasti- cally reduced the costs associated with monitoring inventory, so that schemes such as the one we pro- pose, which rely on close monitoring of service levels, may be increasingly practical.
Another assumption we make is that demand from Loyais and Nonloyals is known to both manufacturer and retailer. This is clearly a strong assumption, although necessary in formal analysis of this type. In reality, mean demand (rather than observed sales) can be difficult to determine even for loyal cus- tomers. Our analysis also relies on knowing the variability in demand. This does not imply, however, that an incentive scheme could not work if the probability distribution of demand were not fully known. Our analysis shows, for example, that an incentive scheme is particularly attractive when Non- loyal demand is both sensitive to service level and has a low variance. One would expect that retail managers would have a reasonable intuitive sense of whether these conditions hold for their products. Given judgmental estimates of these parameters for a variety of products, a motivated manufacturer could undertake some market experiments by offering side-payments and observing the resulting service levels and sales. Thus, our results could be useful to a manufacturer in pointing out the conditions under which it might be advantageous to take action to improve retail service levels.
6. Summary and future research directions
In this paper we have studied the integrated inven- tory problem from a new perspective. Rather than imposing an arbitrary profit-sharing mechanism on the independently-owned manufacturer and retailer, we allow the retailer to choose optimal safety stocks taking into account its inventory holding and stock- out costs, and we allow the manufacturer to offer the retailer a financial incentive to raise service levels above the individually optimal level in order to capture a portion of the lost sales revenue for both manufacturer and retailer. The specific coordination mechanism we propose is the leader-follower game, which we believe is an appropriate concept since the retailer manages thousands of items while the manu- facturer sells relatively few. Another innovation of our approach is an explicit segmentation of con- sumer demand into Loyals who always backorder and Nonloyais whose unmet demand represents lost sales.
Our results show how the nature of consumer
R. Ernst, S.G. Powell / European Journal of Operational Research 104 (1998)437-450 449
demand and the relative power of manufacturer and retailer interact to determine the level of side pay- ment the manufacturer will offer the retailer and the resulting service level and profits. Sensitivity analy- ses show that the resulting service level and profit levels are highly sensitive to the initial variabili ty of demand and to the variability of demand from Non- Ioyals induced to purchase by an increase in the service level. The results are less sensitive to the parameter that measures the sensitivity of mean de- mand to service levels, and to the proportion of the population that backorders. In some of our illustra- tive cases the benefits of a side payment incentive appear to accrue primarily to the retailer, despite the fact that the manufacturer is the leader in the l eader - fo l lower game.
Two areas for future research seem particularly interesting. One is the question of the dynamics of lost sales, specifically, how the entire distribution of demand varies with service levels. We have analyzed a relatively tractable case in which demand is nor- mally distributed and the mean and standard devia- tion change with the service level in a straight-for- ward manner. Whether our results are sensitive to this portion of the model is an open question.
A second area of interest concerns the coordina- tion method between manufacturer and retailer. We have used a l eader - fo l lower framework, which at- tributes a higher level of rationality to the manufac- turer. Our results show, surprisingly, that a prepon- derance of the gains from coordination may go to the retailer. They also show still higher gains available from joint optimization. This raises the question of what determines the relative gains to the retailer and manufacturer from participating in this game. It also raises the question of when firms will follow a l eade r - fo l lower model, when they will effectively integrate (jointly optimize), and when they will fol- low other behavioral patterns.
This research also has interesting implications for empirical research and the practice of management of inventories. Our results suggest that substantial savings could be realized from increasing retail ser- vice levels in situations where demand is sensitive to the service level and demand variabil i ty is signifi- cant. Empirical research is needed to determine in which situations demand is l ikely to be service-sensi- tive, and what its probabil ist ic characteristics are.
This information would then naturally suggest situa- tions in which the manufacturer might consider of- feting an incentive to the retailer to increase service levels. Detai led case studies would be necessary to determine both the benefits and the costs of such an incentive scheme.
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