IIE Transactions (1998) 30, 715±721
On oering economic incentives to backorder GREGORY A. DeCROIX1 and ANTONIO ARREOLA-RISA2 1
Fuqua School of Business, Duke University, Durham, NC 27708-0120, USA E-mail:
[email protected] 2 Department of Information & Operations Management, Lowry Mays College & Graduate School of Business, Texas A&M University, College Station, TX 77843-4217, USA E-mail:
[email protected] Received February 1997 and accepted December 1997
We study a strategy to manage demands that occur when an inventory system is temporarily out of stock: oer the customer facing the unsatis®ed demand an economic incentive to backorder. We explore the bene®ts of this inventory management strategy by analyzing a model of an inventory system with stochastic demand and random supply disruptions, where the probability that a customer facing an un®lled demand will backorder (as opposed to becoming a lost sale) can be in¯uenced by an economic incentive. Our results provide several insights regarding this inventory management strategy and suggest that the bene®ts of oering backorder incentives can be signi®cant.
1. Introduction The classical approach to managing a stochastic-demand inventory system involves identifying the inventory policy that strikes an optimal balance among inventory holding costs, ordering costs and shortage costs (or service levels). In modeling such systems, assumptions are usually made regarding, among other things, the behavior of customers who face a shortage when they arrive. Demands during a shortage are generally assumed to result in either lost sales or the placing of backorders for the product. Some work has pursued the more realistic situation of partial backorders, in which some demands result in lost sales while others result in backorders. However, even in this instance, the partial-backorder behavior is usually assumed to be an exogenous characteristic of the system beyond the control of the decision maker. In practice, however, when organizations ®nd themselves nearing a shortage situation, they frequently oer customers economic incentives to place a backorder rather than risk losing sales. By expanding our concept of an inventory-management policy to include not only stocking levels but also customer incentives to backorder, we should be able to obtain policies that yield a lower total cost of operating the inventory system. This is particularly true for systems where truly exogenous factors beyond the control of the inventory manager ± such as perishable products or unreliable supply ± make it dicult to precisely control the inventory level. In this paper we explore the potential bene®ts of offering economic incentives to backorder as a strategy for 0740-817X
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1998 ``IIE''
inventory management when the system involves an unreliable supply. In particular, the paper addresses the following research questions: (1) when is it optimal to oer backorder incentives? (2) if it is optimal to oer backorder incentives, what is the optimal amount of the incentives? and (3) what are the potential cost savings to be achieved by oering backorder incentives? Previous research on stochastic-demand inventory systems with unreliable supply has focused on solving the traditional inventory-replenishment problem with no allowance for backorder incentives [1±7]. Inventory systems with a perfectly reliable supply, that incorporate partial backorders have been investigated [8±13] but again attention was restricted to solving the inventory-replenishment problem. Assuming a reliable supply, there have been a number of dierent approaches to responding to the problem of shortages. Moinzadeh and Nahmias [14], Cohen et al. [15] and Zhang [16] have analyzed situations where there are two or more supply modes ± e.g., a regular and an expedited mode ± that can be used to replenish the on-hand stock. Moinzadeh and Ingene [17] have explored the use of competitive pricing on an alternate product to encourage customers facing a shortage to substitute this other product for the one originally demanded. Cheung [18] has proposed an explicit economic incentive to persuade customers to backorder. Whenever the on-hand inventory falls below a given level and the time until the next delivery is greater than a prescribed threshold value, a discount is oered to all customers who agree to backorder until the next delivery. It is assumed that the delivered quantity is large enough
716 to ®ll all backorders, and that the time until the next delivery is known. The discount oered and the willingness of customers to backorder are independent of the wait time until the next delivery. Due to the unreliable nature of the supply in our setting, the amount of time remaining until the next delivery is random. As a result, the policy discussed in Cheung [18] cannot be implemented in such a system, since that policy relies heavily on the speci®c arrival time of the next delivery. Therefore, to the best of our knowledge, this paper is the ®rst to consider oering backorder incentives of a type that can be implemented in a system with unreliable supply. The contents of the paper are as follows. In Section 2 a mathematical model of the problem under study is presented. In Section 3 we focus attention on the special case where backorder incentives are not oered until a shortage occurs, and we derive optimization results for this case. Section 4 presents the results of numerical trials exploring the bene®t of oering backorder incentives, and estimating the fraction of that bene®t that can be captured when incentives are not oered until a shortage occurs. Section 5 contains a summary of our results and suggests further research directions.
2. Cost model Consider an inventory system where demand for the product is assumed to follow a Poisson process with average demand rate a. The product's supply is unreliable ± at random points in time the supply becomes unavailable, and remains unavailable for a period of random duration. We assume that both the interarrival time between successive supply disruptions and the durations of successive supply disruptions are independent exponentially distributed random variables with means 1=k and 1=l, respectively. When the supply is available in inventory systems with stochastic supply disruptions, the transportation leadtime tends to be small compared to the average length of a supply disruption. To re¯ect this, and to simplify the analysis, we assume that when the supply is available the inventory-replenishment leadtime is negligible. Costs incurred by the system include a ®xed ordering cost k, a holding cost h per unit per unit time, a lost sales cost p per unit, a backorder cost p0 per unit, and a backorder cost p1 per unit per unit time. In traditional inventory models, the cost parameters p0 and p1 are assumed given, and may include monetary costs such as record-keeping, as well as nonmonetary costs such as loss of customer goodwill. The natural ways of oering an economic incentive for a customer to backorder ± e.g., providing a one-time discount or coupon (as in Cheung [18]), or oering a discount based on the amount of time the customer waits ± can be modeled by making p0 and p1
DeCroix and Arreola-Risa decision variables under the control of the inventory manager. Consistent with previous research, when no economic incentives are oered to customers, a backorder will still result in some unavoidable backorder costs, and we will denote these costs by p0 and p1 . Therefore the backorder incentives are represented by p0 ÿ p0 and p1 ÿ p1 , respectively. For notational simplicity we will work with p0 and p1 instead of p0 ÿ p0 and p1 ÿ p1 , and where no confusion arises we will refer to both pairs as backorder incentives. Due to the unreliable nature of the supply source, inventory is managed according to the following modi®ed (s,S) policy: when the inventory level drops to s and the supply is available, order the necessary amount to bring the inventory level up to S. In deriving and analyzing the cost function we will ®nd it convenient to work with s and D S ÿ s instead of s and S. We will refer to the policy parameters s and D as the inventoryreplenishment parameters. If the supply is unavailable when the inventory level reaches s, the decision maker may wish to begin oering backorder incentives to avoid lost sales. To accommodate this, the modi®ed (s,S) policy is extended to include an additional policy parameter r, with 0 r s, such that once the on-hand inventory drops to r every customer arriving before the next delivery is oered a backorder incentive. We will refer to the policy parameters p0 , p1 and r as the backorder-incentive parameters. Let q
p0 ; p1 be the backorder-response function ± i.e., q
p0 ; p1 represents the probability that a customer who is oered backorder incentives
p0 ; p1 will choose to backorder. Assume that q
p0 ; p1 is nondecreasing on p0 p0 and p1 p1 . To simplify notation, when no confusion results q
; will be denoted by q. Let K denote the expected ordering, holding, backordering and lost sales cost per unit time. Given backorder-incentive parameters
p0 ; p1 ; r and inventoryreplenishment parameters
s; D, we have: K
p0 ; p1 ; r; s; D kE
N hE
OH p0 E
BR p1 E
BL pE
LS;
1
where E(N) = expected number of orders placed per unit time; E(OH) = expected on-hand inventory level; E(BR) = expected number of units backordered per unit time (backorder rate); E(BL) = expected backorder level; E(LS) = expected number of lost sales per unit time. Although the existence of backorders in the system could lead to an optimal s < 0, it seems that such instances would be rare in practice. As a result, we restrict s to be nonnegative. Also, due to the integrality of demand, we assume that r, s and D are integers and that D 1.
On oering economic incentives to backorder
717
Therefore the problem faced by the decision maker is to choose the vector of policy parameters
p0 ; p1 ; r ; s ; D that solves the following problem. Minimize K
p0 ; p1 ; r; s; D; subject to p0 p0 ; p 1 p1 ; r s; r 0; integer; s 0; integer; D 1; integer:
2
We will often ®nd it convenient to represent the problem de®ned in (2) as two mutually dependent subproblems. Given any feasible values of the backorderincentive parameters
p0 ; p1 ; r, de®ne the problem of ®nding the optimal s and D as the inventory-replenishment (IR) problem. Similarly, given any feasible values of the inventory-replenishment parameters
s; D, de®ne the problem of ®nding the optimal p0 , p1 and r as the backorder-incentive (BI) problem. Formal statements of the (BI) and (IR) problems are given below. (BI)
Minimize p0 ;p1 ;r
K
p0 ; p1 ; r; s; D;
subject to p0 p0 ;
Fig. 1. Inventory level transition-rate diagram.
p 1 p1 ;
sÿr q baq E
BR E
N : l sÿr q baq ; E
BL E
N l2
r s; r 0; integer:
IR
Minimize K
p0 ; p1 ; r; s; D; s;D
and
subject to
" #
1 ÿ qr1 qs ba E
LS E
N ;
1 ÿ qqr l
s r; integer; D 1; integer: The transition-rate diagram for the continuous-time Markov chain representing the inventory level of the system is shown in Fig. 1. By deriving the steady-state probabilities for this Markov chain, we can develop an expression for each term of the expected cost function (1). Derivation of these results is straightforward but tedious, and therefore is omitted. The terms in (1) can be expressed as: al : E
N Dl ab D
D 1 D b E
OH E
N s 2a a l s sÿr baq
1 ÿ q qq ÿ 1 : l2
where
" D # k a and q a=
a l: 1ÿ b kl akl
3. Separability and optimization results: r = 0 The results of the preceding section provide closed-form expressions for the expected cost function K. Unfortunately, due to the complexity of these expressions, solving the (BI) and (IR) problems may be quite dicult. However, by focusing on the special case when r 0 ± i.e., the backorder incentives are not oered until a shortage occurs ± the solution of these problems is greatly sim-
718
DeCroix and Arreola-Risa
pli®ed, and a number of insights into the optimal policy can be obtained. In addition to facilitating the analysis, there are other reasons for focusing on the case of r 0. For example, if the product is directly accessible to customers, or if the customer-response dynamics of the system are such that customers would respond unfavorably to being asked to backorder when there is still some product on hand, it may simply be infeasible to choose a value of r > 0. Another reason to restrict attention to r 0 is the potential complexity of implementing a policy with r > 0 in a practical setting. The reduced costs resulting from allowing the general case of 0 r s may not be suciently large to justify this added complexity. In fact, in Section 4 we present the results of a numerical experiment that suggest that in many cases allowing r to be positive provides very little additional bene®t. For the reasons above, we restrict attention in this section to the case of r 0. As a result, we drop r as a decision variable and suppress it in the notation.
described in (2) ± ®rst solve the (BI) problem, or equivalently the (BI0 ) problem, to obtain the optimal backorder-incentive policy, then take this policy as ®xed and solve the (IR) problem. The next two propositions provide insights into the solution of the (BI) problem. The ®rst of these, Proposition 2, states that backorder incentives should only be oered if the incentives result in an expected cost per backorder that is lower than the cost per lost sale.
3.1. The BI problem
The ease with which the (BI0 ) problem can be solved is clearly dependent on the backorder-response function q
p0 ; p1 ± for a q
p0 ; p1 with no particular structure, solving the (BI0 ) problem may require an exhaustive twodimensional search. However, if q
p0 ; p1 is concave on p0 p0 , p1 p1 , solution of the problem becomes quite simple. This is particularly true if only one type of backorder incentive is oered, i.e., p0 p0 or p1 p1 . In this case, concavity of q
p0 ; p1 in the appropriate dimension (with the other variable ®xed at its lower bound) implies convexity of R
p0 ; p1 in the corresponding dimension. As a result, a simple line search with an easily identi®able stopping rule will suce to solve the problem. Note that this approach not only handles the case when, for practical reasons speci®c to the problem, only one type of backorder incentive is considered, but it also serves to identify `boundary' solutions (i.e., p0 p0 and/ or p1 p1 ) to the (BI0 ) problem when both types of incentives are considered. Even if q
p0 ; p1 is jointly concave, it does not seem possible to analytically establish the joint convexity of R
p0 ; p1 . As a result, when both types of incentive are being considered, the behavior of R
p0 ; p1 on the interior of the feasible region (i.e., p0 > p0 and p1 > p1 ) is not completely predictable. However, based on visual analysis of a number of examples, it seems that R
p0 ; p1 is in fact jointly convex when q
p0 ; p1 is concave. As a result, simple search procedures for ®nding
p0 ; p1 should perform well. In addition, ®rst-order optimality conditions lead to the following characterization of any interiorpoint optimal solution to the (BI0 ) problem.
Although the (BI) and (IR) problems are mutually dependent in general, Proposition 1 below establishes that when r 0 the (BI) problem is independent of the (IR) problem, and hence the search for the optimal backorderincentive policy
p0 ; p1 and its associated inventoryreplenishment policy
s ; D is greatly simpli®ed. In establishing this result it is shown that the (BI) problem is equivalent to the (BI0 ) problem, de®ned as:
BI0
Minimize
subject to
p0 ;p1
R
p0 ; p1 ; p0 p0 ; p1 p1 ;
where
p1
1 ÿ q
p0 ; p1 p: R
p0 ; p1 q
p0 ; p1 p0 l
Proposition 1. The optimal backorder-incentive policy
p0 ; p1 is independent of the speci®c values of s and D: Hence the (BI) problem is independent of the (IR) problem. Proof. By ®xing s and D and dropping terms that are independent of p0 and p1 , simple algebra reduces the (BI) problem to the (BI0 ) problem. Since the solution of the (BI0 ) problem is independent of s and D, the same is true for the (BI) problem, and hence both are independent of the (IR) problem. j Notice that while the (BI) problem can be solved independently of the (IR) problem, the reverse is not true since the solution of the (IR) problem will depend on speci®c values of p0 and p1 . Therefore the decision maker should use a nested approach to solving the problem
Proposition 2. If p0 p1 =l > p, then it is optimal not to oer a backorder incentive; otherwise, the optimal back order-incentive policy
p0 ; p1 satis®es p0 p1 l p. Proof. If p0 p1 =l p, then R
p0 ; p1 p, and since p0 p1 =l > p yields R
p0 ; p1 > p any such
p0 ; p1 cannot be optimal. Since R
p0 ; p1 is increasing on
p0 ; p1
p0 ; p1 , if p0 p1 =l > p then clearly
p0 ; p1 is optimal. j
Proposition 3. If p0 > p0 and p1 > p1 , then: @q
p0 ; p1 q
p0 ; p1 @q
p0 ; p1 : l @p0
p ÿ p0 ÿ p1 =l @p1
3
On oering economic incentives to backorder Proof. Follows immediately from the ®rst-order optimality conditions for the (BI0 ) problem. j Equation (3) has some interesting economic interpretations. First, notice that at optimality the marginal increase in the backorder probability q
p0 ; p1 from an increase in p0 should be exactly l times the marginal increase in q
p0 ; p1 from an increase in p1 . This makes intuitive sense, since the marginal decrease in
p ÿ p0 ÿ p1 =l ± i.e., the net savings from a unit of demand that backorders rather than becoming a lost sale, or the unit backorder savings ± due to an increase in p0 is l times the marginal decrease in that quantity due to an increase in p1 . Also, at optimality the marginal increase in q
p0 ; p1 from an increase in p0 should be equal to the ratio of the backorder probability divided by the unit backorder savings. 3.2. The IR problem Once the optimal backorder incentives
p0 ; p1 have been determined, and as a result the optimal backorder fraction q q
p0 ; p1 has been ®xed, the decision maker can solve the (IR) problem. To simplify the mathematical expressions, where there is no risk of confusion we use q in place of q
p0 ; p1 . The problem of determining the optimal inventory-replenishment policy given a ®xed backorder fraction q has been addressed in Arreola-Risa and DeCroix [1]. For completeness we summarize the relevant results here without proof. De®ne s
D arg mins0 K
s; D for D 1; and
D
s arg minD1 K
s; D for s 0:
Result 4. For a ®xed D 1, K
s; D is convex on s 0. Moreover, if : D a h q
p0 l p1
1 ÿ q pl qÿ ln
q > ÿ1 ; b l h 1ÿq then s
D 0; otherwise: ln
ab s
D ; or ln
q
ln
ab s
D ; ln
q
where a
l2 D bal=
ÿba2 ln
q q
p0 l p1
1 ÿ q pl:
and
b h=
h
Result 4 characterizes the optimal reorder point s for a given D. Although an empirical analysis of K
s; D suggests that the function is jointly convex in s and D on s 0, we have been unable to establish this analytically, or even show convexity of K
s; D in D for a ®xed s. However, Result 5 below can be used to ®nd the optimal inventory-replenishment policy
s ; D .
719 Result 5. The following procedure converges to
s ; D in a ®nite number of steps: (a) Perform a line search on D to ®nd D
0. (b) For each D 2 f1; . . . ; D
0g, compute s
D using Result 4. (c) Among the D
0 pairs
s
D; D computed in Step (b), the one that yields the smallest value of K
s; D is the optimal inventory-replenishment policy
s ; D .
4. Numerical studies Now that we have demonstrated our ability to eciently solve the (BI) and (IR) problems for the special case where backorder incentives are not oered until a stockout occurs (r 0), we wish to answer two important questions: (1) to what extent can the use of backorder incentives reduce the expected cost of operating the inventory system? and (2) how much of that cost reduction can be captured if we restrict the policy to the case of r 0 (as opposed to 0 r s)? We now present answers to these questions based on extensive numerical trials. For this numerical study, we use the following two backorder-response (BR) functions. BR function 1: BR function 2:
q1
p0 ; p1 1 ÿ 0:9eÿa
p0 ÿp0 eÿb
p1 ÿp1 a q2
p0 ; p1 1 ÿ 0:45 a p0 ÿ p0 b : b p1 ÿ p1
In addition, we consider all 128 combinations of the following system parameters, for a total of 256 cases: k 200, 500; h 1; p 25, 100; p0 2; p1 10; a 200, 1000; k 1, 4; l 6, 24; a 0:1, 0.5; b 0:05, 0.2. For each case considered, we solve three dierent versions of the problem. Version 1 assumes that no backorder incentives are oered, i.e.,
p0 ; p1
p0 ; p1 . For this case we solve the (IR) problem, denoting the optimal inventory-replenishment policy by
s ; D , and obtain the associated expected cost K
p0 ; p1 ; s ; D . Version 2 includes the possibility of oering backorder incentives, but assumes that any incentives are not oered until a stockout occurs, i.e., r 0. For this version, we solve the (BI) problem and its associated (IR) problem to obtain the optimal policy
p0 ; p1 ; s ; D and its expected cost K
p0 ; p1 ; s ; D . Version 3 allows the oering of backorder incentives at any nonnegative inventory level that does not exceed the reorder point, i.e., 0 r s. Since this problem is quite complex, we use exhaustive search to obtain the optimal policy
p0 ; p1 ; r ; s ; D and its expected cost K
p0 ; p1 ; r ; s ; D .
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DeCroix and Arreola-Risa
To answer the ®rst question posed above, we computed the cost savings achieved by Version 3 relative to Version 1, i.e., the bene®t from oering backorder incentives at any nonnegative inventory level less than the reorder point. In all 256 cases examined, the backorder incentives oered some positive bene®t. The minimum, maximum and average cost savings for the numerical study are summarized in Table 1. To answer the second question, we computed the cost savings achieved by Version 2 relative to Version 1, i.e., the bene®t from oering backorder incentives only during a stockout. The minimum, maximum and average cost savings for this comparison are also summarized in Table 1. The contents of Table 1 indicate that, for this set of test problems, nontrivial cost savings can be obtained by offering economic incentives to backorder. In addition, a comparison of the two columns in Table 1 suggests that very little bene®t is forfeited by waiting until a stockout to oer backorder incentives. The average cost savings under the r 0 case is over 90% of the total cost savings available under the general case of 0 r s. This fact, combined with the analytical and practical simplicity as well as the low implementation cost of the r 0 policy, makes a strong argument in favor of only oering backorder incentives once a shortage occurs. Interestingly, this ®nding contrasts with a ®nding in Cheung [18], namely that granting discounts only when a shortage occurs is not an eective way to manage shortages. In the numerical experiment in Cheung [18], the r 0 approach only captures approximately 50% of the potential cost savings from oering backorder incentives. We believe there are two reasons for this contrast. First, there is additional ¯exibility in our model provided by the backorder-response function in that dierent backorder probabilities q can be obtained by adjusting the incentives p0 and p1 . Second, since the model in Cheung [18] does not include a backorder cost per unit per unit time, the total backorder cost is independent of the duration of each backorder. Because such a cost acts as a disincentive to oering incentives early, the early-incentive strategy is less attractive for our system. Although the r 0 approach performs well on average, there are individual cases where a signi®cant fraction of the potential savings are not captured by the solution resulting from this approach. Such cases seem to arise for problems with small values of l and backorder-response functions that are less responsive to the backorder incentives (i.e., Table 1. Savings of oering economic incentives to backorder
Average Minimum Maximum
when 0 r s (%)
when r=0 (%)
11.65 0.53 36.10
10.55 0.53 31.55
small values of a and b for backorder-response function 1, large values of a and b for backorder-response function 2). The missed cost savings for these cases is not particularly large, the case with the worst performance yielded a cost savings that represented 75.99% of the maximum available. However, in the event that the decision maker wishes to realize the additional savings, the more general case of 0 r s may be worth pursuing.
5. Conclusions Oering an economic incentive to backorder has been proposed as an inventory management strategy to deal with shortages in a system with unreliable supply. For the inventory system under consideration, we established results regarding the optimality of oering backorder incentives and the optimal amount of the backorder incentives to oer. Our analysis suggests that the cost bene®ts of oering backorder incentives can be signi®cant. In addition, our results suggest that most of the available cost savings can be obtained by using a simpli®ed strategy of oering backorder incentives only after a shortage has occurred. This work may be extended in a number of directions. One direction for future research would be to pursue a more detailed development of the customer's decision of whether to backorder in response to a backorder incentive. Such investigation might borrow from the marketing literature to derive backorder-response functions q
p0 ; p1 from consumer choice models. This could lead to quite rich problem settings that incorporate a heterogeneous customer population, customers who become impatient after backordering for too long, and other issues of practical interest.
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Biographies Dr. Tony Arreola-Risa is an Assistant Professor of Operations Management in the Lowry Mays College & Graduate School of Business at Texas A&M University. He received his B.S. in Industrial and Systems Engineering from the Monterrey Institute of Technology (ITESM) in Mexico, his M.S. in Industrial Engineering from the Georgia Institute of Technology, and his M.S. and Ph.D. in Operations Management from Stanford University. His research interests include production and inventory systems, as well as service operations. Dr. Arreola-Risa is a member of DSI, IIE and INFORMS. His research has appeared in European Journal of Operational Research, Management Science and Naval Research Logistics, and will appear in Decision Sciences. Dr. Gregory A. DeCroix is a visiting Associate Professor of Operations Management in the Fugua School of Business at Duke University. He received his B.S. in Mathematics and Statistics from Miami University and his Ph.D. in Operations Research from Stanford University. His research interests include inventory systems and environmental issues in operations management. Dr. DeCroix is a member of INFORMS, and his research has appeared in European Journal of Operational Research, and Management Science, and will appear in Naval Research Logistics. Contributed by Inventory and Supply Chain Management Department.
