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Modeling customer impatience in a newsboy problem with time-sensitive shortages
European Journal of Operational Research 205 (2010) 595–603
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Production, Manufacturing and Logistics
Modeling customer impatience in a newsboy problem with time-sensitive shortages Hwansik Lee, Emmett J. Lodree Jr. * Department of Industrial and Systems Engineering, Auburn University, Auburn, AL 36849-5346, USA Department of Information Systems, Statistics, and Management Science, Culverhouse College of Commerce and Business Administration, The University of Alabama, Tuscaloosa, AL 35487-0226, USA
a r t i c l e
i n f o
Article history: Received 7 December 2007 Accepted 17 January 2010 Available online 25 January 2010 Keywords: Inventory control Customer responsiveness Time-dependent partial backlogging Demand uncertainty Utility theory
a b s t r a c t Customers across all stages of the supply chain often respond negatively to inventory shortages. One approach to modeling customer responses to shortages in the inventory control literature is time-dependent partial backlogging. Partial backlogging refers to the case in which a customer will backorder shortages with some probability, or will otherwise solicit the supplier’s competitors to fulfill outstanding shortages. If the backorder rate (i.e., the probability that a customer elects to backorder shortages) is assumed to be dependent on the supplier’s backorder replenishment lead-time, then shortages are said to be represented as time-dependent partial backlogging. This paper explores various backorder rate functions in a single period stochastic inventory problem in an effort to characterize a diversity of customer responses to shortages. We use concepts from utility theory to formally classify customers in terms of their willingness to wait for the supplier to replenish shortages. Under mild assumptions, we verify the existence of a unique optimal solution that corresponds to each customer type. Sensitivity analysis experiments are conducted in order to compare the optimal actions associated with each customer type under a variety of conditions. Additionally, we introduce the notion of expected value of customer patience information (EVCPI), and then conduct additional sensitivity analyses to determine the most and least opportune conditions for distinguishing between customer behaviors. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Inventory shortage is often an indicator of suboptimal supply chain performance caused by a mismatch between supply and demand. In general, shortages are classified as either backorders or lost sales. Immediate consequences of backlogged shortages include increased administrative costs, the cost of delayed revenue, emergency transportation costs, and the loss of customer goodwill, while lost sales are characterized by the opportunity cost of lost revenue and loss of goodwill. In the long run, inventory shortages can compromise an organization’s market share and negatively affect long term profitability. Conventional stochastic inventory models such as the single period problem (or the newsvendor problem) and its many variants often assume that shortages are either completely backlogged or that all sales are lost (e.g., Khouja, 1999). Although this assumption is sometimes plausible in practice, there are situations in which an alternative approach to modeling shortages is appropriate. This paper explores the implications of incorporating the notion of time-dependent partial backlogging in the single period
* Corresponding author. E-mail addresses: [email protected] (H. Lee), [email protected] (E.J. Lodree Jr.). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.028
stochastic inventory problem. When shortages are partially backlogged, a fraction of shortages incur lost sales penalties while the remaining shortages are backlogged. Therefore, time-dependent partial backlogging implies that the backorder rate (i.e., the fraction of shortages backlogged) depends on the time associated with replenishing the outstanding backorder. In many practical situations, customers are likely to fulfill shortages from a supplier’s competitor who has inventory on hand if the backorder lead-time is extensive. On the other hand, the supplier is more likely to retain the customer’s business and possibly avoid the long term consequences of shortages if the backorder lead-time is reasonably short. From this perspective, the time-dependent partial backlogging approach to modeling inventory shortages is particularly useful to firms who compete in time-sensitive markets and embrace service and responsiveness as a competitive strategy (e.g., Stalk and Hout, 1990). Several variations of inventory models with time-dependent partial backlogging have been discussed in the research literature including (i) models with time-varying demand (Zou et al., 2004); (ii) models with time-varying demand and stock deterioration (e.g., Skouri and Papachristos, 2003; Dye et al., 2006); (iii) models with stock deterioration, ordering decisions, and pricing decisions (Papachristos and Skouri, 2003; Dye, 2007); and (iv) models with time-varying demand, ordering decisions, pricing
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decisions, and stock deterioration (Abad, 1996). In general, the backorder rate is assumed to be a piecewise linear function of the backorder lead-time, except for Papachristos and Skouri (2000) and San José et al. (2006) who consider exponential backorder rate functions. Additionally, the majority of the literature involves continuous review inventory policies in which shortages are replenished at the time of the next scheduled delivery. However, in practice, suppliers may attempt to replenish backlogged shortages before the next scheduled delivery by engaging an emergency replenishment process that involves emergency procurement of component parts, emergency production runs, overtime labor, and expedited delivery. The time-dependent backlogging literature also addresses various demand processes including timevarying, price-dependent, and stock dependent; but the majority ignore demand uncertainty. The latter two issues (demand uncertainty at the time of the inventory decision and emergency replenishment after demand realization) are addressed in Lodree (2007), where the time-dependent partial backlogging approach is used to model shortages in the newsvendor problem. The notion of an emergency supply option in the newsvendor problem seems to have originated with Gallego and Moon (1993) and then later addressed by Khouja (1996), Lodree et al. (2004) and Lodree et al. (2008). In these papers, it is assumed that all shortages are backlogged and the unit cost for each shortage is interpreted as the cost associated with the above-mentioned characteristics of emergency replenishment. A practical example of this is a bookstore who orders textbooks before the beginning of an academic term and then places an emergency order for outstanding shortages upon demand realization. Under these and related circumstances, it is more appropriate to consider the arrival of the emergency order, as opposed to demand realization, as the end of the period. This paper explores linear and nonlinear backorder rate functions in an inventory problem with time-dependent partial backlogging, demand uncertainty, and emergency replenishment in an effort to characterize a diversity of customer responses to shortages. Moreover, we find it convenient to use concepts from utility theory to characterize customer responses to shortage as patient, neutral, or impatient. We also compare the optimal inventory levels associated with each customer type through sensitivity analysis and identify the conditions in which there are minute and significant differences in the optimal levels. Finally, we use this framework to determine the benefit of understanding the dominant market characteristic with respect to customer impatience. In particular, if a firm is uncertain about the characteristics of the market it serves, we define the expected value of customer patience information (EVCPI) so that the firm can assess the value of a study or survey whose results reveal the true dominant market characteristic.
2. General mathematical model In this section, we present a generalization of the newsvendor problem that accounts for time-dependent partial backlogging of shortages. To do so, let b represent the backorder rate, which can be interpreted as the probability a single customer will backorder, the fraction of a shortage a single customer will backorder, or the fraction of customers who choose to backlog shortages. To reflect the time-dependency of b, let L 2 ½0; 1Þ represent the backorder lead-time and b : L#½0; 1 (refer to Table 1 for a list of notations used repeatedly throughout this paper). Observe that if L is fixed, the problem reduces to the partial backlogging case with no time-dependency as in Khouja (1996). Therefore, we allow L to be a random function that depends on the number of shortages. Specifically, let L : X Q #½0; 1Þ for x P Q , where X
Table 1 List of notations. Q L X x f ðxÞ FðxÞ cO cH cB cLS bðx Q Þ M TCðQ Þ Q
Order quantity (the decision variable), where Q P 0 Backorder lead-time (days or hours) Supplier’s demand (the buyer’s order), a continuous random variable Actual value of demand, where x P 0 Probability density function (pdf) of X Cumulative distribution function (cdf) of X Unit ordering/production cost before demand realization Unit cost of holding excess inventory at the of the season less salvage value Unit ordering/production cost after demand realization for backlogged shortages Unit cost of shortages that are lost sales Fraction of shortages that are backlogged, where bðx Q Þ 2 ½0; 1 Lost sales threshold (days or hours), where M P 0 Total expected cost function Optimal quantity that minimizes TCðQ Þ
is demand (a random variable), x is a realization of X, Q is the inventory level before demand realization, and maxfx Q ; 0g is the observed number of shortages. For illustrative purposes, we assume that lead-time is directly proportional to the number of shortages, i.e., LðX Q Þ ¼ X Q . Therefore, the terms ‘‘backorder lead-time” and ‘‘magnitude of shortage” can be used interchangeably for the purposes of this paper and the backorder rate can be expressed as the random function bðLðX Q ÞÞ ¼ bðX Q Þ. Note that a multiplier is assumed that converts the units of X Q (i.e., number of items) into the units of L (i.e., time, such as days or hours). The newsvendor model inherently assumes that demand realization occurs at a single point in time at the end of the period. If demand is interpreted as the order size of a single customer, then the fraction of time this customer chooses to backorder shortages of size x Q is bðx Q Þ and the fraction of time this customer balks is 1 bðx Q Þ. For example, if x Q ¼ 100 and the corresponding backorder rate is bð100Þ ¼ 0:6, then this customer’s average backorder size is 60 whenever the number of shortages is 100. On the other hand, if demand is interpreted as distinct customers (who each order one unit), then our definition of bðX Q Þ suggests that the backorder rate is based on the original number of shortages. So the above example means that 60 customers will choose to backorder and the newsvendor will lose 40 customers. However, one could envision taking the latter case a step further as follows. Since the number of backorder customers is now 60, the resulting backorder lead-time would actually be 60 instead of 100. Consequently, the backorder rate would be bð60Þ resulting in a new number of backorder customers and backorder lead-time (say, 75), and the process would repeat until some equilibrium backorder lead-time and rate are reached. Although our model does not explicitly account for this kind of process, one can think of bðX Q Þ as this equilibrium backorder rate. It can be inferred from the above discussion that the backorder lead-time, LðX Q Þ ¼ X Q , is representative of the production lead-time (or order retrieval lead-time) associated with emergency replenishment as opposed to the actual delivery lead-time. In actuality, the components of delivery lead-time include production, transportation, administrative processing (both shipping and receiving), and other time consuming activities. Therefore, delivery lead-time in practice may not necessarily be directly proportional to the backorder size as in our model, which only represents a subset of the actual components of delivery lead-time. However, this assumption may be valid for larger backorder sizes, in which case production lead-time is likely to be the dominant component of backorder lead-time.
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Now let M be the maximum allowable shortage in the sense that the probability of backlogging is zero if x Q P M. Then assuming X is a continuous random variable, the newsvendor problem with time-dependent partial backlogging can be expressed as follows:
TCðQ Þ ¼ Order Cost þ Expected Holding Cost þ Expected Backorder Cost þ Expected Lost Sales Cost Z Q Z QþM ðQ xÞf ðxÞdx þ cB ðx Q Þbðx Q Þf ðxÞdx ¼ cO Q þ cH 0 Q Z 1 þ cLS ðx QÞ½1 bðx Q Þf ðxÞdx: ð1Þ Q
The backorder rate function should satisfy the following properties. Property 1. M; 1Þ.
dbðxQ Þ dðxQÞ
< 0 8x 2 ½Q ; Q þ MÞ and bðx Q Þ ¼ 0 8x 2 ½Q þ
Fig. 1. Customer impatience and lost sale thresholds.
Property 2. limðxQ Þ!0þ bðx Q Þ ¼ 1. Property 3. limðxQ Þ!1 bðx Q Þ ¼ 0. Property 1 indicates that bðx Q Þ is a decreasing function in the number of observed shortages, which suggests that customers are more likely to wait for backorder replenishment if the backorder lead-time is short, and are less likely to wait if the lead-time is close to the lost sales threshold. The second part of Property 1 is a result of the fact that the probability of backlogging remains zero if x Q P M and Properties 2 and 3 ensure that bðx Q Þ 2 ½0; 1. We wish to determine the inventory level Q that minimizes TCðQ Þ given by Eq. (1), provided an optimum exists. It turns out that the convexity of Eq. (1) cannot be guaranteed in general, but the following theorem identifies a sufficient (but not necessary) condition for the existence of a unique minimizer. Before presenting the theorem, we introduce the following assumption.
Fig. 2. Customer impatience and backorder rates.
Assumption 1. cO < cB < cLS The inequalities cO < cB and cO < cLS are assumed in order to avoid trivial cases and are also reflective of practice. The inequality cB < cLS is also representative of practice, although there are some situations where backorder costs exceed short term lost sale costs such as lost revenue. Additionally, the latter inequality will enable us to prove the next theorem as well as other results presented later. Theorem 1. Let b : ðx Q Þ#½0; 1, where x Q 2 ½0; M, be a continuous and twice differentiable function. Then a sufficient condition for TCðQ Þ to be a convex function for all Q P 0 is
ðx Q Þ b00 ðx Q Þ 6 2b0 ðx QÞ; where
dbðxQ Þ dQ
0
b ðx Q Þ and
db0 ðxQ Þ dQ
Proof. Please refer to Appendix.
ð2Þ 00
b ðx Q Þ. h
3. A classification of customer impatience An obvious candidate for measuring customer impatience is the lost sales threshold, M. For example, if M1 < M2 (where M 1 and M2 are lost sales thresholds for customer 1 and customer 2, respectively), then it seems reasonable to conclude that customer 1 is more impatient than customer 2. However, Fig. 1 challenges this simplistic approach to characterizing customer impatience. In particular, customer 3 seems to be more impatient than customer 2 and customer 2 is sometimes more impatient than customer 1, even though M 1 < M2 < M3 . At the very least, Fig. 1 reveals that
M alone does not distinctively measure customer impatience and that the shape of bðx Q Þ should also be taken into account when distinguishing among varying degrees of customer impatience. Consider the case in which M 1 ¼ M2 ¼ M3 ¼ M as shown in Fig. 2. According to Fig. 2, customer 3 is more impatient than customer 2, and customer 2 is more impatient than customer 1. Fig. 2 also suggests that customer 1 is patient at first, but decreasingly patient because his rate of change in impatience is increasing. On the other hand, customer 3 is impatient at first, but decreasingly impatient. The relative impatience of these customers can be attributed to individual attitudes and personalities, but could also be interpreted as one person’s attitude with respect to waiting for different classes of products. For instance, customer 3 could represent an individual’s willingness to wait for a product, such as a computer mouse pad, in which a suitable substitute can be easily identified. In this case, the customer will most likely purchase the suitable substitute as opposed to waiting for backorder replenishment, even if the backorder lead-time is relatively short. Customer 1 could represent this same customer’s willingness to wait for a different product type, such as specialized computer software, in which a product of comparable utility cannot be as easily identified. In this case, the customer is more likely to wait for backorder replenishment, especially if the backorder lead-time is relatively short. These notions of impatience, increasing impatience, and decreasing impatience for describing customer behavior with respect to waiting for backorder replenishment can be defined more precisely using analogous concepts and terminology from utility theory.
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3.1. Utility theory framework Let uðyÞ represent the utility of the value y. Then the following result is fundamental to utility theory (for example, see Winkler, 2003): Theorem 2. Let uðyÞ be a continuous and twice differentiable function. Then a decision-maker is Risk-averse if and only if u00 ðyÞ < 0. Risk-neutral if and only if u00 ðyÞ ¼ 0. Risk-seeking if and only if u00 ðyÞ > 0. We will classify customers as risk-averse, risk-neutral, or riskseeking based on the backorder rate function bðx Q Þ as opposed to some utility function uðyÞ. In order to develop our classification scheme, we introduce the following definitions all of which are special cases of classical utility theory. Definition 1. A lottery with respect to waiting time, L ¼ ðt 1 ; p1 ; . . . ; t n ; pn Þ, consists of a set of waiting times ft 1 ; . . . ; tn g and a set of probabilities fp1 ; . . . ; pn g such that a decision-maker waits P for ti time units with probability pi , where i ¼ 1; . . . ; n and pi ¼ 1. Definition 2. The certainty equivalent with respect to waiting time of a lottery L ¼ ðt 1 ; p1 ; . . . ; tn ; pn Þ, denoted CEW ðLÞ, is the waiting time such that the decision-maker is indifferent between L and waiting for CEW ðLÞ with certainty. Definition 3. The risk premium with respect to waiting time of a lottery L ¼ ðt 1 ; p1 ; . . . ; tn ; pn Þ, denoted RP W ðLÞ, is defined as
RP W ðLÞ ¼ EW ½L CEW ðLÞ; where EW ½L is the expected value of the lottery L. Definition 4. If L ¼ ðt1 ; p1 ; . . . ; t n ; pn Þ is a lottery with respect to waiting time with n > 1, then a decision-maker is Risk-averse with respect to waiting time if and only if RPW ðLÞ < 0. Risk-neutral with respect to waiting time if and only if RPW ðLÞ ¼ 0. Risk-seeking with respect to waiting time if and only if RPW ðLÞ > 0. The difference in conventional utility theory and utility theory with respect to waiting time as described by Definitions 1–4 is noticeably observable in Definition 4. In particular, the definition of conventional risk aversion is RPW ðLÞ > 0 (see Winkler, 2003, for example), but the definition of risk aversion with respect to waiting time is RP W ðLÞ < 0. Similarly, the inequalities are also reversed in the definitions of conventional risk-seeking and riskseeking with respect to waiting time. The reason that the signs are reversed is due to the fact that unlike traditional utility theory where the argument of a given utility function represents gain, the argument of a utility function within the context of waiting time represents loss, i.e., negative gain. These ideas are illustrated through the following example. Consider two decision-makers, DM1 and DM2, who are presented with a lottery with respect to waiting time related to the time they wait to be seated in a restaurant. In particular, the lottery is defined as L ¼ ð50-minutes; 0:25; 10-minutes;0:75Þ. Suppose DM1 specifies CEW1 ðLÞ ¼ 12 minutes and DM2 specifies CEW2 ðLÞ ¼ 40 minutes. Since EW ½L ¼ 20 minutes, we have RP W1 ðLÞ ¼ 20 12 ¼ 8 minutes and RPW2 ðLÞ ¼ 20 40 ¼ 20 minutes. Since CEW1 ðLÞ is a shorter wait than EW ½L, DM1 considers L to be RPW1 ðLÞ ¼ 8 minutes better than
its expected value, which suggests that DM1 is influenced more by the possibility of the 10-minute wait than the risk of a 50-minute wait. Therefore, DM1 is risk-seeking. On the other hand, since CEW2 ðLÞ is a longer wait than EW ½L, DM2 considers L to be RPW2 ðLÞ ¼ 20 minutes better (i.e., 20 minutes worse) than its expected value, which suggests that DM2 is more influenced by the risk of a 50-minute wait than the chances of a 10-minute wait. Thus DM2 prefers to avoid the lottery’s risk and is therefore riskaverse. This example gives some insight into the logic behind Definition 4. The following theorem relates the backorder rate function, bðx Q Þ, to risk-averse, risk-neutral, and risk-seeking behavior with respect to waiting time. Theorem 3. Let b : y#½0; 1, where y 2 ½0; 1Þ, be a continuous and twice differentiable function. Then a decision-maker is Risk-averse with respect to waiting time if and only if b00 ðyÞ < 0. Risk-neutral with respect to waiting time if and only if b00 ðyÞ ¼ 0. Risk-seeking with respect to waiting time if and only if b00 ðyÞ > 0.
Proof. Please refer to Appendix.
h
Based on Theorem 3, customers 1, 2, and 3 in Fig. 2 are riskaverse, risk-neutral, and risk-seeking, respectively. Thus when customers all have the same lost sales threshold, M, the risk-averse customer is more patient (i.e., more likely to backorder) than the risk-neutral customer, and the risk-neutral customer is more patient than the risk-seeking customer. From the customer’s perspective, risk has to do with lead-time, price, and other sources of uncertainty associated with ordering from the supplier’s competitors. For example, the risk-averse customer is more likely to backorder in order to avoid the risk of lead-time uncertainty. From the perspective of time-dependent partial backlogging, customer impatience as it relates to backorder lead-time can be attributed to factors other than the above-mentioned risks of alternate supplier uncertainty. Such factors may include brand loyalty, information technology compatibility, the setup cost of temporarily changing suppliers, and other factors that have nothing to do with risk or uncertainty. Therefore, we introduce the following definitions in order to reflect this broader notion of customer impatience. Definition 5. Let b : y#½0; 1, where y 2 ½0; 1Þ, be a continuous and twice differentiable function. Then a decision-maker is said to be Patient with respect to waiting time if and only if b00 ðyÞ < 0. Neutral with respect to waiting time if and only if b00 ðyÞ ¼ 0. Impatient with respect to waiting time if and only if b00 ðyÞ > 0. Definition 5 implies that the backorder rate function for the patient customer has the same properties as the backorder rate function for a customer who is risk-averse with respect to waiting time. The difference is that the patient customer’s behavior is governed by a broader range of influences than the risk-averse customer’s behavior. Of course, these generalizations apply to the relationships between neutral and risk-neutral customers as well as between impatient and risk-seeking customers. We will classify customer impatience based on Definition 5 for the remainder of this paper. 3.2. Neutral customer behavior A backorder rate function bðx Q Þ that describes neutral behavior with respect to waiting time (see Definitions 1–5) should satisfy
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Properties 1–3 as well as the second part of Theorem 3. The following proposition defines a representative backorder rate function that satisfies these conditions. Proposition 1. Suppose x Q P 0. Then
xQ bðx Q Þ ¼ max 1 ;0 ; M
ð3Þ
satisfies Properties 1–3 and the second part of Theorem 3. Proof. Since
dbðx Q Þ ¼ dðx QÞ
(
M1 < 0; x 2 ½Q; Q þ MÞ 0;
x 2 ½Q þ M; 1Þ
A backorder rate function bðx Q Þ that describes patient behavior with respect to waiting time should satisfy Properties 1–3 as well as the first part of Theorem 3. The following proposition defines a representative backorder rate function that satisfies these conditions.
2
dðx Q Þ2
Note that the parameter a controls the rate at which bðx Q Þ given by Eq. (4) decreases. In particular, a1 > a2 implies that customer 1 is more impatient than customer 2. The condition a < M2 suggests that Theorem 5 can only guarantee convexity if bðx Q Þ does not decrease too quickly as x Q approaches M. 3.4. Patient customer behavior
:
Property 1 holds. Also,
d bðx Q Þ
8x Q 2 A [ B. In order to verify that TCðQ Þ is convex 8Q P 0, we need to show that convexity holds 8x Q 2 Rþ , which actually reduces to showing that A [ B ¼ Rþ . Let C ¼ ð2a ; MÞ. Then A [ B [ C ¼ Rþ . However, since the condition a 6 M2 is given, we have that C ¼ ; and A [ B [ C ¼ A [ B ¼ Rþ . h
¼ 0;
shows that the second part of Theorem 3 holds. Now
Proposition 3. Suppose x Q P 0. Then
xQ lim max 1 ; 0 ¼ 1; xQ !0 M
( bðx QÞ ¼
shows that Property 2 holds, and
cos
xQ 2M
p ; x 2 ½Q ; Q þ MÞ
0;
x 2 ½Q þ M; 1Þ
;
ð5Þ
xQ ; 0 ¼ maxf1; 0g ¼ 0; lim max 1 xQ !1 M
satisfies Properties 1–3 and the first part of Theorem 3.
shows that Property 3 holds.
Proof. The proof is similar to the proof of Proposition 1. Refer to Appendix for details. h
h
The next result indicates the existence of Q that minimizes TCðQ Þ given by Eq. (1) when bðx Q Þ is defined as in Proposition 1. Theorem 4. Suppose bðx Q Þ is defined as by Eq. (3). Then TCðQ Þ given by Eq. (1) is a convex function. 0
00
Proof. If x 2 ½Q ; Q þ MÞ, then b ðx Q Þ ¼ M1 and b ðx Q Þ ¼ 0. Thus the inequality ðx Q Þb00 ðx Q Þ 6 2b0 ðx Q Þ given by Eq. (2) reduces to M2 P 0. Since the last inequality always holds, it follows that bðx Q Þ satisfies the condition in Theorem 1. If x 2 ½Q þ M; 1Þ, then bðx Q Þ ¼ 0, which also satisfies the condition in Theorem 1. Therefore, Theorem 1 guarantees that TCðQ Þ is a convex function for all Q P 0 whenever bðx Q Þ is defined as in Eq. (3). h 3.3. Impatient customer behavior A backorder rate function bðx Q Þ that describes impatient behavior with respect to waiting time should satisfy Properties 1–3 as well as the third part of Theorem 3. The following proposition defines a representative backorder rate function that satisfies these conditions. Proposition 2. Suppose x Q P 0 and a > 0 is a constant. Then
( bðx Q Þ ¼
eaðxQÞ ; x 2 ½Q ; Q þ MÞ ; 0; x 2 ½Q þ M; 1Þ
ð4Þ
satisfies Properties 1–3 and the third part of Theorem 3. Proof. The proof is similar to the proof of Proposition 1. Refer to Appendix for details. h Theorem 5. Suppose bðx Q Þ is defined by Eq. (4) and a 6 M2 . Then TCðQ Þ given by Eq. (1) is a convex function. Proof. Let A ¼ ð0; 2aÞ and B ¼ ðM; 1Þ. It is straightforward to show that Eq. (2) reduces to x Q 2 A. Since bðx Q Þ ¼ 08x Q 2 B and Eq. (2) holds, it follows from Theorem 1 that TCðQ Þ is convex
Theorem 6. Suppose bðx Q Þ is defined by Eq. (5). Then TCðQ Þ given by Eq. (1) is a convex function. Proof. If x 2 ½Q ; Q þ MÞ, then
xQ p ; 2M 2M p2 xQ 00 : cos p b ðx Q Þ ¼ 2M 4M 2
b0 ðx Q Þ ¼
p
sin
Thus the inequality given by Eq. (2) reduces to
ðx Q Þp xQ p 6 1: cot 2M 4M
ð6Þ
If we let z ¼ ðp=2MÞðx Q Þ, then Eq. (6) becomes ðz=2Þ cotðzÞ 6 1, where z 2 ð0; p=2Þ. Since cotðzÞ > 0 and z=2 < 0 for z 2 ð0; p=2Þ, we have ðz=2Þ cotðzÞ < 0 6 1, which implies that the inequality given by Eq. (6) holds. Thus it follows that bðx Q Þ satisfies the condition in Theorem 1. If x 2 ½Q þ M; 1Þ, then bðx Q Þ ¼ 0, which also satisfies the condition in Theorem 1. Therefore, Theorem 1 guarantees that TCðQ Þ is a convex function for all Q P 0 whenever bðx Q Þ is defined as in Eq. (5). h 4. Sensitivity analysis and management insights This section investigates the effects that various problem parameters have on optimal ordering decisions. The following example data was used for the analysis:
X Nð500; 250Þ; cLS ¼ 100;
cO ¼ 50;
M ¼ 1000:
cH ¼ 20;
cB ¼ 75; ð7Þ
Note that our analyses suggest that it is generally not possible to express the optimal order quantity for any of the three cases in closed form. Therefore, Mathematica™ was used to obtain order quantities for the numerical experiments associated with Figs. 3–9. Based on the results shown in Figs. 3–5, we observe the following:
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Observation 1. Let Q P ; Q N ; and QI be the optimal order quantity associated with the patient, neutral, and impatient cases, respectively. Then Q I P Q N P Q P . This observation is intuitive. It is reasonable to expect the optimal order quantity to be an increasing function of customer impatience.
480
420
Observation 2. Optimal order quantities are always decreasing in cH and M. These results are intuitive and consistent with the results reported in Lodree (2007). Observation 3. According to Fig. 5, differences in optimal order quantities among the three customer types are decreasing functions of the ratio cB =cLS . From a managerial perspective, this means that it is important for the decision-maker to distinguish between customer types when there is a significant difference between cB and cLS , i.e., if cB =cLS is close to 0. If, on the other hand, cB =cLS 1, then there is no need to distinguish between customer types. This result is indeed intuitive. If there is little difference between the unit backorder cost and unit lost sales cost, then the cost associated with shortages is insensitive to whether or not shortages are backorders or lost sales. Similar conclusions can be drawn from Eq. (A.1) in Appendix. In particular, the effect of the backorder rate function bðx Q Þ on the total expected cost function TCðQ Þ becomes more pronounced as the difference cLS cB increases, or equivalently, as the ratio cB =cLS decreases. Thus it is reasonable to expect increasing differences in Q among the three cases as bðx Q Þ assumes a more dominant role in TCðQ Þ. Conversely, as cB =cLS approaches 1, the effect of bðx Q Þ diminishes and TCðQ Þ becomes the newsboy problem. Therefore, the optimal order quantities will approach the newsboy result for each of the three customer types as cB =cLS ! 1. Observation 4. Differences in optimal order quantities among the three customer impatience behaviors are decreasing in cH and increasing in M. However, differences in optimal order quantities are influenced more by the ratio cB =cLS than by cH and M.
360
Neutral Patient Impatient
300 800
900
1000
1100
1200
Fig. 4. The effect of M on the optimal order quantity.
900
Neutral Patient
700
Impatient
500
300 0.15
0.35
0.55
0.75
0.95
Fig. 5. The effect of cB =cLS on the optimal order quantity, with cB ¼ 75.
From a practical perspective, Observation 4 suggests the increasing importance of distinguishing between customer types as holding costs decrease and as the lost sales threshold increases. The underlying intuition is that as holding costs become less significant, the effect of expected shortage costs is magnified. Similarly,
20.2
10,000
450
Assume Neutral
400 350
Neutral
300
Patient
250
Impatient
Assume Impatient
7500
15.2
5000
10.1
2500
5.1
200 150 100 50 0
0 20
120
220
320
Fig. 3. The effect of cH on the optimal order quantity.
420
0.15
0.35
0.55
0.75
0.95
Fig. 6. The effect of cB =cLS on EVCPI if customer is patient.
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3000
6.1
25,000
41.0
Assume Neutral Assume Patient
Assume Impatient
20,000
32.9
Assume Impatient 2000
4.0
1000
15,000
24.7
10,000
16.5
5000
8.2
2.0
0
0 0.15
0.35
0.55
0.75
0.95
Fig. 7. The effect of cB =cLS on EVCPI if customer is neutral.
4500
11.3
Assume Neutral Assume Patient
0.03
0.06
0.09
0.12
0.15
Fig. 9. The effect of cB =cLS on EVCPI if customer is patient.
the patient backorder rate function given by Eq. (5). To illustrate this point, consider the data in Eq. (7) and suppose x Q ¼ 300. Then using Eqs. (3)–(5), bP ð300Þ bN ð300Þ ¼ 0:191 and bN ð300Þ bI ð300Þ ¼ 0:698, where bN ðx Q Þ; bI ðx Q Þ, and bP ðx Q Þ are Eqs. (3)–(5), respectively. Since bN ð300Þ bI ð300Þ > bP ð300Þ bN ð300Þ, it is reasonable to expected that Q I Q N > Q N Q P . 5. Value of customer patience information
3000
7.6
1500
3.8
Suppose the decision-maker can conduct a study to obtain more information about the impatience behavior of a customer or market segment. This section explores the expected value of conducting such a study. To carry out the analysis, let us first assume that the supplier currently orders Q N , which is the optimal order quantity associated with the neutral customer. If the customer is actually patient, then the expected value of a market study is
V ¼ TC P ðQ N Þ TC P ðQ P Þ;
0 0.15
0.35
0.55
0.75
0.95
Fig. 8. The effect of cB =cLS on EVCPI if customer is impatient.
increasing M magnifies the effect of expected shortage costs. Since differences in customer types are reflected only through shortage costs in the decision-maker’s cost structure, the result of magnified shortage costs is larger differences in optimal order quantities. These observations can also be explained by observing Eq. (A.1) in Appendix. Finally, note that changes in cH and M do not significantly affect differences in optimal order quantities, particularly relative to the changes caused by the ratio cB =cLS (compare Figs. 3 and 4 to Fig. 5). Observation 5. Differences in optimal order quantities between the impatient and neutral cases are always greater than the differences in optimal order quantities between the patient and neutral cases. This is necessarily a consequence of the specific backorder rate functions studied in this paper for the patient and impatient cases. More specifically, the impatient backorder rate function given by Eq. (4) is more different than the neutral case when compared to
where TC P ðÞ and Q P are the expected cost function and optimal order quantity, respectively, for the patient customer. From this perspective, V can be defined as the expected value of customer patience information (EVCPI). In general, let V ij equal the expected value of the market study if the supplier orders Q i and the customer patience profile is actually j, where i; j 2 fP; N; Ig (patient, neutral, impatient). Also, let TC i ðÞ and Q i represent the expected total cost function and optimal order quantity, respectively, for case i 2 fP; N; Ig. If i ¼ j, then clearly V ij ¼ 0 for all i 2 fP; N; Ig. Otherwise if i–j, then
V NP ¼ TC P ðQ N Þ TC P ðQ P Þ; V NI ¼ TC I ðQ N Þ TC I ðQ I Þ; V PN ¼ TC N ðQ P Þ TC N ðQ N Þ; V PI ¼ TC I ðQ P Þ TC I ðQ I Þ; V IP ¼ TC P ðQ I Þ TC P ðQ P Þ; V IN ¼ TC N ðQ I Þ TC N ðQ N Þ: The example data from Eq. (7) was used to construct the graphs shown in Figs. 6–9. The following insights can be interpreted from these figures: Observation 6. The expected value of customer patience information (EVCPI) is an increasing function of M, and a decreasing function of cH .
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This observation is intuitive since differences in optimal order quantities among the three customer types are increasing in M, and decreasing in cH (see Observation 2). Note that this observation is supported by the above-mentioned intuition, and that the graphs supporting this idea have not been included in this paper. Observation 7. EVCPI is always a decreasing function of the ratio cB =cLS . The fact that EVCPI is decreasing in cB =cLS is intuitive given that differences in optimal order quantities among the customer types are decreasing in cB =cLS (see Observation 3). Observation 8. In general, EVCPI only seems significant whenever cB =cLS < 0:5. According to Figs. 6–8, the percentage increase in optimal expected cost is always less than 5% whenever 0:5 < cB =cLS 6 1. In fact, one could argue that EVCPI is only significant whenever 0 < cB =cLS < 0:15, although not necessarily (see Fig. 9). The largest percent increase of approximately 33% occurs when the customer is patient, the supplier assumes the customer is impatient, and cB =cLS ¼ 0:03. In this case, the corresponding value of EVCPI is over $20,000. If, on the other hand, the customer is patient, the decision-maker assumes the customer is neutral, and cB =cLS ¼ 0:15, then the percent increase in expected cost and EVCPI are approximately 2% and $1000, respectively. Thus the customer’s actual impatience behavior and the supplier’s assumption regarding the customer’s impatience behavior also have a significant impact on EVCPI, which motivates the next observation. Observation 9. If the supplier has no information regarding the customer’s impatience behavior, then the supplier should assume that the customer is neutral. If the customer is neutral, then assuming a neutral customer profile is obviously the best decision, which results in EVCPI = 0. Now suppose the customer is actually patient. Then according to Fig. 6, assuming a neutral customer profile leads to lower values of EVCPI (and lower percentage expected cost increase) than assuming an impatient customer profile. Similarly, Fig. 8 suggests that assuming a neutral profile when the customer is actually impatient leads to better results than assuming the customer is patient.
the expected benefit of knowing the relative impatience of customers or a market segment. Our results suggest that EVCPI is most significant when there is a large difference between the lost sales cost cLS and backorder cost cB . Our results also indicate that EVCPI increases as the difference cLS cB increases and as the lost sales threshold M increases. This paper can be extended in several ways. A useful extension would be to relax Assumption 1 such that cB > cLS is a possibility. This cost structure is known to happen in practice based on one of the author’s interactions with the production manager of a major manufacturing firm. In particular, this production manager’s primary objective in the event of a shortage is to maintain a positive rapport with the client (especially a major client). He is much less concerned with the short term cost inefficiencies associated with emergency procurement, production, and delivery in light of the long term implications related to compromising a business relationship. Therefore, this manager would replenish backorders at unit cost cB , even in the event that cB far exceeds cLS . This manufacturing firm’s strategic approach to managing shortages suggests another promising extension of this paper, namely an extension that entails both short term and long term management of inventory shortages in a multi-period setting. Another useful extension would be to generalize the relationship between the backorder lead-time, L, and number of shortages, X Q . We assumed that L ¼ X Q , but lead-time functions with more practical relevance should be considered. We would expect consistency with the results described in this paper provided L is an increasing function of X Q . Similarly, the development of other backorder rate functions and the applicability of our findings based on these functions (including their interactions with more general lead-time functions) also warrants exploration. The corresponding analyses in terms of verifying Theorem 1 and Properties 1–3 could be slightly or substantially more difficult than this paper depending on how LðX Q Þ and bðLðX Q ÞÞ are selected. Appendix A. Proof of Theorem 1 TCðQ Þ given by Eq. (1) can be written as
TCðQÞ ¼ cO Q þ cH
Z
Q
ðQ xÞf ðxÞdx þ cLS
0
þ ðcB cLS Þ
Z
1
ðx Q Þf ðxÞdx
Q QþM
ðx Q Þbðx Q Þf ðxÞdx:
ðA:1Þ
Q
6. Summary and future research directions Shortages are often represented as either backorders or lost sales in inventory models and in practice. The time-dependent backlogging approach to characterizing inventory shortages acknowledges that there is some probability associated with whether or not a customer will backorder a shortage, and that this probability (i.e., the backorder rate) is related to the lead-time associated with replenishing the outstanding backorder. While the majority of the research literature considers time-dependent backlogging within the context of continuous review models with deterministic demand, this paper studies time-dependent partial backlogging in the single period inventory problem with stochastic demand. The backorder rate function is used to formally classify customers as either patient, neutral, or impatient with respect to their willingness to wait for the supplier to replenish shortages. Representative backorder rate functions are presented, and the existence of a unique order quantity that minimizes total expected costs is shown for each case. Sensitivity analysis experiments are conducted to examine the similarities and differences in optimal order quantities among the three customer types as a function of various problem parameters. Additionally, the expected value of customer patience information (EVCPI) is defined in order to assess
Z
Recalling from Theorem 1 that b0 ðx Q Þ is defined as the derivative of bðx Q Þ with respect to Q, it follows that the first and second order derivatives of TCðQ Þ are
dTCðQ Þ ¼ cO þ cH dQ
Z
Q
f ðxÞdx cLS
0
þ ðcB cLS Þ
Z
Z
1
f ðxÞdx
Q QþM
½bðx Q Þ þ ðx QÞb0 ðx Q Þf ðxÞdx;
Q
ðA:2Þ 2
d TC 2 ðQ Þ dQ 2
¼ ðcH þ cLS Þf ðQ Þ þ ðcLS cB Þf ðQ Þ þ ðcLS cB Þ
Z
Q þM
½2b0 ðx Q Þ ðx Q Þb00 ðx QÞf ðxÞdx:
ðA:3Þ
Q
The first term in Eq. (A.3) is obviously nonnegative. Since cLS cB > 0 based on Assumption 1, it follows that the second term is also nonnegative. It also follows from Assumption 1 that if the integrand of the third term is nonnegative, we can be sure that the third term is nonnegative. Therefore, a sufficient condition for optimality is given by
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2b0 ðx Q Þ ðx Q Þb00 ðx QÞ P 0 for all x 2 ½Q ; Q þ MÞ: ðA:4Þ Appendix B. Proof of Theorem 3 For illustration, we will show that a decision-maker is riskseeking if and only if his back order rate function is convex. The proofs of parts 1 and 2 of Theorem 3 are similar. ð)Þ By Definitions 3 and 4, we have RPW ðLÞ > 0 and EW ½L CEW ðLÞ > 0. Now recall the second axiom of utility (e.g., Winkler, 2003), which states if a decision-maker is indifferent between a payoff r 1 and a lottery L ¼ ðr 2 ; p; r 3 ; 1 pÞ, then uðr1 Þ ¼ puðr2 Þþ ð1 pÞuðr3 Þ. Similarly, if a decision-maker is indifferent between a lottery with respect to waiting time given by L ¼ ðt 1 ; p1 ; ; t n ; pn Þ and waiting for CEW ðLÞ with certainty, then the likelihood of backordering associated with the lottery L is equal to the likelihood of backordering associated with the certainty equivalent CEW ðLÞ. Hence, above axiom implies that b½CEW ðLÞ ¼ p1 bðt 1 Þ þ . . . þ pn bðt n Þ, where p1 þ þ pn ¼ 1. That is, b½CEW ðLÞ is the convex hull of ft 1 ; . . . ; tn g. Since bðyÞ is a decreasing function by Property 1 and EW ½L CEW ðLÞ > 0, it follows that bðEW ½LÞ bðCEW ðLÞÞ < 0, or equivalently,
bðp1 t 1 þ þ pn t n Þ < p1 bðt 1 Þ þ þ pn bðt n Þ:
ðB:1Þ
Since Eq. (B.1) holds, it follows that bðyÞ is a convex function and b00 ðyÞ > 0 (see, for example, Jeter, 1986, p. 217). ð(Þ Since bðyÞ is a convex function, it follows that Eq. (B.1) holds by Proposition 1 shown in Jeter (1986, p. 217). Since CEW ðLÞ is the convex hull of ft1 ; . . . ; t n g and p1 t1 þ þ pn t n ¼ EW ½L, Eq. (B.1) becomes bðEW ½LÞ bðCEW ðLÞÞ < 0. Based on the last inequality, we know that EW ½L CEW ðLÞ > 0 by Property 1. Therefore, by Definitions 3 and 4, the latter inequality implies risk-seeking. h Appendix C. Proof of Proposition 2 Since
dbðx Q Þ ¼ dðx QÞ
(
aeaðxQÞ < 0; x 2 ½Q ; Q þ MÞ 0;
x 2 ½Q þ M; 1Þ
:
Property 1 holds. Also, 2
d bðx Q Þ dðx Q Þ2
¼ a2 eaðxQ Þ > 0;
shows that the third part of Theorem 3 holds. Now
lim eaðxQ Þ ¼ 1;
xQ !0þ
shows that Property 2 holds, and
lim eaðxQÞ ¼ 0;
xQ !1
shows that Property 3 holds. h Appendix D. Proof of Proposition 3 The derivative is computed as
dbðx Q Þ ¼ dðx Q Þ
(
p sin 2M 0;
xQ 2M
p ; x 2 ½Q ; Q þ MÞ x 2 ½Q þ M; 1Þ
:
If we let z ¼ ðp=2MÞðx Q Þ, then dbðx Q Þ=dðx Q Þ becomes ½z=ðx Q Þ sin z, where z 2 ð0; p=2Þ. Since z=ðx Q Þ < 0 and sin z > 0 in this interval, we have dbðx Q Þ=dðx Q Þ < 0 which shows that Property 1 holds.Also, 2
d bðx Q Þ dðx Q Þ2
¼
p2 4M 2
cos
xQ p < 0; 2M
shows that the first part of Theorem 3 holds. Now
xQ lim þ max cos p ; 0 ¼ 1; 2M xQ !0 shows that Property 2 holds, and
xQ xQ lim max cos p ; 0 ¼ lim cos p ¼ 0; xQ !1 xQ !M 2M 2M shows that Property 3 holds. h References Abad, P., 1996. Optimal pricing and lot-sizing under conditions of perishability and partial backordering. Management Science 42 (8), 1093–1104. Dye, C.-Y., 2007. Joint pricing and ordering policy for a deteriorating inventory with partial backlogging. Omega: International Journal of Management Science 35 (2), 184–189. Dye, C.Y., Chang, H.J., Teng, J.T., 2006. A deteriorating inventory model with timevarying demand and shortage-dependent partial backlogging. European Journal of Operational Research 172 (2), 417–429. Gallego, G., Moon, I., 1993. The distribution free newsboy problem: Review and extensions. Journal of the Operational Research Society 44, 825–834. Jeter, M.W., 1986. Mathematical Programming: An Introduction to Optimization. Marcel Dekker, New York, NY. Khouja, M., 1996. A note on the newsboy problem with an emergency supply option. Journal of the Operational Research Society 47, 1530–1534. Khouja, M., 1999. The single-period (news-vendor) problem: Literature review and suggestions for future research. Omega: International Journal of Management Science 27, 537–553. Lodree, E., Klein, C., Jang, W., 2004. Minimizing customer response time in a twostage supply chain system with variable lead time and stochastic demand. International Journal of Production Research 42 (11), 2263–2278. Lodree, E.J., 2007. Advanced supply chain planning with mixtures of backorders, lost sales, and lost contract. European Journal of Operational Research 181 (1), 168– 183. Lodree, E., Kim, Y., Jang, W., 2008. Time and quantity dependent waiting costs in a newsvendor problem with backlogged shortages. Mathematical and Computer Modeling 47, 60–71. Papachristos, S., Skouri, K., 2000. Optimal replenishment policy for deteriorating items with time-varying demand and partial-exponential type – backlogging. Operations Research Letters 27 (4), 175–184. Papachristos, S., Skouri, K., 2003. An inventory model with deteriorating items, quantity discount, pricing, and time-dependent partial backlogging. International Journal of Production Economics 83 (3), 247–256. San José, L.A., Sicilia, J., García-Lagyna, J., 2006. Analysis of an inventory system with exponential partial backordering. International Journal of Production Economics 100, 76–86. Skouri, K., Papachristos, S., 2003. Four inventory models for deteriorating items with time varying demand and partial backlogging: A cost comparison. Optimal Control Applications and Methods 24 (6), 315–330. Stalk, G., Hout, T., 1990. Competing Against Time: How Time-based Competition is Reshaping Global Markets. Macmillan, New York, NY. Winkler, R.L., 2003. An Introduction to Bayesian Inference and Decision, second ed. Probabilistic Publishing, Gainesville, FL. Zou, Y.-W., Lau, H.-S., Yang, S.-L., 2004. A finite horizon lot-sizing problem with time-varying deterministic demand and waiting-time-dependent partial backlogging. International Journal of Production Economics 91 (2), 109–119.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Modeling customer impatience in a newsboy problem with time-sensitive shortages Hwansik Lee, Emmett J. Lodree Jr. * Department of Industrial and Systems Engineering, Auburn University, Auburn, AL 36849-5346, USA Department of Information Systems, Statistics, and Management Science, Culverhouse College of Commerce and Business Administration, The University of Alabama, Tuscaloosa, AL 35487-0226, USA
a r t i c l e
i n f o
Article history: Received 7 December 2007 Accepted 17 January 2010 Available online 25 January 2010 Keywords: Inventory control Customer responsiveness Time-dependent partial backlogging Demand uncertainty Utility theory
a b s t r a c t Customers across all stages of the supply chain often respond negatively to inventory shortages. One approach to modeling customer responses to shortages in the inventory control literature is time-dependent partial backlogging. Partial backlogging refers to the case in which a customer will backorder shortages with some probability, or will otherwise solicit the supplier’s competitors to fulfill outstanding shortages. If the backorder rate (i.e., the probability that a customer elects to backorder shortages) is assumed to be dependent on the supplier’s backorder replenishment lead-time, then shortages are said to be represented as time-dependent partial backlogging. This paper explores various backorder rate functions in a single period stochastic inventory problem in an effort to characterize a diversity of customer responses to shortages. We use concepts from utility theory to formally classify customers in terms of their willingness to wait for the supplier to replenish shortages. Under mild assumptions, we verify the existence of a unique optimal solution that corresponds to each customer type. Sensitivity analysis experiments are conducted in order to compare the optimal actions associated with each customer type under a variety of conditions. Additionally, we introduce the notion of expected value of customer patience information (EVCPI), and then conduct additional sensitivity analyses to determine the most and least opportune conditions for distinguishing between customer behaviors. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Inventory shortage is often an indicator of suboptimal supply chain performance caused by a mismatch between supply and demand. In general, shortages are classified as either backorders or lost sales. Immediate consequences of backlogged shortages include increased administrative costs, the cost of delayed revenue, emergency transportation costs, and the loss of customer goodwill, while lost sales are characterized by the opportunity cost of lost revenue and loss of goodwill. In the long run, inventory shortages can compromise an organization’s market share and negatively affect long term profitability. Conventional stochastic inventory models such as the single period problem (or the newsvendor problem) and its many variants often assume that shortages are either completely backlogged or that all sales are lost (e.g., Khouja, 1999). Although this assumption is sometimes plausible in practice, there are situations in which an alternative approach to modeling shortages is appropriate. This paper explores the implications of incorporating the notion of time-dependent partial backlogging in the single period
* Corresponding author. E-mail addresses: [email protected] (H. Lee), [email protected] (E.J. Lodree Jr.). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.028
stochastic inventory problem. When shortages are partially backlogged, a fraction of shortages incur lost sales penalties while the remaining shortages are backlogged. Therefore, time-dependent partial backlogging implies that the backorder rate (i.e., the fraction of shortages backlogged) depends on the time associated with replenishing the outstanding backorder. In many practical situations, customers are likely to fulfill shortages from a supplier’s competitor who has inventory on hand if the backorder lead-time is extensive. On the other hand, the supplier is more likely to retain the customer’s business and possibly avoid the long term consequences of shortages if the backorder lead-time is reasonably short. From this perspective, the time-dependent partial backlogging approach to modeling inventory shortages is particularly useful to firms who compete in time-sensitive markets and embrace service and responsiveness as a competitive strategy (e.g., Stalk and Hout, 1990). Several variations of inventory models with time-dependent partial backlogging have been discussed in the research literature including (i) models with time-varying demand (Zou et al., 2004); (ii) models with time-varying demand and stock deterioration (e.g., Skouri and Papachristos, 2003; Dye et al., 2006); (iii) models with stock deterioration, ordering decisions, and pricing decisions (Papachristos and Skouri, 2003; Dye, 2007); and (iv) models with time-varying demand, ordering decisions, pricing
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decisions, and stock deterioration (Abad, 1996). In general, the backorder rate is assumed to be a piecewise linear function of the backorder lead-time, except for Papachristos and Skouri (2000) and San José et al. (2006) who consider exponential backorder rate functions. Additionally, the majority of the literature involves continuous review inventory policies in which shortages are replenished at the time of the next scheduled delivery. However, in practice, suppliers may attempt to replenish backlogged shortages before the next scheduled delivery by engaging an emergency replenishment process that involves emergency procurement of component parts, emergency production runs, overtime labor, and expedited delivery. The time-dependent backlogging literature also addresses various demand processes including timevarying, price-dependent, and stock dependent; but the majority ignore demand uncertainty. The latter two issues (demand uncertainty at the time of the inventory decision and emergency replenishment after demand realization) are addressed in Lodree (2007), where the time-dependent partial backlogging approach is used to model shortages in the newsvendor problem. The notion of an emergency supply option in the newsvendor problem seems to have originated with Gallego and Moon (1993) and then later addressed by Khouja (1996), Lodree et al. (2004) and Lodree et al. (2008). In these papers, it is assumed that all shortages are backlogged and the unit cost for each shortage is interpreted as the cost associated with the above-mentioned characteristics of emergency replenishment. A practical example of this is a bookstore who orders textbooks before the beginning of an academic term and then places an emergency order for outstanding shortages upon demand realization. Under these and related circumstances, it is more appropriate to consider the arrival of the emergency order, as opposed to demand realization, as the end of the period. This paper explores linear and nonlinear backorder rate functions in an inventory problem with time-dependent partial backlogging, demand uncertainty, and emergency replenishment in an effort to characterize a diversity of customer responses to shortages. Moreover, we find it convenient to use concepts from utility theory to characterize customer responses to shortage as patient, neutral, or impatient. We also compare the optimal inventory levels associated with each customer type through sensitivity analysis and identify the conditions in which there are minute and significant differences in the optimal levels. Finally, we use this framework to determine the benefit of understanding the dominant market characteristic with respect to customer impatience. In particular, if a firm is uncertain about the characteristics of the market it serves, we define the expected value of customer patience information (EVCPI) so that the firm can assess the value of a study or survey whose results reveal the true dominant market characteristic.
2. General mathematical model In this section, we present a generalization of the newsvendor problem that accounts for time-dependent partial backlogging of shortages. To do so, let b represent the backorder rate, which can be interpreted as the probability a single customer will backorder, the fraction of a shortage a single customer will backorder, or the fraction of customers who choose to backlog shortages. To reflect the time-dependency of b, let L 2 ½0; 1Þ represent the backorder lead-time and b : L#½0; 1 (refer to Table 1 for a list of notations used repeatedly throughout this paper). Observe that if L is fixed, the problem reduces to the partial backlogging case with no time-dependency as in Khouja (1996). Therefore, we allow L to be a random function that depends on the number of shortages. Specifically, let L : X Q #½0; 1Þ for x P Q , where X
Table 1 List of notations. Q L X x f ðxÞ FðxÞ cO cH cB cLS bðx Q Þ M TCðQ Þ Q
Order quantity (the decision variable), where Q P 0 Backorder lead-time (days or hours) Supplier’s demand (the buyer’s order), a continuous random variable Actual value of demand, where x P 0 Probability density function (pdf) of X Cumulative distribution function (cdf) of X Unit ordering/production cost before demand realization Unit cost of holding excess inventory at the of the season less salvage value Unit ordering/production cost after demand realization for backlogged shortages Unit cost of shortages that are lost sales Fraction of shortages that are backlogged, where bðx Q Þ 2 ½0; 1 Lost sales threshold (days or hours), where M P 0 Total expected cost function Optimal quantity that minimizes TCðQ Þ
is demand (a random variable), x is a realization of X, Q is the inventory level before demand realization, and maxfx Q ; 0g is the observed number of shortages. For illustrative purposes, we assume that lead-time is directly proportional to the number of shortages, i.e., LðX Q Þ ¼ X Q . Therefore, the terms ‘‘backorder lead-time” and ‘‘magnitude of shortage” can be used interchangeably for the purposes of this paper and the backorder rate can be expressed as the random function bðLðX Q ÞÞ ¼ bðX Q Þ. Note that a multiplier is assumed that converts the units of X Q (i.e., number of items) into the units of L (i.e., time, such as days or hours). The newsvendor model inherently assumes that demand realization occurs at a single point in time at the end of the period. If demand is interpreted as the order size of a single customer, then the fraction of time this customer chooses to backorder shortages of size x Q is bðx Q Þ and the fraction of time this customer balks is 1 bðx Q Þ. For example, if x Q ¼ 100 and the corresponding backorder rate is bð100Þ ¼ 0:6, then this customer’s average backorder size is 60 whenever the number of shortages is 100. On the other hand, if demand is interpreted as distinct customers (who each order one unit), then our definition of bðX Q Þ suggests that the backorder rate is based on the original number of shortages. So the above example means that 60 customers will choose to backorder and the newsvendor will lose 40 customers. However, one could envision taking the latter case a step further as follows. Since the number of backorder customers is now 60, the resulting backorder lead-time would actually be 60 instead of 100. Consequently, the backorder rate would be bð60Þ resulting in a new number of backorder customers and backorder lead-time (say, 75), and the process would repeat until some equilibrium backorder lead-time and rate are reached. Although our model does not explicitly account for this kind of process, one can think of bðX Q Þ as this equilibrium backorder rate. It can be inferred from the above discussion that the backorder lead-time, LðX Q Þ ¼ X Q , is representative of the production lead-time (or order retrieval lead-time) associated with emergency replenishment as opposed to the actual delivery lead-time. In actuality, the components of delivery lead-time include production, transportation, administrative processing (both shipping and receiving), and other time consuming activities. Therefore, delivery lead-time in practice may not necessarily be directly proportional to the backorder size as in our model, which only represents a subset of the actual components of delivery lead-time. However, this assumption may be valid for larger backorder sizes, in which case production lead-time is likely to be the dominant component of backorder lead-time.
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597
Now let M be the maximum allowable shortage in the sense that the probability of backlogging is zero if x Q P M. Then assuming X is a continuous random variable, the newsvendor problem with time-dependent partial backlogging can be expressed as follows:
TCðQ Þ ¼ Order Cost þ Expected Holding Cost þ Expected Backorder Cost þ Expected Lost Sales Cost Z Q Z QþM ðQ xÞf ðxÞdx þ cB ðx Q Þbðx Q Þf ðxÞdx ¼ cO Q þ cH 0 Q Z 1 þ cLS ðx QÞ½1 bðx Q Þf ðxÞdx: ð1Þ Q
The backorder rate function should satisfy the following properties. Property 1. M; 1Þ.
dbðxQ Þ dðxQÞ
< 0 8x 2 ½Q ; Q þ MÞ and bðx Q Þ ¼ 0 8x 2 ½Q þ
Fig. 1. Customer impatience and lost sale thresholds.
Property 2. limðxQ Þ!0þ bðx Q Þ ¼ 1. Property 3. limðxQ Þ!1 bðx Q Þ ¼ 0. Property 1 indicates that bðx Q Þ is a decreasing function in the number of observed shortages, which suggests that customers are more likely to wait for backorder replenishment if the backorder lead-time is short, and are less likely to wait if the lead-time is close to the lost sales threshold. The second part of Property 1 is a result of the fact that the probability of backlogging remains zero if x Q P M and Properties 2 and 3 ensure that bðx Q Þ 2 ½0; 1. We wish to determine the inventory level Q that minimizes TCðQ Þ given by Eq. (1), provided an optimum exists. It turns out that the convexity of Eq. (1) cannot be guaranteed in general, but the following theorem identifies a sufficient (but not necessary) condition for the existence of a unique minimizer. Before presenting the theorem, we introduce the following assumption.
Fig. 2. Customer impatience and backorder rates.
Assumption 1. cO < cB < cLS The inequalities cO < cB and cO < cLS are assumed in order to avoid trivial cases and are also reflective of practice. The inequality cB < cLS is also representative of practice, although there are some situations where backorder costs exceed short term lost sale costs such as lost revenue. Additionally, the latter inequality will enable us to prove the next theorem as well as other results presented later. Theorem 1. Let b : ðx Q Þ#½0; 1, where x Q 2 ½0; M, be a continuous and twice differentiable function. Then a sufficient condition for TCðQ Þ to be a convex function for all Q P 0 is
ðx Q Þ b00 ðx Q Þ 6 2b0 ðx QÞ; where
dbðxQ Þ dQ
0
b ðx Q Þ and
db0 ðxQ Þ dQ
Proof. Please refer to Appendix.
ð2Þ 00
b ðx Q Þ. h
3. A classification of customer impatience An obvious candidate for measuring customer impatience is the lost sales threshold, M. For example, if M1 < M2 (where M 1 and M2 are lost sales thresholds for customer 1 and customer 2, respectively), then it seems reasonable to conclude that customer 1 is more impatient than customer 2. However, Fig. 1 challenges this simplistic approach to characterizing customer impatience. In particular, customer 3 seems to be more impatient than customer 2 and customer 2 is sometimes more impatient than customer 1, even though M 1 < M2 < M3 . At the very least, Fig. 1 reveals that
M alone does not distinctively measure customer impatience and that the shape of bðx Q Þ should also be taken into account when distinguishing among varying degrees of customer impatience. Consider the case in which M 1 ¼ M2 ¼ M3 ¼ M as shown in Fig. 2. According to Fig. 2, customer 3 is more impatient than customer 2, and customer 2 is more impatient than customer 1. Fig. 2 also suggests that customer 1 is patient at first, but decreasingly patient because his rate of change in impatience is increasing. On the other hand, customer 3 is impatient at first, but decreasingly impatient. The relative impatience of these customers can be attributed to individual attitudes and personalities, but could also be interpreted as one person’s attitude with respect to waiting for different classes of products. For instance, customer 3 could represent an individual’s willingness to wait for a product, such as a computer mouse pad, in which a suitable substitute can be easily identified. In this case, the customer will most likely purchase the suitable substitute as opposed to waiting for backorder replenishment, even if the backorder lead-time is relatively short. Customer 1 could represent this same customer’s willingness to wait for a different product type, such as specialized computer software, in which a product of comparable utility cannot be as easily identified. In this case, the customer is more likely to wait for backorder replenishment, especially if the backorder lead-time is relatively short. These notions of impatience, increasing impatience, and decreasing impatience for describing customer behavior with respect to waiting for backorder replenishment can be defined more precisely using analogous concepts and terminology from utility theory.
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3.1. Utility theory framework Let uðyÞ represent the utility of the value y. Then the following result is fundamental to utility theory (for example, see Winkler, 2003): Theorem 2. Let uðyÞ be a continuous and twice differentiable function. Then a decision-maker is Risk-averse if and only if u00 ðyÞ < 0. Risk-neutral if and only if u00 ðyÞ ¼ 0. Risk-seeking if and only if u00 ðyÞ > 0. We will classify customers as risk-averse, risk-neutral, or riskseeking based on the backorder rate function bðx Q Þ as opposed to some utility function uðyÞ. In order to develop our classification scheme, we introduce the following definitions all of which are special cases of classical utility theory. Definition 1. A lottery with respect to waiting time, L ¼ ðt 1 ; p1 ; . . . ; t n ; pn Þ, consists of a set of waiting times ft 1 ; . . . ; tn g and a set of probabilities fp1 ; . . . ; pn g such that a decision-maker waits P for ti time units with probability pi , where i ¼ 1; . . . ; n and pi ¼ 1. Definition 2. The certainty equivalent with respect to waiting time of a lottery L ¼ ðt 1 ; p1 ; . . . ; tn ; pn Þ, denoted CEW ðLÞ, is the waiting time such that the decision-maker is indifferent between L and waiting for CEW ðLÞ with certainty. Definition 3. The risk premium with respect to waiting time of a lottery L ¼ ðt 1 ; p1 ; . . . ; tn ; pn Þ, denoted RP W ðLÞ, is defined as
RP W ðLÞ ¼ EW ½L CEW ðLÞ; where EW ½L is the expected value of the lottery L. Definition 4. If L ¼ ðt1 ; p1 ; . . . ; t n ; pn Þ is a lottery with respect to waiting time with n > 1, then a decision-maker is Risk-averse with respect to waiting time if and only if RPW ðLÞ < 0. Risk-neutral with respect to waiting time if and only if RPW ðLÞ ¼ 0. Risk-seeking with respect to waiting time if and only if RPW ðLÞ > 0. The difference in conventional utility theory and utility theory with respect to waiting time as described by Definitions 1–4 is noticeably observable in Definition 4. In particular, the definition of conventional risk aversion is RPW ðLÞ > 0 (see Winkler, 2003, for example), but the definition of risk aversion with respect to waiting time is RP W ðLÞ < 0. Similarly, the inequalities are also reversed in the definitions of conventional risk-seeking and riskseeking with respect to waiting time. The reason that the signs are reversed is due to the fact that unlike traditional utility theory where the argument of a given utility function represents gain, the argument of a utility function within the context of waiting time represents loss, i.e., negative gain. These ideas are illustrated through the following example. Consider two decision-makers, DM1 and DM2, who are presented with a lottery with respect to waiting time related to the time they wait to be seated in a restaurant. In particular, the lottery is defined as L ¼ ð50-minutes; 0:25; 10-minutes;0:75Þ. Suppose DM1 specifies CEW1 ðLÞ ¼ 12 minutes and DM2 specifies CEW2 ðLÞ ¼ 40 minutes. Since EW ½L ¼ 20 minutes, we have RP W1 ðLÞ ¼ 20 12 ¼ 8 minutes and RPW2 ðLÞ ¼ 20 40 ¼ 20 minutes. Since CEW1 ðLÞ is a shorter wait than EW ½L, DM1 considers L to be RPW1 ðLÞ ¼ 8 minutes better than
its expected value, which suggests that DM1 is influenced more by the possibility of the 10-minute wait than the risk of a 50-minute wait. Therefore, DM1 is risk-seeking. On the other hand, since CEW2 ðLÞ is a longer wait than EW ½L, DM2 considers L to be RPW2 ðLÞ ¼ 20 minutes better (i.e., 20 minutes worse) than its expected value, which suggests that DM2 is more influenced by the risk of a 50-minute wait than the chances of a 10-minute wait. Thus DM2 prefers to avoid the lottery’s risk and is therefore riskaverse. This example gives some insight into the logic behind Definition 4. The following theorem relates the backorder rate function, bðx Q Þ, to risk-averse, risk-neutral, and risk-seeking behavior with respect to waiting time. Theorem 3. Let b : y#½0; 1, where y 2 ½0; 1Þ, be a continuous and twice differentiable function. Then a decision-maker is Risk-averse with respect to waiting time if and only if b00 ðyÞ < 0. Risk-neutral with respect to waiting time if and only if b00 ðyÞ ¼ 0. Risk-seeking with respect to waiting time if and only if b00 ðyÞ > 0.
Proof. Please refer to Appendix.
h
Based on Theorem 3, customers 1, 2, and 3 in Fig. 2 are riskaverse, risk-neutral, and risk-seeking, respectively. Thus when customers all have the same lost sales threshold, M, the risk-averse customer is more patient (i.e., more likely to backorder) than the risk-neutral customer, and the risk-neutral customer is more patient than the risk-seeking customer. From the customer’s perspective, risk has to do with lead-time, price, and other sources of uncertainty associated with ordering from the supplier’s competitors. For example, the risk-averse customer is more likely to backorder in order to avoid the risk of lead-time uncertainty. From the perspective of time-dependent partial backlogging, customer impatience as it relates to backorder lead-time can be attributed to factors other than the above-mentioned risks of alternate supplier uncertainty. Such factors may include brand loyalty, information technology compatibility, the setup cost of temporarily changing suppliers, and other factors that have nothing to do with risk or uncertainty. Therefore, we introduce the following definitions in order to reflect this broader notion of customer impatience. Definition 5. Let b : y#½0; 1, where y 2 ½0; 1Þ, be a continuous and twice differentiable function. Then a decision-maker is said to be Patient with respect to waiting time if and only if b00 ðyÞ < 0. Neutral with respect to waiting time if and only if b00 ðyÞ ¼ 0. Impatient with respect to waiting time if and only if b00 ðyÞ > 0. Definition 5 implies that the backorder rate function for the patient customer has the same properties as the backorder rate function for a customer who is risk-averse with respect to waiting time. The difference is that the patient customer’s behavior is governed by a broader range of influences than the risk-averse customer’s behavior. Of course, these generalizations apply to the relationships between neutral and risk-neutral customers as well as between impatient and risk-seeking customers. We will classify customer impatience based on Definition 5 for the remainder of this paper. 3.2. Neutral customer behavior A backorder rate function bðx Q Þ that describes neutral behavior with respect to waiting time (see Definitions 1–5) should satisfy
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Properties 1–3 as well as the second part of Theorem 3. The following proposition defines a representative backorder rate function that satisfies these conditions. Proposition 1. Suppose x Q P 0. Then
xQ bðx Q Þ ¼ max 1 ;0 ; M
ð3Þ
satisfies Properties 1–3 and the second part of Theorem 3. Proof. Since
dbðx Q Þ ¼ dðx QÞ
(
M1 < 0; x 2 ½Q; Q þ MÞ 0;
x 2 ½Q þ M; 1Þ
A backorder rate function bðx Q Þ that describes patient behavior with respect to waiting time should satisfy Properties 1–3 as well as the first part of Theorem 3. The following proposition defines a representative backorder rate function that satisfies these conditions.
2
dðx Q Þ2
Note that the parameter a controls the rate at which bðx Q Þ given by Eq. (4) decreases. In particular, a1 > a2 implies that customer 1 is more impatient than customer 2. The condition a < M2 suggests that Theorem 5 can only guarantee convexity if bðx Q Þ does not decrease too quickly as x Q approaches M. 3.4. Patient customer behavior
:
Property 1 holds. Also,
d bðx Q Þ
8x Q 2 A [ B. In order to verify that TCðQ Þ is convex 8Q P 0, we need to show that convexity holds 8x Q 2 Rþ , which actually reduces to showing that A [ B ¼ Rþ . Let C ¼ ð2a ; MÞ. Then A [ B [ C ¼ Rþ . However, since the condition a 6 M2 is given, we have that C ¼ ; and A [ B [ C ¼ A [ B ¼ Rþ . h
¼ 0;
shows that the second part of Theorem 3 holds. Now
Proposition 3. Suppose x Q P 0. Then
xQ lim max 1 ; 0 ¼ 1; xQ !0 M
( bðx QÞ ¼
shows that Property 2 holds, and
cos
xQ 2M
p ; x 2 ½Q ; Q þ MÞ
0;
x 2 ½Q þ M; 1Þ
;
ð5Þ
xQ ; 0 ¼ maxf1; 0g ¼ 0; lim max 1 xQ !1 M
satisfies Properties 1–3 and the first part of Theorem 3.
shows that Property 3 holds.
Proof. The proof is similar to the proof of Proposition 1. Refer to Appendix for details. h
h
The next result indicates the existence of Q that minimizes TCðQ Þ given by Eq. (1) when bðx Q Þ is defined as in Proposition 1. Theorem 4. Suppose bðx Q Þ is defined as by Eq. (3). Then TCðQ Þ given by Eq. (1) is a convex function. 0
00
Proof. If x 2 ½Q ; Q þ MÞ, then b ðx Q Þ ¼ M1 and b ðx Q Þ ¼ 0. Thus the inequality ðx Q Þb00 ðx Q Þ 6 2b0 ðx Q Þ given by Eq. (2) reduces to M2 P 0. Since the last inequality always holds, it follows that bðx Q Þ satisfies the condition in Theorem 1. If x 2 ½Q þ M; 1Þ, then bðx Q Þ ¼ 0, which also satisfies the condition in Theorem 1. Therefore, Theorem 1 guarantees that TCðQ Þ is a convex function for all Q P 0 whenever bðx Q Þ is defined as in Eq. (3). h 3.3. Impatient customer behavior A backorder rate function bðx Q Þ that describes impatient behavior with respect to waiting time should satisfy Properties 1–3 as well as the third part of Theorem 3. The following proposition defines a representative backorder rate function that satisfies these conditions. Proposition 2. Suppose x Q P 0 and a > 0 is a constant. Then
( bðx Q Þ ¼
eaðxQÞ ; x 2 ½Q ; Q þ MÞ ; 0; x 2 ½Q þ M; 1Þ
ð4Þ
satisfies Properties 1–3 and the third part of Theorem 3. Proof. The proof is similar to the proof of Proposition 1. Refer to Appendix for details. h Theorem 5. Suppose bðx Q Þ is defined by Eq. (4) and a 6 M2 . Then TCðQ Þ given by Eq. (1) is a convex function. Proof. Let A ¼ ð0; 2aÞ and B ¼ ðM; 1Þ. It is straightforward to show that Eq. (2) reduces to x Q 2 A. Since bðx Q Þ ¼ 08x Q 2 B and Eq. (2) holds, it follows from Theorem 1 that TCðQ Þ is convex
Theorem 6. Suppose bðx Q Þ is defined by Eq. (5). Then TCðQ Þ given by Eq. (1) is a convex function. Proof. If x 2 ½Q ; Q þ MÞ, then
xQ p ; 2M 2M p2 xQ 00 : cos p b ðx Q Þ ¼ 2M 4M 2
b0 ðx Q Þ ¼
p
sin
Thus the inequality given by Eq. (2) reduces to
ðx Q Þp xQ p 6 1: cot 2M 4M
ð6Þ
If we let z ¼ ðp=2MÞðx Q Þ, then Eq. (6) becomes ðz=2Þ cotðzÞ 6 1, where z 2 ð0; p=2Þ. Since cotðzÞ > 0 and z=2 < 0 for z 2 ð0; p=2Þ, we have ðz=2Þ cotðzÞ < 0 6 1, which implies that the inequality given by Eq. (6) holds. Thus it follows that bðx Q Þ satisfies the condition in Theorem 1. If x 2 ½Q þ M; 1Þ, then bðx Q Þ ¼ 0, which also satisfies the condition in Theorem 1. Therefore, Theorem 1 guarantees that TCðQ Þ is a convex function for all Q P 0 whenever bðx Q Þ is defined as in Eq. (5). h 4. Sensitivity analysis and management insights This section investigates the effects that various problem parameters have on optimal ordering decisions. The following example data was used for the analysis:
X Nð500; 250Þ; cLS ¼ 100;
cO ¼ 50;
M ¼ 1000:
cH ¼ 20;
cB ¼ 75; ð7Þ
Note that our analyses suggest that it is generally not possible to express the optimal order quantity for any of the three cases in closed form. Therefore, Mathematica™ was used to obtain order quantities for the numerical experiments associated with Figs. 3–9. Based on the results shown in Figs. 3–5, we observe the following:
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Observation 1. Let Q P ; Q N ; and QI be the optimal order quantity associated with the patient, neutral, and impatient cases, respectively. Then Q I P Q N P Q P . This observation is intuitive. It is reasonable to expect the optimal order quantity to be an increasing function of customer impatience.
480
420
Observation 2. Optimal order quantities are always decreasing in cH and M. These results are intuitive and consistent with the results reported in Lodree (2007). Observation 3. According to Fig. 5, differences in optimal order quantities among the three customer types are decreasing functions of the ratio cB =cLS . From a managerial perspective, this means that it is important for the decision-maker to distinguish between customer types when there is a significant difference between cB and cLS , i.e., if cB =cLS is close to 0. If, on the other hand, cB =cLS 1, then there is no need to distinguish between customer types. This result is indeed intuitive. If there is little difference between the unit backorder cost and unit lost sales cost, then the cost associated with shortages is insensitive to whether or not shortages are backorders or lost sales. Similar conclusions can be drawn from Eq. (A.1) in Appendix. In particular, the effect of the backorder rate function bðx Q Þ on the total expected cost function TCðQ Þ becomes more pronounced as the difference cLS cB increases, or equivalently, as the ratio cB =cLS decreases. Thus it is reasonable to expect increasing differences in Q among the three cases as bðx Q Þ assumes a more dominant role in TCðQ Þ. Conversely, as cB =cLS approaches 1, the effect of bðx Q Þ diminishes and TCðQ Þ becomes the newsboy problem. Therefore, the optimal order quantities will approach the newsboy result for each of the three customer types as cB =cLS ! 1. Observation 4. Differences in optimal order quantities among the three customer impatience behaviors are decreasing in cH and increasing in M. However, differences in optimal order quantities are influenced more by the ratio cB =cLS than by cH and M.
360
Neutral Patient Impatient
300 800
900
1000
1100
1200
Fig. 4. The effect of M on the optimal order quantity.
900
Neutral Patient
700
Impatient
500
300 0.15
0.35
0.55
0.75
0.95
Fig. 5. The effect of cB =cLS on the optimal order quantity, with cB ¼ 75.
From a practical perspective, Observation 4 suggests the increasing importance of distinguishing between customer types as holding costs decrease and as the lost sales threshold increases. The underlying intuition is that as holding costs become less significant, the effect of expected shortage costs is magnified. Similarly,
20.2
10,000
450
Assume Neutral
400 350
Neutral
300
Patient
250
Impatient
Assume Impatient
7500
15.2
5000
10.1
2500
5.1
200 150 100 50 0
0 20
120
220
320
Fig. 3. The effect of cH on the optimal order quantity.
420
0.15
0.35
0.55
0.75
0.95
Fig. 6. The effect of cB =cLS on EVCPI if customer is patient.
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3000
6.1
25,000
41.0
Assume Neutral Assume Patient
Assume Impatient
20,000
32.9
Assume Impatient 2000
4.0
1000
15,000
24.7
10,000
16.5
5000
8.2
2.0
0
0 0.15
0.35
0.55
0.75
0.95
Fig. 7. The effect of cB =cLS on EVCPI if customer is neutral.
4500
11.3
Assume Neutral Assume Patient
0.03
0.06
0.09
0.12
0.15
Fig. 9. The effect of cB =cLS on EVCPI if customer is patient.
the patient backorder rate function given by Eq. (5). To illustrate this point, consider the data in Eq. (7) and suppose x Q ¼ 300. Then using Eqs. (3)–(5), bP ð300Þ bN ð300Þ ¼ 0:191 and bN ð300Þ bI ð300Þ ¼ 0:698, where bN ðx Q Þ; bI ðx Q Þ, and bP ðx Q Þ are Eqs. (3)–(5), respectively. Since bN ð300Þ bI ð300Þ > bP ð300Þ bN ð300Þ, it is reasonable to expected that Q I Q N > Q N Q P . 5. Value of customer patience information
3000
7.6
1500
3.8
Suppose the decision-maker can conduct a study to obtain more information about the impatience behavior of a customer or market segment. This section explores the expected value of conducting such a study. To carry out the analysis, let us first assume that the supplier currently orders Q N , which is the optimal order quantity associated with the neutral customer. If the customer is actually patient, then the expected value of a market study is
V ¼ TC P ðQ N Þ TC P ðQ P Þ;
0 0.15
0.35
0.55
0.75
0.95
Fig. 8. The effect of cB =cLS on EVCPI if customer is impatient.
increasing M magnifies the effect of expected shortage costs. Since differences in customer types are reflected only through shortage costs in the decision-maker’s cost structure, the result of magnified shortage costs is larger differences in optimal order quantities. These observations can also be explained by observing Eq. (A.1) in Appendix. Finally, note that changes in cH and M do not significantly affect differences in optimal order quantities, particularly relative to the changes caused by the ratio cB =cLS (compare Figs. 3 and 4 to Fig. 5). Observation 5. Differences in optimal order quantities between the impatient and neutral cases are always greater than the differences in optimal order quantities between the patient and neutral cases. This is necessarily a consequence of the specific backorder rate functions studied in this paper for the patient and impatient cases. More specifically, the impatient backorder rate function given by Eq. (4) is more different than the neutral case when compared to
where TC P ðÞ and Q P are the expected cost function and optimal order quantity, respectively, for the patient customer. From this perspective, V can be defined as the expected value of customer patience information (EVCPI). In general, let V ij equal the expected value of the market study if the supplier orders Q i and the customer patience profile is actually j, where i; j 2 fP; N; Ig (patient, neutral, impatient). Also, let TC i ðÞ and Q i represent the expected total cost function and optimal order quantity, respectively, for case i 2 fP; N; Ig. If i ¼ j, then clearly V ij ¼ 0 for all i 2 fP; N; Ig. Otherwise if i–j, then
V NP ¼ TC P ðQ N Þ TC P ðQ P Þ; V NI ¼ TC I ðQ N Þ TC I ðQ I Þ; V PN ¼ TC N ðQ P Þ TC N ðQ N Þ; V PI ¼ TC I ðQ P Þ TC I ðQ I Þ; V IP ¼ TC P ðQ I Þ TC P ðQ P Þ; V IN ¼ TC N ðQ I Þ TC N ðQ N Þ: The example data from Eq. (7) was used to construct the graphs shown in Figs. 6–9. The following insights can be interpreted from these figures: Observation 6. The expected value of customer patience information (EVCPI) is an increasing function of M, and a decreasing function of cH .
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This observation is intuitive since differences in optimal order quantities among the three customer types are increasing in M, and decreasing in cH (see Observation 2). Note that this observation is supported by the above-mentioned intuition, and that the graphs supporting this idea have not been included in this paper. Observation 7. EVCPI is always a decreasing function of the ratio cB =cLS . The fact that EVCPI is decreasing in cB =cLS is intuitive given that differences in optimal order quantities among the customer types are decreasing in cB =cLS (see Observation 3). Observation 8. In general, EVCPI only seems significant whenever cB =cLS < 0:5. According to Figs. 6–8, the percentage increase in optimal expected cost is always less than 5% whenever 0:5 < cB =cLS 6 1. In fact, one could argue that EVCPI is only significant whenever 0 < cB =cLS < 0:15, although not necessarily (see Fig. 9). The largest percent increase of approximately 33% occurs when the customer is patient, the supplier assumes the customer is impatient, and cB =cLS ¼ 0:03. In this case, the corresponding value of EVCPI is over $20,000. If, on the other hand, the customer is patient, the decision-maker assumes the customer is neutral, and cB =cLS ¼ 0:15, then the percent increase in expected cost and EVCPI are approximately 2% and $1000, respectively. Thus the customer’s actual impatience behavior and the supplier’s assumption regarding the customer’s impatience behavior also have a significant impact on EVCPI, which motivates the next observation. Observation 9. If the supplier has no information regarding the customer’s impatience behavior, then the supplier should assume that the customer is neutral. If the customer is neutral, then assuming a neutral customer profile is obviously the best decision, which results in EVCPI = 0. Now suppose the customer is actually patient. Then according to Fig. 6, assuming a neutral customer profile leads to lower values of EVCPI (and lower percentage expected cost increase) than assuming an impatient customer profile. Similarly, Fig. 8 suggests that assuming a neutral profile when the customer is actually impatient leads to better results than assuming the customer is patient.
the expected benefit of knowing the relative impatience of customers or a market segment. Our results suggest that EVCPI is most significant when there is a large difference between the lost sales cost cLS and backorder cost cB . Our results also indicate that EVCPI increases as the difference cLS cB increases and as the lost sales threshold M increases. This paper can be extended in several ways. A useful extension would be to relax Assumption 1 such that cB > cLS is a possibility. This cost structure is known to happen in practice based on one of the author’s interactions with the production manager of a major manufacturing firm. In particular, this production manager’s primary objective in the event of a shortage is to maintain a positive rapport with the client (especially a major client). He is much less concerned with the short term cost inefficiencies associated with emergency procurement, production, and delivery in light of the long term implications related to compromising a business relationship. Therefore, this manager would replenish backorders at unit cost cB , even in the event that cB far exceeds cLS . This manufacturing firm’s strategic approach to managing shortages suggests another promising extension of this paper, namely an extension that entails both short term and long term management of inventory shortages in a multi-period setting. Another useful extension would be to generalize the relationship between the backorder lead-time, L, and number of shortages, X Q . We assumed that L ¼ X Q , but lead-time functions with more practical relevance should be considered. We would expect consistency with the results described in this paper provided L is an increasing function of X Q . Similarly, the development of other backorder rate functions and the applicability of our findings based on these functions (including their interactions with more general lead-time functions) also warrants exploration. The corresponding analyses in terms of verifying Theorem 1 and Properties 1–3 could be slightly or substantially more difficult than this paper depending on how LðX Q Þ and bðLðX Q ÞÞ are selected. Appendix A. Proof of Theorem 1 TCðQ Þ given by Eq. (1) can be written as
TCðQÞ ¼ cO Q þ cH
Z
Q
ðQ xÞf ðxÞdx þ cLS
0
þ ðcB cLS Þ
Z
1
ðx Q Þf ðxÞdx
Q QþM
ðx Q Þbðx Q Þf ðxÞdx:
ðA:1Þ
Q
6. Summary and future research directions Shortages are often represented as either backorders or lost sales in inventory models and in practice. The time-dependent backlogging approach to characterizing inventory shortages acknowledges that there is some probability associated with whether or not a customer will backorder a shortage, and that this probability (i.e., the backorder rate) is related to the lead-time associated with replenishing the outstanding backorder. While the majority of the research literature considers time-dependent backlogging within the context of continuous review models with deterministic demand, this paper studies time-dependent partial backlogging in the single period inventory problem with stochastic demand. The backorder rate function is used to formally classify customers as either patient, neutral, or impatient with respect to their willingness to wait for the supplier to replenish shortages. Representative backorder rate functions are presented, and the existence of a unique order quantity that minimizes total expected costs is shown for each case. Sensitivity analysis experiments are conducted to examine the similarities and differences in optimal order quantities among the three customer types as a function of various problem parameters. Additionally, the expected value of customer patience information (EVCPI) is defined in order to assess
Z
Recalling from Theorem 1 that b0 ðx Q Þ is defined as the derivative of bðx Q Þ with respect to Q, it follows that the first and second order derivatives of TCðQ Þ are
dTCðQ Þ ¼ cO þ cH dQ
Z
Q
f ðxÞdx cLS
0
þ ðcB cLS Þ
Z
Z
1
f ðxÞdx
Q QþM
½bðx Q Þ þ ðx QÞb0 ðx Q Þf ðxÞdx;
Q
ðA:2Þ 2
d TC 2 ðQ Þ dQ 2
¼ ðcH þ cLS Þf ðQ Þ þ ðcLS cB Þf ðQ Þ þ ðcLS cB Þ
Z
Q þM
½2b0 ðx Q Þ ðx Q Þb00 ðx QÞf ðxÞdx:
ðA:3Þ
Q
The first term in Eq. (A.3) is obviously nonnegative. Since cLS cB > 0 based on Assumption 1, it follows that the second term is also nonnegative. It also follows from Assumption 1 that if the integrand of the third term is nonnegative, we can be sure that the third term is nonnegative. Therefore, a sufficient condition for optimality is given by
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2b0 ðx Q Þ ðx Q Þb00 ðx QÞ P 0 for all x 2 ½Q ; Q þ MÞ: ðA:4Þ Appendix B. Proof of Theorem 3 For illustration, we will show that a decision-maker is riskseeking if and only if his back order rate function is convex. The proofs of parts 1 and 2 of Theorem 3 are similar. ð)Þ By Definitions 3 and 4, we have RPW ðLÞ > 0 and EW ½L CEW ðLÞ > 0. Now recall the second axiom of utility (e.g., Winkler, 2003), which states if a decision-maker is indifferent between a payoff r 1 and a lottery L ¼ ðr 2 ; p; r 3 ; 1 pÞ, then uðr1 Þ ¼ puðr2 Þþ ð1 pÞuðr3 Þ. Similarly, if a decision-maker is indifferent between a lottery with respect to waiting time given by L ¼ ðt 1 ; p1 ; ; t n ; pn Þ and waiting for CEW ðLÞ with certainty, then the likelihood of backordering associated with the lottery L is equal to the likelihood of backordering associated with the certainty equivalent CEW ðLÞ. Hence, above axiom implies that b½CEW ðLÞ ¼ p1 bðt 1 Þ þ . . . þ pn bðt n Þ, where p1 þ þ pn ¼ 1. That is, b½CEW ðLÞ is the convex hull of ft 1 ; . . . ; tn g. Since bðyÞ is a decreasing function by Property 1 and EW ½L CEW ðLÞ > 0, it follows that bðEW ½LÞ bðCEW ðLÞÞ < 0, or equivalently,
bðp1 t 1 þ þ pn t n Þ < p1 bðt 1 Þ þ þ pn bðt n Þ:
ðB:1Þ
Since Eq. (B.1) holds, it follows that bðyÞ is a convex function and b00 ðyÞ > 0 (see, for example, Jeter, 1986, p. 217). ð(Þ Since bðyÞ is a convex function, it follows that Eq. (B.1) holds by Proposition 1 shown in Jeter (1986, p. 217). Since CEW ðLÞ is the convex hull of ft1 ; . . . ; t n g and p1 t1 þ þ pn t n ¼ EW ½L, Eq. (B.1) becomes bðEW ½LÞ bðCEW ðLÞÞ < 0. Based on the last inequality, we know that EW ½L CEW ðLÞ > 0 by Property 1. Therefore, by Definitions 3 and 4, the latter inequality implies risk-seeking. h Appendix C. Proof of Proposition 2 Since
dbðx Q Þ ¼ dðx QÞ
(
aeaðxQÞ < 0; x 2 ½Q ; Q þ MÞ 0;
x 2 ½Q þ M; 1Þ
:
Property 1 holds. Also, 2
d bðx Q Þ dðx Q Þ2
¼ a2 eaðxQ Þ > 0;
shows that the third part of Theorem 3 holds. Now
lim eaðxQ Þ ¼ 1;
xQ !0þ
shows that Property 2 holds, and
lim eaðxQÞ ¼ 0;
xQ !1
shows that Property 3 holds. h Appendix D. Proof of Proposition 3 The derivative is computed as
dbðx Q Þ ¼ dðx Q Þ
(
p sin 2M 0;
xQ 2M
p ; x 2 ½Q ; Q þ MÞ x 2 ½Q þ M; 1Þ
:
If we let z ¼ ðp=2MÞðx Q Þ, then dbðx Q Þ=dðx Q Þ becomes ½z=ðx Q Þ sin z, where z 2 ð0; p=2Þ. Since z=ðx Q Þ < 0 and sin z > 0 in this interval, we have dbðx Q Þ=dðx Q Þ < 0 which shows that Property 1 holds.Also, 2
d bðx Q Þ dðx Q Þ2
¼
p2 4M 2
cos
xQ p < 0; 2M
shows that the first part of Theorem 3 holds. Now
xQ lim þ max cos p ; 0 ¼ 1; 2M xQ !0 shows that Property 2 holds, and
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