Optimal ordering for a probabilistic one-time discount

European Journal of Operational Research 244 (2015) 803–814

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Optimal ordering for a probabilistic one-time discount Yaron Shaposhnik a, Yale T. Herer b,∗, Hussein Naseraldin c,1 a

Operations Research Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel c Department of Industrial Engineering and Management, ORT Braude College, P.O. Box 78, Karmiel 2161002, Israel b

a r t i c l e

i n f o

Article history: Received 23 June 2014 Accepted 10 February 2015 Available online 05 March 2015 Keywords: Inventory One-time discount Special purchasing Economic order quantity,

a b s t r a c t We develop a model for a one-time special purchasing opportunity where there is uncertainty with respect to the materialization of the discounted purchasing offer. Our model captures the phenomenon of an anticipated future event that may or may not lead to a discounted offer. We analyze the model and show that the optimal solution results from a tradeoff between preparing for the special offer and staying with the regular ordering policy. We quantify the tradeoff and find that the optimal solution is one of four intuitive policies. We present numerical illustrations that provide additional insights on the relationship between the different ordering policies.

1. Introduction A basic premise in most inventory management models is that the unit-price is constant. Suppliers, however, are often under pressure to increase sales or reduce inventory levels. As a result, they tend to offer discounts so that retailers will be inclined to order larger quantities than usual. From a retailer’s point-of-view, this may constitute an opportunity to benefit from higher sales margins. For example, consider a retail chain selling a commodity, the behavior of which is relatively static. As the Olympics approach, the retailer identifies a special purchasing opportunity caused by potential overstocking of the commodity at suppliers operating in the vicinity of the hosting city. Such opportunity entails uncertainty, and its materialization is not guaranteed. The retailer faces a tradeoff between reducing stocks prior to the event to benefit from the discounted offer and risking higher inventory costs if the offer does not materialize. It is this tradeoff that we wish to resolve in our work. The supplier also may not know for sure if the discount will be possible. Moreover, she may be reluctant to share information as it will obligate her. Thus, from the retailer’s perspective, quantifying the potential savings that might be gained from preparing for a special purchasing opportunity can shed light on the effort and cost that it is worthwhile to invest in obtaining this information. In Lee, So, and Tang (2000), the authors show the impact the variability of demand ∗

Corresponding author. Tel.: +972 4 823 4423, 972 4 829 4423. E-mail addresses: [email protected] (Y. Shaposhnik), [email protected], [email protected] (Y.T. Herer), [email protected] (H. Naseraldin). 1 Tel.: +972 50 240 1407; fax: +972 4 990 1852.

http://dx.doi.org/10.1016/j.ejor.2015.02.020 0377-2217/© 2015 Elsevier B.V. All rights reserved.

© 2015 Elsevier B.V. All rights reserved.

has on the value of information. In our paper, we show that even in an otherwise deterministic setting, the possible occurrence of a one-time unit-price discount is influential. We use the economic order quantity (EOQ) model as our modeling framework. Even though the model relies on a set of simple and restrictive assumptions, its simplicity and robustness (in the sense that a relatively large deviation in the estimation of the model’s parameters results in a relatively small deviation in the objective function value) makes it a widely used tool. We develop a model that depicts a system in the EOQ setting in which the retailer predicts that at a known point of time, a one-time special purchasing opportunity may occur with a certain probability. We analyze the optimal replenishment policy in such a setting, and characterize its structure. We then show that the optimal replenishment policy is one of four intuitive policies, which can be found and evaluated using closed-form expressions. Finally, we provide additional insights on the relation between the different policies through numerical illustrations. The unit-price discount is extensively studied in the literature (see, e.g., Goyal, Srinivasan, & Arcelus, 1991 and Ramasesh, 2010). The setting we study in this research is unique in the sense that we tackle a one-time probabilistic discount, which can happen at a known time in the future. The next section provides a comprehensive review of this topic. The rest of this work is organized as follows: After reviewing the literature in Section 2, we formulate our problem in Section 3. In Section 4 we analyze the model and derive an optimal solution. In Section 5 the results are visualized and studied using numerical illustrations. In Section 6 we discuss several extensions to our work. Finally, in Section 7, we highlight the main findings and suggest

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further research directions. To aid readability, the technical proofs are presented in Appendix A. 2. Literature review The EOQ model is introduced by Harris (1913) and further developed by Wilson (1934). An elaborate review of the model and its well-known extensions can be found in Nahmias (2008). One of the first works on inventory models with discounts is that of Friend (1960) who study random purchasing opportunities with reduced fixed ordering costs. In their model, demand and discounted offers arrive as a Poisson processes, and the system is modeled as a Markov chain. Naddor (1966) introduce multiple discount models based on the EOQ model, including an immediate price increase. The author assumes that an immediate price increase occurs when the inventory level is zero, which prompts a one-time opportunity to purchase inventory at a discounted unit price. While the body of literature on unit-price discounts in inventory models is quite broad, Friend (1960) and Naddor (1966) exemplify the two main lines of work. The first, on recurrent discounts, usually operates in an infinite horizon setting, and the second, on one-time purchasing opportunity, often occurs in the EOQ environment. We begin with the latter, which is the focus of our work, and for completeness, we also note some of the salient studies on recurrent discounts. One-time discounts appear in the literature mainly in one of three forms: (1) instantaneous discounts in the present or at a future time, (2) discounts over an interval of time, and (3) discounts in the form of an extended credit period. Ardalan (1988) studies a model that is similar to that of Naddor (1966) but allows an arbitrary level of inventory when the special opportunity takes place. Taylor and Bradley (1985) extend the model proposed by Naddor (1966) to account for an announced price increase at a future moment in time. Lev and Weiss (1990) consider the case of a finite horizon EOQ model with a single price change. They base their results on the work of Schwartz (1972), who investigate the EOQ model in a finite horizon setting. They use the total cost as the objective function. Tersine and Schwarzkopf (1989) study a model with an announced special purchasing opportunity for a nonrestrictive time duration in an infinite time horizon setting. AullHyde (1996), Cardenas-Barron, Smith, and Goyal (2010), Al Kindi and Sarker (2011), and Taleizadeh, Pentico, Aryanezhad, and Ghoreyshi (2012) investigate a one-time discount when shortages are allowed. Several authors study similar problems assuming that costs are discounted (for example, see Aucamp & Kuzdrall, 1989; Grubbstrom & Kingsman, 2004). Other types of unit-price discount models include the work of Davis and Gaither (1985) who study an EOQ model with a onetime offer of delayed payments. Ardalan (1994) looks at a one-time discount where the demand is sensitive to the discounted price. Arcelus, Shah, and Srinivasan (2003) look at different one-time price incentives for a model for perishable items. Other work on deteriorating or imperfect inventory include those by Chang, Lin, and Ho (2011), Allah Taleizadeh, Mohammadi, Eduardo Cardenas-Barron, and Samimi (2013), and Kevin Hsu and Yu (2009). An alternative discount model is introduced in Sari, Rusdiansyah, and Huang (2012), where there are several discount offers, with different discounts at different times that can be utilized at most once. Yang, Ouyang, Wu, and Yen (2012) study a one-time discount model where there is capacity constraint, while Peter Chu, Chen, and Thomas Niu (2003) look at a model where the discounted purchased quantity can be selected from a restricted set of values. Sarker and Al Kindi (2006) study multiple types of one-time discounts, including one that is both a one-time and quantity discounts. For an excellent review of one-time discount models, the reader is referred to Ramasesh (2010) and Goyal et al. (1991). One challenge when measuring the performance of policies in an infinite horizon is that one-time disturbances in inventory behavior

do not affect the long-run average cost. This means that the standard EOQ model objective function does not capture these disturbances. The most common approach to overcoming this difficulty is by optimizing the savings comparing to the EOQ model. This can be done in several ways, the most common being to consider the costs in a finite time interval until the system reverts to the EOQ settings, and approximating the EOQ costs during this interval. The conventional method of approximating the EOQ costs is to multiply the duration of the finite interval by the average cost the EOQ model. We denote this method as the Differential Costs approach (also known as the Average Cost Approach in Ramasesh, 2010). The need to approximate the EOQ costs comes from the fact that after the system reverts to the EOQ model, the difference in costs is not constant but rather cyclic and depends on the time when costs are compared. Approximating the EOQ costs, and looking at a finite interval resolves this problem. Two other methods for approximating the costs are offered by Yanasse (1990), who use the worst-case value of the cyclic difference, and Huang, Kulkarni, and Swaminathan (2003) and Lim and Rodrigues (2005), who consider the average difference over the cycle. An alternative approach is to consider discounted costs, a method that is known as the Discounted Cash Flow approach. This method results in a finite cost expression that can then be minimized; see also the discussion in Baker and Hanna (1987). For recurrent discounts, Hurter and Kaminsky (1968b) extend the model presented by Friend (1960) to incorporate discounts on the unit price. Hurter and Kaminsky (1968a) study a different discount model where the system randomly transitions between two states, with high and low costs. They consider a policy that is a mixture of a base-stock policy, and an (s, S) policy for the high and low intervals, respectively. Kalymon (1971) study a periodic review inventory model in which the unit price changes according to a Markov chain, whereas in Golabi (1985), a periodic review model with random costs and fixed demand is analyzed. Silver, Robb, and Rahnama (1993) develop effective heuristics for the model presented by Hurter and Kaminsky (1968b). Zheng (1994) prove the optimality of a threshold-based policy for a continuous review inventory system with Poisson process demand, while Moinzadeh (1997) investigate an EOQ model with random deal offerings. Finally, we note the work by Chaouch (2007) that generalizes Hurter and Kaminsky (1968a) to allow for demand that is affected by cost, and also low- and high-price intervals of different lengths. In this study, we generalize existing work by incorporating a probabilistic one-time discount offer. This generalization is usually overlooked by assuming that the discount offer occurs without notice. Our model captures an ubiquitous phenomenon with considerable economical impact that has not received much attention in the literature, where traditionally it is assumed that one-time discounts materialize with certainty. Moreover, our analysis offers a unified closed-form solution to the problem that is both insightful and intuitive, and can be easily implemented in practice. 3. Model Our model is best thought of as the classical EOQ model with an additional assumption of a one-time special purchasing opportunity. The EOQ parameters are the demand rate D, fixed order cost K, unit cost c, and holding cost rate h. Shortages are not allowed. We add to the classical EOQ model a one-time special purchasing opportunity at a future time ts that may materialize with a known probability p. Without loss of generality, we assume that inventory is zero at the beginning of the planning horizon. The case in which the initial inventory is greater than zero is discussed in Section 6.1. Table 1 summarizes our notation. We use ∗ to denote optimal values. The classical EOQ objective function is the long-run average cost. Thus, any deviation from an ordering policy over a finite-time horizon will not affect the objective function value. This means that a

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Table 1 Notation. Symbol

Description

Input parameters D Demand rate [units/time unit] K Fixed cost for placing an order [$] c Unit price [$/unit] h Inventory holding cost [$/unit/time unit] Unit price with a special purchasing opportunity [$/unit] cs Time that the special purchasing opportunity may ts materialize [time unit] p Probability that the special purchasing opportunity materializes (0 ≤ p ≤ 1) Auxiliary and decision variables  The EOQ order quantity (Q EOQ = 2DK/ h) [units] Q EOQ m Number of orders made in the time interval [0, ts ) (integer) On-hand inventory at ts before an order is placed [units] qs Order size for orders placed in the time interval [0, ts ) [units] Q0 Quantity of inventory purchased at ts when there is a special Qs purchasing opportunity [units] (Q, Q EOQ ) A distance function between a zero and equal-ordering policy from the EOQ policy. This expression captures the long-run average cost of a zero and equal-ordering policy relative to the EOQ policy. x Total costs from time t1 (inclusive) until time t2 (exclusive) costπ [t1 , t2 ) [$] Expected total savings of policy π over the EOQ long-run ETSπ average cost (until depletion of inventory for the first time after ts ) [$]

one-time special purchasing opportunity is ‘invisible’ to the classical EOQ objective function. Intuitively, however, it seems that a discount, even if it is one-time, improves performance. No supply chain manager with near zero inventory would pass up a discount. Clearly, there is a discrepancy between what we intuitively feel is important and the mathematical objective function. A few methods for overcoming this difficulty exist. We refer the reader to Section 2 for more details. Here we take the differential cost approach, which is the most common approach, and the one that we found most appropriate for our analysis. We define our objective function to be the difference between all of the incurred costs and the standard EOQ average cost until inventory runs out for the first time after the time of the special purchasing opportunity ts . At that time the model reverts to the standard EOQ model (see, e.g., Naddor, 1966 and Gallego, 1994). In a deterministic environment, an ordering policy specifies all of the planned orders—sizes and timings. Without loss of generality, these decisions are determined at the beginning of the planning horizon. With a futuristic probabilistic one-time special purchasing opportunity, since there are two possible outcomes for the event, there are also two scenarios that completely describe the ordering policy. We will denote by S the scenario in which the special purchasing opportunity materializes, and by N the scenario in which it does not. Each scenario yields different order sequences—sizes and timings. We refer to the description of all orders under both scenarios as the ordering policy or simply policy. The costs of this ordering policy will be compared to the cost of the standard EOQ policy. We have two possible streams of expenses, each associated with a different scenario. We will denote by costxπ [t1 , t2 ) the total cost of policy π for scenario x ∈ {N, S} for the time period from t1 (inclusive) unS til t2 (exclusive). Note that if t2 ≤ ts , then costN π [t2 , ts ) = costπ [t2 , ts ) since ts is the time at which we find out if the special purchasing opportunity has materialized. We define costxπ [t, t)  0. We denote by Tsx the time duration from ts until the next inventory depletion under scenario x ∈ {N, S} , except for the case when the inventory is zero at ts and there is no special purchasing opportunity,

Fig. 1. Example of an arbitrary ordering policy with two scenarios that deviate from each other for the first time at time ts .

in which case TsN  0. The longer it takes for the inventory to run out after ts , the larger the value of Tsx will be. If the inventory does not run out, we define the value of Tsx to be infinite (Tsx = ∞). Note also that for a given policy, Tsx is known at the beginning of the planning horizon. We denote Ts as the stochastic time duration from ts until the moment of the next inventory depletion. Ts is a random variable since its value depends on a probabilistic event. We use qs to refer to the on-hand inventory at ts before an order is placed. An example of an arbitrary ordering policy is illustrated in Fig. 1. Our differential cost approach compares ordering policies according to two criteria (in the following order): 1. Expected long-run average cost. 2. Expected savings at time ts + Ts denoted by ETSπ , which is calculated relative to the long-run average cost of the EOQ model. An optimal ordering policy is minimal with respect to the first criterion and maximal with respect to the second. Intuitively, we have an EOQ model with a single event that affects costs for a limited period of time. Consequently, we would first like to achieve a long-run average cost as in the EOQ model—we can do no better. Second, we would like to order based on short term savings due to the special purchasing opportunity, i.e., maximize the savings over the EOQ policy. At time ts + Ts , the system reverts to a simple EOQ problem in which the EOQ policy is optimal. EOQ , is The long-run √ average cost of the EOQ policy, denoted by G equal to cD + 2KDh and the optimal ordering quantity is Q EOQ =  2DK/ h. Letting TSxπ be the total savings under scenario x ∈ {N, S} for policy π , we have:

TSxπ = (ts + Tsx )GEOQ − costxπ [0, ts + Tsx ),

(1)

ETSπ = (1 − p)TSNπ + pTSSπ .

(2)

In the next section, we analyze our model and show that the optimal solution can be found by comparing the easily calculated expected total savings of four easily computed policies. Thus, we show that the optimal solution for any problem instance can be found in negligible time. 4. Model analysis and solution In this section we analyze the model presented in Section 3 and derive the optimal policy. We begin by narrowing down the search

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space for an optimal solution by investigating properties of the optimal solution. Using these properties, we present the structure of the optimal policy and the control variables that define it. Finally, we find the optimal values of the control variables. We begin with some general observations, which are related to the long-run characteristics of the optimal ordering policy. 4.1. Observations related to the horizon Our investigation begins with the observations related to the planning horizon. Observation 1. The long-run average cost of any policy under any scenario is equal to the long-run average cost of the same policy under the same scenario starting at any time T. Moreover, the expected long-run average cost of any policy is equal to the expected long-run average cost of the same policy starting at any time T. Indeed, the occurrences until time T have no effect on the long-run average cost. We thus denote the long-run average cost of policy π x , without regard to the time at which under scenario x ∈ N, S by ACπ we start. The implication of Observation 1 is that any event before a finite time (e.g., a special purchasing opportunity) has no effect on the long-run average cost. Observation 2. The long-run average cost of any policy π under scenario x ∈ {N, S} and the expected long-run average cost of any policy π N ≥ GEOQ , AC S ≥ GEOQ , and are greater than or equal to GEOQ ; i.e., ACπ π ACπ ≥ GEOQ . From Observation 1 we learned that the special purchasing opportunity cannot reduce the long-run average cost. This combined with the optimality (in terms of long-run average cost) of the EOQ policy yields Observation 2. 4.2. Properties related to order sizes

Fig. 2. An arbitrary ordering policy that satisfies zero and equal-ordering before time ts .

1. What is the size of orders, denoted by Q0 [units], before time ts ? 2. For a given Q0 , what should the order size be at time ts under scenario S? Let this quantity be denoted as Qs . In Proposition 3 we answer the second question and in Section 4.3 we answer the first. For each value of Q0 there is an associated value of qs , the inventory at time ts . Intuitively, we may choose to order Q0 = Q EOQ before time ts so that at time ts we will have low inventory with the aim of taking advantage of the special purchasing opportunity, if it materializes. Thus, we have a tradeoff between preparing for scenario N, where we keep costs down by keeping Q0 close to Q EOQ , and preparing for scenario S, potentially benefiting from the special purchasing opportunity by keeping qs low so Qs can be large. Proposition 3. If the inventory level at time ts is below the critical value qcs  (c − cs )D/ h, then it is optimal to order-up-to Q EOQ + qcs ; otherwise, it is optimal not to order. That is, the optimal value of Qs as a function of qs is,

 EOQ + qcs − qs Q

qs < qcs

We now proceed to investigate properties related to the ordering policy, when there is no discount at the time of order.

Qs∗ (qs ) =

Proposition 1. [Zero ordering] In the optimal ordering policy, when purchasing cost is regular, orders are placed only upon stock depletion.

Note that the result of Diament (1980) is a special case of our result for qs = 0. Proposition 3 characterizes the ordering decision at time ts for any qs . While there may be several ways to arrive at time ts for a given qs , the ordering decision at time ts is independent of how one arrives at time ts with qs . Reinterpreting Proposition 3, we see that the optimal ordering policy at time ts under scenario S, is an (s, S) policy (with (s, S) = (qcs , qcs + Q EOQ )) and we never order less than Q EOQ at time ts . Note that this proposition allows us to solve the model for the special case of ts = 0. Consequently, unless otherwise mentioned, we assume that ts is greater than zero.

The proof is based on the proof of zero-ordering for the standard EOQ model. This proof along with the proof of most of the subsequent properties can be found in Appendix A. Note that the regular purchasing cost relates to any moment of time other than ts and to time ts under scenario N. Corollary 1. Any ordering policy in which the inventory does not deplete after ts is sub-optimal. Based on Corollary 1, we will only consider policies with finite values of Tsx . The values of TsN and TsS are characterized in the next corollary. Corollary 2. Under any optimal policy, TsN = qs /D and TsS = (Qs + qs )/D. Corollary 3. The EOQ policy is optimal for the time range [ts + Ts , ∞). Without loss of generality, we only consider policies that satisfy zero-ordering, and which order the economic order quantity, Q EOQ , from time ts + Ts and thereafter.

0

Otherwise.

Definition 1. An admissible ordering policy is an ordering policy that cannot be dismissed as non-optimal based on previously proved propositions. Currently, Definition 1 refers to Propositions 1–3, and will progressively include Propositions 4–6. Corollary 4. For an admissible ordering policy, TsS is given by the following expression:



qs /D =  EOQ  Q + qcs /D

qs ≥ qcs

Proposition 2. [Equal-ordering] Under the optimal ordering policy, all orders made before ts are of equal size.

TsS

An example of an ordering policy complying with the properties discussed thus far is presented in Fig. 2. Note that at time ts an order is placed under scenario S, but not under scenario N. Now only two questions remain:

Based on the fact that when qs ≥ qcs , we do not order at time ts , even when the special purchasing order materializes, any deviation from the EOQ ordering policy before time ts has no benefit. This idea is formalized in the following proposition.

Otherwise.

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Proposition 4. Any ordering policy for which qs ≥ qcs , and that is not the EOQ ordering policy, is not optimal.

Lemma 1. The total savings of an admissible ordering policy π under scenario N is:

4.3. The optimal ordering policy

TSNπ =

We now explore the structure of the optimal ordering policy, building on the characterization discussed thus far. Recall that an ordering policy is characterized by all the orders made at all points of time for each scenario. Before continuing with our analysis we review what we already know about the optimal ordering policy. We note that the orders can be separated into three ranges: 1. Orders before ts : All orders made before ts are made when the inventory is zero (Proposition 1) and are of equal size (Proposition 2). Therefore, the entire ordering policy until ts can be described by Q 0 . 2. Orders at ts : If the special purchasing opportunity does not materialize (scenario N) and we have on-hand inventory, then we do not order (Proposition 1). If it does materialize, we order as specified in Proposition 3. 3. Orders after ts : It is optimal not to make any order until inventory runs out (Proposition 1) and use the EOQ policy thereafter (Corollary 3). We thus see that the entire optimal ordering policy can now be described by two decision variables: Q 0 and Q s . However, since Q 0 uniquely determines qs (see Eq. (4)), and qs uniquely determines Q s (Proposition 3), Q 0 alone can be used to describe the entire optimal ordering policy. Defining m to be the number of orders before ts , we have:



m=

ts Q0 D





=

Dts , Q0

(3)

qs = mQ 0 − Dts .

(4)

To maximize our objective function (Eq. (2)), which includes the costs that are due to orders made before ts , we quantify how the order size before ts affects the costs. Consequently, we introduce a measure of the distance of an ordering quantity from the EOQ ordering quantity. This measure is used by Roundy (1985). Definition 2. The distance of a zero and equal-ordering policy with order size Q from the EOQ policy is defined as the ratio of their longrun average costs excluding the purchasing costs, and is equal to:

(Q, Q EOQ ) 

1 2



Q Q EOQ

+

Q EOQ Q

 .

(5)

Observation 3. The function (Q, Q EOQ ) is strictly convex in Q and has a minimum at Q = Q EOQ , where (Q EOQ , Q EOQ ) = 1. Observation 4. The total cost of an admissible ordering policy  π under scenario N in the time interval 0, ts + TsN is equal to √     (Q0 , Q EOQ ) 2DKh ts + TsN + cD ts + TsN . This observation holds since the total cost of the ordering policy  under scenario N in the time interval 0, ts + TsN is the cost of an integer number of complete and identical ordering cycles. The cost of all cycles is the average cycle cost of one cycle, multiplied by the total length of all the cycles, i.e.,

costNπ [0, ts + TsN ) =

K + cQ0 + Q0 D

Q2 h 2D0

  ts + TsN

(6)

     = (Q0 , Q EOQ ) 2DKh ts + TsN + cD ts + TsN . (7) This observation holds since, by definition, there is an integer number of full cycles in the time interval [0, ts + TsN ). Observation 4 leads us to the following two lemmas.



=−

  2DKh ts + TsN (1 − (Q0 , Q EOQ ))

(8)

hm (Q0 − Q EOQ )2 . 2D

(9)

Recall that m denotes the number of orders made before time ts (see Eq. (3)). Note that TSN π is always non-positive, maximal at Q0 = Q EOQ , where the total savings is zero, and decreases as Q0 moves farther away from Q EOQ . Without a special purchasing opportunity the EOQ policy has the lowest long-run average cost. In hopes of taking advantage of the special purchasing opportunity one may want to deviate from the EOQ policy. However, if it does not materialize any change from the EOQ policy can only result in a loss (negative TSN π ). compares the cost of This is due to the fact that our definition of TSN π any policy to the EOQ policy. We now turn our attention to calculating the savings when the special purchasing opportunity materializes (scenario S). Lemma 2. The total savings of an admissible ordering policy π under scenario S is:

⎧ 2 2 hm  h  EOQ ⎪ ⎪ − + qcs − qs Q0 − Q EOQ − K + Q ⎪ ⎪ 2D 2D ⎪ ⎪ ⎨ qs < qcs TSSπ = 2  ⎪ hm ⎪ ⎪ Q0 − Q EOQ − ⎪ ⎪ ⎪ 2D ⎩ qs ≥ qcs .

Corollary 5. The expected total savings of an admissible ordering policy π is:

⎧ 2 2 hm  h  EOQ ⎪ ⎪ − + qcs − qs Q0 − Q EOQ − pK + p Q ⎪ ⎪ 2D 2D ⎪ ⎪ ⎨ qs < qcs ETSπ = 2 ⎪ hm  ⎪ ⎪ Q0 − Q EOQ − ⎪ ⎪ ⎪ 2D ⎩ qs ≥ qcs .

(10) Note that when qs ≥ qcs both the total savings (Lemma 2) and the expected total savings (Corollary 5) are non-positive and negative when Q0 = Q EOQ . This is because if one deviates from the EOQ policy in such a way that when the special purchasing opportunity materializes no order is placed, then no benefit is reaped from the special purchasing opportunity. This would not be a wise policy which is reflected in the value of the savings. We now look at the control variables in Eq. (10) (Q0 and qs ). Similar to our comment after Lemma 1, to maximize ETS, we would like Q0 to be close to the EOQ order size, thus reducing the deviation from the EOQ policy before time ts (the term (hm/2D)(Q0 − Q EOQ )2 ). However, we also would like qs to be as small as possible, thus maximizing the discount utilization if the special opportunity materializes (the term p(h/2D)(Q EOQ + qcs − qs )2 ). Based on these observations we formulate the following proposition, which describes a dominance relationship π between policies. Let qπ s and Qs denote qs and Qs under policy π , respectively. Proposition 5. Given two admissible ordering policies π1 and π2 , if the following conditions are satisfied: π

π

1. qs 2 ≥ qs 1 π π 2. (Q0 2 , Q EOQ ) ≥ (Q0 1 , Q EOQ ), then ETSπ2 ≤ ETSπ1 and we say that policy π1 dominates policy π2 . Corollary 6. For the optimal policy π , the inventory level at ts is always less than or equal to the inventory level at ts achieved with the EOQ policy EOQ ). (i.e., qπ s ≤ qs

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Fig. 3. The inventory level at time ts , qs , as a function of the order size before ts ,Q0 .

Next, we define Q0LB and Q0UB . In Proposition 6 below, we prove that these values are indeed lower and upper bounds on Q0∗ . Definition 3. Define Q0LB and Q0UB as follows:

Q0LB 

Q0UB

Dts

Dts /Q EOQ

,

⎧ EOQ ⎪ ⎨Q  Dts ⎪ ⎩ Dts /Q EOQ

(11)

Dts < Q EOQ otherwise,

(12)

and denote the ordering policies for which Q0 = Q0LB , Q0 = Q0UB , and Q0 = Q EOQ by πLB , πUB , and πEOQ , respectively. Note that when Dts /Q EOQ is an integer, Q0LB = Q0UB = Q EOQ , and that when Dts /Q EOQ < 1, Q0LB = Dts , and Q0UB = Q EOQ . The difference between the EOQ policy and policy πEOQ is that under the EOQ policy, Q EOQ units are ordered if and only if inventory runs out. In contrast, under policy πEOQ , the special opportunity can be exploited if it materializes. The intuitive motivation behind policies πLB and πUB is that policy πLB (and policy πUB , correspondingly, except when Q UB = Q EOQ ) is obtained from the EOQ ordering policy by decreasing (increasing) the order size before ts until the first time that the inventory level at ts reaches zero. For policy πLB , decreasing the order size until depletion at ts will not change the number of orders made before ts . For policy πUB , increasing the order size before ts until the first inventory depletion at ts will decrease the number of orders made before ts by one (as one order will be “squeezed out”). Corollary 7. If Dts > Q EOQ , then the ordering policy πLB outperforms ordering policy πUB if and only if (Q0LB , Q EOQ ) ≤ (Q0UB , Q EOQ ).

Proposition 6. The optimal value of Q0 belongs to the following set:

   Q0UB . Q0∗ ∈ Q0LB , Q EOQ

Proof. Let Q0π denote the order size before ts under the optimal policy π . We prove the proposition by showing that for every order size Q0π  outside the interval [Q0LB , Q EOQ ] {Q0UB }, we can construct an equally good or better solution inside the interval. If Q0π satisfies Q0π < Q0LB , then policy π is dominated by πLB ((Q0π , Q EOQ ) > (Q0LB , Q EOQ ) and π qs LB = 0). Proposition 5 implies that ETSπ ≤ ETSπLB . That is, policy πLB is as good as any ordering policy with a smaller order quantity. Similarly, if Q0π > Q0UB , then π is dominated by πUB . If Q EOQ < Q0π < Q0UB , then policy π is dominated by policy πEOQ .  Graphical interpretation of the solution set Let us now look at the solution set graphically. In Fig. 3, qs is plotted as a function of Q0 ; any point on the graph corresponds to an ordering policy. Horizontal lines represent: the critical inventory level ). Vertical (qcs ), and the inventory level at ts under the EOQ policy (qEOQ s lines (except the one that passes through point C) represent values of Q0 , for which qs = 0. Note that between two such vertical lines, the number of orders made before ts is the same and equal to that of the policy associated with the vertical line to the left. Moreover, the vertical line that passes through point C is associated with the EOQ ordering policy πEOQ . As we increase the order size before ts (i.e., move right on the Q0 axis), the inventory at ts increases, until a point is reached where the number of orders before ts decreases by one, and the inventory at ts drops to zero. Proposition 5 can be interpreted graphically in Fig. 3 as follows: Given two points (i.e., two policies), if the second is higher than the first and geometrically further from the EOQ policy on the

Y. Shaposhnik et al. / European Journal of Operational Research 244 (2015) 803–814

horizontal axis, then the first point is better. Policy πLB , is represented on the graph by point A, i.e., the point (Q0LB ,0); policy πEOQ by point

C, i.e., the point (Q EOQ , qEOQ ); and policy πUB by point D, i.e., the point s (Q0UB ,0). Using this graphical representation it is easy to “graphically prove” Proposition 6. Point A dominates all points to its left. Point D dominates all points to its right and point C dominates all points that are located strictly between points C and D. The optimal solution thus lies between points A and C (inclusive), or at point D. (as the case in Proposition 4 shows that if qcs is lower than qEOQ s Fig. 3), then all policies associated with points between points B and C (exclusive) cannot be optimal. Next, we address the question of finding the optimal solution between points A and C, i.e., Q0LB ≤ Q0 ≤ Q0EOQ . Proposition 7. For a non-dominated ordering policy in which qs < qcs , the expected total savings as a function of Q0 is:

ETS(Q0 ) =

 h  2  Q0 m pm − 1 2D

 Q EOQ h D − mp Q0 Q EOQ + ts D + (c − cs ) − D h p

 h h D 2 −m (Q EOQ )2 + p − pK. ts D + Q EOQ + (c − cs ) 2D 2D h (13)

While in general the number of orders before time ts , i.e. m, is a function of Q0 , in the range of interest (i.e., Q0LB ≤ Q0 ≤ Q0EOQ ) the value of m is fixed and equal to Dts /Q EOQ . Therefore, the function ETS(Q0 ) is quadratic, i.e. is of the form aQ02 + bQ0 + c. Observe that when p Dts /Q EOQ − 1 = 0, ETS(Q0 ) is linear; when p Dts /Q EOQ − 1 < 0, ETS(Q0 ) is concave; and when p Dts /Q EOQ − 1 > 0, ETS(Q0 ) is convex. Definition 4. Let LE stand for a local extremum. Q0LE (and πLE ) is defined as follows:

Q0LE 

pts D − (1 − p)Q EOQ + p(c − cs ) Dh   . p Dts /Q EOQ − 1

(14)

Q0LE is obtained by differentiating Eq. (13) with respect to Q0 and setting the result to zero. Note that since Q0LE is an extreme point of a mathematical function it may be negative. Of course, in this case there would be no practical meaning to this “solution”. The next proposition characterizes when the extreme point Q0LE has a practical meaning of interest. Proposition 8. Among all non-dominated ordering policies that satisfy Q0LB ≤ Q0 ≤ Q EOQ , the optimal order size is Q0LB , Q EOQ , or Q0LE . Moreover, Q0LE is relevant only if Q0LB ≤ Q0LE ≤ Q EOQ and p Dts /Q EOQ < 1. Corollary 8. The optimal solution belongs to the following set:

{Q0LB , Q0LE , Q EOQ , Q0UB }.

We can now present our main result, a simple algorithm for finding the optimal solution for the problem: Algorithm 1 Find the optimal order size before ts Calculate the order sizes pertaining to the admissible policies:Q0LB (using Eq. (11)), Q0LE (using Eq. (14)), Q EOQ (see Table 1), and Q0UB (using Eq. (12)).   2: If Q0LE < Q0LB or Q0LE > Q EOQ or p Dts /Q EOQ > 1, then discard Q0LE . 1:

Calculate the expected total savings for each of the (three or four) policies (using Eq. (13)). 4: The optimal order size before ts is the order size calculated in step 1 associated with the largest expected total savings calculated in step 2. 3:

809

Table 2 Base case model parameters. Model parameters

EOQ policy

D

K

c

h

cs

ts

p

Q EOQ

T EOQ

qEOQ s

GEOQ

5

100

5

0.1

3.5

50

0.3

100

20

50

35

Table 3 The non-dominated solutions of the base case. Policy

Q0

m

qs

Qs

ETS

LB EOQ UB LE

83.33 100 125 −275

3 3 2 N/A

0 50 0 N/A

175 125 175 N/A

53.54 49.37 16.87 N/A

Algorithm 1 tells us that the optimal reaction to a one-time special purchasing opportunity is one of the following intuitive policies: 1. Q0LB or Q0UB – reduce the inventory level at time ts to zero (either by decreasing or increasing the order size Q EOQ ) in order to take full benefit of the potential discount (at the expense of increased replenishment and holding costs before ts ). 2. Q EOQ – avoid the risk of a non-materialized purchasing opportunity by ordering according to the EOQ policy. 3. Q0LE – reduce the order quantity Q EOQ to obtain a positive inventory level at time ts that optimizes the tradeoff between the above strategies. Note that using the fact that the function ETS(Q0 ) is quadratic in the domain [Q0LB , Q EOQ ], and based on the feasibility of Q0LE , we can derive conditions that allow us to reduce the number of admissible solutions from the current four possible solutions. These conditions are simple to check, and so is the evaluation of the points. We discuss them briefly in Section 6.2. 5. Numerical illustration We now present some insights about the dynamics of the problem that were obtained by performing a numerical study. We begin by describing the scope of the study, and then summarize our main findings. We first devise and analyze a ‘base case’ instance (Table 2). We will then vary each model parameter separately in order to learn how they affect each admissible policy πLB , πEOQ , πUB , and πLE (and consequently, the optimal policy). The parameters of the base case are selected to be plausible. Extreme values will be explored via changes to the base case. Table 3 presents the admissible solutions for the base case. We see that the optimal ordering policy is πLB , and that the policy πLE is not feasible. We also see that the order-up-to level, if the special opportunity materializes, is 175 units, i.e., 75percent higher than the EOQ order quantity. We will now vary each problem parameter over a wide range. The associated admissible policies for select parameters is found in Fig. 5. Note that policy πLE is only considered where it is relevant, i.e., Q0LB ≤ Q0LE ≤ Q0EOQ . Based on these graphs we observe the following: 1. qs and ETS versus Q0 . Fig. 4 illustrates how Q0 affects the inventory at time ts (the dotted line) and the objective function (the dashed line). We see the dominance of policies πLB , πUB , and πEOQ (Proposition 6). We also see the discontinuity of the objective function, which explains why there is more than one candidate for the optimal policy. 2. ETS versus ts . Fig. 5a shows how the time of the special purchasing opportunity affects the performance of the admissible policies. As ts increases, we observe that the optimal policy goes backand-forth between policies πLB and πUB . This indicates that the

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Fig. 4. The inventory level at time ts , qs , and the expected total savings, ETS, as functions of the order size before time ts (Q0 ).

optimal solution is sensitive to changes in ts . On the other hand, the relative difference between the expected total savings of the two policies decreases as ts grows, so it is not as critical to choose the best between the two. Also note that the expected total savings is bounded and that this bound is quickly approached as ts grows. That is, the additional savings enabled by knowing that the event would take place far in the future is small. 3. The number of orders before ts . Another consequence of increasing ts is that the number of orders before ts increases. We see in Fig. 5a, that as ts grows, policies πLB and πUB are the only candidates for optimality. This can be explained by the fact that as the number of cycles before ts increases, Q0LB and Q0UB approach QEOQ , which reduces the penalty from deviating from the EOQ policy. Hence, these two policies, which maximally benefit from the special purchasing opportunity, are being minimally penalized for deviating from the EOQ policy. We thus expect policies πEOQ and πLE to be optimal only when the number of order before tS is relatively small. Note that a large number of orders before ts can also be caused by other parameters, i.e., large demand, small fixed order cost, etc. 4. ETS versus p. Fig. 5b shows the objective value for each admissible policy as a function of the probability that the special purchasing opportunity materializes. We see that the objective values of policies πLB , πUB , and πEOQ are linear in p. The reason is that the order size before ts under the three policies does not depend on p, and the dependence of the savings in p is linear (Eq. (13)). Note that the policy πLE is only relevant for a small range of p, and that as p grows, Q0LE shifts from Q0EOQ (p = 0) to Q0LB (p = 0.096), after which it is no longer relevant. 5. ETS versus c − cs . When plotting the dependence of ETS on c and cs (Fig. 5c and d), we notice that we obtain graphs that are nearly mirror images of each other. This can be explained by examining Eq. (13), where we see that ETS depends on the two parameters only through the difference between them. Similar to p, c, and cs do not directly affect the order quantities under policies πLB , πUB , and πEOQ , although they do affect the ordering threshold at time ts (qcs ), which in turn affects the ordering decision of policy πEOQ at time ts .

6. Sensitivity analysis. Finally, we note that the sensitivity analysis has a practical value for making decisions when parameters are difficult to estimate. For example, instead of expending excessive resources to estimate p, we can use sensitivity analysis to find a threshold value of p that would cause us to deviate from the EOQ policy. We can then use it to guide our efforts in estimating p. 6. Extensions We now discuss some simple extensions of our work. 6.1. Initial inventory at the beginning of the planning horizon In the model, we assumed that the inventory level at the beginning of the planning horizon is zero. If we relax this assumption, we can see that all the properties of the optimal solution (Section 4) hold; specifically, due to zero-ordering (Proposition 1), we find that if inventory runs out before time ts , this case is equivalent to starting the planning horizon after the initial inventory runs out. Otherwise it is optimal to wait until ts and then respond according to Propositions 1 and 3. 6.2. Reducing the final set of admissible solutions In Corollary 8, we see that the optimal policy is one of four possibilities. However, this number can be reduced even further. Whenever Dts > Q EOQ , the two ordering policies πLB and πUB both have zero inventory at time ts . This fact is used in Corollary 7 and can be used to dismiss the policy with the larger value of (Q0π , Q EOQ ). We can also use the fact that the objective function is quadratic in Q0 (Eq. (13)) to dismiss sub-optimal solutions. Recall that whether ETS(Q0 ) is convex or concave is determined by the expression pm − 1. When qcs ≥ qEOQ , one can show that if pm − 1 ≥ 0, policy πLB has s greater expected savings than policy πEOQ , and policy πLE is not relevant. If pm − 1 < 0, then policy πEOQ has no greater expected savings than either πLB or πLE . We obtain that if qcs ≥ qEOQ , that is, if the EOQ s policy takes advantage of the purchasing opportunity when it materializes, the EOQ policy is never uniquely optimal. When qcs < qEOQ , s

Y. Shaposhnik et al. / European Journal of Operational Research 244 (2015) 803–814

811

Fig. 5. The expected total savings as a function of the model parameters: (a) ts ; (b) p; (c) c; (d) cs .

then ETS(Q0EOQ ) = 0, and the EOQ policy would be optimal only if the savings under the other policies are negative. Therefore, policy πEOQ is uniquely optimal only when it intersects with the EOQ policy, which happens when there are no savings under any other policy. 6.3. Unit-cost dependent inventory holding cost rate One of our assumptions is a fixed inventory holding cost rate. While we feel that this assumption is the most reasonable one, other related research assumes cost dependent inventory holding costs, that is, h = Ic (where I is the interest rate). For example, Ardalan (1988) consider a special purchasing opportunity at the current time, i.e., ts = 0, p = 1, hs = Ics ). The difference between the two assumptions comes into play when calculating the holding cost of units purchased under the special purchasing opportunity. Incorporating this lower holding cost in our work we find that Propositions 1, 2, and 5 still hold. For the other propositions, we get slightly different expressions, but the overall structure remains. For example, we obtain that the optimal order-up-to level at time ts is (Q EOQ + (c − cs )D/ h)(h/ hs ), and that the critical  value under the new model is qcs = (Q EOQ + (c − cs )D/ h)(h/ hs ) − 2KD/ hs (compare to Proposition 3; note how the two expressions reduce to our previous results when hs = h).

7. Summary In this work, we investigate the impact of information about a probabilistic one-time unit-price discount on the optimal inventory replenishment policy. We quantify this information by calculating the savings due to adapting the inventory policy at the current moment. We prove the following properties of the optimal solution. The replenishment policy at the time of the possible special purchasing opportunity, if it materializes, is an (s, S) policy. We calculate the values for both thresholds. In addition, some EOQ properties are preserved: (1) zero-ordering, except possibly when the special purchasing opportunity materializes, and (2) equal-ordering both before and after the possible special purchasing opportunity. After the time of the possible special purchasing opportunity, this replenishment quantity will be the standard EOQ replenishment quantity. Based on these properties, we characterize the ordering policy as a function of the replenishment quantity at the beginning of the horizon. Combining these properties with the symmetry of the basic EOQ cost function, we are able to use dominance criteria to reduce the number of initial order quantities to four possible values, each with its own intuitive meaning. Closed-form expressions for these values and their corresponding expected total savings are presented. In addition to an algorithm to identify the optimal ordering policy,

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we present a numerical study that provides additional insights into the model dynamics. Finally, we discuss some simple extensions to the model. Future research While exploring the influence of the probabilistic special purchasing opportunity, we assume a momentary drop in price at a knownin-advance time in the future. Natural questions to ask are: What if there was more than one purchasing opportunity? What if the exact time of the discount is not known with certainty? Or, what if we are informed earlier about the materialization of the special opportunity? How would the optimal policy adapt to these changes?

Case 2: Q2 > Q1 . The analysis is similar to Case 1.



A.3. Proposition 3 Proof. In order to find the optimal order size Qs∗ we investigate the expected total savings as a function of Q s for policies that satisfy Propositions 1 and 2, assuming that an order takes place at ts under scenario S. We later consider if an order should be made.

   qs  qs  ETSπ (Q s ) = (1 − p) GEOQ ts + − costNπ 0, ts + D D 

   qs + Qs qs + Qs EOQ S +p G ts + − costπ 0, ts + D D

Acknowledgments The authors would like to express their gratitude to Edward A. Silver who first proposed some of the research questions addressed in this paper. Appendix A. Proofs

(A.1)

 qs  Qs + pGEOQ − costNπ [0, ts ) = GEOQ ts + D D   qs qs + Qs − (1 − p)costNπ [ts , ts + ) − p · costSπ ts , ts + D D (A.2)

A.1. Proposition 1 [Zero ordering] Proof. We prove the proposition by contradiction. Let us assume that the proposition is incorrect. Therefore, there is an optimal policy π and at least one scenario (denoted as scenario x), in which an order is placed at the regular purchasing price at a moment of time (denoted by t) at which the inventory level is positive. We denote as q and Q the inventory level at time t before an order is placed and the order size at time t, respectively. We construct an alternative policy π and show that it is preferable. Policy π is identical to policy π except that under scenario x, the order that is made at time t is postponed to time t + δ . Policies π and π are identical except for time interval [t, t + δ ].   At time t and during t, t + δ , policy π accumulates fewer costs due to the fact that the inventory level is smaller. At time t + δ (under policy π ), we order the same amount that was ordered at time t under policy π (and pay the exact same regular purchasing price for it). After t + δ we accumulate the same costs under both policies, and since at t + δ the total costs are lower for policy π , the cumulative costs for time beyond t + δ are lower for policy π . For the proof to be complete we show that a feasible value of δ exists. Clearly, δ < q/D. Moreover, for the postponement not to interfere with the next order (at time denoted by t2 ), we add the constraint: δ < t2 − t. For the case when t < ts , we also add the constraint δ < ts − t so that the postponement does not interfere with the special purchasing opportunity. Any δ > 0 that is less than these positive quantities will suffice.  A.2. Proposition 2 [Equal-ordering] Proof. We prove by contradiction. Let us assume that the statement is incorrect. Consequently, there is an optimal ordering policy π in which two consecutive orders of different size are made in [0, ts ). Let us denote the two order sizes and timings as Q1 , Q2 , t1 , t2 . Note that the inventory level before the orders arrive at times t1 and t2 is zero (according to Proposition 1). There are two cases. Case 1: Q1 > Q2 We construct a policy π and show that it results in a higher expected total savings. Policy π is identical to policy π except that the order at time t1 is of size Q1 −  , and the next order will be made at t2 −  /D (instead of time t2 ), and of size Q2 +  . Clearly, policy π has slightly lower holding costs than policy π , in particular, ETSπ − ETSπ = Dh  [Q1 − Q2 −  ] > 0. Choosing any  such that  < Q1 and  < Q1 − Q2 completes the proof of Case 1.

 qs  Qs + pGEOQ − costNπ [0, ts ) = GEOQ ts + D D

 q2 (qs + Qs )2 . − (1 − p)h s − p K + cs Qs + h 2D 2D

(A.3)

S Since costN π [0, ts ) = costπ [0, ts ) (both scenarios are identical until ts ), we replace costSπ [0, ts ), with costN π [0, ts ) in Eq. (A.2). In Eq. (A.3) we detail the costs of the time intervals [ts , ts + qs /D) under scenario N and [ts , ts + (qs + Qs )/D) under scenario S. To find the value of Qs that maximizes the expected total savings, ETSπ (Q s ), we note that Eq. (A.3) is a quadratic function in Qs with Qs2 having a negative coefficient; therefore, it has a global maximum that can be obtained simply by solving

pGEOQ dETSπ (Q s ) (qs + Qs ) − pcs − ph =0 = dQ s D D Q EOQ − qs + (c − cs )

D − Qs = 0. h

(A.4)

(A.5)

By solving Eq. (A.5), we find that Qs∗ = Q EOQ + (c − cs ) Dh − qs = Q EOQ + qcs − qs . Thus, if we choose to order, we order-up-to Q EOQ + qcs . If the inventory level at time ts (before ordering) is greater than Q EOQ + qcs , then the optimal order quantity is zero. We now check under what conditions it is better not to order at ts . For this we compare two ordering policies that coincide under scenario N, but differ under scenario S as follows: the first policy orders Qs∗ at ts (policy π ) and the second does not order at ts (policy π ).

q    qs  s − costSπ ts , ts + ETSπ − ETSπ = p GEOQ D D

  ∗ qs + Qs∗ EOQ qs + Qs S −G + costπ ts , ts + D D

(A.6)





ph D −(qs )2 + qs 2Q EOQ + 2(c − cs ) = 2D h 

2 D D − (c − cs ) . − 2Q EOQ (c − cs ) h h

(A.7)

Note that the decision to order at ts in the case when the special purchasing opportunity materializes, depends only on qs . We denote

Y. Shaposhnik et al. / European Journal of Operational Research 244 (2015) 803–814

by η(qs ) the expression inside the square brackets of Eq. (A.7) and present it as a function of qs .



η(qs )  −(qs )2 + qs 2Q EOQ + 2(c − cs )

D h





−

(c − cs )

D h

2

D − 2Q EOQ (c − cs ) . h

A.7. Proposition 5 Proof. Let us look at:

ETSπ1 − ETSπ2 = (1 − p)TSNπ1 + pTSSπ1 − (1 − p)TSNπ2 − pTSSπ2 = (1 − p)(TSNπ1 − TSNπ2 ) + p(TSSπ1 − pTSSπ2 ).

(A.8)

It is thus optimal to place an order if and only if η(qs ) is negative. η(qs ) is a concave quadratic expression in qs and is negative when qs < qcs or when qs > qcs + 2Q EOQ . Since we do not order when qs > Q EOQ + qcs , we find that ordering at ts is better than not ordering at ts if and only if qs < qcs . Combined with the fact that we order-up-to Q EOQ + qcs , we obtain the desired result. 

Proof. Using Proposition 3, we know that an order is not made at ts under scenario S, and therefore, the EOQ policy is strictly better. 

TSNπ1 − TSNπ2 =

π

Proof.

=

    2DKh ts + TsN 1 − (Q0 , Q EOQ ) 

= h ts +

=−

TsN





2Q0 EOQ 1 Q − 2Q0 2



Q EOQ Q0

2

(A.9)

(A.10)

Q2 + 0 Q0



2 hm  Q0 − Q EOQ . 2D

(A.11)

(A.12)

 A.6. Lemma 2

TSSπ = GEOQ (ts + TsS ) − costSπ [0, ts + TsS )

 Q EOQ + qcs = GEOQ ts + − costSπ [0, ts ) − costSπ D   Q EOQ + qcs ts , ts + D

(A.13)

π

π

π

Case 2: qcs ≥ qs 2 ≥ qs 1 Both policies order at ts . From Lemma 2:

2 h  EOQ TSSπ1 − TSSπ2 = TSNπ1 − K + + qcs − qs Q 2D

2  h  EOQ N Q − TSπ2 − K + + qcs − qs 2D 2 h   EOQ N N 1 = TSπ1 − TSπ2 + Q + qcs − qπ s 2D 2   EOQ 2 . − Q + qcs − qπ s π

2 1 N c S Since TSN π1 ≥ TSπ2 and qs ≥ qs ≥ qs ≥ 0, it is easy to see that TSπ1 ≥ S TSπ2 .

π

π

Case 3: qs 2 ≥ qcs ≥ qs 1 π π2 Since qcs ≥ qs 1 , according to Proposition 3, TSSπ1 ≥ TSN π1 since qs ≥ N N qcs , TSSπ2 = TSN π2 . In addition, we know that TSπ1 ≥ TSπ2 ; therefore, TSSπ1 ≥ TSSπ2 . 

Proof. Using Corollary 5: (A.14)

(A.16)

4 and splitting the costs of time interval [0, ts +   Using Corollary Q EOQ + qcs /D) into two time intervals, we get Eq. (A.14). Next, we substitute the costs and arrive at Eq. (A.16). 

(A.19)

A.8. Proposition 7

  2  

Q0 Q EOQ + qcs (qs )2 − m K + cQ0 + h + h = GEOQ ts + D 2D 2D   Q EOQ + qc 2 s h − K − cs Q EOQ + qcs − qs − (A.15) 2D 2 2 hm  h  EOQ Q0 − Q EOQ − K + Q + qcs − qs . 2D 2D

(A.18)

N Since 2 ≥ 1 ≥ 1 and qs 2 ≥ qs 1 , we have TSN π1 ≥ TSπ2 . Let us now consider the total savings under scenario S. There are π π π π π three cases to consider: qs 2 ≥ qs 1 ≥ qcs , qcs ≥ qs 2 ≥ qs 1 , and qs 2 ≥ π qcs ≥ qs 1 : π π Case 1: qs 2 ≥ qs 1 ≥ qcs S N S According to Lemma 2, TSSπ1 = TSN π1 and TSπ2 = TSπ2 ; thus, TSπ1 ≥ S N N TSπ2 (since we showed earlier that TSπ1 ≥ TSπ2 ).

π

Proof. For policy π in which qs ≥ qcs , by Proposition 3, Qs∗ = 0, and c TSSπ = TSN π . For policy π , in which qs < qs , by Proposition 3, an order is made at ts under scenario S, which raises the inventory level to Q EOQ + qcs . The total savings in this case are:

=−

 1    qπ s 2DKh ts + 1 − 1 D 

2    qπ s 1 − 2 − 2DKh ts + D

 2  qπ s 2DKh ts (2 − 1 ) + (2 − 1) D  π1 qs − (1 − 1) . D

=

A.5. Lemma 1 We have:

(A.17)

We now find the difference between the total savings of each scenario. π For convenience, i  (Q0 i , Q EOQ ). For scenario N, we use Lemma 1 to obtain:

A.4. Proposition 4

      TSNπ = GEOQ ts + TsN − (Q0 , Q EOQ ) 2DKh ts + TsN   +cD ts + TsN

813

ETS(Q0 ) = −

2 2 hm  h  EOQ + qcs − qs Q0 − Q EOQ − pK + p Q 2D 2D (A.20)

=

 h  2  Q0 m pm − 1 2D 

Q EOQ h D − mp Q0 Q EOQ + ts D + (c − cs ) − D h p

 h  EOQ 2 h D 2 −m Q ts D + Q EOQ + (c − cs ) +p − pK. 2D 2D h (A.21)

Eq. (A.21) follows from Eq. (A.20) by applying Eq. (4) and simple algebra. 

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A.9. Proposition 8 Proof. From Proposition 7, we see that ETS(Q0 ) can be linear, convex, or concave. In the first two cases, the local maximum is either Q0LB or Q EOQ . If ETS(Q0 ) is concave, then the maximum of the function is at Q0LE . Note that the development of the function ETS(Q0 ) was based on the assumption that the number of orders made before ts is the same as in the EOQ ordering policy. Thus, when Q0LE < Q0LB , Q0LB is the local maximum. In the case when function ETS(Q0 ) is concave, we know that Q0LE ≤ EOQ ; if the other way was possible, due to concavity our results Q would be that Q0 = Q EOQ +  would outperform Q0 = Q EOQ , which is not possible according to Proposition 5.  References Al Kindi, M., & Sarker, B. R. (2011). Optimal inventory system with two backlog costs in response to a discount offer. Production Planning & Control, 22(3), 325– 333. Allah Taleizadeh, A., Mohammadi, B., Eduardo Cardenas-Barron, L., & Samimi, H. (2013). An EOQ model for perishable product with special sale and shortage. International Journal of Production Economics, 145(1), 318–338. Arcelus, F. J., Shah, N. H., & Srinivasan, G. (2003). Retailer’s pricing, credit and inventory policies for deteriorating items in response to temporary price/credit incentives. International Journal of Production Economics, 81-82, 153–162. Ardalan, A. (1988). Optimal ordering policies in response to a sale. IIE Transactions, 20(3), 292–294. Ardalan, A. (1994). Optimal prices and order quantities when temporary price discounts result in increase in demand. European Journal of Operational Research, 72(1), 52–61. Aucamp, D., & Kuzdrall, P. (1989). Order quantities with temporary price reductions. Journal of the Operational Research Society, 40(10), 937–940. Aull-Hyde, R. L. (1996). A backlog inventory model during restricted sale periods. Journal of the Operational Research Society, 47(9), 1192–1200. Baker, R., & Hanna, M. (1987). Cost comparisons for out-of-phase inventory models. The Journal of the Operational Research Society, 38(3), 253–260. Cardenas-Barron, L. E., Smith, N. R., & Goyal, S. K. (2010). Optimal order size to take advantage of a one-time discount offer with allowed backorders. Applied Mathematical Modelling, 34(6), 1642–1652. Chang, H.-J., Lin, W.-F., & Ho, J.-F. (2011). Closed-form solutions for Wee’s and Martin’s EOQ models with a temporary price discount. International Journal of Production Economics, 131(2), 528–534. Chaouch, B. A. (2007). Inventory control and periodic price discounting campaigns. Naval Research Logistics (NRL), 54(1), 94–108. Davis, R. A., & Gaither, N. (1985). Optimal ordering policies under conditions of extended payment privileges. Management Science, 31(4), 499–509. Diament, G. (1980). Production and inventory systems: Planning, managing, and control. Israel Institute of Productivity. Friend, J. K. (1960). Stock control with random opportunities for replenishment. Operational Research Quarterly, 11(3), 130–136. Gallego, G. (1994). When is a base stock policy optimal in recovering disrupted cyclic schedules. Naval Research Logistics, 41(3), 317–333. Golabi, K. (1985). Optimal inventory policies when ordering prices are random. Operations Research, 33(3), 575–588.

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