Optimal order size to take advantage of a one-time discount offer with allowed backorders

Applied Mathematical Modelling 34 (2010) 1642–1652

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Optimal order size to take advantage of a one-time discount offer with allowed backorders Leopoldo Eduardo Cárdenas-Barrón a,b,*, Neale R. Smith c, Suresh Kumar Goyal d a

Department of Industrial and Systems Engineering, School of Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, ITESM, Campus Monterrey, México. E.Garza Sada 2501 Sur, C.P. 64 849, Monterrey, N.L., México b Department of Management, School of Business, Instituto Tecnológico y de Estudios Superiores de Monterrey, ITESM, Campus Monterrey, México, E.Garza Sada 2501 Sur, C.P. 64 849, Monterrey, N.L., México c Centre for Quality and Manufacturing, School of Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, ITESM, Campus Monterrey, México, E.Garza Sada 2501 Sur, C.P. 64 849, Monterrey, N.L., México d Department of Decision Sciences and M.I.S, John Molson School, Concordia University, 1455 de Maisonneuve Blvd., West Montreal, Quebec, Canada H3G 1M8

a r t i c l e

i n f o

Article history: Received 24 November 2008 Received in revised form 2 September 2009 Accepted 7 September 2009 Available online 10 September 2009 Keywords: Economic order quantity Optimal ordering policies Planned backorders Discounts

a b s t r a c t In this paper, we develop an inventory model for determining the optimal ordering policies for a buyer who operates an inventory policy based on an EOQ-type model with planned backorders when the supplier offers a temporary fixed-percentage discount and has specified a minimum quantity of additional units to purchase. A distinguishing feature of the model is that both fixed and linear backorder costs are included, whereas previous works include only the linear backordering cost. A numerical study is performed to provide insight into the behavior of the model. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction It is a common practice for a supplier to offer incentives to customers to motivate them to purchase a larger lot size than the regular lot size on a one-time offer basis. There are several reasons why a supplier might choose to offer customers a temporary discount. These reasons include the following: to increase cash flow, to decrease inventory levels of the products, to boost market share, or simply to retain customers. In this paper, we present an EOQ-type model used to determine the optimal order size when a one-time discount is offered by a supplier and backorders are allowed. A distinguishing feature of the model is that both fixed and linear backorder costs are included, extending previous works. Before describing the model in detail, we will present a brief literature review. The inventory theory literature contains many extensions of the basic EOQ model. Some extensions of the basic EOQ model include inventory models that determine optimal ordering strategies for an announced price increase, a delay in payments, or a one-time discount. Naddor [1] and Whitin [2] derive an EOQ-type inventory model that addressed the situation when there is an announced price increase. According to Baker [3], the inventory policy for products on sale at reorder point is equivalent to the Naddor [1] price increase model. Another paper that addresses the price increase is by Lev and Soyster [4]. Naddor’s approach was used in Silver et al. [5] to develop an EOQ model without backorders to determine a special lot size for a product at reduced unit cost. * Corresponding author. Address Department of Industrial and Systems Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, ITESM, Campus Monterrey, México. E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, N.L., México. Tel.: +52 8183284235; fax: +52 8183284153. E-mail address: [email protected] (L.E. Cárdenas-Barrón). 0307-904X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2009.09.013

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Baker [6] develops an EOQ model without backorders to determine an optimal inventory policy for the situation where a discount is offered at the reorder point. His inventory model considers that the discount price is applied to all units if the order quantity exceeds the regular order quantity (Q*). In this situation the buyer has to choose between continuing with the regular order quantity of Q* units according to the standard EOQ model, thereby foregoing the discount, or making a special larger purchase. In a subsequent article, Baker and Vilcassim [7] introduce an EOQ model where a discount is offered during a regular inventory cycle, between reorder points. Tersine and Leon [8] develop an EOQ inventory model that yields an optimal special lot size depending on a temporary sale discount. Tersine and Gengler [9] suggest two simplified techniques that are useful for determining the order quantity in situations of inflationary price increases and temporary price discounts. Davis and Gaither [10] solve the problem using a different approach. Basically, they presented the optimal ordering policies for companies that are offered extended payments at a reorder point or between reorder points under six different extended payment privileges scenarios. Additionally, the offer could apply to all units ordered or only to a portion of the order. In the same line, Davis and Gaither [11,12] propose an inventory model that determines the optimal ordering policies under three types of offers to delay billing. It is important to point out that such conditions of extended payment privileges are comparable to a price discount. Aucamp and Kuzdrall [13,14] use a present value analysis to derive an inventory model that determines the order quantity that minimizes discounted cash flows for a one-time discount. Their models can also be used to determine the order quantity in the situation of an imminent price increase. Moreover, their inventory model considers the inclusion of initial inventory on hand at the time when the discount is offered. Ardalan [15] also addresses determining optimal ordering policies in response to a discount offer considering two decision variables: order quantity and time. He shows that under certain circumstances a special order may improve the effectiveness of an inventory system. Ardalan [16] relaxes the constant demand assumption and derives optimal ordering policies for a temporary change in both price and demand. In a later paper, Ardalan [17] derives the optimal inventory policies by employing a net present value model and by incorporating the marketing effect on demand. There are some equivalencies among the previous models. For example, the final result to obtain the special order quantity proposed by Baker [6] and Baker and Vilcassim [7] are equivalent to the Tersine [18] and Ardalan [15] results, respectively. Another interesting observation is that although Davis and Gaither [10] solve the problem using a different approach, the expressions to compute the large special order size that they present in their first and fifth scenarios are equivalent to the expressions to calculate the large special order quantity in Baker [6] and in Baker and Vilcassim [7], respectively. Tersine and Schwarzkopf [19] relax the length of the special sale in order to permit more than one order before the offer expires. They derive the optimal order policies for any length of sale time horizon. Goyal [20] proposes a procedure for determining the economic order quantity when the vendor has offered a decrease in price during a given period. Aull-Hyde [21] discusses the optimal ordering rules in response to supplier restrictions on special order size for a temporary discounted price. Tersine [18] deals with a temporary price discount and derives the optimal inventory policies assuming that the temporary unit price discount is available at the regular time for replenishment. Martin [22] identifies a flaw in the Tersine [18] model. He improves the Tersine [18,23]. However, Goyal [24] shows that there is no flaw in the Tersine [18] model. Furthermore, Chen and Min [25] construct and analyze an EOQ model for integrating the inventory policies in response to sales and inventory rules with disposal options. Gaither and Park [26] derive optimal ordering policies for a group of products when the vendor offers a discount for the group. Moreover, their model considers one resource constraint such as budget or space availability. Goyal et al. [27] present a review of the literature available on inventory policies under incentives on a one-time only basis. Wee and Yu [28] deal with the temporary price discount problem under the assumption that some products may deteriorate in storage over time according to an exponential distribution. Chang and Dye [29] extend the Martin [22] model to the case where the products deteriorate according a two-parameter Weibull distribution. Arcelus and Srinivasan [30] present a more general model which incorporates several restrictions on the order quantity in order to apply the discount and on the length of the sale period. Arcelus and Srinivasan [31] extend the Ardalan [16] model. Basically, they consider the one-time only discount problem for a situation in which there is price-sensitive demand. More recently, Arcelus et al. [32] develop an inventory model that considers both the discount price and delay in payments. Their model is based on the profit maximizing buyer with price-dependent demand that permits the buyer to pass a portion of the benefit to his customers in order to induce a higher demand. In a later work, Arcelus et al. [33] present an inventory model and derive the buyer’s profit maximizing promotion strategy when confronted with a supplier’s trade promotion based on a discount in price or a delay in payment for regular and perishable products. Chu et al. [34] prove that the minimum inventory level during a sale period is the optimal replenishment time for the supplier considering restrictions on the special order quantity. Finally, Sarker and Kindi [35,36] derive an EOQ-type inventory model with price discount to obtain the optimal ordering policies for a number of different scenarios. Related to this research is also the paper by CárdenasBarrón [37] where some technical errors appearing in Sarker and Kindi’s [35] paper are corrected, and some interesting and practical extensions can be found in Cárdenas-Barrón [38]. All the above research considers an EOQ-type model without backorders. Economic lot size models considering discounts and backordering have also been studied. For example, the works of Tersine and Barman [39] and Aull-Hyde [40]. The original work presented by Tersine and Barman [39], which consists of a composite EOQ model that can be disaggregated into several traditional EOQ models, shows how to determine the optimal order and backorder quantities in response to a discount with

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allowed backorders. Aull-Hyde [40] develops optimal purchasing policies for an EOQ inventory model with backorders allowed and when a temporary price discount is available between reorder points. Both, Tersine and Barman [39] and Aull-Hyde [40] only consider one type of backorder cost: the linear backordering cost. In this paper, we extend the EOQ inventory model considering discounts and backorders by including the two common types of backorder costs: the fixed and linear costs. The proposed model determines the buyer’s optimal ordering policies when the supplier offers a one-time fixed-percentage price discount for a limited time. The supplier also states a minimum quantity of additional units that must be purchased. The buyer must then decide if the offer will be accepted and how much to order based on the objective to reduce total cost. The remainder of the paper is organized as follows: Section 2 describes the mathematical model, Section 3 presents a numerical example, Section 4 presents numerical experiments and provides some managerial insights, and Section 5 provides conclusions and recommendations for further research. 2. Development of the model This section presents the derivation of the optimal ordering policies for the EOQ model with backorders when the supplier reduces a price temporarily. The model adheres to the following standard static deterministic inventory assumptions: (1) demand is constant and known, (2) lead time is zero, (3) all costs are accurately stated, (4) inventory holding costs are applied to average inventory, (5) inventory storage space and the availability of capital is unlimited, (6) the model is for only one product, and (7) the planning horizon is one year. In addition to the above seven assumptions, the model needs nine additional assumptions: (1) shortages are allowed and all shortages are backordered, (2) backordering costs are applied to average shortages and to the maximum backorder level, (3) unit price is constant, except when the supplier offers a discounted sale price, (4) the supplier’s offer period is very short, (5) the offer period can occur at a reorder point or between reorder points, (6) the fixed-percentage discount is offered one-time, (7) the buyer is currently ordering lots of size Q*, optimal according to the standard EOQ model with backorders, (8) the allowable backorder level is the same in all the cycles, and (9) after a special order, the buyer reverts to the Q* lot size . The following notation will be used in the remainder of the paper. A b* C D h i Q* 0 Q q X* Xmin Xmax

a

p^ p

fixed ordering cost, maximum backorder level, units, product unit cost, demand rate, units per time unit, inventory carrying cost per unit per time unit (h = iC), inventory carrying cost rate, economic order quantity, units, economic special order quantity, units, amount of on hand inventory when special order is placed, units, additional units beyond Q* to purchase to take advantage of discount, minimum quantity beyond Q* specified by the supplier to qualify for the discount, a maximum quantity where the total savings are equal to zero, fixed-percentage discount offered by supplier, backorder cost per unit per time unit, backorder cost per unit, does not depend on backorder duration,

The inventory manager who is offered a one-time discount must decide how to respond. He must determine if a special order will be placed and, if so, of what size. We consider three scenarios which are shown in Table 1. For each scenario two cases are to be considered. Case I represents the situation when the buyer chooses not to take advantage of the sale price and continues to buy Q units. Case II represents the situation when the buyer procures a special large order quantity to take advantage of the discounted unit cost. Figs. 1 and 2 depict the two cases to be evaluated and analyzed in the first and second scenarios. Fig. 3 shows case II for the third scenario.

Table 1 Scenarios for the one-time discount. Scenario 1

Scenario 2

Scenario 3

Assuming that the discount is offered at a reorder point and the discount applies to all units (Q* + X). In addition, the supplier also states a minimum quantity (Xmin) of additional units that must be purchased

Assuming that the discount is offered at a reorder point and the discount applies only to the additional units (X) to be purchased. The supplier also states a minimum quantity (Xmin) of additional units that must be purchased

Assuming that the discount is offered between reorder points and the discount applies only to the additional units (X) to be purchased. The supplier also states a minimum quantity (Xmin) of additional units that must be purchased. The discount only applies to the X additional units because the lot Q* was purchased at the regular price in the beginning of the cycle

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I(t)

Q-b

Q

D Level 0

b

Time

Q/D

Fig. 1. Behavior in case I for the three scenarios.

I(t) X Q-b D Q Level 0

b (Q-b)/D

X/D b/D

(Q+X)/D

Time Fig. 2. Behavior in case II for the first and second scenarios.

X

Q-b q Q

Level 0

b (q+X)/D

b/D

Q/D

(q+X+b) D

Time

Fig. 3. Behavior of case II for scenario 3.

First, the baseline total annual cost, TACI (Q, b), is determined. It is the classical total average cost expression based on EOQ with backordering as documented by Sipper and Bulfin [41]. The TACI (Q, b) includes ordering cost, carrying cost, backorder cost and purchasing costs. For the three scenarios the TACI (Q, b) is the same and is given by:

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TACIðQ ; bÞ ¼

^b AD hðQ  bÞ2 p pbD þ þ CD: þ þ Q Q 2Q 2Q

ð1Þ

The optimal values of order quantity and maximum backorder level are given by the following expressions:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffi ^ 2AD ðpDÞ2 hþp  Q ¼ ; ^ ^ h hðh þ pÞ p 

and 



b ¼

hQ  pD : ^ hþp

For each scenario, the total annual cost with the discount, TACII (Q, b, X), is different and it is formed as the sum of ordering cost, the carrying cost, the backorder cost, and the purchasing costs. The three scenarios are as follows. Scenario 1: In this scenario, the TACII (Q, b, X) is derived as follows. The total ordering costs for the year is the cost A to place the unique order for the (Q* + X) units, plus A times the number of additional orders that are necessary to satisfy the remaining (D  Q  X) units of demand. Thus, the total yearly ordering cost is given by:

total ordering cost ¼ A þ

AðD  Q  XÞ : Q

According to Fig. 2, the total carrying cost for the entire year is based on carrying an average of (Q  b + X)/2 units in bÞ2 for the remaining period inventory for the entire period of (Q  b + X)/D at h(1  a) per unit, plus carrying an average of ðQ2Q of DQD X at h per unit. Therefore, the total carrying costs are given by

total carrying costs ¼

hð1  aÞðQ  b þ XÞ2 hðQ  bÞ2 þ 2D 2Q

!  DQ X : D

According to Fig. 2 and under the assumption that the backorder level is the same in all cycles, the total backordering costs b2 b2 ^ per unit, plus carrying an average of 2Q backorder units at p for the for the entire year is based on carrying an average of 2Q DQ X ^ remaining period of D at p per unit, plus the cost p per unit that occurs each cycle times b; plus p times the number of additional cycles that are necessary to satisfy the remaining (D  Q  X) units of demand. Thus,

total backordering cost ¼

p^ b2 2Q

þ

!    DQ X DQ X þ pb þ ðpbÞ : D Q 2Q

p^ b2

Finally, the total purchasing cost is derived as follows. The total purchasing cost is the sum of the total costs of the units Q + X that are bought at the discount price plus the total costs of the units D  Q  X that are bought at the regular price. Then, the total cost of the units that are bought at the discount price is c(1  a)(Q + X), and the total cost of the units that are bought at the regular price is c(D  Q  X). Thus, the total cost for the inventory system in case II for scenario 1 is given by

!  AðD  Q  XÞ hð1  aÞðQ  b þ XÞ2 hðQ  bÞ2 DQ X p^ b2 TACIIðQ; b; XÞ ¼ A þ þ þ þ þ Q D 2D 2Q 2Q   DQ X þ pb þ ðpbÞ þ cð1  aÞðQ þ XÞ þ cðD  Q  XÞ: Q

p^ b2 2Q

!

DQ X D



Assuming that the optimal values of Q and b are utilized, the resulting function of X is obtained:

! 2 ^ b2 pbD hð1  aÞ 2 hQ hab haQ hb p^ b2 A pb AD hðQ  bÞ2 p Xþ TACIIðXÞ ¼ X þ þ   þ þ CD    aC  þ þ 2D 2D D D Q Q Q 2QD 2QD Q 2Q 2Q þ

2 haQb haQ 2 hab p^ b2     aCQ : D 2D 2D 2D

ð2Þ

Scenario 2: In this scenario, TACII (Q, b, X) is derived as follows. Since the main difference between scenario 1 and scenario 2 is that in scenario 1 the discount applies to all Q + X units while the discount in scenario 2 applies only to the additional units X, the total ordering costs and the total backorder costs are the same in both scenarios. On the other hand, the total inventory carrying cost and the total purchasing cost will be different. Considering Fig. 2, the total carrying costs for the entire year is based on carrying X units in inventory for the entire period of (Q  b)/D at h(1  a) per unit, plus carrying an averbÞ2 units for the remaining period of DX age of X2 units during the period of DX at h(1  a) per unit, plus carrying an average of ðQ2Q D at h per unit. Therefore,

L.E. Cárdenas-Barrón et al. / Applied Mathematical Modelling 34 (2010) 1642–1652

total carrying costs ¼

1647

" #     hð1  aÞðQ  bÞX X X ðQ  bÞ2 D  X þ hð1  aÞ þh : D 2 D D 2Q

The total purchasing cost is derived as follows. The total purchasing cost includes the cost of X units purchased at the discount price and the cost of D  X units purchased at the regular price. Thus, the total cost of the units that are bought at discount is c(1  a)(X), and the total costs of the units that are bought at regular price is c(D  X). Thus, the total cost for the inventory system in case II for scenario 2 is given by:

" #     AðD  Q  XÞ hð1  aÞðQ  bÞX X X ðQ  bÞ2 D  X p^ b2 þh þ þ þ hð1  aÞ Q D 2 D D 2Q 2Q !    2  p^ b DQ X DQ X þ pb þ ðpbÞ þ cð1  aÞðXÞ þ cðD  XÞ: þ D Q 2Q

TACIIðQ ; b; XÞ ¼ A þ

Assuming that the optimal values of Q and b are utilized, the resulting function of X is obtained:

TACIIðXÞ ¼

! 2 hð1  aÞ 2 hQ hab haQ hb p^ b2 A pb X X þ þ      aC  2D 2D D D Q 2QD 2QD Q 2

þ

2

^b AD hðQ  bÞ2 p pbD p^ b þ þ CD  þ þ : Q Q 2Q 2Q 2D

ð3Þ

Scenario 3: In this scenario, the TACII (Q, b, X) is derived as follows. The total yearly ordering cost is 2A because there are two orders: the last order and the special order for X additional units (see Fig. 3) plus A times the number of additional orders that are necessary to place in order to satisfy the remaining D  Q  X demand. The total costs of backordering are the same as for scenario 1. Since the percent discount applies only to the X additional units, the total purchasing costs are the same as in scenario 2. On the other hand, the total inventory carrying cost for the year is based on carrying X units in inventory for the entire period of q/D at h(1  a) per unit, plus carrying an average of X2 units during the period of DX at h(1  a) per unit, plus bÞ2 bÞ2 units for the remaining period of DQD X at h per unit, plus carrying an average of ðQ2Q for the carrying an average of ðQ2Q Q entire cycle D in which is accepted the price discount at h per unit. Thus,

" # " #      ðQ  bÞ2 ðD  Q  XÞ ðQ  bÞ2 Q X X hð1  aÞXq þh þ hð1  aÞ þ total carrying costs ¼ h : D D 2 D D 2Q 2Q

And, the total cost for the inventory system in case II for scenario 2 is given by

" # " #      AðD  Q  XÞ ðQ  bÞ2 ðD  Q  XÞ ðQ  bÞ2 Q X X hð1  aÞXq þh þ hð1  aÞ þ TACIIðQ ; b; XÞ ¼ 2A þ þh Q D D 2 D D 2Q 2Q !    p^ b2 p^ b2 D  Q  X DQ X þ þ pb þ ðpbÞ þ cð1  aÞðXÞ þ cðD  XÞ: þ D Q 2Q 2Q Assuming that the optimal values of Q and b are utilized, the resulting function of X is obtained:

! 2 ^ b2 pbD hð1  aÞ 2 hb hq hQ hb p^ b2 A pb AD hðQ  bÞ2 p Xþ TACIIðXÞ ¼ X þ þ ð1  aÞ   þ    aC  þ þ 2D D D 2D 2QD 2QD Q Q Q Q 2Q 2Q þ CD þ A 

p^ b2 2D

ð4Þ

:

The difference representing the savings resulting from accepting the discount is D(X) = TACI  TACII. The difference functions for each scenario are shown in Table 2. Eqs. (5–7) represent the difference functions for scenarios 1, 2, and 3, respectively. Although TACI is a constant in the decision variable X, formulating D(X) serves to make the results more understandable to practitioners. Thus, D(X) will be attractive when it is a positive number. The maximum of D(X) can be found using classical optimization techniques. The first derivative of Eqs. (5–7) are given in the third column of Table 2. The second derivatives of these functions are negative for all values of X, which implies that there is a global local maximum X* that maximizes D(X). The second derivative of Eqs. (8–10) is

D00 ðXÞ ¼ 

hð1  aÞ : D

ð11Þ

Please note that Eqs. (5) and (6) differ only by a constant. This implies that both have the same maximum given by

X ¼

  a  D  Q  b þ : i 1a

ð12Þ

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Table 2 Difference functions for each scenario. Scenario Difference function D(X) 1

aÞ 2 DðXÞ ¼  hð1 2D X þ



A Q

2

 2 hb p^ b2 þ pb  hQ  hab X þ haDQ þ aC þ 2QD þ 2QD 2D Q D 2

2

aÞ 2 DðXÞ ¼  hð1 2D X þ 2 2 b b þ p^2D  p^2Q

3

aÞ 2 DðXÞ ¼  hð1 2D X þ 2



A Q

Eq. no.

(5) D0 ðXÞ ¼  hð1aÞX þ A þ haQ þ aC þ hb2 þ p^ b2 þ pb  hQ  hab D 2D 2QD 2QD D Q Q D

(8)

(6) D0 ðXÞ ¼  hð1aÞX þ A þ haQ þ aC þ hb2 þ p^ b2 þ pb  hQ  hab ; D 2D D 2QD 2QD Q Q D

(9)

2

Q b b þaCQ þ ha2D þ hab þ p^2D  p^2Q  haDQb  2D  2 haQ A hb p^ b2 pb hQ hab þ þ a C þ D 2QD þ 2QD þ Q  2D  D X Q

2

Eq. First derivative of difference function D0 (X) no.

 2 (7) D0 ðXÞ ¼  hð1aÞX þ A þ hQ þ aC þ hb2 þ p^ b2 þ pb  hb  hq ð1  aÞ: (10) hb p^ b2 pb hb hq þ hQ 2D D 2QD 2QD D Q Q D 2D þ aC þ 2QD þ 2QD þ Q  D  D ð1  aÞ X

2

b b  p^2Q A þ p^2D

Eq. (12) represents the optimal X* additional units that the buyer needs to purchase to maximize savings when D(X) > 0. This result applies to scenarios 1 and 2. It is important to point out that if the supplier does not offer a discount (a = 0) then Eq. (12) suggests purchasing zero additional units (X* = 0). This means that the special order Q0 = Q* + X* is equal to Q0 = Q* which is the regular Q* given by the EOQ with backorders, as would be expected. Setting Eq. (10) to zero and solving for X* yields:

X ¼

  1 aD   q: Q  b þ 1a i



ð13Þ

Eq. (13) represents the optimal X* additional units that the buyer needs to purchase to maximize savings when D(X) > 0. This result applies to scenario 3. In this case, when the supplier does not apply the discount (a = 0), Eq. (13) suggests purchasing X* = Q*  b*  q additional units. If one has q = Q*  b* units on hand, X* = 0 and no additional units are purchased, as would be expected. On the other hand, if one has q = b* units on hand, X* = Q* and the regular lot size is purchased, as would be expected. The larger special order size in scenarios 1 and 2 is given by:

Q0 ¼



   1 D  b Q þ a : 1a i

ð14Þ

For scenario 3, the larger special order size is given by:

Q0 ¼

  1 aD   q: Q  b þ 1a i



ð15Þ

When the on hand inventory level is q = b*, Eq. (15) reduces to Eq. (14). That is to say, scenario 3 reduces to scenario 1 when the on hand inventory level is q = b*. Furthermore, when the on hand inventory level is q = Q*  b* Eq. (15) reduces to Eq. (12). In other words, scenario 3 reduces to scenario 2 when the on hand inventory level is q = Q*  b*. Also, we can derive closed forms for the total average cost savings by substituting the expression for the optimal value of X* into the difference function D(X). The closed forms for the total average cost savings are shown in Table 3. Now, we specify the optimal ordering policies. First, the value of Xmax is obtained by setting the difference function, Eqs. (5–7), equal to zero and determining the larger of its two characteristic roots. Then, X* is computed using Eqs. (12) or (13). Now, (a) If Xmin < X*, then X* additional units should be ordered and the large special order size is calculated using Eqs. (14) or (15), with the total average cost savings given by Eqs. (5), (6), or (7). (b) If X* 6 Xmin < Xmax, then Xmin additional units should be ordered and the large special order size is Q0 = Q* + Xmin with the total average cost savings given by Eqs. (5), (6), or (7). (c) If Xmin P Xmax, then zero additional units should be ordered and the buyer continues to order the regular Q* units.

Table 3 Total average cost savings for each scenario. Scenario 1 2 3

Total average cost savings D(X*) 

DðX Þ ¼

ha2 2Dð1aÞ







D 2 i

Eq. no. h

i

2  2 ha þ p^2b D1  Q1 þ aCQ  þ 2D ½Q   b 

Q b þ h i 

2  ha2 D 2 DðX  Þ ¼ 2Dð1 þ p^2b D1  Q1 aÞ Q  b þ i i

2 p^ b2 h 1  aÞ  1   D DðX  Þ ¼ hð1 þ 2 D  Q1  A 2D 1a Q  b þ i  q

(16) (17) (18)

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The papers that propose models that do not consider backorders are those by Baker [6], Tersine and Leon [8], Davis and Gaither [10], Tersine [18], Baker and Vilcassim [7], Ardalan [15], and Sarker and Kindi [35]. Regarding scenario 1, the reader may verify that with some slight modification of the notation, Eq. (12) transforms immediately to the equation for the special order in Tersine and Leon [8], to Eq. (5) in Baker [6], to Eq. (4) in Davis and Gaither [10], and to Eq. (5) in Sarker and Kindi [35] when b* = 0. Also, Eq. (14) for the special order quantity is equivalent to the expression given by Tersine [18 , p. 115] and to Eq. (11) given by Sarker and Kindi [35]. Regarding scenario 2, the reader may verify that (12) is equivalent to Eq. (7) proposed by Davis and Gaither [10] when b* = 0. Regarding scenario 3, the reader may verify that Eq. (13) reduces to Eq. (5) in Baker and Vilcassim [7] and to Eq. (10) in Davis and Gaither [10] when b* = 0. Also, Eq. (15) for the special order (Q0 = X*) is equivalent to Eq. (3) given by Ardalan [15] and to the expression proposed by Tersine [18, p. 116]. Also, our model reduces to the model of Davis and Gaither [10] when b* = 0. That is to say, our scenario 1 reduces to scenarios 1 and 2 of Davis and Gaither [10]. Furthermore, our scenario 2 reduces to scenarios 3 and 4 of Davis and Gaither [10] and our scenario 3 reduces to scenarios 5 and 6 of Davis and Gaither [10]. The models with backorders of Tersine and Barman [39] and Aull-Hyde [40] are equivalent. Both models apply when a discount is offered between the reorder points and consider only one type of backorder cost: the linear backordering cost, p^ . Thus, as to be expected, Eq. (13) of scenario 3, is equivalent to Eqs. (17) and (18) in Aull-Hyde [40] and to the expressions given in Tersine and Barman [39]. 3. Numerical example For scenarios 1 and 2, as described in Section 2, Eq. (12) applies. Consider the following scenario. A supplier is offering a 10% discount on a product that has a demand rate of 50,000 units per year. The inventory carrying cost rate is 20% per year, the economic order quantity is 28,121 units, and the maximum allowed backorder level is 27,993. Eq. (12) is used to calculate the optimal number of additional units to order. For this example, the optimal number of additional units to order is 27,792. If Xmin had been specified at 28,000, then 28,000 additional units would be ordered as long as Xmax were verified to be greater than 28,000. If Xmax were found to be less than 28,000 then no additional units would be ordered. For scenario 3, Eq. (13) applies. In this case, one additional piece of information must be supplied: the amount of on hand inventory when the special order is placed. Let’s assume this to be near the end of an inventory cycle, with only 51 units on hand. Applying Eq. (13) yields that the optimal number of additional units is 27,869. The same logic with respect to Xmin and Xmax would apply as in scenarios 1 and 2. 4. Numerical experiments and managerial insights In order to gain insight into the behavior of the model, a numerical study was performed. The study consisted of a series of designed experiments with varying parameter ranges. In the case of scenarios 1 and 2, for each experiment, the main effects of each parameter on the values of b*, Q*, X*, and the difference values given by Eqs. (5) and (6) were computed. In the case of scenario 3, the main effects of each parameter on the values of b*, Q*, X*, and the difference value given by Eq. (7) was computed. Four experiments were performed for scenarios 1, 2, and 3. Table 4 Parameter values for first experiment, scenarios 1, 2, and 3.

D i

a C A

p

p^

Low value

High value

100 0.2 10% 20 20 0.2 1

250 0.4 40% 60 150 0.8 4

Table 5 Parameter values for second experiment, scenarios 1, 2, and 3.

D i

a C A

p

p^

Low value

High value

1100 0.2 10% 100 120 0.2 1

2000 0.4 40% 300 250 0.8 4

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The parameter ranges for scenarios 1, 2, and 3 are given in Tables 4–7. The main effects for each of these experiments are shown in Tables 8–11. The main effects for the scenario 3 experiments are shown in Tables 12–15. For scenario 3 experiments, the low value of q was computed as (Q*  b*)  0.25 and the high value was computed as (Q*  b*)  0.75. These four experiments are not meant to capture every possible behavior of the model. Instead, they serve as a small sample of possible behavior. Some observations are the following: 1. Savings are higher as the demand rate, D, increases. 2. Savings are lower as inventory carrying cost rate, i, increases.

Table 6 Parameter values for third experiment, scenarios 1, 2, and 3.

D i

a C A

p

p^

Low value

High value

10,000 0.2 10% 200 200 0.2 0.1

20,000 0.4 40% 400 300 0.8 0.6

Low value

High value

50,000 0.2 10% 500 800 0.2 0.1

200,000 0.4 40% 600 1000 0.8 0.6

Table 7 Parameter values for fourth experiment, scenarios 1, 2, and 3.

D i

a C A

p

p^

Table 8 Main effects for first experiment, scenarios 1 and 2.

*

b Q* X* D(X) (5) D(X) (6)

D

i

a

C

A

p

p^

0.2376 0.3472 1.5009 14.1967 10.7177

0.5977 0.2573 8.8394 67.2654 185.683

0 0 12.7235 169.0058 120.274

0.4554 0.1989 0.2584 74.4555 45.7796

0.7680 0.9445 0.0686 10.1754 0.9693

25.9459 12.5334 5.2159 17.6725 71.0115

24.8496 21.7137 1.2195 220.306 4.2451

Table 9 Main effects for second experiment, scenarios 1 and 2.

*

b Q* X* D(X) (5) D(X) (6)

D

i

a

C

A

p

p^

0.1567 0.1763 1.4659 61.2546 52.3613

1.2913 0.0768 75.8793 2725.63 2722.42

0 0 108.4459 6117.915 5008.135

0.1962 0.0118 0.0809 541.7936 403.5791

1.5375 1.6130 0.0293 80.8863 1.0876

66.1152 32.2832 13.1568 409.357 791.3436

122.169 117.351 1.8739 5915.22 19.5201

Table 10 Main effects for third experiment, scenarios 1 and 2.

*

b Q* X* D(X) (5) D(X) (6)

D

i

a

C

A

p

p^

0.1439 0.1513 1.4612 90.2760 78.3840

23.5329 19.5171 730.728 37954.7 39214.5

0 0 1043.946 88737.39 72155.7

2.3532 1.9517 0.1561 5418.214 3906.365

13.5023 13.5503 0.0186 1000.037 1.0870

1780.25 1577.41 78.8848 95548.2 7702.32

9169.04 9140.26 11.1907 694039.0 71.1032

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3. 4. 5. 6.

Savings are higher as the discount increases. Savings are higher as the unit cost increases. Savings are higher as the setup cost increases. However, the effect is much smaller in scenario 2. The effect of i on Q* is negative in some cases and positive in others. Since the range of i was not changed, this indicates an interaction with one or more other factors. 7. The effect of D on X* is nearly constant for very different ranges of D. 8. The effect of p on D(X) (5) is negative and on D(X) (6) positive. ^ on D(X) (5) is negative and on D(X) (6) positive. 9. The effect of p These last two observations show that although these models seem similar at fist glance, they can display significantly different behavior in specific instances. Observations 1 through 7 listed for scenarios 1 and 2 also apply to scenario 3. In addition it was observed that 1. Savings are lower when the special order is placed when inventory is high (shortly after a regular order is received).

Table 11 Main effects for fourth experiment, scenarios 1 and 2.

b* Q* X* D(X) (5) D(X) (6)

D

i

a

C

A

p

p^

0.097584 0.101172 1.459729 157.4592 143.4889

157.8789 142.3045 6082.45 578,666 597,986

0 0 8689.346 12,54,419 11,00,298

9.21559 8.370837 0.32851 40699.3 32552.44

20.71989 20.75574 0.013939 2847.967 1.168659

11875.8 11039.9 325.0777 14,37,159 64573.98

47,001 47926.8 28.85431 6461091.0 303.5563

Table 12 Main effects for first experiment, scenario 3.

b* Q* X* D(X) (7)

D

i

a

C

A

p

p^

q

0.2376 0.3472 1.5557 11.1596

0.597 0.257 9.266 68.69

0 0 12.7235 125.3795

0.4554 0.1989 0.5906 46.1897

0.7680 0.9445 0.1569 0.7162

15.5675 7.5200 7.1532 78.7531

24.8496 21.7137 2.7874 19.1535

0 0 30.82 255.27

Table 13 Main effects for second experiment, scenario 3.

*

b Q* X* D(X) (7)

D

i

a

C

A

p

p^

q

0.1567 0.1763 1.4757 52.7377

1.2913 0.0768 76.5634 2748.97

0 0 108.4459 5041.579

0.1962 0.0118 0.185 403.7928

1.5375 1.6130 0.0671 1.5322

39.6691 19.3699 18.0436 871.5358

122.169 117.351 4.283 115.821

0 0 43.578 1672.17

Table 14 Main effects for third experiment, scenario 3.

b* Q* X* D(X) (7)

D

i

a

C

A

p

p^

q

0.1439 0.1513 1.4649 78.6343

23.5329 19.5171 732.736 39349.4

0 0 1043.946 72320.68

2.3532 1.9517 0.3569 3906.955

13.5023 13.5503 0.0427 1.6665

1068.15 946.443 108.1849 8714.022

9169.04 9140.26 25.5788 1054.96

0 0 123.075 8249.1

Table 15 Main effects for fourth experiment, scenario 3.

b* Q* X* D(X) (7)

D

i

a

C

A

p

p^

q

0.0975 0.1011 1.4615 143.7339

157.8789 142.3045 6090.23 599050.02

0 0 8689.346 1101594.2

9.2155 8.3708 0.7508 32553.62

20.7198 20.7557 0.0318 2.6108

7125.49 6623.94 445.8208 73020.18

47001.03 46926.8 65.9527 5382.284

0 0 474.681 64824.8

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^ on D(X) (7) are positive. This coincides with the corresponding effects on D(X) (6), as would be 2. The effects of p and p expected due to the similarity in discount policy.

5. Conclusions and future research This paper presents an EOQ-type model with backorders to determine the optimal ordering policies when a temporary discount is offered. Several scenarios are analyzed and the optimal order quantity is derived in each case. A distinguishing feature of the model is that the two most common types of backorder cost are included. Numerical studies are performed in order to provide insight into the behavior of the model in a sample of instances. Some suggestions for future research are to optimize the time an order is placed within an order cycle when it is possible to do so and optimize the backorder level following a special order. The problem could also be modeled as a bi-level optimization problem with the leader being the supplier and the follower being the purchaser. Both the supplier and the purchaser wish to maximize their profits but an increase in profits for one does not necessarily reduce the profits for the other. One additional suggestion is to include the possibility of defective products in the delivered quantity. Acknowledgments This research was partially supported by the School of Business and the Tecnológico de Monterrey research fund number CAT128. The authors would like to thank the two anonymous referees for their constructive comments and suggestions that enhanced this paper. A special acknowledgment to Sarai Rodríguez Payán for her valuable support during the development of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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