Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 928932 | 19 pages | https://doi.org/10.1155/2009/928932

Optimal Inventory Policy Involving Ordering Cost Reduction, Back-Order Discounts, and Variable Lead Time Demand by Minimax Criterion

Academic Editor: Wei-Chiang Hong
Received24 Dec 2008
Revised20 Apr 2009
Accepted24 May 2009
Published30 Jul 2009

Abstract

This paper allows the backorder rate as a control variable to widen applications of a continuous review inventory model. Moreover, we also consider the backorder rate that is proposed by combining Ouyang and Chuang (2001) (or Lee (2005) with Pan and Hsiao (2001) to present a new form. Thus, the backorder rate is dependent on the amount of shortages and backorder price discounts. Besides, we also treat the ordering cost as a decision variable. Hence, we develop an algorithmic procedure to find the optimal inventory policy by minimax criterion. Finally, a numerical example is also given to illustrate the results.

1. Introduction

In most of literatures dealing with inventory problems, either in deterministic or probabilistic model, lead time is viewed as a prescribed constant or a stochastic variable, and is not subject to control (see, e.g., Naddor [1], Liberatore [2], Magson [3], Kim and Park [4], Silver and Peterson [5], Foote et al. [6], Azoury and Brill [7] and Chiu [8]). However, as pointed out by Tersine [9], lead time usually comprises several components, such as setup time, process time, wait time, move time, and queue time. In many practical situations, lead time can be reduced using an added crash cost; in other words, it is controllable. By shortening the lead time, we can lower the safety stock, reduce the stock-out loss and improve the service level to the customer so as to increase the competitive edge in business. Firstly, Liao and Shyu [10] presented a probabilistic model in which the order quantity is predetermined and the lead time is the only decision variable. Secondly, Ben-Daya and Raouf [11] have also extended the model of Liao and Shyu [10] by considering both the lead time and the order quantity as decision variables and the situation of shortage is neglected. Subsequently, Ouyang et al. [12] considered an inventory model with a mixture of backorders and lost sales to generalized Ben-Daya and Raouf's [11] model, where the backorder rate is fixed.

In this study, we consider to allow the backorder rate as a control variable. Under most market behaviors, we can often observe that many products of famous brands or fashionable commodities may lead to a situation in which customers prefer their demands to be backordered while shortages occur. Certainly, if the quantity of shortages is accumulated to a degree that exceeds the waiting patience of customers, some may refuse the backorder case. However, the supplier can offer a price discount on the stock-out item in order to secure more backorders. In the real market as unsatisfied demands occur, the longer the length of lead time is, the larger the amount of shortages is, the smaller the proportion of customers can wait, and hence the smaller the backorder rate would be. But, the larger the backorder discount is, the larger the backorder rate would be. Thus, the backorder rate is dependent on the amount of shortages and backorder price discounts. Therefore, we also consider the backorder rate that is proposed by combining Ouyang and Chuang [13] (or Lee [14]) with Pan and Hsiao [15] to present a new form. In addition, there are many authors that (Porteus [16–18], Billington [19], Nasri et al. [20], Kim et al. [21], Paknejad et al. [22], Sarker and Coates [23], Ouyang et al. [24, 25], Moon and Choi [26], Chuang et al. [27], Lin and Hou [28], and Chang et al. [29]) have investigated the effects of investing in reducing ordering cost. Hence, we treat the ordering cost as a decision variable in this study.

Because the demand of different customers is not identical in the lead time, we cannot only use a single distribution (such as [13]) to describe the demand of the lead time. It is more reasonable that mixture distribution is applied to describe the lead time demand than single distribution is used. Besides, in many practical situations, the probability distributional information of lead time demand is often quite limited. Since Lee et al. [30] consider that the lead time demand follows a mixture of normal distributions, we relax the assumption about the form of the mixture of distribution functions of the lead time demand. Therefore, we consider that any mixture of distribution functions (d.f.s); say πΉβˆ—=𝑝𝐹1+(1βˆ’π‘)𝐹2, of the lead time demand has only known finite first and second moments (and hence, mean and standard deviations are also known and finite) but we make no assumption on the distribution form of πΉβˆ—. That is, 𝐹1 and 𝐹2 of πΉβˆ— belong to the class Ξ© of all single d.f.s' with finite mean and standard deviation. Our goal is to solve a mixture inventory model by using the minimax criterion. This is, the minimax criterion (such as Wu et al. [31]) for our model is to find the most unfavorable d.f.s 𝐹1 and 𝐹2 in πΉβˆ— for each decision variable and then to minimize over the decision variables. Finally, one numerical example is also given to illustrate that when 𝑝=0 or 1, the model considers only one kind of customers' demand; when 0<𝑝<1, the model considers two kinds of customers' demand for the fixed backorder parameters πœ€ and 𝛿. It implies that the minimum expected total annual costs of two kinds of customers' demand are larger than the minimum expected total annual cost of one kind of customers' demand. Thus, the minimum expected total annual cost increases as the distance between 𝑝 and 0 (or 1) increases for the fixed backorder parameters πœ€ and 𝛿. Hence, if the true distribution of the lead time demand is a mixture of normal distributions, we use a single distribution (such as [13]) to substitute the true distribution of the lead time demand then the minimum expected total annual cost will be underestimated.

2. Model Formulation

To establish the mathematical model, the notation and assumptions employed throughout the paper are as follows:

𝐴: odering cost per order,𝐷: average demand per year,β„Ž: inventory holding cost per item per year,𝐿: length of lead time,𝑄: order quantity,π‘Ÿ: reorder point,𝑋: lead time demand with the mixtures of distribution function,𝛽: fraction of the demand backordered during the stock-out period, π›½βˆˆ[0,1],πœ‹0∢ gross marginal profit per unit,πœ‹π‘₯: back-order price discount offered by the supplier per unit, 0β‰€πœ‹π‘₯β‰€πœ‹0,𝛿,πœ€: back-order parameters, 0≀𝛿≀1,0β‰€πœ€β‰€βˆž,𝑝: the weight of the component distributions, 0≀𝑝≀1,π‘₯+: maximum value of π‘₯and 0, that is, π‘₯+=max{π‘₯,0},π‘₯βˆ’: maximum value of βˆ’π‘₯and 0, that is, π‘₯βˆ’=max{βˆ’π‘₯,0}: 𝐡(π‘Ÿ)=𝐸(π‘‹βˆ’π‘Ÿ)+: the expected shortage quantity at the end of cycle,π‘ž: the allowable stock-out probability during 𝐿,π‘˜: the safety factor which satisfies 𝑃(𝑋>π‘Ÿ)=π‘ž,πœ‡βˆ—: the mean of lead time demand with the mixture of distributions,πœŽβˆ—: the standard deviation of lead time demand with the mixture of distributions,𝐴0: original ordering cost,𝐼(𝐴): capital investment required to achieve ordering cost 𝐴,0<𝐴≀𝐴0,πœƒ: fractional opportunity cost of capital per unit time,πœ‰: percentage decrease in ordering cost 𝐴 per dollar increase in investment 𝐼(𝐴).

The assumptions of the model are exactly the same as those in Ouyang and Chuang [13] who expect the following assumptions: the reorder point π‘Ÿ = expected demand during the lead time + safety stock (SS), and SS = π‘˜Γ— (standard deviation of lead time demand), that is, π‘Ÿ=πœ‡βˆ—πΏ+π‘˜πœŽβˆ—βˆšπΏ, where πœ‡βˆ—=π‘πœ‡1+(1βˆ’π‘)πœ‡2, πœŽβˆ—=(1+𝑝(1βˆ’π‘)πœ‚2)1/2𝜎, πœ‡1=πœ‡βˆ—βˆš+(1βˆ’π‘)πœ‚πœŽ/𝐿, πœ‡2=πœ‡βˆ—βˆšβˆ’π‘πœ‚πœŽ/𝐿 (it means that πœ‡1βˆ’πœ‡2√=πœ‚πœŽ/𝐿,πœ‚βˆˆπ‘…), and π‘˜ is the safety factor. Moreover, the mixtures of distribution functions are unimodal for all 𝑝 if (πœ‡1βˆ’πœ‡2)2<27𝜎2/(8𝐿) (or βˆšπœ‚<27/8) (see [32]). Besides, the reorder point must satisfy the following equation which implies a service level constraint 𝑃(𝑋>π‘Ÿ)=π‘ž, where π‘ž represents the allowable stock-out probability during 𝐿.

In addition, we assume that the capital investment, 𝐼(𝐴), in reducing ordering cost is a logarithmic function of the ordering cost 𝐴. That is,

This function is consistent with the Japanese experience as reported in Hall [33], and has been utilized in many articles (see [16, 17, 20–24, 34], etc.).

In this study, we relax the restriction about the form of the mixtures of d.f. of lead time demand, that is, we assume here that the lead time demand X has the mixtures of d.f. πΉβˆ—=𝑝𝐹1+(1βˆ’π‘)𝐹2, where 𝐹1 has finite mean πœ‡1𝐿 and standard deviation 𝜎√𝐿 and 𝐹2 has finite mean πœ‡2𝐿 and standard deviation 𝜎√𝐿, πœ‡1βˆ’πœ‡2√=πœ‚πœŽ/𝐿, πœ‚βˆˆπ‘…. Then the expected shortage at the end of the cycle is defined by 𝐡(π‘Ÿ)=𝐸(π‘‹βˆ’π‘Ÿ)+. Thus, the expected number of backorders per cycle is 𝛽𝐡(π‘Ÿ) and the expected lost sales per cycle is (1βˆ’π›½)𝐡(π‘Ÿ). Hence, the expected annual stock-out cost is (𝐷/𝑄)[πœ‹π‘₯𝛽+πœ‹0(1βˆ’π›½)]𝐡(π‘Ÿ).

The expected net inventory level just before the order arrives is

and the expected net inventory level at the beginning of the cycle is

Therefore, the expected annual holding cost is

Finally, the total expected annual cost (EAC) can be expressed as follows:

In practical situations, as shortage occurs, the longer the length of lead time is, the larger the amount of shortage is, the smaller the proportion of customers can wait, and hence the smaller the backorder rate would be; in addition, the larger backorder price discount is, hence the larger the backorder rate would be. Therefore, we also consider the backorder rate that is proposed by combining Ouyang and Chuang [13] (or Lee [14]) with Pan and Hsiao [15] at the same time. Thus, we define 𝛽=𝛽0πœ‹π‘₯/πœ‹0, where

Hence, the total expected annual cost (2.6) reduces to Besides, we also consider that the ordering cost can be reduced through capital investment and the ordering cost 𝐴 as a decision variable. Hence, we seek to minimize the sum of capital investment cost of reducing ordering cost 𝐴 and the inventory costs (as expressed in (2.8) by optimizing over 𝑄, 𝐴, πœ‹π‘₯, and 𝐿 constrained on 0<𝐴≀𝐴0. That is, the objective of our problem is to minimize the following total expected annual cost: subject to 0<𝐴≀𝐴0.

Now, we attempt to use a minimax criterion to solve this problem. If we let Ω be the class of all single c.d.f. (included 𝐹1 and 𝐹2) with finite mean and standard deviation, then the minimax criterion for our problem is to find the most unfavorable c.d.f.s 𝐹1 and 𝐹2 in Ω for each decision variable and then to minimize over the decision variables; that is, our problem is to solve

In addition, we also need the following Proposition which was asserted by Gallego and Moon [35] to solve the above problem.

Proposition 2.1. For any 𝐹∈Ω, Moreover, the upper bound (2.11) is tight. In other words, we can always find a distribution in which the above bound is satisfied with equality for every π‘Ÿ.

Using the inequality (2.11) for 𝐹1 and 𝐹2, we obtain

where

Then, the problem (2.10) is equivalent to minimize where subject to 0<𝐴≀𝐴0.

In order to solve this nonlinear programming problem, we first ignore the restriction 0<𝐴≀𝐴0 and take the first partial derivatives of EAC(𝑄,𝐴,πœ‹π‘₯,𝐿) with respect to 𝑄,𝐴,πœ‹π‘₯ and 𝐿∈(𝐿𝑖,πΏπ‘–βˆ’1), respectively. We can obtain where πœƒ2=(πœ‹π‘₯/πœ‹0)(1βˆ’πœ‹π‘₯/πœ‹0)𝛿, Ξ”(𝐿)=πœ€πΈ(π‘‹βˆ’π‘Ÿ)+=πœ€π΅(π‘Ÿ),, 𝐡(π‘Ÿ) and 𝐡(π‘Ÿ) is expressed as (2.15).

Since 𝐡(π‘Ÿ)>0 is the expected shortage quantity at the end of cycle, we know that 𝐡(π‘Ÿ)=0, if shortages occur; 𝐡(π‘Ÿ) otherwise. It is clear that EAC(𝑄,𝐴,πœ‹π‘₯,𝐿) is positive. By examining the second-order sufficient conditions (SOSCs), it can be easily verified that (𝑄,𝐴,πœ‹π‘₯,𝐿) is not a convex function of 𝑄. However, for fixed 𝐴,, πœ‹π‘₯ and EAC(𝑄,πœ‹π‘₯,𝐿), 𝐿∈(𝐿𝑖,πΏπ‘–βˆ’1) is concave in πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πΏ21=βˆ’4β„Žπ‘˜πΏ3/2ξ‚ƒβˆš1+𝑝(1βˆ’π‘)πœ‚2πœŽξ‚„βˆ’14𝜎𝐿3/2Γ—ξƒ―β„Žξ€·1βˆ’πœƒ1ξ€Έ[]+[]1+3Ξ”(𝐿)3+Ξ”(𝐿)][Ξ”(𝐿)2[]1+Ξ”(𝐿)3+π·π‘„πœ‹0ξ€·1βˆ’πœƒ2ξ€Έ[]+[]1+3Ξ”(𝐿)3+Ξ”(𝐿)][Ξ”(𝐿)2[]1+Ξ”(𝐿)3×𝐡(π‘Ÿ)𝜎√𝐿.(2.18) because

Therefore, for fixed 𝐴,, πœ‹π‘₯ and (𝐿𝑖,πΏπ‘–βˆ’1), the minimum total EAC will occur at the end points of the interval 𝐿∈(𝐿𝑖,πΏπ‘–βˆ’1). On the other hand, for a given value of 𝑄=2π·β„Žξ‚΅ξ‚Έπœ‹π΄+2π‘₯πœ‹0𝛽0+πœ‹0ξ‚΅πœ‹1βˆ’π‘₯πœ‹0𝛽0𝐡(π‘Ÿ)+𝑅(𝐿)ξ‚Άξ‚Ό1/2,(2.19)𝐴=πœƒπ‘£π‘„π·πœ‹,(2.20)π‘₯=12ξ‚΅β„Žπ‘„π·+πœ‹0ξ‚Ά,(2.21), by setting (2.16) equal to zero, we obtain where 𝑄 and 𝐴, is expressed as (2.15).

Theoretically, for fixed πœ‹π‘₯, from (2.19)–(2.21), we can get the values of π‘„βˆ—, π΄βˆ—, and πœ‹βˆ—π‘₯ (we denote these values by (π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯), 𝐿∈(𝐿𝑖,πΏπ‘–βˆ’1) and (π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯)).

Moreover, it can be shown that the SOSCs are satisfied since the Hessian matrix is positive definite at point 0<𝐴≀𝐴0. (see the appendix for the proof). Hence, for a fixed π΄βˆ—, the point π΄βˆ—<𝐴0 is the local optimal solution such that the total expected annual cost has minimum value.

We now consider the constraint (π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯) From (2.20), we note that 𝐿∈(𝐿𝑖,πΏπ‘–βˆ’1) is positive. Also, if π΄βˆ—β‰₯𝐴0, then π΄βˆ—=𝐴0 is an interior optimal solution for given 𝑄=πœƒπ‘£+(πœƒπ‘£)2ξ€Ίξ€·+2β„Žπ·1βˆ’β„Žπ›½0/2π·πœ‹0ξ€Έπœ‹π΅(π‘Ÿ)ξ€»ξ€½0ξ€Ί1βˆ’(1/4)𝛽0𝐡(π‘Ÿ)+𝑅(𝐿)β„Žξ€Ίξ€·1βˆ’β„Žπ›½0/2π·πœ‹0𝐡(π‘Ÿ),(2.22). However, if 𝛽0=𝛿/(1+πœ€πΈ(π‘‹βˆ’π‘Ÿ)+), then it is unrealistic to invest in changing the current ordering cost level. For this special case, the optimal ordering cost is the original ordering cost, that is, 𝐡(π‘Ÿ), and our model reduces to (2.8) (i.e., the model of Lee et al. [36] with any mixture of distribution functions, not just mixture of normal distributions).

Substituting (2.20) and (2.21) into (2.19), we get

where 𝐷 and β„Ž is expressed as (2.15).

Theoretically, for fixed πœ‹0, 𝜎, πœ‚, 𝛿, 𝑝, π‘ž, πœ€, πœƒ, 𝑣, 𝐿𝑖(𝑖=1,2,…,𝑛),, (𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖,𝐿𝑖), 𝐿𝑖 and each EAC(𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖,𝐿𝑖) the optimal min𝑖=0,1,2,…,𝑛EAC(𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖,𝐿𝑖) pair given 𝑓𝑋 can be obtained by solving (2.22) iteratively until convergence. The convergence of the procedure can be shown. Furthermore, using (2.9), we can obtain the corresponding total expected annual cost 𝑋. Hence, the minimum total expected annual cost is π‘ž. However, in practice, since the p.d.f. π‘˜ of the lead time demand π‘˜ is unknown, even if the value of π‘Œ is given, we cannot get the exact value of π‘“π‘Œ(𝑦). Thus, in order to find the value of πœ‡πΏ, we need the following proposition.

Proposition 2.2. Let 𝜎√𝐿(>0) be a random variable which has a p.d.f. 𝑐>πœ‡πΏ with finite mean πœŽπ‘ƒ(𝑋>𝑐)≀2𝐿𝜎2𝐿+(π‘βˆ’πœ‡πΏ)2.(2.23) and standard deviation πΉβˆ—=𝑝𝐹1+(1βˆ’π‘)𝐹2, then for any real number π‘Ÿ=πœ‡βˆ—πΏ+π‘˜πœŽβˆ—βˆšπΏ,,

So, by using π‘ž=𝑃(𝑋>π‘Ÿ), the recorder point √0β‰€π‘˜β‰€(1/π‘ž)βˆ’1+|πœ‚| and Proposition 2.2, we get Further, it is assumed that the allowable stock-out probability q during lead time is given, that is, βˆšπ‘˜βˆˆ[0,(1/π‘ž)βˆ’1+|πœ‚|], then from (2.24), we get 𝑄. It is easy to verify that 𝐴 has a smooth curve for πœ‹π‘₯,. Hence, we can establish the following algorithm to obtain the suitable k and hence the optimal 𝐿, 𝐴0, 𝐷 and β„Ž.

Algorithm 2.3. Step 1. Input the values of πœ‚, 𝜎, πœ‹0, πœƒ, 𝑣, 𝑝, π‘ž, 𝛿, πœ€, π‘Žπ‘–, 𝑏𝑖,, 𝑐𝑖, 𝑖=1,2,…,𝑛, √[0,(1/π‘ž)βˆ’1+|πœ‚|] and π‘š, π‘š.Step 2. For a given q, we divide the interval π‘˜0=0 into π‘˜π‘=√(1/π‘ž)βˆ’1+|πœ‚| equal subintervals, where π‘˜π‘—=π‘˜π‘—βˆ’1+(π‘˜π‘βˆ’π‘˜0)/π‘š is large enough. And we let 𝑗=1,2,…,π‘šβˆ’1., π‘Žπ‘– and 𝑏𝑖, 𝑐𝑖,Step 3. Use the 𝐿𝑖, 𝑖=1,2,…,𝑛., 𝐿𝑖 to compute 𝑖=1,2,…,𝑛, 𝑄𝑖Step 4. For each π‘˜π‘—βˆˆ{π‘˜0,π‘˜1,…,π‘˜π‘š}, 𝑗=1,2,…,π‘š, compute 𝐴𝑖 by using (2.22) for given πœ‹π‘₯𝑖, πœ‹π‘₯𝑖. Then, compute πœ‹0 and πœ‹π‘₯𝑖<πœ‹0 by using (2.20) and (2.21).Step 5. Compare πœ‹π‘₯𝑖 and πœ‹π‘₯𝑖β‰₯πœ‹0. If πœ‹π‘₯𝑖=πœ‹0, then take 𝐴𝑖 into Step 6; if 𝐴0, then take 𝐴𝑖<𝐴0 into Step 6. Compare 𝐴𝑖 and 𝐴𝑖β‰₯𝐴0. If 𝐴𝑖=𝐴0, then take (𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖,𝐿𝑖) into Step 6; if π‘˜π‘—βˆˆ{π‘˜0,π‘˜1,…,π‘˜π‘š}, then take EACπ‘˜π‘—(𝑄𝑖,πœ‹π‘₯𝑖,𝐿𝑖) into Step 6.Step 6. For each pair 𝑖=1,2,…,𝑛 and minπ‘˜π‘—βˆˆ{π‘˜0,π‘˜1,…,π‘˜π‘š}EACπ‘˜π‘—(𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖,𝐿𝑖), compute the corresponding total expected annual cost EACπ‘˜βˆ—π‘†(𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖,𝐿𝑖)=minπ‘˜π‘—βˆˆ{π‘˜0,π‘˜1,…,π‘˜π‘š}EACπ‘˜π‘—(𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖𝐿𝑖), min𝑖=0,1,2,…,𝑛EACπ‘˜βˆ—π‘†(𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖𝐿𝑖).Step 7. Find EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—)=min𝑖=0,1,…,𝑛EACπ‘˜βˆ—π‘†(𝑄𝑖,𝐴𝑖,πœ‹π‘₯𝑖,𝐿𝑖). If (π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—), then find π‘˜π‘ (𝑖). If EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—), then π‘˜βˆ— is the optimal solution; the value of 𝐷=600 such that 𝐴0=$200 exists is the optimal safety factor, and we denote it by β„Ž=$20.Step 8. Stop.

3. A Numerical Example

In order to illustrate the above solution procedure, let us consider an inventory system with the following data: πœ‹0=$150 units/year, πœ‡βˆ—=11 per order, 𝜎=7, πœƒ=0.1, 𝑣=5800. units/week, 𝑖 units/week, and the lead time has three components with data shown in Table 1. Besides, for ordering cost reduction, we take 𝑏𝑖 per dollar per year and π‘Žπ‘–


Lead time componentNormal durationMinimum durationUnit crashing cost
𝑐 𝑖 𝑝 = 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 (days) π‘ž = 0 . 2 (days) 𝛿 = 0 , 0 . 5 , 1 ($/day)

12060.4
22061.2
31695.0

We assume here that the lead time demand follows a mixture of distribution functions and want to solve the case when πœ‚=0.7,, πœ€=0,0.5,1,10,20,40,80,100,∞., 𝐹1, 𝐹2 and πΉβˆ—

If we knew the form of the c.d.f.s EAC𝑛𝑄,𝐴,πœ‹π‘₯𝐴,𝐿=πœƒπ‘£ln0𝐴𝐷+𝐴𝑄𝑄+β„Ž2√+πœŽπΏξƒ―π‘ξƒ¬π‘Ÿ1Ξ¦ξƒ©πœ‡βˆ—βˆšπΏπœŽξƒͺξƒ©πœ‡+(1βˆ’π‘)πœ‚βˆ’πœ™βˆ—βˆšπΏπœŽξƒ¬π‘Ÿ+(1βˆ’π‘)πœ‚ξƒͺξƒ­+(1βˆ’π‘)2Ξ¦ξƒ©πœ‡βˆ—βˆšπΏπœŽξƒͺξƒ©πœ‡βˆ’π‘πœ‚βˆ’πœ™βˆ—βˆšπΏπœŽβˆšβˆ’π‘πœ‚ξƒͺξƒ­ξƒ°+(1βˆ’π›½)πœŽξ€·π‘ŸπΏΞ¨1,π‘Ÿ2ξ€Έξƒͺ+𝐷,π‘π‘„ξ€Ίπœ‹π‘₯𝛽+πœ‹0ξ€»πœŽβˆš(1βˆ’π›½)ξ€·π‘ŸπΏΞ¨1,π‘Ÿ2ξ€Έ+𝐷,𝑝𝑄𝑅(𝐿).(3.1) and (𝑄𝑛,𝐴𝑛,πœ‹π‘₯𝑛,𝐿𝑛), we could solve the problem optimally for that particular distribution. For example, if EAC𝑛(𝑄𝑛,𝐴𝑛,πœ‹π‘₯𝑛,𝐿𝑛) is c.d.f. of mixture of normal distributions, then the total expected annual cost is

We can obtain the optimal (π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—) by the standard procedure and incur an expected cost (𝑄𝑛,𝐴𝑛,πœ‹π‘₯𝑛,𝐿𝑛) (see Lee et al. [30]). For fixed EAC𝑛(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—), if we use πΉβˆ— instead of the optimal EAC𝑛(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—)βˆ’EAC𝑛(𝑄𝑛,𝐴𝑛,πœ‹π‘₯𝑛,𝐿𝑛) for mixture of normal distributions, then we can get an expected cost πΉβˆ—. Hence, as the c.d.f. π‘„βˆ— is mixture cumulative of normal distributions, the added cost by using the minimax mixture of distributions free procedure instead of the standard procedure is π΄βˆ—. This is the largest amount that we would be willing to pay for the knowledge of d.f. πœ‹βˆ—π‘₯,. This quantity can be regarded as the expected value of additional information (EVPI). The results of the solution procedure are solved by using the subroutine ZREAL of IMSL from the computer software Compaq Visual Fortran V6.0 (Inclusive of IMSL) [37] and summarized in Table 2. From Table 2, we note that (i) the order quantity EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—), the ordering cost πœ€, the backorder price discount 𝛽 and the minimum total expected annual costs 𝛿=0.5,1.0 increase as 𝑝 increases (i.e., the back-order rate EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—) decreases) for πœ€ and the fixed 𝛿; (ii) the minimum total expected annual cost 𝛿=0.5,1.0 increase and then decrease as p increases for the fixed πœ€ and 𝑝=0, thus for 0<𝑝<1 and the fixed EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—), when EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—) or 1, the model considers only one kind of customers' demand; when π‘„βˆ—, the model considers two kinds of customers' demand. It implies that π΄βˆ— of two kinds of customers' demand are larger than πœ‹βˆ—π‘₯ of one kind of customers' demand, thus if the true distribution of the lead time demand is a mixture of normal distributions, we use a single distribution (such as [13]) to substitute the true distribution of the lead time demand then the minimum expected total annual cost will be underestimated; (iii) the order quantity EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—), the ordering cost 𝛿, the backorder price discount πœ€ and the minimum total expected annual cost πΏβˆ— decrease as 𝛿=0(0.5)1 increases for the fixed πœ€=0,0.5,1,10,20,40,80,100,∞; and p; (iv) no matter what values of p, the optimal lead time 𝑝 is approached to a certain value (3 weeks) for πœ€ and 𝛽 (v) while for the fixed 𝛿=0.5, EVAI increases and then decreases as 𝑝 increases (i.e., the backorder rate πœ€=0 decreases) for 𝛿=1.0 and fixed EACξ…žπ‘›; (vi) EVAI increases and then decreases as p increases except EAC𝑛, when πœ€; (vii) the cost penalty 𝛽/𝑝 of using the distribution free operating policy instead of the optimal one is increasing and then decreasing as 𝛿=0.0 increases (i.e., the backorder rate πœ€=0 decreases) for the fixed 𝑝=0.6,0.8,1.0 except 𝛿=1.0 and 𝐿𝑖, πœ€, when (π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,πΏβˆ—). Conveniently, we organize the above (i)–(vii) in Table 3.


E A C ξ…ž 𝑛 ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) E A C 𝑛 E A C ξ…ž 𝑛 / E A C 𝑛 𝑝 = 0 . 0 ∞ EVAI 𝑝 = 0 . 2

∞
0(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
0.5(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
1(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
10(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
20(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
40(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
80(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
100(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

∞
0(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
0.5(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
1(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
10(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
20(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
40(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
80(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
100(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
∞ (148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220

p = 0.4
0(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
0.5(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
1(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
10(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
20(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
40(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
80(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
100(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
∞ (148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237

p = 0.6
0(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
0.5(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
1(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
10(148,143, 77.463, 3)3833.2413590.772(160,154., 77.659, 3)3583.1597.6131.00212
20(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
40(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
80(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
100(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
∞ (148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212

p = 0.8
0(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
0.5(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
1(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
10(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
20(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
40(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
80(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
100(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
∞ (148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
p = 1.0
0(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
0.5(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
1(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
10(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
20(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
40(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
80(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
100(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.0
0(145,140, 77.420, 3)3731.3883430.876(151,146, 77.521, 3)3428.8302.0461.00060
0.5(145,141, 77.424, 3)3765.7603475.731(154,149, 77.562, 3)3471.9103.8211.00110
1(146,141, 77.431, 3)3781.2843494.414(155,150, 77.580, 3)3490.0194.3951.00126
10(148,143, 77.460, 3)3816.6673532.005(157,152, 77.615, 3)3527.2634.7411.00134
20(148,143, 77.464, 3)3820.2273535.434(157,152, 77.619, 3)3530.7094.7251.00134
40(148,143, 77.466, 3)3822.1253537.240(157,152, 77.620, 3)3532.5244.7161.00134
80(148,143, 77.467, 3)3823.1053538.166(157,152, 77.621, 3)3533.4564.7101.00133
100(148,143, 77.467, 3)3823.3033538.354(157,152, 77.621, 3)3533.6444.7101.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.2
0(145,140, 77.415, 3)3739.2223477.626(154,148, 77.559, 3)3473.4904.1361.00119
0.5(145,140, 77.421, 3)3773.2823526.203(156,151, 77.603, 3)3519.5576.6461.00189
1(146,141, 77.428, 3)3788.7243545.816(157,152, 77.621, 3)3538.3477.4691.00211
10(147,142, 77.457, 3)3824.0423583.914(159,154, 77.656, 3)3575.9737.9401.00222
20(148,143, 77.460, 3)3827.6063587.301(160,154, 77.660, 3)3579.3867.9141.00221
40(148,143, 77.462, 3)3829.5053589.078(160,154, 77.661, 3)3581.1807.8981.00221
80(148,143, 77.463, 3)3830.4873589.989(160,154, 77.662, 3)3582.1007.8891.00220
100(148,143, 77.463, 3)3830.6863590.173(160,154, 77.662, 3)3582.2867.8871.00220
∞ (148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220

p = 0.4
0(145,140, 77.414, 3)3741.9233488.279(154,149, 77.565, 3)3483.6824.5971.00132
0.5(145,140, 77.419, 3)3775.9153537.551(157,151, 77.609, 3)3530.2457.3061.00207
1(146,141, 77.427, 3)3791.3383557.216(158,152, 77.627, 3)3549.1448.0721.00227
10(147,142, 77.454, 3)3826.6413595.519(160,154, 77.663, 3)3586.8288.6911.00242
20(148,143, 77.459, 3)3830.2053598.779(160,155, 77.666, 3)3590.2378.5421.00238
40(148,143, 77.461, 3)3832.1053600.552(160,155, 77.668, 3)3592.0278.5251.00237
80(148,143, 77.462, 3)3833.0873601.460(160,155, 77.669, 3)3592.9458.5151.00237
100(148,143, 77.462, 3)3833.2863601.643(160,155, 77.669, 3)3593.1308.5131.00237
∞ (148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237

p = 0.6
0(145,140, 77.415, 3)3740.9863477.898(153,148, 77.555, 3)3473.9823.9161.00113
0.5(145,140, 77.421, 3)3775.0373526.154(156,151, 77.599, 3)3519.7696.3851.00181
1(146,141, 77.427, 3)3790.4763545.698(157,152, 77.617, 3)3538.4977.2001.00203
10(147,142, 77.456, 3)3825.7923583.761(159,154, 77.653, 3)3576.0907.6711.00215
20(148,143, 77.460, 3)3829.3563587.152(159,154, 77.656, 3)3579.5067.6461.00214
40(148,143, 77.461, 3)3831.2563588.932(159,154, 77.658, 3)3581.3027.6301.00213
80(148,143, 77.462, 3)3832.2373589.844(160,154, 77.658, 3)3582.2227.6221.00213
100(148,143, 77.463, 3)3832.4363590.028(160,154, 77.659, 3)3582.4097.6201.00213
∞ (148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6121.00212
p = 0.8
0(145,140, 77.417, 3)3737.2853456.812(152,147, 77.539, 3)3453.8492.9631.00086
0.5(145,140, 77.422, 3)3771.4723503.469(155,150, 77.582, 3)3498.3465.1231.00146
1(146,141, 77.430, 3)3786.9483522.514(156,151, 77.599, 3)3516.7835.7311.00163
10(147,143, 77.458, 3)3822.2913560.458(158,153, 77.635, 3)3554.2166.2431.00176
20(148,143, 77.461, 3)3825.8543563.868(158,153, 77.638, 3)3557.6466.2221.00175
40(148,143, 77.463, 3)3827.7533565.660(158,153, 77.640, 3)3559.4516.2091.00174
80(148,143, 77.464, 3)3828.7343566.579(158,153, 77.641, 3)3560.3776.2021.00174
100(148,143, 77.465, 3)3828.9333566.666(158,153, 77.641, 3)3560.5646.1021.00171
∞ (148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171

p = 1.0
0(145,140, 77.420, 3)3731.3883430.876(151,146, 77.521, 3)3428.8312.0461.00060
0.5(145,141, 77.424, 3)3765.7603475.731(154,149, 77.562, 3)3471.9103.8211.00110
1(146,141, 77.431, 3)3781.2843494.414(155,150, 77.580, 3)3490.0204.3941.00126
10(148,143, 77.460, 3)3816.6673532.005(157,152, 77.615, 3)3527.2644.7411.00134
20(148,143, 77.464, 3)3820.2273535.435(157,152, 77.619, 3)3530.7094.7261.00134
40(148,143, 77.466, 3)3822.1253537.240(157,152, 77.620, 3)3532.5244.7161.00134
80(148,143, 77.467, 3)3823.1053538.167(157,152, 77.621, 3)3533.4564.7111.00133
100(148,143, 77.467, 3)3823.3033538.354(157,152, 77.621, 3)3533.6454.7091.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.0
0(142,137, 77.367, 3)3630.3183319.556(142,137, 77.364, 4)3306.32913.2271.00400
0.5(143,138, 77.379, 3)3705.6593411.048(150,145, 77.501, 3)3408.0193.0291.00089
1(144,139, 77.394, 3)3737.6913449.035(152,147, 77.536, 3)3444.9404.0951.00119
10(147,142, 77.452, 3)3809.2043524.967(157,151, 77.608, 3)3520.1044.8621.00138
20(148,143, 77.460, 3)3816.3423531.755(157,152, 77.615, 3)3527.0094.7461.00135
40(148,143, 77.464, 3)3820.1413535.369(157,152, 77.618, 3)3530.6424.7271.00134
80(148,143, 77.466, 3)3822.1033537.223(157,152, 77.620, 3)3532.5074.7161.00134
100(148,143, 77.466, 3)3822.5003537.598(157,152, 77.621, 3)3532.8844.7141.00133
∞ (148,143, 77.468, 3)3824.1073539.109(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.2
0(142,137, 77.362, 3)3638.7033361.079(144,139, 77.405, 4)3354.2456.8341.00204
0.5(142,138, 77.374, 3)3713.3243460.282(152,147, 77.541, 3)3454.6685.6131.00162
1(143,139, 77.391, 3)3745.1873500.043(155,149, 77.577, 3)3492.9717.0721.00202
10(147,142, 77.448, 3)3816.5713577.010(159,154, 77.650, 3)3568.8948.1161.00227
20(147,142, 77.455, 3)3823.7153583.795(159,154, 77.656, 3)3575.7338.0621.00225
40(148,143, 77.460, 3)3827.5193587.240(160,154, 77.660, 3)3579.3247.9161.00221
80(148,143, 77.462, 3)3829.4833589.063(160,154, 77.661, 3)3581.1647.8981.00221
100(148,143, 77.462, 3)3829.8813589.431(160,154, 77.662, 3)3581.5367.8951.00220
∞ (148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220

p = 0.4
0(142,137, 77.362, 3)3641.5203370.810(145,140, 77.411, 4)3365.7245.0851.00151
0.5(142,138, 77.374, 3)3715.9883471.297(153,148, 77.547, 3)3465.2006.0971.00176
1(143,139, 77.390, 3)3747.8133511.377(155,150, 77.584, 3)3503.7247.6531.00218
10(147,142, 77.447, 3)3819.1673588.520(159,154, 77.657, 3)3579.7628.7581.00245
20(147,142, 77.454, 3)3826.3133595.290(160,154, 77.663, 3)3586.5918.6991.00243
40(148,143, 77.459, 3)3830.1183598.719(160,155, 77.666, 3)3590.1748.5451.00238
80(148,143, 77.461, 3)3832.0833600.536(160,155, 77.668, 3)3592.0118.5261.00237
100(148,143, 77.461, 3)3832.4813600.903(160,155, 77.668, 3)3592.3828.5221.00237
∞ (148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237

p = 0.6
0(142,137, 77.362, 3)3640.4733361.860(144,139, 77.401, 4)3355.7856.0751.00181
0.5(142,138, 77.375, 3)3715.0833460.331(152,147, 77.537, 3)3454.9705.3611.00155
1(143,139, 77.390, 3)3746.9403499.948(154,149, 77.574, 3)3493.1476.8021.00195
10(147,142, 77.447, 3)3818.3203576.848(159,153, 77.646, 3)3569.0037.8451.00220
20(147,142, 77.454, 3)3825.4653583.641(159,154, 77.652, 3)3575.8487.7921.00218
40(148,143, 77.459, 3)3829.2693587.091(159,154, 77.656, 3)3579.4437.6481.00214
80(148,143, 77.461, 3)3831.2333588.916(159,154, 77.658, 3)3581.2857.6311.00213
100(148,143, 77.462, 3)3831.6313589.285(159,154, 77.658, 3)3581.6587.6271.00213
( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) (148,143, 77.463, 3)3833.2403590.772(160,154, 77.659, 3)3583.1597.6131.00212

p = 0.8
0(142,137, 77.364, 3)3636.5343343.016(143,138, 77.384, 4)3334.1118.9051.00267
0.5(143,138, 77.377, 3)3711.4563438.123(151,146, 77.520, 3)3433.9704.1531.00121
1(143,139, 77.391, 3)3743.3883477.029(153,148, 77.556, 3)3471.5575.4721.00158
10(147,142, 77.449, 3)3814.8243553.487(158,152, 77.628, 3)3547.0956.3921.00180
20(147,143, 77.457, 3)3821.9653560.217(158,153, 77.635, 3)3553.9686.2491.00176
40(148,143, 77.461, 3)3825.7673563.804(158,153, 77.638, 3)3557.5816.2231.00175
80(148,143, 77.463, 3)3827.7313565.644(158,153, 77.640, 3)3559.4346.2091.00174
100(148,143, 77.463, 3)3828.1283566.015(158,153, 77.640, 3)3559.8086.2071.00174
𝐹 βˆ— (148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171

p = 1.0
0(142,137, 77.367, 3)3630.3183319.556(142,137, 77.364, 4)3306.32913.2271.00400
0.5(143,138, 77.379, 3)3705.6593411.049(150,145, 77.501, 3)3408.0193.0291.00089
1(144,139, 77.394, 3)3737.6913449.036(152,147, 77.536, 3)3444.9404.0951.00119
10(147,142, 77.452, 3)3809.2043524.967(157,151, 77.608, 3)3520.1054.8621.00138
20(148,143, 77.460, 3)3816.3423531.755(157,152, 77.615, 3)3527.0094.7461.00135
40(148,143, 77.464, 3)3820.1413535.369(157,152, 77.618, 3)3530.6424.7271.00134
80(148,143, 77.466, 3)3822.1023537.223(157,152, 77.620, 3)3532.5074.7161.00134
100(148,143, 77.466, 3)3822.5003537.598(157,152, 77.621, 3)3532.8844.7141.00133
E A C 𝑛 ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

Note: we obtain the optimal ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) by the standard procedure, E A C ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) is mixture of normal distribution, and incur an expected annual cost ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) . ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) stands for the optimal order quantity, the ordering cost, the back-order price discount, and the optimal lead time, respectively, that the demand in the lead time is mixture of free distribution; E A C 𝑛 ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) is the minimum total expected annual cost. We use E A C ξ…ž = E A C ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) instead of the optimal E A C ξ…ž 𝑛 = E A C 𝑛 ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) , for E A C 𝑛 = E A C 𝑛 ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) . In other word, 𝑝 , 𝛿 = 0 . 5 , 1 . 0 and πœ€ ↑ .

Fixed parameterVariable parametersOutcomes
Decision variablesObjective

( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ ) , ↑ E A C ↑ 𝛿 πœ€ 𝑝 ↑
E A C β†— β†˜ , 𝑝 πœ€ 𝛿 ↑
( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ ) , ↓ E A C ↓ 𝛿 𝑝 πœ€
𝐿 βˆ— = 3 , 𝑝 , 𝛿 = 0 . 5 πœ€ ↑
β†— β†˜ , 𝛿 = 1 . 0 πœ€ β‰  0 EVAI 𝑝 ↑
β†— β†˜ , 𝑃 ( 𝑋 > π‘Ÿ ) = π‘ž 𝐿 EVAI 𝐻

4. Concluding Remarks

In this article, we consider that the backorder rate is dependent on the amount of the shortages and the backorder price discount by using the idea of Ouyang and Chuang [13] and Pan and Hsiao [15]. Hence, we regard the backorder rate as controllable variable. In addition, the ordering cost can be controlled and reduced through various efforts such as worker training, procedural changes, and specialized equipment acquisition. So, we also treat the ordering cost as a decision variable. Moreover, we make no assumption about the form of the mixtures of distribution functions of the lead time demand and apply the minimax criterion to solve the problem. We also develop an algorithmic procedure to find the optimal inventory policy.

In this study, we consider the service level constraint to satisfy the equation as βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πœ•π»=2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π‘„2πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ•π‘„πœ•π΄2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π‘„πœ•πœ‹π‘₯πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ•π΄πœ•π‘„2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π΄2πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π΄πœ•πœ‹π‘₯πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ‹π‘₯πœ•πœ•π‘„2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ‹π‘₯πœ•πœ•π΄2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ‹2π‘₯⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(A.1). But, the reorder point r is a controllable variable. Hence, it would be interesting to treat the reorder point as a decision variable in future research.

Appendix

For a given value of πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π‘„2=2𝐴𝐷𝑄3𝐷+2𝑄3πœ‹0ξ‚΅πœƒ1βˆ’2ξ‚Ά1+Ξ”(𝐿)Ξ”(𝐿)πœ€+2𝐷𝑄3πœ•π‘…(𝐿)>0,2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π΄2=πœƒπ‘£π΄2πœ•>0,2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ‹2π‘₯𝐷=2𝑄𝛿/πœ‹01+Ξ”(𝐿)Ξ”(𝐿)πœ€πœ•>0,2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,𝐿=πœ•πœ•π‘„πœ•π΄2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπ·πœ•π΄πœ•π‘„=βˆ’π‘„2,πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π‘„πœ•πœ‹π‘₯=πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ‹π‘₯=π·πœ•π‘„π‘„2ξ‚Έπœ‹1βˆ’2π‘₯πœ‹0𝛿1+Ξ”(𝐿)Ξ”(𝐿)πœ€,πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•π΄πœ•πœ‹π‘₯=πœ•2ξ€·EAC𝑄,𝐴,πœ‹π‘₯ξ€Έ,πΏπœ•πœ‹π‘₯πœ•π΄=0.(A.2), we first obtain the Hessian matrix 𝐻 as follows:

where

Then we proceed by evaluating the principal minor of ||𝐻11||=πœ•2𝑄EACβˆ—,π΄βˆ—,πœ‹βˆ—π‘₯ξ€Έ,πΏπœ•π‘„βˆ—2>0.(A.3) at point 𝐻. The first principal minor of 𝐷/π‘„βˆ—=πœƒπœˆ/π΄βˆ— is The second principal minor of 𝐻 is (note that from (2.16) ||𝐻33||=πœ•2𝑄EACβˆ—,π΄βˆ—,πœ‹βˆ—π‘₯ξ€Έ,πΏπœ•πœ‹π‘₯βˆ—2β‹…||𝐻22||βˆ’ξƒ¬πœ•2EAC(π‘„βˆ—,π΄βˆ—,πœ‹βˆ—π‘₯,𝐿)πœ•π‘„βˆ—πœ•πœ‹βˆ—π‘₯ξƒ­2β‹…πœ•2𝑄EACβˆ—,π΄βˆ—,πœ‹βˆ—π‘₯ξ€Έ,πΏπœ•π΄βˆ—2𝐷=22π‘„βˆ—4ξ‚Έπœƒπ‘£π΄βˆ—+2πœƒπ‘£π΄βˆ—2𝑅(𝐿)𝛿/πœ‹0Ξ”1+Ξ”(𝐿)(𝐿)πœ€+𝐷2π‘„βˆ—4πœƒπ‘£π΄βˆ—2𝛿1+Ξ”(𝐿)Ξ”(𝐿)πœ€ξ‚Ά2𝛿4βˆ’ξ‚Ή1+Ξ”(𝐿)>0,(A.5)) The third principal minor of Ξ”(𝐿)=πœ€π΅(π‘Ÿ) is where (𝑄,π‘Ÿ), (𝑠,𝑄).

Therefore, from (A.3)–(A.5), it is clearly seen that the Hessian matrix H is positive definite at point .

Acknowledgments

The authors wish to thank the referees for valuable suggestions which led to the improvement of this paper. This research was partially supported by the National Science Council, Taiwan (Plan no. NSC 96-2221-E-309-001 and NSC 97-2221-E-309-008).

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