(Q,r,L) Inventory model with defective items

Computers & Industrial Engineering 39 (2001) 173±185

www.elsevier.com/locate/dsw

(Q,r,L) Inventory model with defective items Kun-Shan Wu a,*, Liang-Yuh Ouyang b a

Department of Mathematical Statistics and Actuarial Sciences, Aletheia University, Tamsui, Taipei 251, Taiwan, ROC b Department of Management Sciences, Tamkang University, Tamsui, Taipei 251, Taiwan, ROC

Abstract This paper assumes that an arrival order lot may contain some defective items, and the number of defective items is a random variable. We derive a modi®ed mixture inventory model with backorders and lost sales, in which the order quantity, the reorder point and the lead time are decision variables. In our studies, we ®rst assume that the lead time demand follows a normal distribution, and then relax the assumption about the form of the distribution function of the lead time demand and apply the minimax distribution-free procedure to solve the problem. We develop an algorithm procedure to obtain the optimal ordering strategy. Furthermore, the effects of parameters are also included. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Inventory; Defective items; Lead time; Crashing cost; Minimax distribution-free procedure

1. Introduction Lead time plays an important role and has been a topic of interest for many authors in inventory management (Das, 1975; Foote, Kebriaei & Kumin, 1988; Magson, 1979). In most of the early literature dealing with inventory problems, either in deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable, which therefore, is not subject to control (Naddor, 1966; Silver & Peterson, 1985). In fact, lead time usually consists of the following components: order preparation, order transit, supplier lead time, delivery time, and setup time (Tersine, 1982). In many practical situations, lead time can be reduced at an added crashing cost; in other words, it is controllable. By shortening the lead time, we can lower the safety stock, reduce the loss caused by stockout, improve the service level to the customer, and increase the competitive ability in business. Recently, some models considering lead time as a decision variable have been developed. Liao and Shyu (1991) ®rst presented a probabilistic model in which the order quantity is predetermined and lead * Corresponding author. E-mail address: [email protected] (K.-S. Wu). 0360-8352/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 0 ) 0 0 07 7 - 2

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Nomenclature D A h h0 p p0 v b p g…p† Q r L X

Expected demand per year Setup cost per setup Nondefective holding cost per unit per year Defective holding cost per unit per year, h 0 , h Shortage cost per unit short Marginal pro®t per unit Unit inspection cost Fraction of the demand during the stockout period will be backordered, b [ ‰0; 1Š Defective rate in an order lot (independent of lot size), p [ ‰0; 1† and is a random variable The probability density function (pdf) of p Lot size (order quantity), decision variable Reorder point, decision variable Length of the lead time, decision variable p The lead time demand with ®nite mean mL and standard deviation s L….0†; a random variable Y Number of defective items in a lot, a random variable E…´† Mathematical expectation E2 …´† ˆ ‰E…´†Š2 x1 Maximum value of x and 0; i.e. x1 ˆ max{x; 0} time is a unique decision variable. In their model, lead time can be decomposed into n components, each of which has a different crashing cost for reducing lead time. Ben-Daya and Raouf (1994) extended Liao and Shyu's model by considering both the lead time and the order quantity as decision variables where shortages were neglected. Later, Ouyang, Yeh and Wu (1996) generalized Ben-Daya and Raouf's model by allowing shortages and considering that the lead time demand follows a normal distribution. Recently, Ouyang and Wu (2001) relaxed the assumption about the form of the probability distribution of lead time demand and applied the minimax distribution-free procedure to solve the distribution-free model. In the above models (Ben-Daya & Raouf, 1994; Liao & Shyu, 1991; Ouyang & Wu, 2001; Ouyang et al., 1996), the reorder point was not taken into consideration, and merely focused on the relationship between lead time and order quantity, that is, they neglected the possible impact of the reorder point on the economic ordering strategy. On the other hand, the above body of literature does not describe the possible relationship between the order lot and quality. As a result of imperfect production of the supplier, and/or damage in transit, an arrival order lot often contains some defective items. If there are defective items in orders, there will be an impact on the on-hand inventory level, the number of shortages and the frequency of orders in the inventory system. Therefore, ordering policies determined by conventional inventory models may be inappropriate for the situation in which an arrival lot contains some defective items. Recently, Paknejad, Nasri and Af®sco (1995) presented a quality-adjusted lot-sizing model with stochastic demand and constant lead time. In their article, the shortages are allowed and fully backordered and the defective rate in an order lot is ®xed. In this paper, we develop an inventory model with a mixture of backorders and lost sales in which the order quantity, the reorder point and the lead time are

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decision variables. We assume that an arrival order lot may contain some defective items, and the number of defective items in an arrival order of size Q is a random variable. We also assume, in the spirit of just-in-time (JIT) purchasing, that the purchaser inspects all the items. Inspection is assumed to be non-destructive and error-free; and all defective items are assumed to be discovered and will be returned to the vendor at the time of delivery of the next lot. Besides, inventory is continuously reviewed, an order of size Q is made whenever the inventory level (based on the number of non-defective items) falls to the reorder point r. In this article, we ®rst assume that the lead time demand follows a normal distribution, and ®nd the optimal ordering strategy. And then we consider any distribution function (df), say F, of the lead time demand has only known ®nite ®rst and second moments (and hence, mean and variance are also known). In this situation, for convenience, we let F denote the class of df Fs with ®nite mean m L and variance s2 L (where L is the length of the lead time; and m and s 2 are the mean and variance of the demand per unit time, respectively). Our goal is to solve a mixed inventory model with defective items by using the minimax distribution-free approach. That is, the purpose of this problem is to develop an algorithm procedure to ®nd the most unfavorable df F in F for each decision variable …Q; r; L† and then to minimize over the decision variables. This paper is organized as follows. In the next section, we state our assumptions. The continuous review model containing a random number of defective items with 100% inspection is developed in Section 3. A similar model for the minimax distribution-free procedure is also discussed in Section 3. Effects of parameters are included in Section 4. Section 5 summarizes the paper and contains some concluding remarks. 2. Assumptions We make the following assumptions: 1. The reorder point r ˆ expected demand during leadptime  1 safety stock (SS), and SS ˆ k £ (standard deviation of lead time demand), i.e. r ˆ mL 1 ks L where k is the safety factor. 2. An arrival order may contain some defective items. We assume that the number of defective items in an arriving order of size Q is a binomial random variable with parameters Q and p, where p …0 # p # 1† represents the defective rate in an order lot. Upon arrival of an order, all the items are inspected and defective items in each lot will be returned to the vendor at the time of delivery of the next lot. 3. Inventory is continuously reviewed. Replenishments are made whenever the inventory level (based on the number of non-defective items) falls to the reorder point r. 4. The lead time L has n mutually independent components. The ith component has a minimum duration ai and normal duration bi ; and a crashing cost per unit time ci : Further, for convenience, we rearrange ci such that c1 # c2 # ¼ # cn : 5. The components of lead time are crashed one at a time starting with component 1 (because it has the minimum unit crashing cost), and then component 2, and so on.

3. (Q, r, L) model The inventory model considered in this paper is the continuous review model with a mixture of

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backorders and lost sales, and assumes that each lot contains a random number of defectives. Upon order arrival, the purchaser inspects all the items and all defective items are assumed to be discovered and will be returned to the vendor at the time of delivery of the next lot. Therefore, the model will have an extra cost for the inspection of each lot and an extra cost for holding the defective items in stock until they are returned to the supplier. In this paper, we have assumed p that shortages are allowed and the leadptime  demand X has ®nite mean m L and standard deviation s L….0†; and the reorder point r ˆ mL 1 ks L; where k is the safety factor. Then the expected shortage at the end of the cycle is given by E…X 2 r†1 : Thus, the expected number of backorders per cycle is bE…X 2 r†1 and the expected loss in sales per cycle is …1 2 b†E…X 2 r†1 : Hence, the stockout cost per cycle is ‰p 1 p0 …1 2 b†ŠE…X 2 r†1 : The expected net inventory level just before the order arrives is r 2 mL 1 …1 2 b†E…X 2 r†1 and the expected net inventory level at the beginning of the cycle, given that there are y defective items in an arriving order of size Q, is …Q 2 y† 1 r 2 mL 1 …1 2 b†E…X 2 r†1 : Therefore, the expected holding cost per cycle is   Q2y Q2y 1 h 1 r 2 mL 1 …1 2 b†E…X 2 r† : D 2 P Next, if we let L0 ; njˆ1 bj and Li be the length of lead time with components 1, 2,¼, i crashed to their minimum duration, then Li can be expressed as Li ˆ

n X jˆ1

bj 2

i X jˆ1

…bj 2 aj †;

i ˆ 1; 2; ¼; n;

and the lead time crashing cost R…L† per cycle for a given L [ ‰Li ; Li21 Š; is given by R…L† ˆ ci …Li21 2 L† 1

iX 21 jˆ1

cj …bj 2 aj †

where ai and bi are the minimum and normal durations of component i, respectively, and ci is a crashing cost per unit time. Therefore, the cost per cycle given that there are y defective items in an arriving order of size Q is C…y† ˆ ordering cost 1 nondefective holding cost 1 defective holding cost 1 stockout cost 1 inspecting cost 1 lead time crashing cost   Q2y Q2y Q2y 1 1 r 2 mL 1 …1 2 b†E…X 2 r† 1 h 0 y ˆA1h D 2 D

…1†

1‰p 1 p0 …1 2 b†ŠE…X 2 r†1 1 vQ 1 R…L† The expected length of the cycle time and expected cycle cost under the lot of size Q are, respectively, E…T† ˆ

E…Q 2 Y† D

…2†

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and E…C† ; E‰C…Y†Š ˆ A 1

h h E…Q 2 Y†2 1 ‰r 2 mL 1 …1 2 b†E…X 2 r†1 ŠE…Q 2 Y† 2D D

h0 h0 1 QE…Y† 2 E…Y 2 † 1 ‰p 1 p0 …1 2 b†ŠE…X 2 r†1 1 vQ 1 R…L† D D

…3†

Hence, the expected total annual cost is EAC…Q; r; L† ˆ

AD hE…Q 2 Y†2 E…Y† 1 1 h‰r 2 mL 1 …1 2 b†E…X 2 r†1 Š 1 h 0 Q E…Q 2 Y† E…Q 2 Y† 2E…Q 2 Y† 2 h0

E…Y 2 † D‰p 1 p0 …1 2 b†ŠE…X 2 r†1 DvQ D 1 1 R…L† 1 E…Q 2 Y† E…Q 2 Y† E…Q 2 Y† E…Q 2 Y† …4†

For a given defective rate p in the entire lot, we assume that the number of defective items is a random variable (Y), which has a binomial distribution: Pr…Yup† ˆ CyQ py …1 2 p†Q2y ;

y ˆ 0; 1; 2; ¼; Q

…5†

In this case, E…Yup† ˆ Qp;

…6†

Var…Yup† ˆ Qp…1 2 p†

…7†

and Hence, unconditioning on p, we have E…Y† ˆ

Z1 0

E…Yup†g…p† dp ˆ QE…p†

…8†

and E…Y 2 † ˆ

Z1 0

E…Y 2 up†g…p† dp ˆ

Z1 0

{‰E…Yup†Š2 1 Var…Yup†}g…p† dp ˆ Q2 E…p2 † 1 QE‰p…1 2 p†Š …9†

Substituting Eqs. (8) and (9) into Eq. (4), we get ( ) AD h Q‰E…p2 † 2 E2 …p†Š E‰p…1 2 p†Š EAC…Q; r; L† ˆ 1 Q‰1 2 E…p†Š 1 1 Q‰1 2 E…p†Š 2 1 2 E…p† 1 2 E…p† 1 h{r 2 mL 1 …1 2 b†E…X 2 r†1 } 1 1 h 0 …Q 2 1†

D‰p 1 p0 …1 2 b†ŠE…X 2 r†1 Q‰1 2 E…p†Š

E‰p…1 2 p†Š Dv R…L†D 1 1 1 2 E…p† 1 2 E…p† Q‰1 2 E…p†Š

…10†

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3.1. Normal distribution model In this section, wepassume that the lead time  p demand X has a normal df F…x† with mean m L and standard deviation s L: Since r ˆ mL 1 ks L; we can also consider the safety factor k as a decision variable instead of r. Hence, the expected demand short at the end of the cycle is given by E…X 2 r†1 ˆ p R1 r …x 2 r† dF…x† ˆ s LC…k†; where C…k† ; f…k† 2 k‰1 2 F…k†Š; and f , F are the standard normal pdf and df, respectively. Therefore, model (10) is reduced to ( ) AD h Q‰E…p2 † 2 E2 …p†Š E‰p…1 2 p†Š 1 Q‰1 2 E…p†Š 1 1 EAC…Q; k; L† ˆ Q‰1 2 E…p†Š 2 1 2 E…p† 1 2 E…p† p p p D‰p 1 p0 …1 2 b†Šs LC…k† 1 h{ks L 1 …1 2 b†s LC…k†} 1 Q‰1 2 E…p†Š 1 h 0 …Q 2 1†

E‰p…1 2 p†Š Dv R…L†D 1 1 1 2 E…p† 1 2 E…p† Q‰1 2 E…p†Š

…11†

For ®xed Q and k, EAC…Q; k; L† is a concave function of L [ …Li ; Li21 †; because   22 EAC…Q; k; L† 21 1 D‰p 1 p0 …1 2 b†Š 23=2 ˆ 2 sL23=2 C…k† , 0 hksL h…1 2 b† 1 4 4 Q‰1 2 E…p†Š 2L2 Hence, for ®xed …Q; k†; the minimum expected total annual cost will occur at the end points of the interval ‰Li ; Li21 Š: On the other hand, it can be easily shown that, for a given value of L [ ‰Li ; Li21 Š; EAC…Q; k; L† is a convex function of …Q; k†: Therefore, for ®xed L [ ‰Li ; Li21 Š; the minimum value of EAC…Q; k; L† will occur at the point …Q; k† which satis®es ( ) 2EAC…Q; k; L† AD h E…p2 † 2 E2 …p† ˆ0ˆ2 2 1 2 E…p† 1 1 2Q 2 1 2 E…p† Q ‰1 2 E…p†Š 2

D‰p 1 p0 …1 2 b†Š p E‰p…1 2 p†Š R…L†D 2 2 s LC…k† 1 h 0 2 1 2 E…p† Q ‰1 2 E…p†Š Q ‰1 2 E…p†Š

p p p 2EAC…Q; k; L† D‰p 1 p0 …1 2 b†Šs L ˆ 0 ˆ hs L 2 h…1 2 b†s LPz …k† 2 Pz …k† 2k Q‰1 2 E…p†Š where Pz …k† ; Pr…Z $ k†; and Z is the standard normal random variable. Solving Eqs. (12) and (13) for Q and Pz …k†; respectively, we obtain " #1=2 p 2D{A 1 ‰p 1 p0 …1 2 b†Šs LC…k† 1 R…L†} Qˆ ; L [ …Li ; Li21 † hd Pz …k† ˆ

h D‰p 1 p0 …1 2 b†Š 1 h…1 2 b† Q‰1 2 E…p†Š

…12†

…13†

…14†

…15†

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Table 1 Lead time data Lead time component, i

Normal duration, bi (days)

Minimum duration, ai (days)

Unit crashing cost, ci ($/day)

1 2 3

20 20 16

6 6 9

0.4 1.2 5.0

where

d ˆ 1 2 2E…p† 1 E…p2 † 1

2h 0 E‰p…1 2 p†Š h

Explicit general solutions for Q and k are not possible because the evaluation of each of Eqs. (14) and (15) requires a knowledge of the value of the other. Consequently, we must establish the following iterative algorithm to ®nd the optimal …Q; k; L†: Algorithm 1. Step 1. For each Li ; i ˆ 0; 1; 2; ¼; n; perform (i)±(v). (i) Start with ki1 ˆ 0 and get C…ki1 † ˆ 0:3989 by checking the table from Silver and Peterson (1985) (p. 699±708). (ii) By substituting the value of C…ki1 † into Eq. (14), evaluate Qi1 : (iii) By using Qi1 ; determine Pz …ki2 † from Eq. (15). (iv) By checking Pz …ki2 † from Silver and Peterson (1985), ®nd ki2 and C…ki2 †: (v) Repeat (ii)±(iv) until no change occurs in the values of Qi and ki : Step 2. For each …Qi ; ki ; Li †; compute the corresponding expected total annual cost EAC…Qi ; ki ; Li †; i ˆ 0; 1; 2; ¼; n: †; then Step 3. Find miniˆ0;1;¼;n EAC…Qi ; ki ; Li †:If EAC…Qp ; kp ; Lp † ˆ miniˆ0;1;¼;n EAC…Qi ; ki ; Lip p p p p p p p …Q ; k ; L † is the optimal solution. And hence, the optimal reorder point is r ˆ mL 1 k s L : Example 1. In order to illustrate the above solution procedure, let us consider an inventory system with the following data: D ˆ 600 units=year; A ˆ $200 per order, h ˆ $20; h 0 ˆ $12; v ˆ $1:6; p ˆ $50; p0 ˆ $150; s ˆ 7 units=week; the lead time has three components with data shown in Table 1 and the defective rate p has a Beta distribution with parameters s ˆ 1 and t ˆ 4; that is, the pdf of p is given by: g…p† ˆ 4…1 2 p†3 ;

0,p,1

Hence, E…p† ˆ s=…s 1 t† ˆ 1=5; and E…p2 † ˆ s…s 1 1†=……s 1 t†…s 1 t 1 1†† ˆ 1=15: Applying the proposed Algorithm 1 procedure yields the results shown in Table 2 for b ˆ 0; 0.5, 0.8, and 1. From Table 2, the optimal inventory policy can be found by comparing EAC…Qi ; ri ; Li † for i ˆ 0; 1, 2, 3, and a summary is presented in Table 3.

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Table 2 Results of the optimal procedure …Li in weeks)

b

LI

R(Li)

Qi

ri

EAC(Qi,ri,Li)

0

8 6 4 3

0 5.6 22.4 57.4

127 128 132 141

134 105 76 60

4678.08 4579.10 4519.96 4619.99

0.5

8 6 4 3

0 5.6 22.4 57.4

128 129 132 141

130 102 73 57

4608.68 4518.95 4470.70 4576.91

0.8

8 6 4 3

0 5.6 22.4 57.4

129 129 133 142

126 98 70 55

4538.34 4457.97 4420.67 4533.18

1.0

8 6 4 3

0 5.6 22.4 57.4

129 130 133 142

121 94 67 52

4458.02 4388.38 4363.56 4483.15

3.2. Distribution-free model The information about the form of the probability distribution of lead time demand is often limited in practice. In this section, we relax the restriction about the form of the probability distribution of lead time demand, i.e. we only p assume here that the df F of the lead time demand has a known ®nite mean m L and  standard deviation s L but make no assumption on the distributional form of F. Now, we try to use a minimax distribution-free procedure to solve this problem. The minimax approach for this model is to ®nd the most unfavorable df F in F for each …Q; r; L† and then to minimize the expected total annual cost over …Q; r; L†: That is, our problem is to solve: min

max EAC…Q; r; L†

…16†

Q.0;r.0;L.0 F[F

To this end, we need the following proposition (Moon & Gallego, 1994). Table 3 Summary of the results of the optimal procedure …Li in weeks)

b

Lp

Qp

rp

EAC(Q p,r p,L p)

0.0 0.5 0.8 1.0

4 4 4 4

132 132 133 133

76 73 70 67

4519.96 4470.70 4420.67 4363.56

K.-S. Wu, L.-Y. Ouyang / Computers & Industrial Engineering 39 (2001) 173±185

Proposition. For any F [ F;  q E…X 2 r†1 # 12 s2 L 1 …r 2 mL†2 2 …r 2 mL†

181

…17†

Moreover, the upper bound (17) is tight. p Because r 2 mL ˆ ks L and for any df F of the lead time demand X, the above inequality always holds. Then, using inequality (17), the model in Eq. (10) is reduced to minimize ( ) 2 2 AD h Q‰E…p † 2 E …p†Š E‰p…1 2 p†Š 1 Q‰1 2 E…p†Š 1 1 EACu …Q; k; L† ˆ Q‰1 2 E…p†Š 2 1 2 E…p† 1 2 E…p†  p    pp 1 2 1 h ks L 1 12b s L 11k 2k 2   p p2 D‰p 1 p0 …1 2 b†Šs L 1 1 k 2 k E‰p…1 2 p†Š Dv 1 1 h 0 …Q 2 1† 1 1 2 E…p† 1 2 E…p† 2Q‰1 2 E…p†Š 2 3 iX 21 D 4ci …Li21 2 L† 1 1 cj …bj 2 aj †5; Q‰1 2 E…p†Š jˆ1 L [ ‰Li ; Li21 Š

(18)

where EACu …Q; k; L† is the least upper bound of EAC…Q; k; L†: As discussed in the previous section, it can be shown that EACu …Q; k; L† is a concave function of L [ ‰Li ; Li21 Š for ®xed …Q; k†: Hence, the minimum upper bound of the expected total annual cost will occur at the end point of the interval ‰Li ; Li21 Š for ®xed …Q; k†: In addition, it can be shown that EACu …Q; k; L† is a convex function in Q and k for ®xed L. Therefore, the ®rst-order conditions are necessary and suf®cient conditions for optimality. The ®rst-order conditions are: " #1=2 p p 2D{A 1 12 ‰p 1 p0 …1 2 b†Šs L… 1 1 k2 2 k† 1 R…L†} ; L [ …Li ; Li21 † …19† Qˆ hd p 2 1 1 k2 D‰p 1 p0 …1 2 b†Š p ˆ …1 2 b† 1 2 hQ‰1 2 E…p†Š 11k 2k Therefore, we can establish the following iterative algorithm to ®nd the optimal …Q; k; L†: Algorithm 2. Step 1. For each Li ; i ˆ 0; 1; 2; ¼; n; perform (i)±(iv). (i) Start with ki1 ˆ 0: (ii) By substituting the value of ki1 into Eq. (19), evaluate Qi1 :

…20†

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Table 4 Results of the optimal procedure …Li in week)

b

Li

Qi

ri

EAC u(Qi,ri,Li)

0

8 6 4 3

198 189 182 183

151 122 90 72

6440.33 6128.79 5794.25 5697.95

0.5

8 6 4 3

183 177 171 174

140 111 81 64

5894.78 5647.88 5394.66 5354.01

0.8

8 6 4 3

172 167 163 168

130 102 74 58

5464.47 5269.42 5082.14 5086.25

1.0

8 6 4 3

163 159 157 162

121 95 67 52

5087.56 4939.06 4810.65 4853.85

(iii) By using Qi1 ; determine ki2 from Eq. (20). (iv) Repeat (ii) and (iii) until no change occurs in the values of Qi and ki : Step 2. For each …Qi ; ki ; Li †; compute the corresponding minimum upper bound of the expected total annual cost EACu …Qi ; ki ; Li †; i ˆ 0; 1; 2; ¼; n: Step 3. Find miniˆ0;1;¼;n EACu …Qi ; ki ; Li †:If EACu …Qp ; kp ; Lp † ˆ miniˆ0;1;¼;n EACu …Qi ; ki ; Lp i †; then …Qp ; kp ; Lp † is the optimal solution. And hence, the optimal reorder point is rp ˆ mLp 1 kp s Lp : Example 2. The data is as in Example 1. We assume here that the probability distribution of the lead time demand is free. Applying the proposed Algorithm 2 procedure yields the results shown in Table 4. A summary of these optimal results is presented in Table 5. The expected total annual cost EAC…Qp ; rp ; Lp † is obtained by substituting Qp, rp and Lp into Eq. (11) when the lead time demand is normally distributed. The expected value of additional information (EVAI) is the largest amount that one is willing to pay for the knowledge of the form of the lead time demand distribution and is equal to EAC…Qp ; rp ; Lp † 2 EAC…Qp ; r p ; Lp †: Moreover, the cost penalty is the ratio of the approximate expected cost to the optimal one. It can be observed from Table 6 that the cost performance of the distribution-free approach is improving as b gets larger. 4. Effects of the parameters 1. Notice that if b ˆ 1; Eq. (11) reduces to the expected total annual cost of the backorders case, and

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Table 5 Summary of the results of the optimal procedure …Li in week)

b

Lp

Qp

rp

EAC u(Qp,rp,Lp)

0.0 0.5 0.8 1.0

3 3 4 4

183 174 163 157

72 64 74 67

5697.95 5354.01 5082.14 4810.65

hence Eq. (14) becomes " Qbˆ1 ˆ

#1=2 p 2D{A 1 ps LC…k† 1 R…L†} hd

…21†

When b ˆ 0; Eq. (11) reduces to that of the lost sales case, and thus Eq. (14) becomes " Qbˆ0 ˆ

#1=2 p 2D{A 1 ‰p 1 p0 Šs LC…k† 1 R…L†} hd

…22†

Hence, for ®xed L and k, comparing Eqs. (21) and (22), we get Qbˆ0 . Qbˆ1 : That is, the order quantity per cycle in the lost sales case is greater than in the backorders case (the distribution-free model has the same result). 2. The effect of b on the minimum expected total annual cost, say EACpb ; may be examined. It has the minimum value when b ˆ 1 (backorders case) and the maximum value when b ˆ 0 (lost sales case). Hence, for 0 , b , 1; EACpbˆ1 , EACpb , EACpbˆ0 : 3. For ®xed b , k and L [ …Li ; Li21 †; taking the derivative of Eq. (14) with respect to L, we obtain   dQ D 1 p1 2 c i ˆ dL Qhd 2 where p1 ; ‰p 1 p0 …1 2 b†Š…E…X 2 r†1 †=L is the stockout cost per unit time during lead time L. Because D, Q, h and h 0 are all positive, if p1 . 2ci ; we get dQ=dL . 0; this means increasing the lead time L increases the order quantity Q; and if p1 , 2ci ; we have dQ=dL , 0; this implies increasing the lead time L decreases the order quantity Q. 4. We assume that the df F of the lead time demand is a normal distribution. Under this situation, for Table 6 Comparison of the two procedures

b

EAC(Qp,rp,Lp)

EAC(Q p,r p,L p)

EVAI

Cost penalty

0.0 0.5 0.8 1.0

4897.60 4712.92 4572.64 4402.43

4519.96 4470.70 4420.67 4363.56

377.64 242.22 151.97 38.87

1.084 1.054 1.034 1.009

184

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®xed b and L [ …Li ; Li21 †; taking the derivative of Eq. (14) with respect to k, we have p dQ 2D‰p 1 p0 …1 2 b†Šs L ˆ Pz …k† , 0 Qhd dk This means that Q and k have negative relative relation. It implies that decreasing k (or, equivalently, decreasing the reorder point r) increases the order quantity Q (the distribution-free model has the same result). 5. When the supplier promises that the arriving order contains no defective itemsp(i.e. p ˆ 0; and hence  v ˆ 0; h 0 ˆ 0† and if s . 0 and k is suf®ciently large, we get C…k† ! 0 and 1 1 k2 2 k ! 0; then Eqs. (11) and (18), respectively, can be reduced to the model of Ben-Daya and Raouf (1994).

5. Concluding remarks The primary purpose of this paper is to examine the effect of defective items on a mixture of backorders and lost sales inventory model for minimizing the sum of the ordering cost, holding cost (including the defective and non-defective items), stockout cost, inspection cost and lead time crashing cost, where the order quantity, reorder point and lead time are considered as the decision variables. In this article, we ®rst assume that the lead time demand follows a normal distribution, and assume that the number of defective items in an arrival order is a random variable. Then, we relax the assumption about the form of the probability distribution of the lead time demand and apply the minimax distribution-free procedure to solve the problem. We develop an algorithm procedure to ®nd the optimal solution. Furthermore, the effects of parameters are also studied. In future research on this problem, it would be of interest to examine a non-linear relationship that exists between the crashing cost and the lead time. Possible extension of this work includes developing models dealing with variable lead time in sub-lot sampling inspection policy. Acknowledgements The authors wish to thank the reviewer for his useful suggestions on the manuscript. References Ben-Daya, M., & Raouf, A. (1994). Inventory models involving lead time as decision variable. Journal of the Operational Research Society, 45, 579±582. Das, C. (1975). Approximate solution to the (Q,r) inventory model for gamma lead time demand. Management Science, 22, 273±282. Foote, B., Kebriaei, N., & Kumin, H. (1988). Heuristic policies for inventory ordering problems with long and randomly varing lead time. Journal of Operations Management, 7, 115±124. Liao, C. J., & Shyu, C. H. (1991). An analytical determination of lead time with normal demand. International Journal of Operations Production Management, 11, 72±78. Magson, D. (1979). Stock control when the lead time cannot be considered constant. Journal of the Operational Research Society, 30, 317±322.

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