Extend (r, Q) Inventory Model Under Lead Time and Ordering Cost Reductions When the Receiving Quantity is Different from the Ordered Quantity

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1 Quality & Quantity 38: , Kluwer Academic Publishers. Printed in the Netherlands. 771 Extend (r, Q) Inventory Model Under Lead Time and Ordering Cost Reductions When the Receiving Quantity is Different from the Ordered Quantity KUN-SHAN WU 1 and I-CHUAN LIN 2 1 Department of Business Administration, Tamkang University, Tamsui, Taipei 251, Taiwan; 2 Graduate of Management Sciences, Aletheia University, Tamsui, Taipei 251, Taiwan Abstract. This paper investigates the continuous review inventory model involving variable lead time with partial backorders, where the amount received is uncertain. The options of investing in ordering cost reduction is included, and lead time can be shortened at an extra crashing cost. The objective of this article is to simultaneously optimize the order quantity, reorder point, ordering cost and lead time. We first assume that the lead time demand follows a normal distribution and develop an algorithm to find the optimal solution. Then, we relax the assumption of normality to consider a distribution free case where only the mean and standard deviation of lead time demand are known. We apply the minimax distribution free procedure to solve this problem. For both cases, we also show that the objective cost function to be minimized is jointly convex in the decision variables. Furthermore, two numerical examples are given to illustrate the results. Key words: inventory, ordering cost reduction, lead time, minimax distribution free. 1. Introduction Among the modern production management, the Japanese successful experiences of using Just-In-Time (JIT) production show that the advantages and benefits associated with the efforts to control the lead-time can be clearly perceived. The goal of JIT inventory management philosophies is the focus that emphasizes high quality, keeps low inventory level and lead-time to a practical minimum. Shortening the lead time is recognized as the feasible and effective way to achieve the goal of JIT. In traditional, most deterministic and stochastic inventory models assume that the lead-time is a given parameter or a random variable (therefore it is uncontrollable), and determines the optimal operating policy on the basis of this unrealistic assumption (Naddor, 1966; Silver and Peterson, 1998). In fact, in many practical situations, lead-time is not a given parameter or a random variable; it can be controlled and reduced at an added cost. Recently, some models considering lead-time as a decision variable have been developed. Liao and Shyu (1991) have initiated a study on lead-time reduction by presenting an inventory model in which lead- Author for correspondence. kunshan@mail.tku.edu.tw

2 772 KUN-SHAN WU AND I-CHUAN LIN time is a decision variable and the order quantity is predetermined. Ben-Daya and Raouf (1994) developed a model that considered both lead-time and order quantity as decision variables. Later, Ouyang et al. (1996) and Ouyang and Wu (1997, 1998, 1999) have generalized the Ben-Daya and Raouf (1994) model by considering shortages in which the lead-time demand is considered a normal distribution or distribution free. Later, Moon and Choi (1998), Hariga and Ben-Daya (1999) have extended the Ouyang et al. (1996) model to relax the assumption of a given service level and treat the reorder point as a decision variable. Other papers related to this area are Lan et al. (1999), Ouyang and Chuang (1998), Ouyang and Chang (2001), Pan and Hsiao (2001), and others. However, all models previously mentioned have assumed that the quantity received is the same as the quantity ordered. The quantity received may not match the quantity ordered due to various reasons such as rejection during inspection, damage or breakage during transportation, etc. Silver (1976) extended the simple Wilson lot-size model including the case when the quantity received is not necessary equal to the quantity ordered. Two separate cases were considered to account for the variability in the quantity received. Karlo and Gohil (1994) extended Silver s model by allowing shortages when the amount received is uncertain. Gor and Shah (1994) developed a lot-size model for deteriorating items by allowing shortages. Warrier and Shah (1999) presented a lot-size model with partial backordering and partial lost sales when the quantity received is uncertain and units in the inventory are subject to deterioration at a constant rate. On the other hand, accompanying the growth of Just-In-Time (JIT) production, which has evidenced that many benefits can be obtained from reducing setup cost, the issue of investment in setup cost reduction has received a great deal of attention. Porteus (1985) first introduced the concept and developed a framework of setup cost reduction on the classical economic order quantity (EOQ) model. Billington (1987) considered the economic production quantity (EPQ) model without backorders and included the setup cost as a function of capital expenditure. Nasri et al. (1990) investigated the effects of setup cost reduction on the EOQ model with stochastic lead time. Kim et al. (1992) presented several classes of setup cost reduction functions and described a general solution procedure on the EPQ model. Paknejad et al. (1995) presented a quality-adjusted lot-sizing model with stochastic demand and constant lead time and further studied the benefits of lower setup cost in this model. Sarker and Coates (1997) extended EPQ model with setup cost reduction under stochastic lead time and finite number of investment possibilities to reduce setup cost. From the above literature review, we find though there is no shortage of those studying lead time reduction, setup cost (ordering cost) reduction and the amount received is uncertain, little work has been done on considering them simultaneously. Hence, combining all of the above-mentioned factors, we propose a continuous review inventory model in which shortages are allowed with partial

3 EXTEND (r, Q) INVENTORY MODEL UNDER LEAD TIME AND ORDERING COST REDUCTIONS 773 backorders and the amount received is uncertain. We simultaneously optimize the order quantity, reorder point, ordering cost, and lead time with the objective of minimizing the total relevant costs. We first consider the case where lead time demand follows a normal distribution and develop an algorithm to find the optimal solution. Then, we relax the assumption on the distributional form of lead time demand and merely assume that the first and second moments are known and finite. For this case, we solve the problem by applying the minimax distribution free approach, originally proposed by Scraf (1958) and disseminated by Gallego and Moon (1993). 2. Notations and Assumptions In this paper, we propose a continuous review inventory model in which shortages are allowed with partial backorders and the amount received is uncertain. We simultaneously optimize the order quantity, reorder point, ordering cost, and lead time with the objective of minimizing the total relevant costs. The model is developed using the following notations and assumptions 2.1. NOTATIONS D A h π π 0 average demand per year, ordering cost per order, inventory holding cost per item per year, fixed penalty cost per unit short, marginal profit per unit, β fraction of the demand backordered during the shortage period, 0 β 1, r reorder point, Q order quantity, L length of lead time, X the lead time demand with finite mean DL and standard deviation σ L, where σ is the standard deviation per unit time demand, f X (x) the probability distribution function (p.d.f.) of random variable X, Y the quantity received, a random variable, E( ) expected value, E 2 ( ) [E( )] 2, x + maximum value of x and 0, i.e., x + = max{x,0}.

4 774 KUN-SHAN WU AND I-CHUAN LIN 2.2. ASSUMPTIONS (1) The reorder point r = expected demand during lead time + safety stock (SS), and SS = k (standard deviation of lead time demand), i.e., r = DL + kσ L where k is the safety factor. (2) Inventory is continuously reviewed. Replenishments are made whenever the inventory level falls to the reorder point r. (3) The lead time L has n mutually independent components. The ith component has a minimum duration a i and normal duration b i, and a crashing cost per unit time c i. Furthermore, we assume that c 1 c 2 c n. Then, it is clear that the reduction of lead time should be first on component 1 (because it has the minimum unit crashing cost), and then component 2, and so on. (4) If we let L 0 = n j=1 b j and L i be the length of lead time with components 1, 2,..., i crashed to their minimum duration, then L i can be expressed as L i = n j=1 b j i j=1 (b j a j ), i = 1, 2,...,n; and hence the lead time crashing cost R(L) per cycle for a given L [L i,l i 1 ],isgivenby i 1 R(L) = c i (L i 1 L) + c j (b j a j ). j=1 3. Basic Model In this study, the quantity received is uncertain and depends on the quantity ordered. If a quantity Q is ordered each time, the expected quantity received will be E(Y Q) = αq,whereα is the bias factor (0 α 1 when the expected quantity received is less than or equal to the order quantity, as is the usual case). When the expected quantity received is greater than the order quantity due to various reasons such as counting errors, good production runs, etc., leading to large amount of inventory, α>1. The variance of the quantity received is given by Var(Y Q) = σ σ 2 1 Q2, (1) where σ0 2 and σ 1 2 are non-negative constant. If σ 1 2 = 0, then the standard deviation of the quantity received is independent of the quantity ordered; and if σ0 2 = 0, then the standard deviation of the quantity received is proportional to the quantity ordered. Under the assumptions described above, the total cost per cycle with a variable lead time can be obtained as Moon and Choi (1998) or Hariga and Ben-Daya (1999) given that Y units are received is C(Y,r,L) = A + h Y [ ] Y + r DL + (1 β)e(x r)+ D 2 + πe(x r) + + R(L), (2)

5 EXTEND (r, Q) INVENTORY MODEL UNDER LEAD TIME AND ORDERING COST REDUCTIONS 775 where E(X r) + is the expected number of shortages per cycle and π = π + (1 β)π 0. Therefore, the expected total cost per cycle with variable lead time when the amount received is uncertain is E[C(Y,r,L) Q] = A + h αq D [r DL + (1 β)e(x r)+ ] Moreover, the expected cycle time is E(Y Q) D + h 2D [σ (σ α2 )Q 2 ]+ πe(x r) + + R(L),(3) = αq D. (4) The total expected annual cost with variable lead time when the amount received is uncertain, denoted by EAC(Q,r,L), is then given by E[C(Y,r,L) Q] divided by E(Y Q)/D. Using Equations (3) and (4), we have EAC(Q,r,L) = AD αq + h[r DL + (1 β)e(x r)+ ] + h 2αQ [σ (σ α2 )Q 2 ]+ πd αq E(X r)+ + R(L)D αd. (5) As mentioned above, this formulation assumes that the ordering cost, A, is constant. In the following section, we extend the model (5) by considering the investment in reduced ordering cost. 4. The Ordering Cost Reduction Model In this section, we investigate the possibility of investing in the reduction of the ordering cost parameter, A, of the basic model presented in the previous section. The underlying assumption in above model of Equation (5) is that the ordering cost, A, is a fixed constant and not subject to control. However, as stated earlier, in some practical situations the ordering cost can be controlled by additional investment. How much is worth or most economic for the investment is the decision-making problem that we wish study. Let I(A) denote the capital amount required to reduce ordering cost from original level, A 0, to a target level, A; and I(A) is the one-time investment cost whose benefits will extend indefinitely in to future. Also, let θ denote the fractional cost of capital investment per unit time. Thus, the cost such an investment per unit time is θi(a). In the literatures, several investment-cost functions have been adopted formulate the setup (ordering) cost reduction model. Among them, the logarithmic function is widely utilized since it is consistent with the Japanese experience as reported in Hall (1983), and has been utilized by Porteus (1985, 1986) and others. We employ this function to describe

6 776 KUN-SHAN WU AND I-CHUAN LIN the relationship between ordering cost and ordering cost reduction investment. That is, ( ) A0 I(A) = b ln for 0 <A A 0, where b = 1 A δ, where δ represents the percentage decrease in A per dollar increase in I(A).Therefore, when the ordering cost is no longer considered to be a fixed parameter but a decision variable, we then seek to minimize the sum of the capital investment cost of reducing ordering cost and the inventory costs (as expressed in (5)) by optimizing over Q, A, r, andl constrained on 0 <A A 0. That is, the objective of our problem is to minimize the following total expected annual cost. EAC(Q,A,r,L) = θi(a)+ AD αq + h[r DL + (1 β)e(x r)+ ] + h 2αQ [σ (σ α2 )Q 2 ]+ πd αq E(X r)+ + R(L)D αq ( ) A0 = θbln + AD A αq + h[r DL + (1 β)e(x r)+ ] + h 2αQ [σ (σ α2 )Q 2 ]+ πd E(X r)+ αq + R(L)D αq, (6) where 0 <A A 0. We first assume that the lead time demand X has a normal p.d.f. f X (x) with mean DL and standard deviation σ L.Sincer = DL + kσ L, we can also consider the safety factor k as a decision variable instead of r. Hence, the expected demand shortage at the end of the cycle is given by E(X r) + = r (x r)f X (x) dx = σ L (k), where (k) φ(k) k[1 (k)] and φ, denote the standard normal density and distribution function respectively. Therefore, the model (6) can be transformed to ( ) A0 min EAC(Q,A,k,L) = θbln + AD A αq + h[kσ L + (1 β)σ L (k)] + h 2αQ [σ (σ α2 )Q 2 ] + πd αq σ L (k) + R(L)D αq (7)

7 EXTEND (r, Q) INVENTORY MODEL UNDER LEAD TIME AND ORDERING COST REDUCTIONS 777 subject to 0 <A A 0. In order to solve this nonlinear programming problem, we first ignore the restriction 0 <A A 0 and take the first partial derivatives of EAC(Q,A,k,L) with respect to Q, A, k and L (L i,l i 1 ), respectively. We have EAC(Q,A,k,L) Q = AD αq 2 hσ 2 0 2αQ 2 + h 2α (σ α2 ) πd αq 2 σ L (k) R(L)D αq 2, (8) EAC(Q,A,k,L) A = θb A + D αq, (9) and EAC(Q,A,k,L) k EAC(Q,A,k,L) L = hσ L h(1 β)σ L(1 (k)) πd αq σ L(1 (k)) (10) = 1 2 hkσ L 1/ h(1 β)σl 1/2 (k) πd αq σl 1/2 (k) D αq c i. (11) By examining the second order sufficient conditions, it can be easily verified that EAC(Q,A,k,L) is not a convex function of (Q,A,k,L).However,forfixed Q, A and k, EAC(Q,A,k,L) is a concave function of L [L i,l i 1 ], because 2 EAC(Q,A,k,L) L 2 [ 1 = h 4 kσl 3/2 1 ] 4 (1 β)σl 3/2 (k) 1 πd 4 αq σl 3/2 (k) < 0. Hence, for fixed (Q,A,k), the minimum total expected annual cost would occur at the end point of the interval [L i,l i 1 ]. On the other hand, for a given value of L [L i,l i 1 ], by setting Equations (8), (9) and (10) equal to zero, we obtain 2D Q = [ A + h ] 2D σ πσ L (k) + R(L) h(σ α2 ), (12)

8 778 KUN-SHAN WU AND I-CHUAN LIN A = αθbq D (13) and (k) = 1 hαq h(1 β)αq + D π. (14) Theoretically, for fixed L [L i,l i 1 ], from Equations (12) (14), we can get the values of Q, A and k (we denote these values by Q, A, k ). The following proposition asserts that, for fixed L [L i,l i 1 ], when the constraint 0 <A A 0 is ignored, the point (Q,A,k ) is the optimal solution such that the expected annual total cost has a minimum. PROPOSITION 1. For fixed L [L i,l i 1 ], the Hessian matrix for EAC(Q,A,k,L) (as expressed in Equation (7)) is positive definite at point (Q,A,k ). Proof. See Appendix. We now consider the constraint 0 <A A 0. From Equation (13), we note that A is positive. Also, if A <A 0,then(Q,A,k ) is an interior optimal solution for a given L [L i,l i 1 ].However,ifA A 0, then it is unrealistic to invest in changing the current ordering cost level. For this special case, the optimal order cost is the original ordering cost, that is, A = A 0. Next, the explicit general solution for Q, A and k are not possible because the evaluation of each of the Equations (12) (14) requires a knowledge of the value of the other. Consequently, we must establish the following iterative algorithm to find the optimal solutions for the order quantity, ordering cost, reorder point and lead time. Algorithm 1 Step 1. For each L i, i = 0, 1,...,n, perform (i) to (v). (i) Start with A i1 = A 0 and k i1 = 0 (it implies (k i1 ) = , which can be obtained by checking the standard normal table from Silver and Peterson (1998), and φ(k i1 ) = and (k i1 ) = 0.5). (ii) By substituting the values of A i1 and (k i1 ) into (12), to evaluate Q i1. (iii) By using Q i1, determine A i2 from (13). (iv) By using Q i1 determines (k i2 ) from (14), then finds k i2 and (k i2 ) by the standard normal table from Silver and Peterson (1998). (v) Repeat (ii) to (iv) until no changes occurs in the values of Q i, A i and k i. Step 2. Compare A i and A 0. (i) If A i <A 0, then the solution found in Step 1 is optimal for given L i.we denote the solution by (Q i,a i,k i ).

9 EXTEND (r, Q) INVENTORY MODEL UNDER LEAD TIME AND ORDERING COST REDUCTIONS 779 Table I. Lead time data Lead time Normal duration Minimum duration Unit crashing cost component i b i (days) a i (days) c i ($/day) (ii) If A i A 0, then for given L i,takea i = A 0, and the corresponding (Q i,k i ) can be obtained by substituting A i = A 0 into (12) and then solving (12) and (14) iteratively until convergence (the solution procedure is similar to that giveninstep1). Step 3. For each (Q i,a i,k i,l i), compute the corresponding total expected annual cost EAC(Q i,a i,k i,l i), i = 0, 1,...,n. Step 4. Find min i=0,1,...,n EAC(Q i,a i,k i,l i). If EAC(Q N,A N,k N,L N ) = min i=0,1,...,n EAC(Q i,a i,k i,l i), then(q N, A N, k N, L N ) is the optimal solution. And hence, the optimal reorder point r N = DL N + k N σ L N. Example 1 In order to illustrate the proceeding solution procedure, we consider an inventory system with the following data used in Ouyang et al. (1996): D = 600 units/year, A = $200 per order, h = $20, π = $50, π 0 = $150, σ = 7 units/week, except we put σ0 2 = 100, σ 1 2 = 0.1 andα = 0.9. Besides, for ordering cost reduction, we take θ = 0.1 per dollar per year and b = 5800 (i.e., δ = 0.02%. The lead time has three components with data shown in Table I. We assume that the lead time demand follows a normal distribution. Applying the proposed Algorithm 1 procedure yields the results shown in Table II for β = 0, 0.5, 0.8 and 1. From Table II, the optimal inventory policy can be found by comparing EAC(Q i,a i,k i,k i ) for i = 0, 1, 2, 3, and a summary is presented in Table III. 5. Distribution Free Model Information about the distributional form of lead time demand might be limited in practical situations, In this section, we relax the assumption that lead time demand is normally distributed and only assume that the d.f. of lead time demand belongs to the class F of d.f. s with finite mean DL and standard deviation σ L. Since the form of lead time demand distribution is unknown, the expected shortages per order

10 780 KUN-SHAN WU AND I-CHUAN LIN Table II. Results of the optimal procedure (L i in week) β L i R(L i ) A i Q i r i EAC(Q i,r i,a i,l i) Table III. Summary of the results of the optimal procedure (L i in week) β L A Q r EAC(Q,r,A,L ) cycle, E(X r) + cannot be determined. Therefore, the minimax distribution free procedure, originally proposed by Scraf (1958) and disseminated by Gallego and Moon (1993), is utilized to find the least favorable d.f. in F for each (Q,A,r,L) and then to minimize the expected annual total cost over Q, A, r, andl. Our problem is then to solve: subject to min max EAC(Q,A,r,L) (15) Q>0,r>0,L>0 0 <A A 0. f X F

11 EXTEND (r, Q) INVENTORY MODEL UNDER LEAD TIME AND ORDERING COST REDUCTIONS 781 We note that to find the least favorable d.f. in F for (15) is equivalent to find the worst case for E(X r) + in model (6). Fortunately, this task can be achieved by utilizing the following proposition that was asserted by Gallego and Moon (1993). PROPOSITION 2. For any f X F, E(X r) + 1 ( σ 2 L + (r DL) 2 (r DL)). (16) 2 Moreover, the upper bound (16) is tight. Because r DL = kσ L and for any p.d.f. f X of the lead time demand X, the above inequality always holds. Then, using Proposition 2, our problem is to minimize the cost function for the worst distribution ( ) min EAC u A0 (Q,A,k,L) = θbln + AD A αq [ +h kσ L (1 β)σ L( ] 1 + k 2 k) + h 2αQ [σ (σ α2 )Q 2 ] subject to 0 <A A 0, + πd 2αQ σ L( 1 + k 2 k) + R(L)D αq (17) where EAC u (Q,A,k,L)is the least upper bound of EAC(Q,A,k,L). As discussed in the previous section, we first ignore the constraint 0 <A A 0, and then it can be shown that EAC u (Q,A,k,L) is a concave function of L [L i,l i 1 ] for fixed (Q,A,k). Hence, the minimum upper bound of the total expected annual cost will occur at the end point of the interval [L i,l i 1 ] for fixed (Q,A,k). The first conditions are: [ 2D A + h 2D σ πσ L( ] 1 + k 2 k) + R(L) Q = h(σ1 2 +, (18) α2 ) and A = αθbq D (19) k = 1 hαq 1 + k 2 h(1 β)αq + D π. (20)

12 782 KUN-SHAN WU AND I-CHUAN LIN Table IV. Results of the optimal procedure (L i in week) β L i R(L i ) A u i Q u i r u i EAC u (Q u i,ru i,au i 0,L i) u u u u Again, we note that as in the normal distribution case, for a given L, it can be shown that the values of Q, A and k determining from Equations (18) (20) (we denote these values by Q D, A D and k D ) satisfy the second order sufficient condition, and hence it is a relative minimum. The similar algorithm procedure as proposed in the previous section can be performed to obtain the optimal solutions for the order quantity, ordering cost, reorder point and lead time. Example 2 The data is the same as in Example 1, except that the probability distribution of the lead time demand is free. Applying the similar procedure as Algorithm 1, we obtain the optimal solutions, which are summarized in Table IV. The optimal results of fixed ordering cost model are listed in the same table for comparison. The total expected annual cost EAC(Q D,A D,r D,L D ) is obtained by substituting Q D, A D, r D and L D into (7) 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(Q D,A D,r D,L D ) EAC(Q N,A N,r N,L N ). Moreover, the cost penalty is the ratio of the approximate expected cost to the optimal one. It can be obtained from Table VI that the cost performance of the distribution free approach is improving as β gets larger.

13 EXTEND (r, Q) INVENTORY MODEL UNDER LEAD TIME AND ORDERING COST REDUCTIONS 783 Table V. Summary of the results of the optimal procedure (L i in week) β L u A u Q u r u EAC u (Q u,r u,a u,l u ) Table VI. Comparison of the two procedures β EAC(Q u,r u,a u,l u ) EAC(Q N,r N,A N,L N ) EVAI Conclusion This paper studied the effect of ordering cost reduction for the continuous review inventory system with variable lead time when the amount received is uncertain. Two models with objective of minimizing the total relevant costs are formulated and analyzed. The first model considers the lead time demand follows a normal distribution. The second model 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. Also, in each case, we show that the objective cost function is jointly convex in the decision variables. Moreover, two numerical examples are presented to illustrate the important issues related to the proposed models. Note that if σ0 2 = σ 1 2 = 0andα = 1 (i.e., the quantity received is equal to the quantity ordered), then our model will degenerate to the model proposed by Ouyang et al. (1996). Moreover, if A = A 0, σ0 2 = σ 1 2 = 0andα = 1, then our model will degenerate to the model proposed by Moon and Choi (1998). Acknowledgements This research was support by the National Science Council of Taiwan under Grant NSC H

14 784 KUN-SHAN WU AND I-CHUAN LIN Appendix For a given value of L, we first obtain the Hessian matrix H as follows: 2 EAC(Q,A,k,L) 2 EAC(Q,A,k,L) 2 EAC(Q,A,k,L) Q 2 Q A Q 2 2 EAC(Q,A,k,L) 2 EAC(Q,A,k,L) 2 EAC(Q,A,k,L) A Q A 2 A k 2 EAC(Q,A,k,L) 2 EAC(Q,A,k,L) 2 EAC(Q,A,k,L) k Q k A k 2, where 2 EAC(Q,A,k,L) Q 2 = 2D [ A + πσ L (k) + R(L) + hσ 2 ] 0, αq 3 2D 2 EAC(Q,A,k,L) Q A 2 EAC(Q,A,k,L) Q k 2 EAC(Q,A,k,L) A 2 = 2 EAC(Q,A,k,L) A Q = 2 EAC(Q,A,k,L) k Q = θb A 2, = D αq 2, = πd αq 2 σ L(1 (k)), 2 EAC(Q,A,k,L) A k 2 EAC(Q,A,k,L) k 2 = = 2 EAC(Q,A,k,L) = 0, k A [ h(1 β) + πd αq ] σ Lφ(k). Then we proceed by evaluating the principal minor of H at point (Q,A,k ). The first principal minor of H is H 11 = 2D [ A + πσ L (k ) + R(L) + hσ 2 ] 0 > 0. (A1) α(q ) 3 2D The second principal minor of H is (note that from Equation (13), θb/a = D/αQ ) H 22 = θb [ 2D A + πσ L (k ) + R(L) + hσ 2 ] ( ) 0 D 2 (A ) 2 α(q ) 3 2D α(q ) 2 = D2 α 2 (Q ) + θb [ 2D πσ L (k ) + R(L) + hσ 2 ] 0 > 0. (A2) 4 (A ) 2 α(q ) 3 2D

15 EXTEND (r, Q) INVENTORY MODEL UNDER LEAD TIME AND ORDERING COST REDUCTIONS 785 The third principal minor of H is H 33 =h(1 β)σ { [ θb Lφ(k 2D ) A + π L (k ) + R(L) + hσ 2 ] 0 (A ) 2 α(q ) 3 2D } D2 + πd2 α 2 (Q ) 4 α 2 (Q ) σ Lφ(k ) 4 { [ 2θb A + π L (k ) + R(L) + hσ 2 ] 0 (A ) 2 2D D αq θb } πσ L[1 (k )] 2 (A ) 2 φ(k ) = h(1 β)σ Lφ(k ) H 22 + π 2 D 2 α 2 (Q ) σ Lφ(k ) θb 4 A { 1 + 2R(L) + hσ 0 2 A 2A D + πσ } LG(k ), (A3) A φ(k ) where G(k ) = 2φ(k ) (k ) [1 (k )] 2 > 0, k [0, )(we prove it later). Hence, H 33 > 0. Therefore, from Equations (A1) (A3), it is clearly seen that the Hessian matrix H is positive definite at point (Q,A,k ). We now show k [0, ), G(k ) is positive. Taking the derivative of G(k ), we have dg(k )/dk = 2k φ(k ) (k )<0. Moreover, by checking the standard normal table, we have lim k 0 G(k ) = 2(0.3989)(0.3989) (1 0.5) 2 = and lim k G(k ) = 0. Hence, G(k ) is a decreasing function of k,andg(k ) is distributed in the first quadrant, which is positive. References Ben-Daya, M. & Raouf, A. (1994). Inventory models involving lead time as decision variable. Journal of Operational Research Society 45: Billington, P. J. (1987). The classic economic production quantity model with setup cost as a function of capital expenditure. Decision Sciences 18: Gallego, G. & Moon, I. (1993). The distribution free newsboy problem: review and extensions. Journal of the Operational Research Society 44: Gor, A. S. & Shah, N. H. (1994). Order level lot-size inventory model for deteriorating items under random supply. Industrial Engineering 23: Hall, R. W (1983). Zero Inventories. Dow Jones-Irwin: Homewood: Illinois. Hariga, M. & Ben-Daya, M. (1999). Some stochastic inventory models with deterministic variable lead time. European Journal of Operational Research 113: Karlo, A. & Gohil, M. (1994). A lot-size model with backlogging when the amount received is uncertain. International Journal of Production Research 20:

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