A continuous review inventory model with the controllable production rate of the manufacturer

Transcription

1 Intl. Trans. in O. Res. 12 (2005) INTERNATIONAL TRANSACTIONS IN OERATIONAL RESEARCH A continuous review inventory model with the controllable roduction rate of the manufacturer I. K. Moon and B. C. Cha Deartment of Industrial Engineering, usan National University, Busan , Korea ikmoon@usan.ac.kr Received January 2004; received in revised form July 2004; acceted Setember 2004 Abstract Otimal oerating olicy in most deterministic and stochastic inventory models is based on the unrealistic assumtion that lead-time is a given arameter. In this article, we develo an inventory model where the relenishment lead-time is assumed to be deendent on the lot size and the roduction rate of the manufacturer. At the time of contract with a manufacturer, the retailer can negotiate the lead-time by considering the regular roduction rate of the manufacturer, who usually has the otion of increasing his regular roduction rate u to the maximum (designed) roduction caacity. If the retailer intends to reduce the lead-time, he has to ay an additional cost to accomlish the increased roduction rate. Under the assumtion that the stochastic demand during lead-time follows a Normal distribution, we study the lead-time reduction by changing the regular roduction rate of the manufacturer at the risk of aying additional cost. We rovide a solution rocedure to obtain the efficient ordering strategy of the develoed model. Numerical examles are resented to illustrate the solution rocedure. Keywords: lot sizing; lead-time reduction; roduction rate 1. Introduction The flexibility and usefulness of models can be imroved by a structured relaxation of their inherent assumtions. Silver (1993) suggested a wide variety of ossible imrovements during the oerations of manufacturing, including setu reduction, increasing quality, changing roduction rates, etc. Given arameters are assumed to be fixed or given, including setu time, setu cost, roduction rate, lead-time, etc. In recent years, there has been a growing literature devoted to the effects of changing the givens (setu time, lead-time, quality level, etc.) for the inventory r 2005 International Federation of Oerational Research Societies. ublished by Blackwell ublishing Ltd.

2 248 I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) control decisions. A comrehensive review of this literature u to 1997 can be found in Silver et al. (1998). In most deterministic and robabilistic inventory models, lead-time is viewed as a rescribed constant or a stochastic variable, which, therefore, is not subject to control. But in many ractical situations, lead-time can be reduced at an added cost; in other words, it is controllable. By shortening the lead-time, we can lower the safety stock, reduce the loss caused by stockout, imrove the service level to the customer, and therefore, increase the cometitiveness in business. Through the Jaanese exerience of using JIT roduction, the advantages and benefits associated with the efforts to control the lead-time can be clearly erceived. Liao and Shyu (1991) resented a robabilistic model in which the order quantity was redetermined and lead-time was a unique decision variable. Later, Ben-Daya and Raouf (1994) extended Liao and Shyu s (1991) model by considering both lead-time and the order quantity as decision variables where shortages were neglected. Ouyang et al. (1996) allowed shortages and extended Ben-Daya and Raouf s (1994) model by adding the stockout cost. In addition, the total amount of stockout was considered a mixture of backorders and lost sales during the stockout eriod. Moon and Choi (1998) and Hariga and Ben-Daya (1999) imroved the model of Ouyang et al. (1996) by simultaneously otimizing the order quantity, the reorder oint and lead-time. Ouyang et al. (1999) incororated ordering cost reduction into the model of Moon and Choi (1998), where the ordering cost can be reduced by caital investment. Kim and Benton (1995) established a linear relationshi between lead-time and lot size based on the observations of Karmarkar (1987). They incororated this lead-time/lot size relation in the classical stochastic continuous review model. In a recent aer, Hariga (2000) develoed a stochastic setu cost reduction model with lead-time, lot size and setu time interaction. an et al. (2002) considered the lead-time crashing cost as a function of both the order quantity and the reduced lead-time. Ouyang and Chang (2002) generalized the model of Moon and Choi (1998) by modifying the lead-time crashing cost function and treating the setu cost as one of the decision variables. In this aer, the lead-time is exressed by the relation between the lot size and the roduction rate of the manufacturer. Under the assumtion that the stochastic demand during lead-time follows a Normal distribution, our objective is to reduce the relenishment lead-time by increasing the roduction rate of the manufacturer subject to the additional cost to be aid by the retailer. A retailer can negotiate the lead-time by considering the regular roduction rate of the manufacturer, since the manufacturer can increase his roduction rate u to the maximum roduction caacity. If the retailer wants the manufacturer to increase his roduction rate, the retailer has to accet an additional cost. In this case, the additional cost function is easily induced by the difference between the negotiated roduction rate and the regular roduction rate. The remainder of this aer is organized as follows. We rovide a basic continuous review model in Section 2. In Section 3, the lead-time reduction model is addressed and a solution algorithm is resented. Numerical examles to illustrate the solution rocedure are shown in Section 4. Finally, Section 5 resents the conclusion of this aer.

3 2. The basic continuous review model The notation used in this aer is as follows: l average demand er year m L exected lead-time demand A fixed ordering cost er order h cost of holding a unit er year cost er unit short S additional cost er unit for increasing roduction rate Z(r) exected number of units short er cycle Q order quantity (decision variable) roduction quantity er day (decision variable) 0 regular roduction quantity er day max maximum roduction quantity er day r reorder oint k safety factor L relenishment lead-time In the continuous review inventory model, an order of size Q is laced when the inventory level reaches the reorder oint r, i.e., there is no overshoot of the reorder oint. Following the normal aroximation, the exected total annual inventory cost TC can be reresented as follows: TCðQ; rþ ¼ la Q þ h Q 2 þ r m L þ l Q ZðrÞ: ð2:1þ In this basic formulation, it is assumed that the lead-time is deterministic and indeendent of the lot size. Kim and Benton (1995) exressed their concerns over this fixed lead-time assumtion. They introduced the lead-time as a function of the lot size, setu time and unit roduction time as follows: L ¼ðsþQÞd; ð2:2þ where s is the setu time, is the run time er unit and d is the sho floor queuing factor. Here, the lead-time L is exressed as the sum of the setu and rocess time of a roduction lot. The sho floor queuing factor (d) means some, maybe a much larger, ortion of the lead-time is sent in queues or material handling rocesses (d41). Since the lead-time demand X follows a Normal distribution with mean m L and standard deviation s ffiffiffi L, the reorder oint r is the sum of the exected lead-time demand and the Safety Stock (SS). The SS is k times the standard deviation of the lead-time demand, i.e., r ¼ m L þ ks ffiffiffi L ; where k is the safety factor and s is the standard deviation of the daily demand. The safety factor k is determined from FðkÞ ¼ Z 1 k I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) fðzþ dz;

4 250 where F(k) is the comlementary cumulative distribution of a standard Normal distribution that gives the robability of stockout er cycle at the safety factor k and f(z) is the robability density function of the standard Normal distribution. The exected number of units short er cycle is given by ZðrÞ ¼ Z 1 r ðx rþf ðxþ dx ¼ s ffiffiffi L CðkÞ; where C(k) is the unit linear loss integral and is given by CðkÞ ¼ Z 1 k I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) ðz kþfðzþ dz ¼ Z 1 k zfðzþ dz kfðkþ ¼fðkÞ kfðkþ: In this case, the exected total annual inventory cost is as follows: TCðQ; kþ ¼ la Q þ h Q 2 þ sk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðQ; sþ þ s l Q CðkÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðQ; sþ: ð2:3þ Hariga (2000) extended this basic continuous review inventory model by considering the investment in reduced setu time and the ractical relationshi between the lead-time, lot size and setu time. According to him, the setu time can often be reduced at very low cost by a simle rocedure such as the single-minute exchange of dies and without using high technology. He considered a certain amortized investment cost iv(s), where i is the cost of caital and V(s) is the one-time investment cost. He also assumed that the setu cost A can be reresented by a scalar multile of the setu time, i.e. A(s) 5 as, where a reresents the daily direct labor cost associated with the setu oeration. In this case, the exected total annual inventory cost is given by TCðQ; k; sþ ¼ als Q þ h Q 2 þ s hk þ l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q CðkÞ LðQ; sþ þ ivðsþ ð2:4þ subject to LðQ; sþ ¼ðsþQÞd; 0 < s)s 0 ; where s 0 is the original setu time. Hariga (2000) assumed that the one-time investment cost is a convex and strictly decreasing function of s. 3. The lead-time reduction model 3.1. Model formulation In this model, the lead-time, the lot size and the roduction rate of a manufacturer are connected by the following relation: LðQ; Þ ¼ Q ; 0 < 0)) max ; where 0 is the regular roduction rate and max is the maximum roduction rate of the manufacturer.

5 I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) M 0 Q R Q L Time Fig. 1. Behavior of the inventory system. Inventory is continuously reviewed and relenishments are made whenever the inventory level falls to the reorder oint r. When the retailer laces an order, the manufacturer starts to roduce at the roduction rate (less than or equal to the maximum roduction rate) according to the contract with the retailer. The behavior of the inventory system over time is shown in Fig. 1. Our aim is to reduce the relenishment lead-time by increasing the roduction rate of the manufacturer. If the retailer tells the manufacturer to increase his roduction rate, the retailer will be asked by the manufacturer for an additional cost in order to accomlish this by overtime, etc. In this case, the additional cost that is induced by the difference between the desired roduction rate and the regular roduction is given by l ð Q 0ÞLðQ; ÞS ¼ 1 0 ls: Therefore, the exected total annual cost to be minimized is TCðQ; k; Þ ¼ la Q þ h Q 2 þ s hk þ l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q CðkÞ LðQ; Þ subject to LðQ; Þ ¼ Q ; 0 < 0)) max : þ 1 0 ls ð3:1þ Taking the artial derivatives of TC(Q, k, ) with resect to Q, k and, we ¼ h 2 lða þ scðkþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðQ; ÞÞ s Q 2 þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hk þ l LðQ; Þ Q CðkÞ ¼ ¼ s h l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q FðkÞ LðQ; Þ ¼ 0; ð3:2þ ð3:3þ

6 ¼ I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 2 LðQ; Þ 2 hk þ l Q CðkÞ þ 0lS 2 : ð3:4þ It is difficult to rove the convexity of TC(Q, k, ). However, we can show that for fixed k and, the exected total annual inventory cost function is not convex in Q but the otimal solution can be determined uniquely. Moreover, for fixed Q and k, TC(Q, k, ) is not convex in. But for fixed Q and, TC(Q, k, ) is convex in k. The roofs are given in the following lemmas: Lemma 1. For fixed k, and, TC(Q, k, ) is not convex in Q. But the otimal solution can be determined uniquely and have either the ossible minimum lot size Q min or Q that ¼ 0. roof. The second-order artial derivative of TC(Q, k, ) with resect to Q for fixed k and is given 2 2 ¼ 2lAQ 3 þ 3slCðkÞ 4 ffiffiffi Q 5=2 shk 4 ffiffiffi Q 3=2 ¼ Q 3=2 2lA Q 3=2 þ 3slCðkÞ 4 ffiffiffi Q 1 shk 4 ffiffiffi : ð3:5þ As the first and second terms within the bracket converge to zero for large Q, it is obvious < 0 for large Q and 2 ¼ 0. Therefore, TC(Q, k, ) is not convex in Q. & To rove the existence of the unique solution, we consider the following. The first-order artial derivative of TC(Q, k, ) with resect to Q for fixed k and is given ¼ h 2 þ shk 2 ffiffiffi Q 1=2 laq 2 slcðkþ 2 ffiffiffi Q 3=2 ¼ h 2 þ shk Q 1=2 2 ffiffiffi la Q 3=2 slcðkþ 2 ffiffiffi Q 1 : ð3:6þ In equation (3.6), the last two terms within the bracket converge to zero for large Q and it is certain ¼ h 2 > 0. ¼ h 2 þ shk 2 ffiffiffi la slcðkþ 2 ffiffiffi *0; *0, for all Q, this indicates that TC is strictly increasing function for Q and the otimal solution is the ossible minimum lot size Q ¼ h 2 þ shk 2 ffiffiffi la slcðkþ 2 ffiffiffi < 0, it is clear that the sign of the first-order artial derivative changes from negative to ositive only once for 14Qo1. This means ¼ 0

7 I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) has a unique solution for fixed k and. The sign of the first-order derivative indicates that TC function gradually increases after a oint of inflection. Thus, for fixed k and, there exists an otimal solution that can be determined uniquely in Q. Lemma 2. For fixed Q and k, TC(Q, k, ) is strictly increasing or decreasing or a concave function when 0 44 max. Therefore, the otimal roduction rate minimizing TC(Q, k, ) is either 0 or max. roof. For fixed Q and k, equation (3.4) can be easily written ¼ 1 2 ðz ffiffiffi 1 Z 2 Þ; where Z1 ¼ 0 ls; Z 2 ¼ s ffiffiffiffi Q hk þ l 2 Q CðkÞ : As Z 1, Z 2 40, the trend of the total cost function TC(Q, k, ) deends on the sizes of Z 1 and Z 2. Case 1 ffiffiffiffiffi Z 1 Z 2 0)0: TC(Q, k, ) is strictly decreasing as increases in the interval 0 44 max.in this case, max is the otimal roduction rate minimizing TC(Q, k, ). Case 2 ffiffiffiffiffiffiffiffiffiffi Z 1 Z 2 max *0: TC(Q, k, ) is strictly increasing as increases in the interval 0 44 max. In this case, 0 is the otimal roduction rate minimizing TC(Q, k, ). Case 3 ffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi Z 1 Z 2 0*0 and Z1 Z 2 max )0: TC(Q, k, ) is a concave function in the interval 0 44 max. In this case, the otimal roduction rate that minimizes TC(Q, k, ) for fixed Q and k can be selected as either 0 or max by comaring TC(Q, k, 0 ) and TC(Q, k, max ). We know that the otimal roduction rate is always 0 or max. from Lemma 2. It is easy to see that for fixed Q and, the exected total annual inventory cost function is convex in k as the second-order artial 2 2 ¼ sl ffiffiffi Q 1=2 fðkþ*0 for all k After some maniulations, equations (3.2) and (3.3) can be rewritten as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l A þ scðkþ LðQ; Þ Q ¼ ( s h 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ CðkÞ ) ; ð3:7þ u t LðQ; Þ FðkÞ FðkÞ ¼ hq l : ð3:8þ 3.2. Solution rocedure We now resent an iterative rocedure similar to the one used in most inventory textbooks (e.g. Hadley and Whitin, 1963) to solve the classical continuous review roblem.

8 254 I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) Algorithm & rffiffiffiffiffiffiffiffi 2Al Ste 1. Set Q ¼.IfQoQ min, Q 5 Q min. h Comute k from equation (3.8) and decide from Lemma 2. Here, dxe means the smallest integer larger than or equal to x. Ste 2. Check the sign ¼ h 2 þ shk 2 ffiffiffi la slcðkþ 2 ffiffiffi. *0, Q 0 5 Q min. Otherwise, comute Q 0 from equation (3.7). If Q 0 oq min, Q 0 5 Q min. Otherwise, set Q 0 ¼ dq 0 e: Comute k 0 from equation (3.8), decide 0 from Lemma 2. Ste 3. If Q 0 5 Q, sto iteration and calculate TC(Q, k, ) and TC(Q 0, k 0, 0 ). If TC(Q, k, )otc(q 0, k 0, 0 ), then the near-otimal solution is (Q, k, ). Otherwise, the near-otimal solution is (Q 0, k 0, 0 ). Else if Q 0 6¼Q, set Q Q 0. k k 0, 0. Go to Ste 2. The roosed iterative rocedure outlined above starts with an EOQ solution. The first value of the safety factor is found by solving equation (3.8) and the roduction rate is selected as either 0 or max by Lemma 2. In the next iteration, a new value for Q (called Q 0 ) is comuted from equation (3.7) and Lemma 1 using the lot size, the safety factor, and the roduction rate of the first iteration. The modified values of k and are determined from equation (3.8) and Lemma 2 using Q 0. The iterative rocedure stos when two successive values of the lot size are equal. To find the near-otimal solution, we have to comare the TCs of the last two iterations. 4. Numerical examles To illustrate the solution rocedure, we consider the examle roblem. The data for this examle roblem are as follows: Average demand rate l units er year, standard deviation of demand s 5 50 units er day, fixed ordering cost A 5 $4000 er order, inventory holding cost h 5 $500 er unit er year, shortage cost 5 $1000 er unit short, regular roduction rate units er day, maximum roduction rate max units er day and minimum lot size Q min Tables 1 and 2 resent the results of the solution rocedures for each case of the additional cost S 5 $1.8 er unit and $1.9 er unit. Using a comlete enumeration for different integer values of Q, it is examined that our solution rocedure finds the otimal solution in each case. Figures 2 and 3 show resectively the grah of the total cost function for iterations 1 and 2 in Table 2 when Q and k are fixed. Clearly, the behavior of the cost function verifies Lemma 2. The two cases indicate case 3 in Lemma 2. The grahs are drawn using Mathematica 4.0. Note that Figs 2 and 3 indicate that the otimal roduction rates of iterations 1 and 2 are 300 and 200, resectively. And the two figures show that the otimal strategy for can be reached at

9 Table 1 Results for S 5 $1.8 er unit Iteration Q Q 0 k L SS r TC $509, $507, $507, $507, indicates the otimal values of decision variables. I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) Table 2 Results for S 5 $1.9 er unit Iteration Q Q 0 k L SS r TC $510, $508, $508, $508, $508, indicates the otimal values of decision variables * max Fig. 2. Grah of TC for iteration 1 in Table 2 (otimal solution is max ). the maximum roduction rate of the manufacturer. For iteration 4 in Table 2, the grahs of the TC and its first-order derivative with resect to Q are shown in Figs 4 and 5. These figures show the case ¼ h 2 þ shk 2 ffiffiffi la slcðkþ 2 ffiffiffi < 0: Both Tables 1 and 2 show that if we shorten the lead-time, we can lower the SS. Moreover, as exected from the total annual cost function, if the additional cost S is high, we should

10 256 I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) * 0 max Fig. 3. Grah of TC for iteration 2 in Table 2 (otimal solution is 0 ) Fig. 4. Grah of TC for iteration 4 in Table Fig. 5. Grah of =@Q for iteration 4 in Table 2. not increase the roduction rate. Table 3 shows the ercentage changes in the total cost function TC for the changes of additional cost S. From Table 3 it is clear that the ercentage saving in the total cost is low, as exected, for higher additional cost is required to increase the roduction rate.

11 Table 3 Results for the different values of S I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) S Q k L SS r TC Saving (%) $508, $508, $507, $504, $498, $491, Conclusion In this aer, we develo a stochastic lead-time reduction model with the lead-time deending on the ordering lot size and the roduction rate of the manufacturer. We consider the situation where the relenishment lead-time can be reduced by increasing the roduction rate of the manufacturer. At the time of contract with a manufacturer, the retailer can negotiate the lead-time by considering the regular roduction rate of the manufacturer, since the manufacturer usually has the otion of increasing his regular roduction rate u to the maximum roduction caacity. If the retailer wants the manufacturer to increase his roduction rate, he will be asked by the manufacturer for an additional cost in order to accomlish this by overtime, etc. If the manufacturer asks for a relatively high additional cost, the retailer s decision of increasing the roduction rate is not rofitable as evident from the numerical exeriment. Acknowledgements The authors are very grateful to the four anonymous referees who rovided useful comments to imrove the manuscrit. This work was suorted by Grant no. R from the Basic Research rogram of the Korea Science Engineering Foundation (KOSEF). References Ben-Daya, M., Raouf, A., Inventory models involving lead-time as decision variable. Journal of the Oerational Research Society, 45, Hadley, G., Whitin, T., Analysis of Inventory Systems. rentice-hall, New Jersey. Hariga, M., Setu cost reduction in (Q, r) olicy with lot size, setu time and lead-time interactions. Journal of the Oerational Research Society, 51, Hariga, M., Ben-Daya, M., Some stochastic inventory models with deterministic variable lead-time. Euroean Journal of Oerational Research, 113, Karmarkar, U., Lot sizes, lead-times and in-rocess inventories. Management Science, 33, Kim, J., Benton, W., Lot size deendent lead-times in a (Q, R) inventory system. International Journal of roduction Research, 33,

12 258 I. K. Moon and B. C. Cha / Intl. Trans. in O. Res. 12 (2005) Liao, C., Shyu, C., An analytical determination of lead-time with normal demand. International Journal of Oerations roduction Management, 11, Moon, I., Choi, S., A note on lead-time and distributional assumtions in continuous review inventory models. Comuters and Oerations Research, 25, Ouyang, L., Chang, H., Lot size reorder oint inventory model with controllable lead time and set-u cost. International Journal of Systems Science, 33, Ouyang, L., Cheng, C., Chang, H., Lead time and ordering cost reductions in continuous review inventory systems with artial backorders. Journal of the Oerational Research Society, 50, Ouyang, L., Yeh, N., Wu, K., Mixture inventory model with backorders and lost sales for variable lead-time. Journal of the Oerational Research Society, 47, an, J., Hsiao, Y., Lee, C., Inventory models with fixed and variable lead time crashing costs considerations. Journal of the Oerational Research Society, 53, Silver, E., ersectives in Oerations Management. Kluwer Academic ublishers, Massachusetts. Silver, E., yke, D., eterson, R., Inventory Management and roduction lanning and Scheduling. John Wiley & Sons, New York.