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Optimal order policy for deteriorating items with inflation induced demand
ARTICLE IN PRESS
Int. J. Production Economics 103 (2006) 707–714 www.elsevier.com/locate/ijpe
Optimal order policy for deteriorating items with inflation induced demand Chandra K. Jaggi, K.K. Aggarwal, S.K. Goel Department of Operational Research, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India Received 5 October 2004; accepted 27 January 2006 Available online 29 March 2006
Abstract This paper presents the optimal inventory replenishment policy of deteriorating items under inflationary conditions using a discounted cash flow (DCF) approach over a finite planning horizon. A DCF approach permits a proper recognition of the financial implication of the opportunity cost and out-of-pocket costs in inventory analysis. It also permits an explicit recognition of the exact timing of cash-flows associated with an inventory system. The demand rate is assumed to be a function of inflation; shortages are allowed and completely backlogged. Optimal solution for the proposed model is derived and the effects of deterioration and inflation on the optimal inventory replenishment policy are studied with the help of numerical example. r 2006 Elsevier B.V. All rights reserved. Keywords: Inventory; Deterioration; Inflation; Discounted cash flow (DCF)
1. Introduction One of the important problems faced in inventory management is how to control and maintain the inventories of deteriorating items. Food items, pharmaceuticals, chemicals and blood are a few examples of such items. Ghare and Schrader (1963) first proposed an inventory model having a constant rate of deterioration and constant rate of demand over a finite-planning horizon. Covert and Phillip (1973) extended Ghare and Schrader’s model by considering variable rate of deterioration. Dave and Patel (1981) studied an inventory model with deterministic but linearly changing demand rate and constant deterioration over a finiteplanning horizon. Hollier and Mak (1983) presented inventory replenishment policies for deteriorating items in a declining market. Sachan (1984) extended Dave and Patel’s model to allow shortages. Datta and Pal (1988) developed an EOQ model by introducing a variable deterioration rate and power demand pattern. Chung and Ting (1994) determined the replenishment schedules for deteriorating items with time proportional demand. Hariga and Benkherouf (1994) developed an inventory replenishment model for deteriorating items with exponential time varying demand. This work was extended by Hargia (1995) to allow shortages.
Corresponding author. Tel./fax: +91 11 27666672.
E-mail address: [email protected] (C.K. Jaggi). 0925-5273/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.01.004
ARTICLE IN PRESS C.K. Jaggi et al. / Int. J. Production Economics 103 (2006) 707–714
708
Chakrabarti and Chaudhuri (1997) studied an inventory model with linearly changing demand rate, constant deterioration rate and shortages in all cycles over a finite-planning horizon. All the above articles are based on the assumption that the cost involved in inventory system remain constant over the planning horizon. This assumption may not be true in the real life, as many countries experience high annual inflation rate. Besides this, inflation also influences demand of certain products. As inflation increases the value of money goes down which erodes the future worth of savings and forces one for more current spending. Usually, these spendings are on peripherals and luxury items that give rise to demand of these items. As a result, while determining the optimal inventory policy, the effect of inflation and time value of money cannot be ignored. The fundamental result in the development of EOQ model with inflation is that of Buzacott (1975) who discussed EOQ model with inflation subject to different types of pricing policies. Bose et al. (1995) presented a paper on deteriorating items with linear time dependent rate and shortages under inflation and time discounting. Wee and Law (1999) addressed the problem with finite replenishment rate of deteriorating items taking account of time value of money. Chang (2004) proposes an inventory model for deteriorating items under inflation under a situation in which the supplier provides the purchaser a permissible delay of payments if the purchaser orders a large quantity. In this paper an attempt has been made to develop an inventory model with shortages, in which units are deteriorating at a constant rate and the demand rate is increasing exponentially due to inflation, over a finite planning horizon using the DCF approach for the analysis of optimal inventory replenishment policy. Optimal solution for the proposed model is derived and a comprehensive sensitivity analysis has also been performed to observe the effects of deterioration and inflation on the optimal inventory replenishment policies. 2. Assumptions and notations 1. The demand rate is exponentially increasing and is reqpresented by lðtÞ ¼ l0 eat ; where 0pap1 is a constant inflation rate and l0 is the initial demand rate. 2. Rate of replenishment is infinite and lead-time is negligible. 3. Shortages, if any, are made up as soon as the fresh stock arrives. 4. A0, C0, C10 and C20 are the ordering cost for an order, the unit purchase cost of an item, the out-of-pocket inventory carrying cost per unit per unit time and shortage cost per unit per unit time, respectively, at time zero. 5. A constant fraction yð0pyp1Þ of the on-hand inventory deteriorates per unit time. 6. There is no repair or replenishment of the deteriorated items during the inventory cycle. 7. A Discounted Cash Flow (DCF) approach is used to consider the various costs at various times, rðr4aÞ is the discount rate. 8. L is the length of the finite planning horizon. 3. Mathematical formulation Assuming continuous compounding of inflation, the ordering cost, unit cost of the item, out-of-pocket inventory carrying cost and shortage cost at any time t are AðtÞ ¼ A0 eat , CðtÞ ¼ C 0 eat , C 1 ðtÞ ¼ C 10 eat C 2 ðtÞ ¼ C 20 eat
and (Buzacott, (1975))
ð1Þ
The planning horizon (L) has been divided into n equal cycles of length T (i.e. T ¼ L=n). Let us consider the ith cycle, i.e. ti1 ptpti , where t0 ¼ 0, tn ¼ L, ti ti1 ¼ T and ti ¼ iT ði ¼ 1; 2; . . . ::; nÞ. At the beginning of the ith cycle, a batch of qi units enters the inventory system from which si units are delivered towards backorders leaving a balance of I0i units as the initial inventory level of the ith cycle, i.e. qi ¼ I 0i þ si . Thereafter, as time passes, the inventory level gradually decreasing mainly due to demand and partly due to
ARTICLE IN PRESS C.K. Jaggi et al. / Int. J. Production Economics 103 (2006) 707–714
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deterioration and reaches zero at time ti1 (Fig. 1). Further, demands during the remaining period of the cycle, i.e. from ti1 to ti, are backlogged and are fulfilled by a new procurement. Now ti1 ¼ ti kT ¼ ði kÞL=n, ði ¼ 1; 2; . . . ::; nÞ, ð0pkp1Þ, where kT is the fraction of the cycle having shortages. Let Ii (t) be the inventory level of the ith cycle at time tðti1 ptpti , i ¼ 1; 2; . . . ; nÞ. The differential equations describing the instantaneous states of Ii(t) over (ti1, ti) are dI i ðtÞ þ yI i ðtÞ ¼ lðtÞ ¼ l0 eat ; dt dI i ðtÞ ¼ lðtÞ ¼ l0 eat ; dt
ti1 ptpti1 ;
i ¼ 1; 2; :::; n;
ti1 ptpti ;
i ¼ 1; 2; :::; n:
(2)
The solution of the above differential equations along with the boundary conditions I i ðti1 Þ ¼ I 0i and I i ðti1 Þ ¼ 0 are 8 > l0 eyt ðyþaÞt > yðt tÞ > e eðyþaÞti1 ; ti1 ptpti1 ; < I 0i e i1 ða þ yÞ (3a,3b) I i ðtÞ ¼ > l 0 > at ati1 > ; t e e ptpt ; i ¼ 1; 2; :::; n: i1 i : a Since I i ðti1 Þ ¼ 0 and I i ðti Þ ¼ si , Eqs. (3a) and (3b) give I 0i ¼
l0 eyti1 ðyþaÞti1 e eðyþaÞti1 ; ða þ yÞ
i ¼ 1; 2; :::; n
(4)
and l0 ati e eati1 ; i ¼ 1; 2; :::; n. a Substituting I0i from Eq. (4), Eq. (3a) becomes si ¼
I i ðtÞ ¼
l0 eyt ðyþaÞti1 e eðyþaÞt ; ða þ yÞ
(5)
ti1 ptpti1 ;
i ¼ 1; 2; :::; n.
(6)
Further, batch size qi for the ith cycle is qi ¼ I 0i þ si . From Eqs. (4) and (5), we get qi ¼
l0 at l0 eyti1 ðyþaÞti1 e e i eati1 ; eðyþaÞti1 þ ða þ yÞ a
i ¼ 1; 2; :::; n.
(7)
Now at the beginning of each cycle there will be cash out flow of ordering cost and purchase cost. Further, since the inventory carrying cost is proportional to the value of the inventory, the out-of-pocket (physical
Ioi
si
t0
t11
T
t1
ti-1
ti1
T Fig. 1.
ti
tn-1
tn1
T
tn
ARTICLE IN PRESS C.K. Jaggi et al. / Int. J. Production Economics 103 (2006) 707–714
710
storage) inventory carrying cost per unit time at time t is I(t)C1(t). Similarly, the shortage cost can also be obtained. By using DCF approach, the present worth of the various costs for the ith cycle are as follows: 1. Present worth of ordering cost for the ith cycle, Ai, is Ai ¼ Aðti1 Þerti1 ¼ A0 eðarÞti1 ;
i ¼ 1; 2; :::; n.
(8)
2. Present worth of the purchase cost for the ith cycle, Pi, is Pi ¼ qi Cðti1 Þerti1 ¼ qi C 0 eðarÞti1 ;
i ¼ 1; 2; :::; n.
(9)
3. Present worth of the inventory carrying cost for the ith cycle, Hi, is Z ti1 rti1 H i ¼ C 1 ðti1 Þe I i ðtÞert dt ti1
Z l0 C 10 eðarÞti1 ti1 ðyþaÞti1 yt ¼ e e eat ert dt ðusing Eq: ð6ÞÞ ða þ yÞ ti1 ðarÞt ðrþyÞti1 ðaþyÞti1 i1 e eðarÞti1 e eðarÞti1 l0 C 10 e ¼ eðarÞti1 ; ðr þ yÞ ða rÞ ða þ yÞ
i ¼ 1; 2; . . . ; n.
ð10Þ
4. Present worth of the shortage cost for the ith cycle, pi, is Z ti pi ¼ C 2 ðti1 Þerti1 I i ðtÞert dt ti1
Z l0 C 20 eðarÞti1 ti at ¼ e eati1 ert dt ðusing Eq: ð3bÞÞ a ti1 l0 C 20 eðarÞti eðarÞti1 eati1 rti þ ¼ e erti1 eðarÞti1 ; ða rÞ a r
i ¼ 1; 2; :::; n.
ð11Þ
Therefore, the present worth of the total variable cost for the ith cycle, PWi, is the sum of the ordering cost (Ai), purchase cost (Pi), inventory carrying cost (Hi), and shortage cost (pi), i.e. PW i ¼ Ai þ Pi þ H i þ pi ;
i ¼ 1; 2; :::; n.
(12)
The present worth of the total variable cost of the system during the entire time horizon L is given by PW L ðk; nÞ ¼
n X i¼1
PW i ¼
n X ðAi þ Pi þ H i þ pi Þ.
(13)
i¼1
Substituting the values of Ai, Pi, Hi and pi from Eqs. (8), (9), (10) and (11), respectively, in Eq. (13) and after simplification, we get C l A0 ð1 eðarÞL Þ C 0 l0 ðaþyÞð1kÞL=n 0 0 aL=n ð1kÞaL=n e e 1 þ e PW L ðk; nÞ ¼ þ ða þ yÞ a ð1 eðarÞL=n Þ C 10 l0 C 10 l0 eðaþyÞð1kÞL=n eðarÞð1kÞL=n eðarÞð1kÞL=n 1 þ ða þ yÞðy þ rÞ ða þ yÞða rÞ C l C 20 l0 ðarÞL=n 20 0 þ e ekaaL=n0 eðarÞL=n eðarÞð1kÞL=n eðarÞð1kÞL=n þ aða rÞ ar ð2arÞL 1e ð14Þ 1 eð2arÞL=n
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Our problem is to determine the optimal values of k and n which minimize PWL(k,n). Since the cost function, PWL(k,n), is a function of two variables k and n, where k is a continuous and n is a discrete variable, therefore, for any given value of n ¼ n0 (say), the necessary condition for PWL(k, n0) to be minimum is dPW L ðk; n0 Þ ¼0 dk )
C0e
aL=n0
eðarÞL=n0 kaaL=n0 C 10 C 20 ð1kÞðarÞL=n0 þ C 20 þ e e r yþr r C 10 C0 þ eð1kÞðyþaÞL=n0 ¼ 0. yþr
ð15Þ
Further, PWL(k,n0) is a convex function in k (Appendix A). Therefore, Eq. (15) will provide the optimal values of k for n ¼ n0 . 4. Solution procedure In order to obtain the values of k and n which minimize PWL(k,n), the following procedure is adopted. Step 1: Solve Eq. (15) for k by substituting n ¼ np and n ¼ np þ 1, the corresponding values of k are knp and knp þ1 ; respectively, ðnp ¼ 1; 2; . . . ::Þ. Step 2: Compute PW L ðknp ; np Þ and PW L ðknp þ1 ; np þ 1Þ: Step 3: If PW L ðknp ; np ÞpPW L ðknp þ1 ; np þ 1Þ; then the optimal values of k and n are k ¼ knp and n ¼ np : The optimal value of T can be obtained using the relation T ¼ L=n while the optimal value of PWL(k, n) can be obtained by substituting k* and n* in Eq. (14) and lot sizes ðqi Þ for i ¼ 1; 2; . . . ; n can be obtained from Eq. (7). Else, go to Step 4. Step 4: Replace np by np+1 and go to Step 1. 5. Special case When k ¼ 0 (no shortages), Eq. (14) reduces to A0 ð1 eðarÞL Þ C 0 l0 ðaþyÞL=n C 10 l0 PW L ðnÞ ¼ e 1 þ þ ða þ yÞ ða þ yÞðy þ rÞ ð1 eðarÞL=n Þ 1 eð2arÞL C 10 l0 ðaþyÞL=n ðarÞL=n ðarÞL=n e e 1 e . ða þ yÞða rÞ 1 eð2arÞL=n
ð16Þ
6. Numerical example Let A0 ¼ $700, l0 ¼ 1100 units per year, r ¼ 0:12 per year, C 0 ¼ $250, C 10 ¼ $25, C 20 ¼ $100, L ¼ 2 year, a ¼ 0:05 per year and y ¼ 0:05 per year. Using the solution procedure (Fig. 2), we get n ¼ 12, k ¼ 0:2758, T ¼ 61 days, PW L ðk ; n Þ ¼ $5; 55; 089 and lot sizes qi for i ¼ 1; 2; . . . ; 12 have been shown in Table 1. Further, sensitivity analysis has been performed to study the impact of inflation (a) and deterioration (y) on the optimal number of cycles (n*), optimal cycle length (T*), present worth of total variable cost (PWL(k*, n*)) and lot size (q*) (Table 2). Findings are very much consistent with the reality, i.e.: 1. For any fixed y, if the value of a increases then the optimal number of cycles n* decreases (i.e. cycle length T* increases) while the present worth of total variable cost PWL(k*, n*) and lot size q* increase substantially. 2. For any fixed a, if the value of y increases then the optimal number of cycles n* increases (i.e. cycle length T* decreases) while the present worth of total variable cost PWL(k*, n*) and lot size q* increase marginally.
ARTICLE IN PRESS C.K. Jaggi et al. / Int. J. Production Economics 103 (2006) 707–714
712
558500 n=10
558000
n*=12
n=14
n=16
pw of cost
557500 557000 556500 556000 555500 555000 554500 0
0.05
0.1
0.15
0.2
0.25
0.3 k
0.35
0.4
0.45
0.5
0.55
0.6
Fig. 2.
Table 1 Cycle no (i)
1
2
3
4
5
6
7
8
9
10
11
12
Total
qi (in units)
185
186
188
189
191
192
194
196
197
199
200
202
q ¼ 2319
Table 2 Impact of a and y on the optimal replenishment policy ak
y-
0
0.10
0.20
0.30
Nearly 0
n* T* (days) PWL(k*,n*) ($) q* (units)
13 56 504,778 2200
14 52 506,738 2207
15 49 508,016 2210
16 46 508,917 2210
0.05
n* T* (days) PWL(k*,n*) ($) q* (units)
11 66 553,735 2314
13 56 556,131 2322
14 52 557,650 2324
15 49 558,707 2325
0.10
n* T* (days) PWL(k*,n*) ($) q* (units)
9 81 609,233 2435
12 61 612,300 2444
13 56 614,145 2447
14 52 615,417 2448
0.15
n* T* (days) PWL(k*,n*) ($) q* (units)
7 104 671,668 2566
10 73 675,992 2577
11 66 678,378 2581
12 61 679,960 2581
7. Conclusion In reality the value or utility of goods decreases over time for deteriorating items, which in turn suggests smaller cycle length, whereas presence of inflation in cost and its impact on demand suggests larger cycle length. In this article, inventory model has been developed considering both the opposite characteristics (deterioration and inflation) of the items, with shortages over a finite planning horizon. The study has been conducted under the Discounted Cash Flow (DCF) approach as it permits a proper recognition of the financial implication of the opportunity cost in inventory analysis.
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713
Acknowledgements The authors would like to thank the anonymous referees for their valuable suggestions and comments, which helped in improving the paper. Appendix A. The convexity In this section, convexity of the cost function, PWL(k, n), on appropriate domain is shown. Theorem 1. PWL(k, n0) is a convex function of k for any positive value of n ¼ n0 (say).
Proof. We treat PWL(k, n0) to be defined on 0pko1 for any positive value of n0. C l A0 ð1 eðarÞL Þ C 0 l0 ðaþyÞð1kÞL=n0 0 0 aL=n0 ð1kÞaL=n0 e e 1 þ e PW L ðk; n0 Þ ¼ þ ða þ yÞ a ð1 eðarÞL=n0 Þ C 10 l0 C 10 l0 þ eðaþyÞð1kÞL=n0 eðarÞð1kÞL=n0 eðarÞð1kÞL=n0 1 ða þ yÞðy þ rÞ ða þ yÞða rÞ C 20 l0 C 20 l0 kaL=n0 ðarÞL=n0 þ eðarÞL=n0 eðarÞð1kÞL=n0 þ e e eðarÞð1kÞL=n0 aða rÞ ar ð2arÞL 1e ; 1 eð2arÞL=n0 PW 0L ðk; n0 Þ
L eðarÞL=n0 kaL=n0 C 10 C 20 aL=n0 þ ¼ l0 C 0 e C 20 þ e n0 r yþr r C 10 eð1kÞðarÞL=n0 C 0 þ eð1kÞðyþaÞL=n0 , yþr 1 eð2arÞL 1 eð2arÞL=n0
n o 1 eð2arÞL L2 ð1kÞyL=n0 ¼ l C ðy þ aÞe a eð1kÞaL=n0 þ ðr aÞ 0 0 1 eð2arÞL=n0 n20 o C 10 C 20 ð1kÞðarÞL=n0 n a ðarÞL=n0 kaL=n0 ð1kÞðyþaÞL=n0 þ þ C 20 e e þ C 10 e e r yþr r n o
1eð2arÞL Since r4a and 0pkp1, ) 1e 40 and ðy þ aÞeð1kÞyL=n0 a 40 ) PW 00 ðk; n0 Þ40 for any integer ð2arÞL=n0 n0 40. Hence, PWL(k, n0) is a convex function of k for any positive integer, n. PW 0L ðk; n0 Þ
References Bose, S., Goswami, A., Choudhuri, K.S., 1995. An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under inflation and time discounting. Journal of the Operational Research Society 46, 771–782. Buzacott, J.A., 1975. Economic order quantity with inflation. Operations Research Qtr. 26 (3), 553–558. Chakrabarti, T., Chaudhuri, K.S., 1997. An EOQ for deteriorating items with a linear trend in demand and shortages in all cycles. International Journal of Production Economics 49, 205–213. Chang, C.-T., 2004. An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. International Journal of Production Economics 88, 307–316. Chung, K.J., Ting, P.S., 1994. On replenishment schedule for deteriorating items with time-proportional demand. Production Planning and Control 5 (4), 392–396. Covert, R.P., Phillip, G.C., 1973. An EOQ model with weibull distribution deterioration. AIIE Transactions 5, 323–326. Datta, T.K., Pal, A.K., 1988. Order level inventory system with power demand pattern for items with variable rate of deterioration. Indian Journal of Pure and Applied Mathematics 19, 1043–1053. Dave, U., Patel, L.K., 1981. (T, Si) policy inventory model for deteriorating items with time proportional demand. Journal of Operational Research Society 32, 137–142.
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Ghare, P.M., Schrader, S.K., 1963. A model for exponentially decaying inventory. Journal of Industrial Engineering 14 (5), 238–243. Hariga, M.A., 1995. An EOQ model for deteriorating items with shortages and time-varying demand. Journal of the Operational Research Society 46, 398–404. Hariga, M.A., Benkherouf, L., 1994. Optimal and heuristic inventory replenishment models for deteriorating items with time-varying demand. European Journal of Operational Research 79, 123–137. Hollier, R.H., Mak, K.L., 1983. Inventory replenishment policies for deteriorating items in a declining market. International Journal of Production Research 21 (6), 813–826. Sachan, R.S., 1984. On (T, Si) inventory policy model for deteriorating items with time proportional demand. Journal of Operational Research Society 35, 1013–1019. Wee, H.M., Law, S.T., 1999. Economic production lot size for deteriorating items taking account of time value of money. Computers and Operations Research 26, 545–558.
Int. J. Production Economics 103 (2006) 707–714 www.elsevier.com/locate/ijpe
Optimal order policy for deteriorating items with inflation induced demand Chandra K. Jaggi, K.K. Aggarwal, S.K. Goel Department of Operational Research, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India Received 5 October 2004; accepted 27 January 2006 Available online 29 March 2006
Abstract This paper presents the optimal inventory replenishment policy of deteriorating items under inflationary conditions using a discounted cash flow (DCF) approach over a finite planning horizon. A DCF approach permits a proper recognition of the financial implication of the opportunity cost and out-of-pocket costs in inventory analysis. It also permits an explicit recognition of the exact timing of cash-flows associated with an inventory system. The demand rate is assumed to be a function of inflation; shortages are allowed and completely backlogged. Optimal solution for the proposed model is derived and the effects of deterioration and inflation on the optimal inventory replenishment policy are studied with the help of numerical example. r 2006 Elsevier B.V. All rights reserved. Keywords: Inventory; Deterioration; Inflation; Discounted cash flow (DCF)
1. Introduction One of the important problems faced in inventory management is how to control and maintain the inventories of deteriorating items. Food items, pharmaceuticals, chemicals and blood are a few examples of such items. Ghare and Schrader (1963) first proposed an inventory model having a constant rate of deterioration and constant rate of demand over a finite-planning horizon. Covert and Phillip (1973) extended Ghare and Schrader’s model by considering variable rate of deterioration. Dave and Patel (1981) studied an inventory model with deterministic but linearly changing demand rate and constant deterioration over a finiteplanning horizon. Hollier and Mak (1983) presented inventory replenishment policies for deteriorating items in a declining market. Sachan (1984) extended Dave and Patel’s model to allow shortages. Datta and Pal (1988) developed an EOQ model by introducing a variable deterioration rate and power demand pattern. Chung and Ting (1994) determined the replenishment schedules for deteriorating items with time proportional demand. Hariga and Benkherouf (1994) developed an inventory replenishment model for deteriorating items with exponential time varying demand. This work was extended by Hargia (1995) to allow shortages.
Corresponding author. Tel./fax: +91 11 27666672.
E-mail address: [email protected] (C.K. Jaggi). 0925-5273/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.01.004
ARTICLE IN PRESS C.K. Jaggi et al. / Int. J. Production Economics 103 (2006) 707–714
708
Chakrabarti and Chaudhuri (1997) studied an inventory model with linearly changing demand rate, constant deterioration rate and shortages in all cycles over a finite-planning horizon. All the above articles are based on the assumption that the cost involved in inventory system remain constant over the planning horizon. This assumption may not be true in the real life, as many countries experience high annual inflation rate. Besides this, inflation also influences demand of certain products. As inflation increases the value of money goes down which erodes the future worth of savings and forces one for more current spending. Usually, these spendings are on peripherals and luxury items that give rise to demand of these items. As a result, while determining the optimal inventory policy, the effect of inflation and time value of money cannot be ignored. The fundamental result in the development of EOQ model with inflation is that of Buzacott (1975) who discussed EOQ model with inflation subject to different types of pricing policies. Bose et al. (1995) presented a paper on deteriorating items with linear time dependent rate and shortages under inflation and time discounting. Wee and Law (1999) addressed the problem with finite replenishment rate of deteriorating items taking account of time value of money. Chang (2004) proposes an inventory model for deteriorating items under inflation under a situation in which the supplier provides the purchaser a permissible delay of payments if the purchaser orders a large quantity. In this paper an attempt has been made to develop an inventory model with shortages, in which units are deteriorating at a constant rate and the demand rate is increasing exponentially due to inflation, over a finite planning horizon using the DCF approach for the analysis of optimal inventory replenishment policy. Optimal solution for the proposed model is derived and a comprehensive sensitivity analysis has also been performed to observe the effects of deterioration and inflation on the optimal inventory replenishment policies. 2. Assumptions and notations 1. The demand rate is exponentially increasing and is reqpresented by lðtÞ ¼ l0 eat ; where 0pap1 is a constant inflation rate and l0 is the initial demand rate. 2. Rate of replenishment is infinite and lead-time is negligible. 3. Shortages, if any, are made up as soon as the fresh stock arrives. 4. A0, C0, C10 and C20 are the ordering cost for an order, the unit purchase cost of an item, the out-of-pocket inventory carrying cost per unit per unit time and shortage cost per unit per unit time, respectively, at time zero. 5. A constant fraction yð0pyp1Þ of the on-hand inventory deteriorates per unit time. 6. There is no repair or replenishment of the deteriorated items during the inventory cycle. 7. A Discounted Cash Flow (DCF) approach is used to consider the various costs at various times, rðr4aÞ is the discount rate. 8. L is the length of the finite planning horizon. 3. Mathematical formulation Assuming continuous compounding of inflation, the ordering cost, unit cost of the item, out-of-pocket inventory carrying cost and shortage cost at any time t are AðtÞ ¼ A0 eat , CðtÞ ¼ C 0 eat , C 1 ðtÞ ¼ C 10 eat C 2 ðtÞ ¼ C 20 eat
and (Buzacott, (1975))
ð1Þ
The planning horizon (L) has been divided into n equal cycles of length T (i.e. T ¼ L=n). Let us consider the ith cycle, i.e. ti1 ptpti , where t0 ¼ 0, tn ¼ L, ti ti1 ¼ T and ti ¼ iT ði ¼ 1; 2; . . . ::; nÞ. At the beginning of the ith cycle, a batch of qi units enters the inventory system from which si units are delivered towards backorders leaving a balance of I0i units as the initial inventory level of the ith cycle, i.e. qi ¼ I 0i þ si . Thereafter, as time passes, the inventory level gradually decreasing mainly due to demand and partly due to
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deterioration and reaches zero at time ti1 (Fig. 1). Further, demands during the remaining period of the cycle, i.e. from ti1 to ti, are backlogged and are fulfilled by a new procurement. Now ti1 ¼ ti kT ¼ ði kÞL=n, ði ¼ 1; 2; . . . ::; nÞ, ð0pkp1Þ, where kT is the fraction of the cycle having shortages. Let Ii (t) be the inventory level of the ith cycle at time tðti1 ptpti , i ¼ 1; 2; . . . ; nÞ. The differential equations describing the instantaneous states of Ii(t) over (ti1, ti) are dI i ðtÞ þ yI i ðtÞ ¼ lðtÞ ¼ l0 eat ; dt dI i ðtÞ ¼ lðtÞ ¼ l0 eat ; dt
ti1 ptpti1 ;
i ¼ 1; 2; :::; n;
ti1 ptpti ;
i ¼ 1; 2; :::; n:
(2)
The solution of the above differential equations along with the boundary conditions I i ðti1 Þ ¼ I 0i and I i ðti1 Þ ¼ 0 are 8 > l0 eyt ðyþaÞt > yðt tÞ > e eðyþaÞti1 ; ti1 ptpti1 ; < I 0i e i1 ða þ yÞ (3a,3b) I i ðtÞ ¼ > l 0 > at ati1 > ; t e e ptpt ; i ¼ 1; 2; :::; n: i1 i : a Since I i ðti1 Þ ¼ 0 and I i ðti Þ ¼ si , Eqs. (3a) and (3b) give I 0i ¼
l0 eyti1 ðyþaÞti1 e eðyþaÞti1 ; ða þ yÞ
i ¼ 1; 2; :::; n
(4)
and l0 ati e eati1 ; i ¼ 1; 2; :::; n. a Substituting I0i from Eq. (4), Eq. (3a) becomes si ¼
I i ðtÞ ¼
l0 eyt ðyþaÞti1 e eðyþaÞt ; ða þ yÞ
(5)
ti1 ptpti1 ;
i ¼ 1; 2; :::; n.
(6)
Further, batch size qi for the ith cycle is qi ¼ I 0i þ si . From Eqs. (4) and (5), we get qi ¼
l0 at l0 eyti1 ðyþaÞti1 e e i eati1 ; eðyþaÞti1 þ ða þ yÞ a
i ¼ 1; 2; :::; n.
(7)
Now at the beginning of each cycle there will be cash out flow of ordering cost and purchase cost. Further, since the inventory carrying cost is proportional to the value of the inventory, the out-of-pocket (physical
Ioi
si
t0
t11
T
t1
ti-1
ti1
T Fig. 1.
ti
tn-1
tn1
T
tn
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storage) inventory carrying cost per unit time at time t is I(t)C1(t). Similarly, the shortage cost can also be obtained. By using DCF approach, the present worth of the various costs for the ith cycle are as follows: 1. Present worth of ordering cost for the ith cycle, Ai, is Ai ¼ Aðti1 Þerti1 ¼ A0 eðarÞti1 ;
i ¼ 1; 2; :::; n.
(8)
2. Present worth of the purchase cost for the ith cycle, Pi, is Pi ¼ qi Cðti1 Þerti1 ¼ qi C 0 eðarÞti1 ;
i ¼ 1; 2; :::; n.
(9)
3. Present worth of the inventory carrying cost for the ith cycle, Hi, is Z ti1 rti1 H i ¼ C 1 ðti1 Þe I i ðtÞert dt ti1
Z l0 C 10 eðarÞti1 ti1 ðyþaÞti1 yt ¼ e e eat ert dt ðusing Eq: ð6ÞÞ ða þ yÞ ti1 ðarÞt ðrþyÞti1 ðaþyÞti1 i1 e eðarÞti1 e eðarÞti1 l0 C 10 e ¼ eðarÞti1 ; ðr þ yÞ ða rÞ ða þ yÞ
i ¼ 1; 2; . . . ; n.
ð10Þ
4. Present worth of the shortage cost for the ith cycle, pi, is Z ti pi ¼ C 2 ðti1 Þerti1 I i ðtÞert dt ti1
Z l0 C 20 eðarÞti1 ti at ¼ e eati1 ert dt ðusing Eq: ð3bÞÞ a ti1 l0 C 20 eðarÞti eðarÞti1 eati1 rti þ ¼ e erti1 eðarÞti1 ; ða rÞ a r
i ¼ 1; 2; :::; n.
ð11Þ
Therefore, the present worth of the total variable cost for the ith cycle, PWi, is the sum of the ordering cost (Ai), purchase cost (Pi), inventory carrying cost (Hi), and shortage cost (pi), i.e. PW i ¼ Ai þ Pi þ H i þ pi ;
i ¼ 1; 2; :::; n.
(12)
The present worth of the total variable cost of the system during the entire time horizon L is given by PW L ðk; nÞ ¼
n X i¼1
PW i ¼
n X ðAi þ Pi þ H i þ pi Þ.
(13)
i¼1
Substituting the values of Ai, Pi, Hi and pi from Eqs. (8), (9), (10) and (11), respectively, in Eq. (13) and after simplification, we get C l A0 ð1 eðarÞL Þ C 0 l0 ðaþyÞð1kÞL=n 0 0 aL=n ð1kÞaL=n e e 1 þ e PW L ðk; nÞ ¼ þ ða þ yÞ a ð1 eðarÞL=n Þ C 10 l0 C 10 l0 eðaþyÞð1kÞL=n eðarÞð1kÞL=n eðarÞð1kÞL=n 1 þ ða þ yÞðy þ rÞ ða þ yÞða rÞ C l C 20 l0 ðarÞL=n 20 0 þ e ekaaL=n0 eðarÞL=n eðarÞð1kÞL=n eðarÞð1kÞL=n þ aða rÞ ar ð2arÞL 1e ð14Þ 1 eð2arÞL=n
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Our problem is to determine the optimal values of k and n which minimize PWL(k,n). Since the cost function, PWL(k,n), is a function of two variables k and n, where k is a continuous and n is a discrete variable, therefore, for any given value of n ¼ n0 (say), the necessary condition for PWL(k, n0) to be minimum is dPW L ðk; n0 Þ ¼0 dk )
C0e
aL=n0
eðarÞL=n0 kaaL=n0 C 10 C 20 ð1kÞðarÞL=n0 þ C 20 þ e e r yþr r C 10 C0 þ eð1kÞðyþaÞL=n0 ¼ 0. yþr
ð15Þ
Further, PWL(k,n0) is a convex function in k (Appendix A). Therefore, Eq. (15) will provide the optimal values of k for n ¼ n0 . 4. Solution procedure In order to obtain the values of k and n which minimize PWL(k,n), the following procedure is adopted. Step 1: Solve Eq. (15) for k by substituting n ¼ np and n ¼ np þ 1, the corresponding values of k are knp and knp þ1 ; respectively, ðnp ¼ 1; 2; . . . ::Þ. Step 2: Compute PW L ðknp ; np Þ and PW L ðknp þ1 ; np þ 1Þ: Step 3: If PW L ðknp ; np ÞpPW L ðknp þ1 ; np þ 1Þ; then the optimal values of k and n are k ¼ knp and n ¼ np : The optimal value of T can be obtained using the relation T ¼ L=n while the optimal value of PWL(k, n) can be obtained by substituting k* and n* in Eq. (14) and lot sizes ðqi Þ for i ¼ 1; 2; . . . ; n can be obtained from Eq. (7). Else, go to Step 4. Step 4: Replace np by np+1 and go to Step 1. 5. Special case When k ¼ 0 (no shortages), Eq. (14) reduces to A0 ð1 eðarÞL Þ C 0 l0 ðaþyÞL=n C 10 l0 PW L ðnÞ ¼ e 1 þ þ ða þ yÞ ða þ yÞðy þ rÞ ð1 eðarÞL=n Þ 1 eð2arÞL C 10 l0 ðaþyÞL=n ðarÞL=n ðarÞL=n e e 1 e . ða þ yÞða rÞ 1 eð2arÞL=n
ð16Þ
6. Numerical example Let A0 ¼ $700, l0 ¼ 1100 units per year, r ¼ 0:12 per year, C 0 ¼ $250, C 10 ¼ $25, C 20 ¼ $100, L ¼ 2 year, a ¼ 0:05 per year and y ¼ 0:05 per year. Using the solution procedure (Fig. 2), we get n ¼ 12, k ¼ 0:2758, T ¼ 61 days, PW L ðk ; n Þ ¼ $5; 55; 089 and lot sizes qi for i ¼ 1; 2; . . . ; 12 have been shown in Table 1. Further, sensitivity analysis has been performed to study the impact of inflation (a) and deterioration (y) on the optimal number of cycles (n*), optimal cycle length (T*), present worth of total variable cost (PWL(k*, n*)) and lot size (q*) (Table 2). Findings are very much consistent with the reality, i.e.: 1. For any fixed y, if the value of a increases then the optimal number of cycles n* decreases (i.e. cycle length T* increases) while the present worth of total variable cost PWL(k*, n*) and lot size q* increase substantially. 2. For any fixed a, if the value of y increases then the optimal number of cycles n* increases (i.e. cycle length T* decreases) while the present worth of total variable cost PWL(k*, n*) and lot size q* increase marginally.
ARTICLE IN PRESS C.K. Jaggi et al. / Int. J. Production Economics 103 (2006) 707–714
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558500 n=10
558000
n*=12
n=14
n=16
pw of cost
557500 557000 556500 556000 555500 555000 554500 0
0.05
0.1
0.15
0.2
0.25
0.3 k
0.35
0.4
0.45
0.5
0.55
0.6
Fig. 2.
Table 1 Cycle no (i)
1
2
3
4
5
6
7
8
9
10
11
12
Total
qi (in units)
185
186
188
189
191
192
194
196
197
199
200
202
q ¼ 2319
Table 2 Impact of a and y on the optimal replenishment policy ak
y-
0
0.10
0.20
0.30
Nearly 0
n* T* (days) PWL(k*,n*) ($) q* (units)
13 56 504,778 2200
14 52 506,738 2207
15 49 508,016 2210
16 46 508,917 2210
0.05
n* T* (days) PWL(k*,n*) ($) q* (units)
11 66 553,735 2314
13 56 556,131 2322
14 52 557,650 2324
15 49 558,707 2325
0.10
n* T* (days) PWL(k*,n*) ($) q* (units)
9 81 609,233 2435
12 61 612,300 2444
13 56 614,145 2447
14 52 615,417 2448
0.15
n* T* (days) PWL(k*,n*) ($) q* (units)
7 104 671,668 2566
10 73 675,992 2577
11 66 678,378 2581
12 61 679,960 2581
7. Conclusion In reality the value or utility of goods decreases over time for deteriorating items, which in turn suggests smaller cycle length, whereas presence of inflation in cost and its impact on demand suggests larger cycle length. In this article, inventory model has been developed considering both the opposite characteristics (deterioration and inflation) of the items, with shortages over a finite planning horizon. The study has been conducted under the Discounted Cash Flow (DCF) approach as it permits a proper recognition of the financial implication of the opportunity cost in inventory analysis.
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Acknowledgements The authors would like to thank the anonymous referees for their valuable suggestions and comments, which helped in improving the paper. Appendix A. The convexity In this section, convexity of the cost function, PWL(k, n), on appropriate domain is shown. Theorem 1. PWL(k, n0) is a convex function of k for any positive value of n ¼ n0 (say).
Proof. We treat PWL(k, n0) to be defined on 0pko1 for any positive value of n0. C l A0 ð1 eðarÞL Þ C 0 l0 ðaþyÞð1kÞL=n0 0 0 aL=n0 ð1kÞaL=n0 e e 1 þ e PW L ðk; n0 Þ ¼ þ ða þ yÞ a ð1 eðarÞL=n0 Þ C 10 l0 C 10 l0 þ eðaþyÞð1kÞL=n0 eðarÞð1kÞL=n0 eðarÞð1kÞL=n0 1 ða þ yÞðy þ rÞ ða þ yÞða rÞ C 20 l0 C 20 l0 kaL=n0 ðarÞL=n0 þ eðarÞL=n0 eðarÞð1kÞL=n0 þ e e eðarÞð1kÞL=n0 aða rÞ ar ð2arÞL 1e ; 1 eð2arÞL=n0 PW 0L ðk; n0 Þ
L eðarÞL=n0 kaL=n0 C 10 C 20 aL=n0 þ ¼ l0 C 0 e C 20 þ e n0 r yþr r C 10 eð1kÞðarÞL=n0 C 0 þ eð1kÞðyþaÞL=n0 , yþr 1 eð2arÞL 1 eð2arÞL=n0
n o 1 eð2arÞL L2 ð1kÞyL=n0 ¼ l C ðy þ aÞe a eð1kÞaL=n0 þ ðr aÞ 0 0 1 eð2arÞL=n0 n20 o C 10 C 20 ð1kÞðarÞL=n0 n a ðarÞL=n0 kaL=n0 ð1kÞðyþaÞL=n0 þ þ C 20 e e þ C 10 e e r yþr r n o
1eð2arÞL Since r4a and 0pkp1, ) 1e 40 and ðy þ aÞeð1kÞyL=n0 a 40 ) PW 00 ðk; n0 Þ40 for any integer ð2arÞL=n0 n0 40. Hence, PWL(k, n0) is a convex function of k for any positive integer, n. PW 0L ðk; n0 Þ
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