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Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost
ELSEVIER
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH European Journal of Operational Research 105 (1998) 467-474
Theory and Methodology
Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost B.C. Girl *, K.S. Chaudhuri Department of Mathematics, JadaL'pur University, Calcutta, 700 032, India
Received 1 March 1996; accepted 1 January 1997
Abstract This paper deals with an extended EOQ-type inventory model for a perishable product where the demand rate is a function of the on-hand inventory. The traditional parameters of unit item cost and ordering cost are kept constant; but the holding cost is treated as (i) a nonlinear function of the length of time for which the item is held in stock, and (ii) a functional form of the amount of the on-hand inventory. The approximate optimal solution in both the cases are derived. Computational results are presented indicating the effects of nonlinearity in holding costs. 9 1998 Elsevier Science B.V.
Ke)~vords: Inventory;Deterioration; Stock-dependentdemand rate; Nonlinear holding cost functions
1. I n t r o d u c t i o n Many supermarket managers have observed that, for some items, the demand rate is directly related to the amount of inventory displayed. According to Levin et al. [7], " a t times, the presence of inventory has a motivational effect on the people around it. It is a c o m m o n belief that large piles of goods displayed in a supermarket will lead the customer to buy more." Silver and Peterson [15] have also noted that sales at the retail level tend to be proportional to inventory displayed. These observations have attracted many marketing researchers and practitioners to investigate the modelling aspects of this phenomenon. In the last few years, researchers like Baker and Urban [2], Mondal and Phaujder [8], Datta and Pal [4], Pal et al. [12] have focused on the analysis of the inventory system which describes the demand rate as a power function, dependent on the level of the on-hand inventory; the holding cost per unit item per unit of time is taken as a constant in all these models. The variability in the holding cost was introduced first in the model developed by Muhlemann and Valtis-Spanopoulous [9]. In an EOQ model with a constant demand rate, they expressed the holding cost as a
' Correspondingauthor. 0377-2217/98/S19.00 9 1998 Elsevier Science B.V. All rights reserved. Pll S0377-2217(97)00086-6
468
B.C. Giri, .K.S. Chaudhuri/ European Journal of Operational Research 105 (1998) 467-474
percentage of the average value of capital invested in the stock. Van der Veen [18] developed an inventory system with constant demand rate, taking the holding cost as a nonlinear function of inventory. Weiss [19] studied the same model treating holding cost per unit as a nonlinear function of the length of time for which thd item was held in stock. Naddor [10] gave a detailed derivation of the total inventory cost for a constantdemand-rate lot-size-system by considering the holding cost as q ' t " , q being the amount of stock held for a time t and m , n being positive integers. Naddor's presentation can be regarded as a generalization of both Van der Veen's [18] and Weiss's [19] work. Later on, Goh [6] discussed the model of Baker and Urban [2] relaxing the assumption of a constant holding cost. In his model, he employed the holding cost as (i) a nonlinear function of the length of time the item is held in stock (as in Weiss [19]), and (ii) a nonlinear function of the amount of the on-hand inventory (as in Van der Veen [18]). The nonlinearity in time for the holding cost is justified for the inventory system in which not only the cost of holding an item in stock increases but also the value of the unsold inventory decreases with each passing day. The quantity-dependent functional form for the holding cost can be encountered when the value of the inventory item is very high and many precautionary steps are to be taken to ensure its safety and quality. In perishable inventory literature, perishability is classified in terms of fixed lifetime and random lifetime (Nahmias [11]). Fixed life-time product (e.g. human blood for use in transfusion) has a deterministic shelf-life while the random lifetime scenario assumes that the useful life of each unit is a random variable. The random lifetime scenario is closely related to the case of an inventory which experiences continuous physical depletion due to deterioration or decay. Ghare and Schrader [5], Covert and Philip [3], Philip [13], Tadikamalla [16], Shah [14], and Aggarwal [1] have presented deterministic EOQ models for deteriorating items under this framework. In the present paper, we have extended Goh's [6] model to cover an inventory of a deteriorating item where the rate of deterioration at any instant is a constant fraction 0(0 < 0 << 1) of the on-hand inventory. We have studied both the models with nonlinear time-dependent and stock-dependent holding costs. The objective, in both the cases, is to minimize the total cost function of the inventory system over a long period of time. The effects of nonlinearity in the holding cost on the near-optimal solution have been observed with the help of a numerical example.
2. Assumptions and notations We adopt the following assumptions and notations for the models to be discussed. Assumption 1. Item cost does not vary with order size. Assumption 2. The delivery lead time is zero. Assumption 3. Replenishments are instantaneous. Assumption 4. Replenishment cost is known and constant. Assumption 5. The inventory system consists of only one item. Assumption 6. There is only one stocking point in each cycle. Assumption 7. The time horizon of the inventory system is infinite. Only a typical planning schedule of length T is considered, all remaining cycles are identical.
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
469
Assumption 8. The demand rate is deterministic and is a known function of the instantaneous level of inventory q. The functional relationship between the demand rate R(q) and the instantaneous inventory level q(t) is given by the following expression:
R(q)=Dq tj,
D>0,
0
q>0,
where/3 denotes the shape parameter and is the measure of responsiveness of the demand rate to changes in the level of the on-hand inventory and D denotes the scale parameter. The variable q is assumed continuous in time. Assumption 9. A constant fraction 0, assumed small, of the on-hand inventory gets deteriorated per unit time. Notations:
Q: T: q(t): K: C: HC: DC: TCU:
Order quantity of the item. Cycle time. On-hand inventory level at any time t. Ordering cost per order. Cost per unit item. Holding cost per cycle. Deterioration cost per cycle. Total relevant inventory cost per unit time.
3. Mathematical model At the beginning of each cycle, the inventory level decreases rapidly because the quantity demanded is greater at a higher level of inventory. As the inventory is depleted, the rate of decrease of inventory level slows down. Ultimately the inventory reaches the zero level at the end of the cycle time T. The graphical representation of the inventory system is depicted in Fig. 1. The instantaneous states of q(t) over the cycle time T is given by the following first order nonlinear differential equation dq(t)
dt
+Oq(t)
z
-D(q(t))t3; O
(1)
with the initial condition q(0) = Q. Solving (1), we get
,1
,Ot=ln(, + o
,.(1+
Oq.]
On expansion of the right hand side, the first order approximation of 0 gives t=
or-D-
1-'~
(Q'~ + q ' )
where et = 1 - / 3 , 0 < ot < 1. Since
a~
~ /
r=---~ 1- 2D ]'
q(T) =
O
(2)
' 0, the cycle time T is given by
(3)
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
470
INVENTORY LEVEL
o
2T
TIME
Fig. 1. The inventory system with inventory-level-dependentdemand rate.
3.1. Model A: Nonlinear time-dependentholding cost For this model, we assume that the cost of holding an amount dq of the item up to and including time t is where n E Z+\{1}, h > 0; n = 1 implies linear time-dependent holding cost. Therefore,
ht" dq
HC =
f;ht" dq.
(4)
Substitution of (2) in (4) yields
HC=
anD"--~ 1--~Q
f;(Qa-q'~)"dq f;
nO
2D
(5)
q~(Q"-q")"dq
(to the first order approximation of 0). To evaluate the above integrals in the right hand side, we put Q'~ - q" = Q a Z and obtain h HC
2 ~n+ I o n + l
ha+Or+
1
~
B n+
,
(6)
where B denotes the Beta function. The deterioration cost in (0,T) is given by
DC = C[Q- f;Dqt3 dt]. Using (2) in (7) one can easily find
DC=
COQ~+1 (a+
1)D"
(s}
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
471
The total relevant inventory cost per unit time is, therefore, given by K + HC + DC
TCU =
T
(9)
Our problem is to determine the order quantity Q* which minimizes TCU of the inventory system. For fixed n, the necessary condition for TCU to be minimum is d - - (TCU) = O, dQ which gives
dT H C + DC) d Q
-(K+
(10)
0.
As 0 ~ 0, the above equation leads to K D n a " +2
(Q,)""+'
h ( n a - a + l)B
( ') n+l,--
or
which is the same result as derived in Goh [6] In the limit as /3 ~ 0, i.e., ot --* 1 and n ~ 1, Q ~ --* 7r2KD/h which is just the classical EOQ result as expected. It is quite impossible to find an explicit expression for Q by solving (10). However, solving (10) numerically by any one-dimensional search technique, one can easily find Q ", the approximate optimal order quantity. Then the corresponding cycle time T" and the average total inventory cost TCU* can be found from (3) and (9) respectively.
3.2. Model B: Nonlinear stock-dependent holding cost Holding cost is considered here as a power function of the on-hand inventory: d dt(HC)=hq",
n>l.
(11)
To find the holding cost per order, we integrate (11) over time t between the limits t = 0 and t = T. Hence HC
(12)
= rjoThq, dt.
Making use of (2) in (12), we obtain
HC=h
Q,+~ (n+-d-)D
OQ n+2a
]
(n+2ot)D 2 "
(13)
B.C. Giri, .K.S. Chaudhuri / European Journal o f Operational Research 105 (1998) 467-474
472
Replacing this expression for HC in (10) and proceeding in the same way as in the Model A, the value of Q" and the corresponding values of T* and TCU" can be determined numerically.
4. Numerical example In this section, we present computational results that yield some insight into the behaviour of Q*, T* and TCU" as n varies. In both the models the parameter values are taken as D = 2.0,
C = $10.0 per unit,
h = S0.5 per unit,
K = $200 per order, and 0 = 0.03.
Tables 1 and 2 present the effects of the shape parameter fl and the nonlinearity factor n on the approximate optimal solution ( Q ' , T * , T C U * ). It has been checked separately in both the cases shown in the Tables 1 and 2 that the appropriate sufficienl condition for minimization of TCU is satisfied. It is observed from Tables 1 and 2 that (i) for a particular/3, Q * and T" generally decreases but TCU * increases as the degree of nonlinearity in the holding cost increases;
Table 1 Effects o f / 3 and n on (Q ~ ,T" ,TCU" ) for Model A
n 2
3
4
5
6
7
8
9
10
Q~ T" TCU" Q~ T" TCU" Q~ T" TCU" Q~ T~ TCU ~ Q~ T~ TCU ~ Q~ T" TCU ~ Q~ T" TCU" Q~ T" TCU ~ Q* T~ TCU ~
0.1
0.3
0.5
0.7
0.9
15.96 6.11 49.04 9.49 3.97 67.50 6.91 3.03 82.92 5.60 2.52 95.45 4.82 2.22 105.68 4.31 2.01 114.14 3.95 1.8 121.22 3.68 1.75 127.23 3.48 1.67 132.39
20.44 5.54 51.72 12.03 3.90 67.06 8.48 3.08 80.23 6.65 2.61 91.25 5.56 2.31 100.47 4.84 2.11 108.26 4.34 1.96 114.91 3.98 1.84 120.65 3.70 1.75 125.65
27.41 5.03 53.69 ! 6.19 3.90 64.80 10.98 3.23 74.93 8.20 2.80 83.83 6.54 2.51 91.59 5.47 2.30 98.39 4.72 2.14 104.37 4.18 2.01 109.66 3.77 1.91 114.37
36.77 4.81 51.67 22.77 4.18 57.41 14.82 3.68 63.40 10.26 3.30 69.23 7.51 3.01 74.72 5.76 2.78 79.84 4.58 2.60 84.59 3.76 2.45 88.98 3.16 2.33 93.05
33.17 7.02 31.14 23.89 6.80 31.97 15.22 6.50 33.22 9.06 6.17 34.76 5.26 5.85 36.50 3.05 5.54 38.36 1.80 5.26 40.28 1.08 5.00 42.22 0.67 4.77 44.18
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
473
Table 2 Effects of fl and n on (Q ~,T ~,TCU-") for Model B /3 0.1 2
3
4
5
6
7
8
9
to
Q" T" TCU" Q" T9 TCU" Q" T" TCU Q" T" TCU Q" T" TCU Q" T* TCU Q"
T" TCU Q" T" TCU Q" T" TCU
0.3
0.5
0.7
0.9
11.00
11.60
11.91
11.35
4.49 64.24 5.80 2.61 99.64 4.00 1.89 129.81 3.14 1.53 154.68 2.65 1.31 175.15 2.34 1.17 192.18 2.12 1.08 206.49
3.81 71.04 5.87 2.40 102.80 3.99 1.84 127.52 3.11 1.55 146.69 2.62 1.38 161.83 2.31 1.26 174.01 2.09 1.18 184.01
3.36 74.65 5,79 2.36 98.91 3.89 1.94 115.89 3.03 1.72 128.20 2.55 1.58 137.49 2.25 1.48 144.71 2.04 1.41 150.49
3.40 67.91 5.41 2.73 80.73 3.64 2.43 88.67 2.84 2.26 94.03 2.41 2.15 97.87 2.13 2.07 100.75 1.95 2.02 103.00
8.01 6.10 34.52 4.12 5.71 36.25 2.92 5.52 37.20 2.37 5.41 37.79 2.05 5.33 38.19 1.86 5.28 38.48 1.72 5.24 38.71
1.96 1.01
1.94 1.12
1.89 1.36
1.81 1.97
1.62 5.21
218.65
192.34
155.21
104.80
38.88
1.85
1.82
1.78
1.71
1.54
0.95 229.11
1.07 199.39
1.32 159.13
! .94 106.27
5.18 39.02
(ii) for a particular n, Q" and T " of Model A are larger than those of Model B as fl increases; (iii) TCU* is much high when the nonlinear stock-dependent holding cost is encountered in the inventory system.
5. C o n c l u d i n g r e m a r k s In this article, a perishable inventory model with stock-dependent demand rate and nonlinear holding cost functions is developed for an infinite planning horizon. In Model A, the holding cost is regarded as a nonlinear function of the length of time the item is held in stock. This type of assumption is quite appropriate when the value of the unsold items decreases with time. Retailers in supermarket face this problem while selling products like green vegetables, fruits and breads whose quality drops with each passing day. As a result, increasing holding costs are incurred to arrange better storage facilities to prevent spoilage and to maintain freshness of the items in stock. The functional form of nonlinear time-dependent holding cost is quite realistic from that point of view. The assumption in Model B that the holding cost is a nonlinear function of the on-hand stock is justified for the products such as electronic components, radioactive substances, volatile liquids etc. which are not only costly but also require more sophisticated arrangements for their security and safety.
474
B.C. Giri, .K.S. Chaudhuri / European Journal o f Operational Research 105 (1998) 467-474
!
In the i n v e n t o r y model with the stock-dependent d e m a n d rate d e v e l o p e d by Baker and U r b a n [2], it w a s a s s u m e d that the i n v e n t o ~ level would be zero at the e n d o f the scheduling period. T h e same terminal condition ! was used by Datta and Pal [4] also. Later, U r b a n [17] relaxed the terminal condition of z e r o - e n d i n g inventoryl~ a l l o w i n g the " p o s s i b i l i t y - not the requirement - of h a v i n g i n v e n t o r y r e m a i n i n g at the e n d o f the order c y c l e " and obtained results better than those of Datta and Pal [4]. In the light of this work of U r b a n [17], we also analyzed our model by relaxing the terminal c o n d i t i o n o f z e r o - e n d i n g inventory; but it yielded results which were in n o w a y better than the results of the model presented here. W e surmise that the terminal condition of z e r o - e n d i n g i n v e n t o r y yields optimal solution in the stock-dependent demand-rate model w h e n the h o l d i n g costs are nonlinear.
Acknowledgements The authors express their sincerest thanks to the a n o n y m o u s referees for their constructive c o m m e n t s and suggestions on the earlier version o f the paper. T h i s research is supported by the University Grants C o m m i s s i o n , New Delhi and J a d a v p u r University, Calcutta.
References [1] S.P. Aggarwal, Note on: An order level lot size inventory model for detefiomtlng items by Y.K. Shah, AIIE Transactions 11 (1979) 344-346. [2] R.C. Baker, T.L. Urban, A deterministic inventory system with an inventory level dependent demand rate, Journal of the Operational Research Society 39 (1988) 823-831. [3] R.P. Covert, G.C. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Transactions ~5(1973) 323-326. [4] T.K. Datta, A.K. Pal, A note on an inventory model with inventory level dependent demand rate, Journal of the Operational Research Society 41 (1990) 971-975. [5] P.M. Ghare, G.F. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering 14 (1963) 238-243. [6] M. Goh, EOQ models with general demand and holding cost functions, European Journal of Operational Research 73 (1994) 50-54. [7] R.I. Levin, C.P. McLaughlin, R.P. Lamone, J.F. Kottas, Productions/Operations Management: Contemporary Policy for Managing Operating Systems, McGraw-Hill, New York, 1972, p. 373. [8] B.N. Mondal, S. Phaujder, An inventory model for deteriorating items and stock-dependent consumption rate, Journal of the Operational Research Society 40 (5) (1989) 483-488. [9] A.P. Muhlemann, N.P. Valtis Spanopoulos, A variable holding cost rate EOQ model, European Journal of Operational Research 4 (1980) 132-135. [I0] E. Naddor, Inventory Systems, Wiley, New York, 1966. [11] S. Nahmias, Perishable inventory theory: A review, Operations Research 30 (1982) 680-708. [12] S. Pal, A. Goswami, K. Chaudhuri, A deterministic inventory model for deteriorating items with stockodependent demand rate, Journal of Production Economics 32 (1993) 291-299. [13] G.C. Philip, A generalised EOQ model for items with Weibull distribution deterioration, AIIE Transactions 6 (1974) 159-162. [14] Y.K. Shah, An order-level lot size inventory model for deteriorating items, AIIE Transactions 9 (1977) 108-112. [15] E.A. Silver, R. Peterson, Decision Systems for Inventory Management and Production Planning, 2nd edition, Wiley, New York. [16] P.R. Tadikamalla, An EOQ model for items with gamma distribution deterioration, AIIE Transactions 10 (1978) 100-103. [17] T.L. Urban, An inventory model with an inventory-level-dependent demand rate and relaxed terminal conditions, Joumal of the Operational Research Society 43 (7) (1992) 721-724. [18] B. Van der Veen, Introduction to the Theory of Operational Research, Philips Technical Library, Springer-Verlag, New York, 1967. [19] H.J. Weiss, Economic order quantity models with nonlinear holding cost, European Joumal of Operational Research 9 (1982) 56-60.
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH European Journal of Operational Research 105 (1998) 467-474
Theory and Methodology
Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost B.C. Girl *, K.S. Chaudhuri Department of Mathematics, JadaL'pur University, Calcutta, 700 032, India
Received 1 March 1996; accepted 1 January 1997
Abstract This paper deals with an extended EOQ-type inventory model for a perishable product where the demand rate is a function of the on-hand inventory. The traditional parameters of unit item cost and ordering cost are kept constant; but the holding cost is treated as (i) a nonlinear function of the length of time for which the item is held in stock, and (ii) a functional form of the amount of the on-hand inventory. The approximate optimal solution in both the cases are derived. Computational results are presented indicating the effects of nonlinearity in holding costs. 9 1998 Elsevier Science B.V.
Ke)~vords: Inventory;Deterioration; Stock-dependentdemand rate; Nonlinear holding cost functions
1. I n t r o d u c t i o n Many supermarket managers have observed that, for some items, the demand rate is directly related to the amount of inventory displayed. According to Levin et al. [7], " a t times, the presence of inventory has a motivational effect on the people around it. It is a c o m m o n belief that large piles of goods displayed in a supermarket will lead the customer to buy more." Silver and Peterson [15] have also noted that sales at the retail level tend to be proportional to inventory displayed. These observations have attracted many marketing researchers and practitioners to investigate the modelling aspects of this phenomenon. In the last few years, researchers like Baker and Urban [2], Mondal and Phaujder [8], Datta and Pal [4], Pal et al. [12] have focused on the analysis of the inventory system which describes the demand rate as a power function, dependent on the level of the on-hand inventory; the holding cost per unit item per unit of time is taken as a constant in all these models. The variability in the holding cost was introduced first in the model developed by Muhlemann and Valtis-Spanopoulous [9]. In an EOQ model with a constant demand rate, they expressed the holding cost as a
' Correspondingauthor. 0377-2217/98/S19.00 9 1998 Elsevier Science B.V. All rights reserved. Pll S0377-2217(97)00086-6
468
B.C. Giri, .K.S. Chaudhuri/ European Journal of Operational Research 105 (1998) 467-474
percentage of the average value of capital invested in the stock. Van der Veen [18] developed an inventory system with constant demand rate, taking the holding cost as a nonlinear function of inventory. Weiss [19] studied the same model treating holding cost per unit as a nonlinear function of the length of time for which thd item was held in stock. Naddor [10] gave a detailed derivation of the total inventory cost for a constantdemand-rate lot-size-system by considering the holding cost as q ' t " , q being the amount of stock held for a time t and m , n being positive integers. Naddor's presentation can be regarded as a generalization of both Van der Veen's [18] and Weiss's [19] work. Later on, Goh [6] discussed the model of Baker and Urban [2] relaxing the assumption of a constant holding cost. In his model, he employed the holding cost as (i) a nonlinear function of the length of time the item is held in stock (as in Weiss [19]), and (ii) a nonlinear function of the amount of the on-hand inventory (as in Van der Veen [18]). The nonlinearity in time for the holding cost is justified for the inventory system in which not only the cost of holding an item in stock increases but also the value of the unsold inventory decreases with each passing day. The quantity-dependent functional form for the holding cost can be encountered when the value of the inventory item is very high and many precautionary steps are to be taken to ensure its safety and quality. In perishable inventory literature, perishability is classified in terms of fixed lifetime and random lifetime (Nahmias [11]). Fixed life-time product (e.g. human blood for use in transfusion) has a deterministic shelf-life while the random lifetime scenario assumes that the useful life of each unit is a random variable. The random lifetime scenario is closely related to the case of an inventory which experiences continuous physical depletion due to deterioration or decay. Ghare and Schrader [5], Covert and Philip [3], Philip [13], Tadikamalla [16], Shah [14], and Aggarwal [1] have presented deterministic EOQ models for deteriorating items under this framework. In the present paper, we have extended Goh's [6] model to cover an inventory of a deteriorating item where the rate of deterioration at any instant is a constant fraction 0(0 < 0 << 1) of the on-hand inventory. We have studied both the models with nonlinear time-dependent and stock-dependent holding costs. The objective, in both the cases, is to minimize the total cost function of the inventory system over a long period of time. The effects of nonlinearity in the holding cost on the near-optimal solution have been observed with the help of a numerical example.
2. Assumptions and notations We adopt the following assumptions and notations for the models to be discussed. Assumption 1. Item cost does not vary with order size. Assumption 2. The delivery lead time is zero. Assumption 3. Replenishments are instantaneous. Assumption 4. Replenishment cost is known and constant. Assumption 5. The inventory system consists of only one item. Assumption 6. There is only one stocking point in each cycle. Assumption 7. The time horizon of the inventory system is infinite. Only a typical planning schedule of length T is considered, all remaining cycles are identical.
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
469
Assumption 8. The demand rate is deterministic and is a known function of the instantaneous level of inventory q. The functional relationship between the demand rate R(q) and the instantaneous inventory level q(t) is given by the following expression:
R(q)=Dq tj,
D>0,
0
q>0,
where/3 denotes the shape parameter and is the measure of responsiveness of the demand rate to changes in the level of the on-hand inventory and D denotes the scale parameter. The variable q is assumed continuous in time. Assumption 9. A constant fraction 0, assumed small, of the on-hand inventory gets deteriorated per unit time. Notations:
Q: T: q(t): K: C: HC: DC: TCU:
Order quantity of the item. Cycle time. On-hand inventory level at any time t. Ordering cost per order. Cost per unit item. Holding cost per cycle. Deterioration cost per cycle. Total relevant inventory cost per unit time.
3. Mathematical model At the beginning of each cycle, the inventory level decreases rapidly because the quantity demanded is greater at a higher level of inventory. As the inventory is depleted, the rate of decrease of inventory level slows down. Ultimately the inventory reaches the zero level at the end of the cycle time T. The graphical representation of the inventory system is depicted in Fig. 1. The instantaneous states of q(t) over the cycle time T is given by the following first order nonlinear differential equation dq(t)
dt
+Oq(t)
z
-D(q(t))t3; O
(1)
with the initial condition q(0) = Q. Solving (1), we get
,1
,Ot=ln(, + o
,.(1+
Oq.]
On expansion of the right hand side, the first order approximation of 0 gives t=
or-D-
1-'~
(Q'~ + q ' )
where et = 1 - / 3 , 0 < ot < 1. Since
a~
~ /
r=---~ 1- 2D ]'
q(T) =
O
(2)
' 0, the cycle time T is given by
(3)
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
470
INVENTORY LEVEL
o
2T
TIME
Fig. 1. The inventory system with inventory-level-dependentdemand rate.
3.1. Model A: Nonlinear time-dependentholding cost For this model, we assume that the cost of holding an amount dq of the item up to and including time t is where n E Z+\{1}, h > 0; n = 1 implies linear time-dependent holding cost. Therefore,
ht" dq
HC =
f;ht" dq.
(4)
Substitution of (2) in (4) yields
HC=
anD"--~ 1--~Q
f;(Qa-q'~)"dq f;
nO
2D
(5)
q~(Q"-q")"dq
(to the first order approximation of 0). To evaluate the above integrals in the right hand side, we put Q'~ - q" = Q a Z and obtain h HC
2 ~n+ I o n + l
ha+Or+
1
~
B n+
,
(6)
where B denotes the Beta function. The deterioration cost in (0,T) is given by
DC = C[Q- f;Dqt3 dt]. Using (2) in (7) one can easily find
DC=
COQ~+1 (a+
1)D"
(s}
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
471
The total relevant inventory cost per unit time is, therefore, given by K + HC + DC
TCU =
T
(9)
Our problem is to determine the order quantity Q* which minimizes TCU of the inventory system. For fixed n, the necessary condition for TCU to be minimum is d - - (TCU) = O, dQ which gives
dT H C + DC) d Q
-(K+
(10)
0.
As 0 ~ 0, the above equation leads to K D n a " +2
(Q,)""+'
h ( n a - a + l)B
( ') n+l,--
or
which is the same result as derived in Goh [6] In the limit as /3 ~ 0, i.e., ot --* 1 and n ~ 1, Q ~ --* 7r2KD/h which is just the classical EOQ result as expected. It is quite impossible to find an explicit expression for Q by solving (10). However, solving (10) numerically by any one-dimensional search technique, one can easily find Q ", the approximate optimal order quantity. Then the corresponding cycle time T" and the average total inventory cost TCU* can be found from (3) and (9) respectively.
3.2. Model B: Nonlinear stock-dependent holding cost Holding cost is considered here as a power function of the on-hand inventory: d dt(HC)=hq",
n>l.
(11)
To find the holding cost per order, we integrate (11) over time t between the limits t = 0 and t = T. Hence HC
(12)
= rjoThq, dt.
Making use of (2) in (12), we obtain
HC=h
Q,+~ (n+-d-)D
OQ n+2a
]
(n+2ot)D 2 "
(13)
B.C. Giri, .K.S. Chaudhuri / European Journal o f Operational Research 105 (1998) 467-474
472
Replacing this expression for HC in (10) and proceeding in the same way as in the Model A, the value of Q" and the corresponding values of T* and TCU" can be determined numerically.
4. Numerical example In this section, we present computational results that yield some insight into the behaviour of Q*, T* and TCU" as n varies. In both the models the parameter values are taken as D = 2.0,
C = $10.0 per unit,
h = S0.5 per unit,
K = $200 per order, and 0 = 0.03.
Tables 1 and 2 present the effects of the shape parameter fl and the nonlinearity factor n on the approximate optimal solution ( Q ' , T * , T C U * ). It has been checked separately in both the cases shown in the Tables 1 and 2 that the appropriate sufficienl condition for minimization of TCU is satisfied. It is observed from Tables 1 and 2 that (i) for a particular/3, Q * and T" generally decreases but TCU * increases as the degree of nonlinearity in the holding cost increases;
Table 1 Effects o f / 3 and n on (Q ~ ,T" ,TCU" ) for Model A
n 2
3
4
5
6
7
8
9
10
Q~ T" TCU" Q~ T" TCU" Q~ T" TCU" Q~ T~ TCU ~ Q~ T~ TCU ~ Q~ T" TCU ~ Q~ T" TCU" Q~ T" TCU ~ Q* T~ TCU ~
0.1
0.3
0.5
0.7
0.9
15.96 6.11 49.04 9.49 3.97 67.50 6.91 3.03 82.92 5.60 2.52 95.45 4.82 2.22 105.68 4.31 2.01 114.14 3.95 1.8 121.22 3.68 1.75 127.23 3.48 1.67 132.39
20.44 5.54 51.72 12.03 3.90 67.06 8.48 3.08 80.23 6.65 2.61 91.25 5.56 2.31 100.47 4.84 2.11 108.26 4.34 1.96 114.91 3.98 1.84 120.65 3.70 1.75 125.65
27.41 5.03 53.69 ! 6.19 3.90 64.80 10.98 3.23 74.93 8.20 2.80 83.83 6.54 2.51 91.59 5.47 2.30 98.39 4.72 2.14 104.37 4.18 2.01 109.66 3.77 1.91 114.37
36.77 4.81 51.67 22.77 4.18 57.41 14.82 3.68 63.40 10.26 3.30 69.23 7.51 3.01 74.72 5.76 2.78 79.84 4.58 2.60 84.59 3.76 2.45 88.98 3.16 2.33 93.05
33.17 7.02 31.14 23.89 6.80 31.97 15.22 6.50 33.22 9.06 6.17 34.76 5.26 5.85 36.50 3.05 5.54 38.36 1.80 5.26 40.28 1.08 5.00 42.22 0.67 4.77 44.18
B.C. Giri, .K.S. Chaudhuri / European Journal of Operational Research 105 (1998) 467-474
473
Table 2 Effects of fl and n on (Q ~,T ~,TCU-") for Model B /3 0.1 2
3
4
5
6
7
8
9
to
Q" T" TCU" Q" T9 TCU" Q" T" TCU Q" T" TCU Q" T" TCU Q" T* TCU Q"
T" TCU Q" T" TCU Q" T" TCU
0.3
0.5
0.7
0.9
11.00
11.60
11.91
11.35
4.49 64.24 5.80 2.61 99.64 4.00 1.89 129.81 3.14 1.53 154.68 2.65 1.31 175.15 2.34 1.17 192.18 2.12 1.08 206.49
3.81 71.04 5.87 2.40 102.80 3.99 1.84 127.52 3.11 1.55 146.69 2.62 1.38 161.83 2.31 1.26 174.01 2.09 1.18 184.01
3.36 74.65 5,79 2.36 98.91 3.89 1.94 115.89 3.03 1.72 128.20 2.55 1.58 137.49 2.25 1.48 144.71 2.04 1.41 150.49
3.40 67.91 5.41 2.73 80.73 3.64 2.43 88.67 2.84 2.26 94.03 2.41 2.15 97.87 2.13 2.07 100.75 1.95 2.02 103.00
8.01 6.10 34.52 4.12 5.71 36.25 2.92 5.52 37.20 2.37 5.41 37.79 2.05 5.33 38.19 1.86 5.28 38.48 1.72 5.24 38.71
1.96 1.01
1.94 1.12
1.89 1.36
1.81 1.97
1.62 5.21
218.65
192.34
155.21
104.80
38.88
1.85
1.82
1.78
1.71
1.54
0.95 229.11
1.07 199.39
1.32 159.13
! .94 106.27
5.18 39.02
(ii) for a particular n, Q" and T " of Model A are larger than those of Model B as fl increases; (iii) TCU* is much high when the nonlinear stock-dependent holding cost is encountered in the inventory system.
5. C o n c l u d i n g r e m a r k s In this article, a perishable inventory model with stock-dependent demand rate and nonlinear holding cost functions is developed for an infinite planning horizon. In Model A, the holding cost is regarded as a nonlinear function of the length of time the item is held in stock. This type of assumption is quite appropriate when the value of the unsold items decreases with time. Retailers in supermarket face this problem while selling products like green vegetables, fruits and breads whose quality drops with each passing day. As a result, increasing holding costs are incurred to arrange better storage facilities to prevent spoilage and to maintain freshness of the items in stock. The functional form of nonlinear time-dependent holding cost is quite realistic from that point of view. The assumption in Model B that the holding cost is a nonlinear function of the on-hand stock is justified for the products such as electronic components, radioactive substances, volatile liquids etc. which are not only costly but also require more sophisticated arrangements for their security and safety.
474
B.C. Giri, .K.S. Chaudhuri / European Journal o f Operational Research 105 (1998) 467-474
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In the i n v e n t o r y model with the stock-dependent d e m a n d rate d e v e l o p e d by Baker and U r b a n [2], it w a s a s s u m e d that the i n v e n t o ~ level would be zero at the e n d o f the scheduling period. T h e same terminal condition ! was used by Datta and Pal [4] also. Later, U r b a n [17] relaxed the terminal condition of z e r o - e n d i n g inventoryl~ a l l o w i n g the " p o s s i b i l i t y - not the requirement - of h a v i n g i n v e n t o r y r e m a i n i n g at the e n d o f the order c y c l e " and obtained results better than those of Datta and Pal [4]. In the light of this work of U r b a n [17], we also analyzed our model by relaxing the terminal c o n d i t i o n o f z e r o - e n d i n g inventory; but it yielded results which were in n o w a y better than the results of the model presented here. W e surmise that the terminal condition of z e r o - e n d i n g i n v e n t o r y yields optimal solution in the stock-dependent demand-rate model w h e n the h o l d i n g costs are nonlinear.
Acknowledgements The authors express their sincerest thanks to the a n o n y m o u s referees for their constructive c o m m e n t s and suggestions on the earlier version o f the paper. T h i s research is supported by the University Grants C o m m i s s i o n , New Delhi and J a d a v p u r University, Calcutta.
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