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EOQ models with general demand and holding cost functions
50
European Journal of Operational Research 73 (1994) 50-54 North-Holland
Theory and Methodology
EOQ models with general demand and holding cost functions M. Goh Department of Decision Sciences, Faculty of Business Administration, National University of Singapore, Singapore 0511, Singapore Received November 1991; revised July 1992
Abstract: We consider the continuous, deterministic, infinite horizon, single item inventory system, within the setting of a retail sector, in which the demand rate for an item is dependent on the existing inventory level. The traditional p a r a m e t e r of the replenishment cost is kept constant but the carrying cost per unit is allowed to vary. Two possibilities of variation are considered: (a) a non-linear function of the length of time the item is held in inventory and (b) a non-linear function of the amount of on-hand inventory. We find the optimal policies and decision rules and show that the classical E O Q model is obtained as a limiting case. Keywords: Inventory control; D e p e n d e n t demand rate; Holding cost
I. Introduction Deterministic inventory models have been developed for which the d e m a n d rate is either a constant, a function of the length of time an item is held in inventory or a known function of the on-hand stock. In the instance when the demand rate is constant, the effects of variability of the holding cost on the total inventory cost functions of such models have also been considered. Muhlemann and Valtis-Spanopoulos [7], for example, investigated the constant demand rate E O Q model but with a variable holding cost which is expressed as a percentage of the average value of capital invested in stock. This can be perceived
Correspondence to: Dr. M. Goh, Department of Decision Sciences, Faculty of Business Administration, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511, Singapore.
either as the opportunity cost of not having the capital required for stocking other items or interest payable on a loan that had to be raised. An E O Q inventory system with the holding cost as a non-linear function of inventory has also been presented by Van der Veen [9]. Weiss [10], too, studied the traditional E O Q model but with the holding cost per unit modified as a non-linear function of the length of time an item is held in stock. He mentioned that such a model is applicable to any inventory system where the value of the item decreases non-linearly the longer it is kept in stock, and he showed that the optimal order amount is given by an (n + D-st root Wilson lot size formula. Naddor [8] gives a detailed derivation of the total inventory cost for a constant demand rate lot size system when the holding on hand is cost as qmtn, where q denotes the amount of stock held and t the length of time it is kept, m and n being positive integers. Naddor's
0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved
SSDI 0377-2217(92)00040-V
M. Goh / EOQ models with generalfunctions
presentation can be regarded as a generalization of both Weiss' and Van der Veen's work. The recent works of Baker and U r b a n [2] and D a t t a and Pal [3] have focused on the deterministic inventory system with an inventory level dependent demand rate and with the holding cost held at a constant rate of h per unit per unit time. They analyzed the effect of inventory on sales. According to Baker and Urban, who asserted that, "to keep sales higher, the inc,entory
level would need to remain higher; of course, this would also result in higher holding ... costs". It is for this very reason that this present p a p e r looks at the impact of a non-linear holding cost on the total cost function. We consider the model developed by Baker and Urban [2] and relax the assumption of a constant holding cost. The holding cost is, instead, treated as (a) a polynomial function of the length of time spent in holding (as in Weiss) and (b) a functional form of the amount of on-hand stock (as in Van der Veen [9]). By using a polynomial representation in time for holding cost, we are adopting the perspective that the value of keeping any remaining or unsold inventory decreases with time. This type of inventory problem is often faced by retailers in the supermarket setting where products such as bread have their selling price decreased markedly with each passing day, as a result of the loss in "freshness". Holding on to such products will result in lost revenue. Viewed from this standpoint, the temporal functional form expression for holding cost is justified. The type of inventory problem which carries with it a quantity functional form expression for holding cost can be enountered in carrying luxury items like expensive jewelry and designer watches. As the on-hand stock volume grows, some firms holding such products not only employ more security but also higher dimensions of security such as hidden cameras and infrared sensors, if the value of the inventory is high enough. In such cases, the holding cost increases markedly. The trade-off would then be to balance the customers' desire to choose and, hence the quantity of the on-hand stock, against the carrying cost. The p a p e r is organized as follows. In the next section, we establish the underlying assumptions and notations used for the modified E O Q problem. The following section then presents a general solution to model the relationship between
51
the total inventory cost and the various parameters of the model, with regard to the two different forms of the modified holding cost. Two models are considered. The objective, in both cases, is to minimize the total cost function, based on the values of the inventory level dependent demand rate, the order quantity and the inventory carrying cost over a long period of time. Integral and differential calculus are used to determine the optimal policy and decision rules. An illustrative example is employed in Section 4 to show the impact of the parameters on the optimal quantity and the optimum total inventory cost for the models in question. Finally, some concluding remarks and future directions for research are detailed in the last section.
2. Assumptions The following notation is adopted: = Order quantity of item, assumed continuous. k = R e p l a c e m e n t cost. T = Cycle time. t = Length of time spent in inventory. D = Constant annual demand rate. q = On-hand inventory level. H C = Holding cost per cycle. T I C = Total relevant inventory cost per unit time. To construct a mathematical model for this problem, we assume the following: 1. Item cost does not vary with order size. 2. No backorders are allowed. 3. The time horizon of the inventory system is infinite. 4. The inventory system involves only one item. 5. The inventory system has only one stocking point. 6. The replinishment cost is known and constant. 7. The demand rate is deterministic and is a known function of the level of inventory. It is strictly differentiable, with a concave polynomial functional form Q
R ( q ) = D q t~, 0 < / 3 < 1 ,
O<_q<_O,
(1)
where the shape p a r a m e t e r /3 measures the inventory level elasticity.
52
M. Goh / EOQ models with general functions
Assumption 7 allows the inventory level to decrease rapidly initially, as the quantity demanded will be at a higher level of inventory. As more inventory is depleted, the quantity demanded R(q) will decrease resulting in a slowdown of inventory depletion. In the extreme case of a totally inelastic demand (/3 = 0), (1) reverts the model to the traditional constant demand rate inventory problem.
3. Model development
We now assume, for this model, that the cost on holding an item dq up to and including time t is given by ht ~ dq, where n ~ Z+\{1}, h > 0. For the inventory dependent demand rate problem, the differential equation governing the model is
Dnan+-------7B
- - = -Dq t3,
n + 1,
with initial condition q(0) = Q. The solution to this differential equation is given by
q l - t ~ = - D ( 1 - f l ) t + Q 1-t~, O < t < T ,
(3)
with the corresponding cycle time T given by 01-~ D(1 - f l ) "
The order quantity Q which minimizes the total relevant inventory cost per unit time can be found from the inventory cost equation, k HC TIC = T + T '
(4)
with the holding cost per cycle written as the integral of ht n dq from t = 0 to time t - T. Writing this mathematically, we have
£Ohtn dq.
(5)
Rearranging (3) and substituting into (5) yields
kDnan+2 h(an +fl)B(n + 1, 1/a) "
(8)
As a ---, 1 with n arbitrary, n
+~
Q* ~ But as
kD n hnB(n + 1,1)
B(n
+ 1, 1) =
"
1/(n + 1),
it follows that
n+~/_~( 1 ) 1+
,
(9)
which is Weiss' [10] result. In particular, the limit as a ~ 1, n ~ 1 yields Q* =
~/ 2kD h
'
(
1
D(1-[3)t )
dt.
(6)
(10)
which is just the E O Q result, as expected. When n = l and 0 < a < l , i.e. the holding cost increases linearly with time, the optimal order quantity Q* has the root type formula (the root being the ( a + 1)-st root), namely
Q* =
1+~/(1 + a ) k D a V h
(11)
which is the same result proven in Gob [4]. In general, the corresponding minimum total inventory cost can be found by substituting (7) into (4) giving TIC * =
aDk(an + 1) an + l - a (Q*)-~ k(an + 1)
= (an + 1 -a)T*'
DhQt3aortnf
(7)
(2)
dt
HC =
.
Substituting (7) into (4) and optimizing the resultant equation with respect to Q, we obtain
Q* ~
dq
HC =
HC
(Q*)""+' =
Model A: Instantaneous replenishment with nonlinear time dependent holding cost
T=
We now put a = 1 - ft. Through substitution techniques, (6) is shown to have a closed form solution which can be expressed as a Beta function, namely
(12)
with Q* given by (8). From (12), we note that the choice of the non-linearity factor n determines the degree to
M. Goh / EOQ models with generalfunctions
which the TIC*-value will be influenced. As n o~, i.e. it becomes exorbitantly expensive to hold any stock more than a unit of time, T I C * - - , k / T * . An example of such a scenario includes goods with some degree of obsolescence where the cost of carrying these outmoded lines explosively with time. Also (12), being a function of/3, suggests the apparent importance of the inventory level elasticity factor/3 on the optimal total inventory cost.
Model B: Instantaneous replenishment with nonlinear stock dependent carrying cost In this formulation, we treat the holding cost rate as a power function of the on-hand inventory namely, d(HC) dt
hq",
n>l,
hQ "+,~ TOnah = - D(n +4) (n +a)
(14)
Combining (14) with (4), and differentiating the resultant equation once with respect to Q, gives the optimal order quantity as
n+(/ kDa2( n + 4) Q* =
V
(15)
hn
this minimum total inventory cost per unit time is written as T I C * : k(1 + (a2/n)) T*
(n +a2)h
4(n +4)
(Q*)" (16)
Thus for items which require an expensive insuring system and obey an inventory dependent use rate, the optimal inventory cost is the product of the redefined holding cost rate and the optimal order quantity raised to the power n. Also, in the limit as n ~ ~, i.e. it becomes exorbitantly expensive to hold more than one unit of stock at any one time, (16) approaches the same asymptotic result as in Model A above.
4. Numerical example
(13)
where carrying stock on hand is costed at a rate of h per [value] n. To derive the holding cost per order, we integrate (13) with respect to time, taking limits from t = 0 to t = T. Hence, HC -
53
Having established Q*, (4) can be re-used to yield the optimal TIC. In general, for 4 E (0,1),
To illustrate the models, let us suppose that the parameters for a given problem are assigned the following values: D=I,
k=$10,
h---$0.50.
To observe the effects of 13 and n on Q* and T I C * , with D, k and h fixed for Model A, (8) and (12) are used. The results are exhibited in Table 1. The cells with blank entries denote values of Q* less than 1. Likewise, to observe the impact of/3 and n on Q* and T I C * , with fixed D-, k- and h-values for Model B, (15) and (16) are used. The results are as shown in Table 2. The results obtained for the illustrative example provide certain insights about the problem studied. Some of them are as follows:
Table 1 Effects of/3 and n on (Q*, TIC*) for Model A
n 1 2 4 6 8 10 20 30
0.1
0.3
0.5
0.7
0.9
(6.07, 3.37) (3.17, 4.70) (1.88, 6.33) (1.52, 7.19) (1.35, 7.72) (1.25, 8.08) (1.07, 8.92) (1.01, 9.24)
(5.23, 3.73) (3.24, 4.34) (1.79, 5.72) (1.36, 6.54) (1.15, 7.08) (1.04, 7.47)
(3.83, 383) (3.16, 3.75) (1.55, 4.81) (1.05, 5.56)
(1.92, 3.20) (2.67, 2.75) (1.06, 3.41)
(1.16, 1.08)
54
M. Goh / EOQ models with generalfunctions
Table 2 Effects of 13 and n on (Q*, TIC*) for Model B
n 1 2 4 6 8 10 20 30
/3 0.1
0.3
(6.07, 3.21) (2.70, 4.75) (1.84, 6.25) (1.53, 6.98) (1.38, 7.40) (1.30, 7.68) (1.15, 8.29) (1.10, 8.52)
(5.23 (2.60 (1.68 (1.43 (1.31 (1.25 (1.12 (1.08
3.27) 4.46) 5.46) 4.57) 6.14) 6.30) 6.63) 6.75)
0.5
0.7
(3.83, 3.19) (2.08, 3.90) (1.47, 4.39) (1.30, 4.57) (1.22, 4.67) (1.17, 4.74) (1.08, 4.87) (1.06, 4.91)
(1.92, 2.69) (1.37, 2.85) (1.17, 2.93) (1.11, 2.96) (1.08, 2.97) (1.06, 2.97) (1.03, 2.99) (1.02, 2.99)
1. As 13 increases from 0.1 to 0.9, with n fixed, Q* and TIC* generally decrease. 2. For a particular/3, as the non-linearity factor n increases, Q* decreases while TIC* generally increases. 3. The effect of n is more pronounced on Q* for items with a higher/3-value.
5. Conclusion The integration of a non-linear holding cost component in the presence of an inventory dependent demand rate, either with or without continual replenishment, is an important practical problem, often neglected in the inventory literature of the retail industry. This paper represents an attempt to fill this gap, by showing the effect on the total inventory cost. Future research work will consider the added effect of a non-linear ordering cost, due to varying freight charges.
Acknowledgement The author would like to thank the anonymous referees for their helpful comments and suggestions.
References [1] Abramovitz, M., and Stegun, I.A. (eds.), Handbook of Mathematical Functions, Dover, New York, 1984. [2] Baker, R.C., and Urban, T.L., "A deterministic inventory system with an inventory level dependent demand rate", Journal of the Operational Research Society 39 (1988) 823-831. [3] Datta, T.K., and Pal, A.K., "A note on an inventory model with inventory level dependent demand rate", Journal of the Operational Research Society 41 (1990) 971-975. [4] Goh, M., "Some results in inventory models having inventory level dependent demand rate", International Journal of Production Economics 27/2 (1992) 155-160. [5] Levin, R.I., McLaughlin, C.P., Lamone, R.P., and Kottas, J.F., Productions/ Operations Management: Contemporary Policyfor Managing Operating Systems, McGraw-Hill, New York, 1972, p. 373. [6] Misra, R.B., and Wortham, R.B., "EOQ model with continuous compounding", Omega 5/1 (1988) 98-99. [7] Muhlemann, A.P. and Valtis-Spanopoulos, N.P., "A variable holding cost rate EOQ model", European Journal of Operational Research 4 (1980) 132-135. [8] Naddor, E., Inventory Systems, Wiley, New York, 1966. [9] Van der Veen, B., Introduction to the Theory of Operational Research, Philips Technical Library, SpringerVerlag, New York, 1967. [10] Weiss, H.J., "Economic order quantity models with nonlinear holding cost", European Journal of Operational Research 9 (1982) 56-60.
European Journal of Operational Research 73 (1994) 50-54 North-Holland
Theory and Methodology
EOQ models with general demand and holding cost functions M. Goh Department of Decision Sciences, Faculty of Business Administration, National University of Singapore, Singapore 0511, Singapore Received November 1991; revised July 1992
Abstract: We consider the continuous, deterministic, infinite horizon, single item inventory system, within the setting of a retail sector, in which the demand rate for an item is dependent on the existing inventory level. The traditional p a r a m e t e r of the replenishment cost is kept constant but the carrying cost per unit is allowed to vary. Two possibilities of variation are considered: (a) a non-linear function of the length of time the item is held in inventory and (b) a non-linear function of the amount of on-hand inventory. We find the optimal policies and decision rules and show that the classical E O Q model is obtained as a limiting case. Keywords: Inventory control; D e p e n d e n t demand rate; Holding cost
I. Introduction Deterministic inventory models have been developed for which the d e m a n d rate is either a constant, a function of the length of time an item is held in inventory or a known function of the on-hand stock. In the instance when the demand rate is constant, the effects of variability of the holding cost on the total inventory cost functions of such models have also been considered. Muhlemann and Valtis-Spanopoulos [7], for example, investigated the constant demand rate E O Q model but with a variable holding cost which is expressed as a percentage of the average value of capital invested in stock. This can be perceived
Correspondence to: Dr. M. Goh, Department of Decision Sciences, Faculty of Business Administration, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511, Singapore.
either as the opportunity cost of not having the capital required for stocking other items or interest payable on a loan that had to be raised. An E O Q inventory system with the holding cost as a non-linear function of inventory has also been presented by Van der Veen [9]. Weiss [10], too, studied the traditional E O Q model but with the holding cost per unit modified as a non-linear function of the length of time an item is held in stock. He mentioned that such a model is applicable to any inventory system where the value of the item decreases non-linearly the longer it is kept in stock, and he showed that the optimal order amount is given by an (n + D-st root Wilson lot size formula. Naddor [8] gives a detailed derivation of the total inventory cost for a constant demand rate lot size system when the holding on hand is cost as qmtn, where q denotes the amount of stock held and t the length of time it is kept, m and n being positive integers. Naddor's
0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved
SSDI 0377-2217(92)00040-V
M. Goh / EOQ models with generalfunctions
presentation can be regarded as a generalization of both Weiss' and Van der Veen's work. The recent works of Baker and U r b a n [2] and D a t t a and Pal [3] have focused on the deterministic inventory system with an inventory level dependent demand rate and with the holding cost held at a constant rate of h per unit per unit time. They analyzed the effect of inventory on sales. According to Baker and Urban, who asserted that, "to keep sales higher, the inc,entory
level would need to remain higher; of course, this would also result in higher holding ... costs". It is for this very reason that this present p a p e r looks at the impact of a non-linear holding cost on the total cost function. We consider the model developed by Baker and Urban [2] and relax the assumption of a constant holding cost. The holding cost is, instead, treated as (a) a polynomial function of the length of time spent in holding (as in Weiss) and (b) a functional form of the amount of on-hand stock (as in Van der Veen [9]). By using a polynomial representation in time for holding cost, we are adopting the perspective that the value of keeping any remaining or unsold inventory decreases with time. This type of inventory problem is often faced by retailers in the supermarket setting where products such as bread have their selling price decreased markedly with each passing day, as a result of the loss in "freshness". Holding on to such products will result in lost revenue. Viewed from this standpoint, the temporal functional form expression for holding cost is justified. The type of inventory problem which carries with it a quantity functional form expression for holding cost can be enountered in carrying luxury items like expensive jewelry and designer watches. As the on-hand stock volume grows, some firms holding such products not only employ more security but also higher dimensions of security such as hidden cameras and infrared sensors, if the value of the inventory is high enough. In such cases, the holding cost increases markedly. The trade-off would then be to balance the customers' desire to choose and, hence the quantity of the on-hand stock, against the carrying cost. The p a p e r is organized as follows. In the next section, we establish the underlying assumptions and notations used for the modified E O Q problem. The following section then presents a general solution to model the relationship between
51
the total inventory cost and the various parameters of the model, with regard to the two different forms of the modified holding cost. Two models are considered. The objective, in both cases, is to minimize the total cost function, based on the values of the inventory level dependent demand rate, the order quantity and the inventory carrying cost over a long period of time. Integral and differential calculus are used to determine the optimal policy and decision rules. An illustrative example is employed in Section 4 to show the impact of the parameters on the optimal quantity and the optimum total inventory cost for the models in question. Finally, some concluding remarks and future directions for research are detailed in the last section.
2. Assumptions The following notation is adopted: = Order quantity of item, assumed continuous. k = R e p l a c e m e n t cost. T = Cycle time. t = Length of time spent in inventory. D = Constant annual demand rate. q = On-hand inventory level. H C = Holding cost per cycle. T I C = Total relevant inventory cost per unit time. To construct a mathematical model for this problem, we assume the following: 1. Item cost does not vary with order size. 2. No backorders are allowed. 3. The time horizon of the inventory system is infinite. 4. The inventory system involves only one item. 5. The inventory system has only one stocking point. 6. The replinishment cost is known and constant. 7. The demand rate is deterministic and is a known function of the level of inventory. It is strictly differentiable, with a concave polynomial functional form Q
R ( q ) = D q t~, 0 < / 3 < 1 ,
O<_q<_O,
(1)
where the shape p a r a m e t e r /3 measures the inventory level elasticity.
52
M. Goh / EOQ models with general functions
Assumption 7 allows the inventory level to decrease rapidly initially, as the quantity demanded will be at a higher level of inventory. As more inventory is depleted, the quantity demanded R(q) will decrease resulting in a slowdown of inventory depletion. In the extreme case of a totally inelastic demand (/3 = 0), (1) reverts the model to the traditional constant demand rate inventory problem.
3. Model development
We now assume, for this model, that the cost on holding an item dq up to and including time t is given by ht ~ dq, where n ~ Z+\{1}, h > 0. For the inventory dependent demand rate problem, the differential equation governing the model is
Dnan+-------7B
- - = -Dq t3,
n + 1,
with initial condition q(0) = Q. The solution to this differential equation is given by
q l - t ~ = - D ( 1 - f l ) t + Q 1-t~, O < t < T ,
(3)
with the corresponding cycle time T given by 01-~ D(1 - f l ) "
The order quantity Q which minimizes the total relevant inventory cost per unit time can be found from the inventory cost equation, k HC TIC = T + T '
(4)
with the holding cost per cycle written as the integral of ht n dq from t = 0 to time t - T. Writing this mathematically, we have
£Ohtn dq.
(5)
Rearranging (3) and substituting into (5) yields
kDnan+2 h(an +fl)B(n + 1, 1/a) "
(8)
As a ---, 1 with n arbitrary, n
+~
Q* ~ But as
kD n hnB(n + 1,1)
B(n
+ 1, 1) =
"
1/(n + 1),
it follows that
n+~/_~( 1 ) 1+
,
(9)
which is Weiss' [10] result. In particular, the limit as a ~ 1, n ~ 1 yields Q* =
~/ 2kD h
'
(
1
D(1-[3)t )
dt.
(6)
(10)
which is just the E O Q result, as expected. When n = l and 0 < a < l , i.e. the holding cost increases linearly with time, the optimal order quantity Q* has the root type formula (the root being the ( a + 1)-st root), namely
Q* =
1+~/(1 + a ) k D a V h
(11)
which is the same result proven in Gob [4]. In general, the corresponding minimum total inventory cost can be found by substituting (7) into (4) giving TIC * =
aDk(an + 1) an + l - a (Q*)-~ k(an + 1)
= (an + 1 -a)T*'
DhQt3aortnf
(7)
(2)
dt
HC =
.
Substituting (7) into (4) and optimizing the resultant equation with respect to Q, we obtain
Q* ~
dq
HC =
HC
(Q*)""+' =
Model A: Instantaneous replenishment with nonlinear time dependent holding cost
T=
We now put a = 1 - ft. Through substitution techniques, (6) is shown to have a closed form solution which can be expressed as a Beta function, namely
(12)
with Q* given by (8). From (12), we note that the choice of the non-linearity factor n determines the degree to
M. Goh / EOQ models with generalfunctions
which the TIC*-value will be influenced. As n o~, i.e. it becomes exorbitantly expensive to hold any stock more than a unit of time, T I C * - - , k / T * . An example of such a scenario includes goods with some degree of obsolescence where the cost of carrying these outmoded lines explosively with time. Also (12), being a function of/3, suggests the apparent importance of the inventory level elasticity factor/3 on the optimal total inventory cost.
Model B: Instantaneous replenishment with nonlinear stock dependent carrying cost In this formulation, we treat the holding cost rate as a power function of the on-hand inventory namely, d(HC) dt
hq",
n>l,
hQ "+,~ TOnah = - D(n +4) (n +a)
(14)
Combining (14) with (4), and differentiating the resultant equation once with respect to Q, gives the optimal order quantity as
n+(/ kDa2( n + 4) Q* =
V
(15)
hn
this minimum total inventory cost per unit time is written as T I C * : k(1 + (a2/n)) T*
(n +a2)h
4(n +4)
(Q*)" (16)
Thus for items which require an expensive insuring system and obey an inventory dependent use rate, the optimal inventory cost is the product of the redefined holding cost rate and the optimal order quantity raised to the power n. Also, in the limit as n ~ ~, i.e. it becomes exorbitantly expensive to hold more than one unit of stock at any one time, (16) approaches the same asymptotic result as in Model A above.
4. Numerical example
(13)
where carrying stock on hand is costed at a rate of h per [value] n. To derive the holding cost per order, we integrate (13) with respect to time, taking limits from t = 0 to t = T. Hence, HC -
53
Having established Q*, (4) can be re-used to yield the optimal TIC. In general, for 4 E (0,1),
To illustrate the models, let us suppose that the parameters for a given problem are assigned the following values: D=I,
k=$10,
h---$0.50.
To observe the effects of 13 and n on Q* and T I C * , with D, k and h fixed for Model A, (8) and (12) are used. The results are exhibited in Table 1. The cells with blank entries denote values of Q* less than 1. Likewise, to observe the impact of/3 and n on Q* and T I C * , with fixed D-, k- and h-values for Model B, (15) and (16) are used. The results are as shown in Table 2. The results obtained for the illustrative example provide certain insights about the problem studied. Some of them are as follows:
Table 1 Effects of/3 and n on (Q*, TIC*) for Model A
n 1 2 4 6 8 10 20 30
0.1
0.3
0.5
0.7
0.9
(6.07, 3.37) (3.17, 4.70) (1.88, 6.33) (1.52, 7.19) (1.35, 7.72) (1.25, 8.08) (1.07, 8.92) (1.01, 9.24)
(5.23, 3.73) (3.24, 4.34) (1.79, 5.72) (1.36, 6.54) (1.15, 7.08) (1.04, 7.47)
(3.83, 383) (3.16, 3.75) (1.55, 4.81) (1.05, 5.56)
(1.92, 3.20) (2.67, 2.75) (1.06, 3.41)
(1.16, 1.08)
54
M. Goh / EOQ models with generalfunctions
Table 2 Effects of 13 and n on (Q*, TIC*) for Model B
n 1 2 4 6 8 10 20 30
/3 0.1
0.3
(6.07, 3.21) (2.70, 4.75) (1.84, 6.25) (1.53, 6.98) (1.38, 7.40) (1.30, 7.68) (1.15, 8.29) (1.10, 8.52)
(5.23 (2.60 (1.68 (1.43 (1.31 (1.25 (1.12 (1.08
3.27) 4.46) 5.46) 4.57) 6.14) 6.30) 6.63) 6.75)
0.5
0.7
(3.83, 3.19) (2.08, 3.90) (1.47, 4.39) (1.30, 4.57) (1.22, 4.67) (1.17, 4.74) (1.08, 4.87) (1.06, 4.91)
(1.92, 2.69) (1.37, 2.85) (1.17, 2.93) (1.11, 2.96) (1.08, 2.97) (1.06, 2.97) (1.03, 2.99) (1.02, 2.99)
1. As 13 increases from 0.1 to 0.9, with n fixed, Q* and TIC* generally decrease. 2. For a particular/3, as the non-linearity factor n increases, Q* decreases while TIC* generally increases. 3. The effect of n is more pronounced on Q* for items with a higher/3-value.
5. Conclusion The integration of a non-linear holding cost component in the presence of an inventory dependent demand rate, either with or without continual replenishment, is an important practical problem, often neglected in the inventory literature of the retail industry. This paper represents an attempt to fill this gap, by showing the effect on the total inventory cost. Future research work will consider the added effect of a non-linear ordering cost, due to varying freight charges.
Acknowledgement The author would like to thank the anonymous referees for their helpful comments and suggestions.
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