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Inventory models with the demand rate dependent on stock and shortage levels
hbmatkmal journalof
production economics
ELSEVIER
Int. J. Production
Economics
40 (1995) 21-28
Inventory models with the demand rate dependent on stock and shortage levels Timothy L. Urban Department
of Quantitative Methods and MIS, The University of Tulsa, 600 South College Avenue, Tulsa, OK 74104.3189, USA Received
20 June 1994; accepted
for publication
14 March 1995
Abstract This paper investigates an inventory system in which the demand rate during stockout periods differs from that during the in-stock period by a given amount. The previous analysis conducted in this area formulated the model as a cost-minimization model. This type of formulation will intentionally try to decrease demand, by increasing the stockout period, in order to decrease costs. This paper appropriately considers the model as a profit-maximization model and develops a closed-form solution. The model is then further generalized to incorporate the effect of stock-level-dependent demand rates and considers initial-stock-dependent demand as well as instantaneous-stock-dependent demand rates.
1. Introduction
were suggested as the solution methodology as the typical calculus methods do not provide closedform solutions. Since then, extensions of this model have been developed for deteriorating items [4] and stochastic demand [S]. Baker and Urban [6] investigated the situation in which the demand rate is a function of the instantaneous stock level. In this case, the demand rate is continuously changing as the inventory level de&eases. Nonlinear (separable) programming was suggested as the solution methodology as we can no longer guarantee global concavity of the profit function. A great deal of research has since been conducted on extensions of this model, including models incorporating deteriorating items with linear stock-dependent demand [7] and nonlinear stock-dependent demand [S], fixed order-level models [9], finite replenishment and backorder models [lo], and models where the holding cost also varies with the inventory level
Traditional inventory models assume that demand is constant, insensitive to either the stock level or the shortage level. However, in some situations (e.g., retail sales), the demand rate is expected to be dependent on the stock level. Schary and Becker [l] discuss the impact of product availability for stimulating demand and Wolfe [2] presented empirical evidence of the demand rate changing as the stock level changes. Particularly in situations in which a retailer runs out of on-hand stock, we would expect to see the demand rate change, ranging somewhere between 100% backorders to 100% lost sales. Several models have recently been developed to incorporate the effect of demand dependence on the stock level. Gupta and Vrat [3] developed a model in which the demand rate is a function of the order quantity (the initial stock level). Search techniques 0925-5273/95/$09.50 0 1995 Elsevier SSDI 0925-5273(95)00036-4
Science B.V. All rights
Cl 11. reserved
22
T.L. UrbanlInt. J. Production Economics 40 (1995) 21-28
In a recent article by Mandal and Ghosh [12], a model was developed in which a different demand rate is observed during the ‘in-stock’ period, when inventory is available for the customer, and the stockout period, when there is no stock on hand. This model has obvious application to retail sales in which some customers may want to see the product before deciding whether or not to make a purchase; thus, the demand rate may be different when there is no stock on hand. The model was developed as a cost-minimization model in which the objective is to minimize the total cost of holding, shortage, ordering, and purchase cost of the items. A practical drawback of this formulation is that increasing the time of the stockout period can effectively reduce the total demand which, in turn, will reduce the overall costs. That is, it will be desirable to reduce demand by increasing the time during which shortages are incurred, in order to reduce the purchase cost of the items, even if the item is profitable. Of course, the objective is not merely to minimize cost, we also want to increase sales and profit. Traditional inventory models (i.e., EOQ models) can ignore the effect of revenue, as it is not dependent on the decision variables; it is a fixed amount (price times demand). However, in the in-stock/ stockout model, the demand is dependent on the decision variables; thus, a sound formulation of the model would need to incorporate the effect of revenue. In fact, it can be shown that the cost-minimization model is equivalent to assuming that the price markup is zero; that is, the product will be given away. For inventory models in which the decision variables - in this case, the order quantity and the reorder point - affect the demand rate, the model should be formulated as a profit-maximization model. The purpose of this paper is then (1) to formulate the in-stock/stockout model as a profit-maximization model to account for the profit-generating potential of an item, and (2) to generalize the model to include the effect of a stock-dependent demand rate, considering both initial-stock-dependent demand and instantaneous-stock-dependent demand models.
2. The basic in-stock/stockout
model
2. I. Model development
The model presented is the continuous, deterin-stock/stockout ministic, profit-maximization, model. The assumptions of this model are the same as the typical EOQ models with backorders - with the exceptions that the demand rate is different during in-stock and stockout periods and there is a constant markup on the items-and are as follows: 1. Replenishments are instantaneous with a known, constant lead time. 2. The unit cost, C, and the price markup, m, are known and constant. 3. The holding cost per unit time, C1, the shortage cost per unit time, CZ, and the procurement cost, C3, are known and constant. 4. The demand rate is known with a constant rate, D, during the in-stock period which is different from the rate during the stockout period by a constant stockout factor, k. 5. The time horizon is infinite. 6. The inventory system involves only one item and one stocking point. In this situation, the demand rate is not a function of the inventory level (this assumption will be relaxed in Section 3) and will decrease at a constant rate from the order-up-to point, S, as shown in Fig. 1. The entire stock is then depleted at time tr = S/D, at which point the demand rate changes, and the shortages reach the reorder point, s, at time T = tl + t2 = S/D - s/kD (note that tl and t2 are the number of time periods of the in-stock and stockout periods, respectively). We can then easily determine the average inventory level, f kS’/(kS - s), the average shortage level, fs’/(kS - s), and the number of orders per unit time, kD/(kS - s). Given this, the objective will be to maximize the average net profit; the profit function would be as follows: average net profit
rz(S, s) =
=
average gross profit
-
kD(m - l)C(S
average holding cost
-
average shortage cost
- s) - jkSZCl
-
average procurement cost
- is2C2
- kDC3
kS - s (1)
23
T.L. UrbanlInt.J. ProductionEconomics40 (1995) 21-28
‘I---+---‘*------3 Fig. 1. Inventory
curve t’or the in-stock/stockout
model.
The typical differential calculus methods can be used to find the optimal solution for the profitmaximization model. The optimal order-up-topoint, S*, and the optimal reorder point, s*, are as follows:
(2) and kDv
s* =
kc2 - qC1’
where 1=
- 2kCIC2C3 + vJkC1C2D[2C3(CI
+ kCZ) - v*D]
c1 [VZD - 2C,C3]
Note, rupees (Rs.) and paise are units of currency for several countries; there are 100 paise per rupee. Holding cost Cr = 5 paise per unit per week. Shortage cost CZ = 20 paise per unit per week. Ordering cost C3 = Rs. 10 per order. Unit cost C = Rs. 5 per unit. Demand parameters D = 8 units per week and k = 0.9. The solution to this problem using the cost-minimization approach would be an order quantity of 45 units and a reorder point of - 25 units resulting in an average cost of Rs. 41.0 per week. More time is spent in a stockout condition (t2 = 3.5 weeks) than in an in-stock condition (tI = 2.5 weeks) resulting in decreased total demand. The average demand rate using this solution will be 7.54 units per week. Now suppose that there is a markup of 50% percent (m = 1.5) on the cost of the item for a selling price of Rs. 7.50. The solution to this problem using the profit-maximization approach would be q* = 60 units and s* = - 4 units. Here, the average weekly sales are 7.94 units, an increase of 0.4 units per week, due to the reduced stockout time (cl = 7.0 weeks, tz = 0.6 weeks). This results in average revenues of Rs. 59.6 per week and costs of Rs. 42.4 per week providing a weekly profit of Rs. 17.2. Note that the total cost increased by about 6% over the cost-minimization approach, but the additional revenues, due to the increased demand, increase
the total profit by approximately
11%.
and v = (1 - k)(l - m)C. The optimal order quantity (q* = S* - s*) can easily be calculated from Eqs. (2) and (3) and the optimal value of IZ*(S, s) from Eq. (1). Due to the complexity of the model, we have not been able to prove quasiconcavity for all values of the parameters; however, all of the examples that we considered display this condition.
2.2. Numeric example To illustrate the effect of this model, consider the numeric example from Mandal and Ghosh [12].
2.3. EfSect of stockout factor and price markup
To gain further insight into the consequences of using this model, we will evaluate the effect of two parameters on the solution: the stockout factor, k, and the price markup, m. Consider first the stockout factor. As previously stated, the effect of k on the cost-minimization model is to encourage stockouts. This is depicted in the numeric example above, as more time was spent in a stockout condition than in an in-stock condition. If there was no change in the demand rate during the stockout period (i.e., k = 1, the equivalent of the classical EOQ model with backorders), we would realize demand of 8 units per week during the entire cycle
T.L. UrbanlInt. J. Production
24
Economics
40 (1995) 21-28
and have inventory available for 6.3 weeks of the 7.9-week cycle time. Of course, the effect of the stockout factor would be greater if the value of k were smaller; in fact, reducing k to 0.868 drives t1 to zero resulting in no inventory being held at any time (q* = - s* = 26.4 units) and an average demand of kD = 6.95 units per week. The effect of changes in the value of the demand parameter k will be just the opposite in the profitmaximization formulation as it is in the cost-minimization model. Now, a decrease in k will further encourage limiting the stockout time as the effect on demand will be greater. For k = 0.859, there will be no stockout period (t2 = 0). Increasing k to a value greater than one - implying the demand rate increases during stockout periods, as may be the case for some status items - provides more demand when there are shortages and will result in a longer stockout period. When k = 1.326, there will be no stock on hand (ti = 0), all demand will be during the stockout period. Fig. 2 illustrates the sensitivity of the solution (the in-stock period, ti, and the stockout period, tz) to changes in the value of k for our example. While the effect of the stockout factor on the in-stock and stockout periods is what we would expect, another interesting consequence that was
observed was its effect on the optimal cycle time and optimal order quantity. It was found that the largest values of the cycle time and order quantity occur when k is very near one; in fact, the maximum value of the cycle time for this example, T = tl + t2 = 7.91 weeks, occurred at k = 0.990 and the maximum value of the order quantity, q = 63.3 units, occurred at k = 1.013. As the impact of the stockout factor becomes more pronounced (that is, as k moves from 1.0 in either direction), the optimal order quantity and cycle time decrease. The effect that the price markup, m, has on the time the item is in stock versus the time it is out of stock is illustrated in Fig. 3 for the numeric example. Again, this reflects the importance of increasing the demand for profit maximization. Note that as the price markup increases, the stockout time decreases. By increasing the relative amount of time that there is stock on hand, the demand increases which allows increased profits. If the item is marked down (na < l), then the model does encourage stockouts by extending the stockout time, so the number of units sold is minimized. In fact, for m = 0, we realize the same solution as the costminimization model. On the other hand, there are no shortages (t2 = 0) at a markup of m = 1.71 (assuming that we can sell all of the units at this price),
Fig. 2. Effect of the demand and stockout periods.
Fig. 3. Effect of price stockout periods.
parameter,
k, on length of in-stock
markup,
m, on length
of in-stock
and
T.L. UrbanlInt. J. Production Economics 40 (1995) 21-28
and the solution is simply the classical EOQ solution without backorders (4* = 56.6 units, s* = 0 units). Again, it is interesting to note which values of m result in the largest cycle time and order quantity. The maximum value of the cycle time (T = 7.99 weeks) occurs when m = 1; thus, as the unit price deviates from the unit cost of the item, the optimal cycle time decreases. The effect of the price markup on the optimal order quantity is somewhat different; the maximum value of the order quantity (4 = 56.6 units) occurs when m = 1.064. It is at this value of the price markup that the revenue on the unit covers all of the stated costs; that is, the unit cost, holding cost, shortage cost, and procurement cost. Any deviation from zero net profit results in decreasing the order quantity. An interesting extension to the model would be to include the price markup as a decision variable.
3. Incorporating the stock-dependent demand model In this section, we will briefly consider the situation in which the demand rate is a function of the stock level as well as having the demand pattern change with an in-stock versus a stockout condition.
3.1. Demand as a function of the initial stock level First, suppose that the demand of an item is dependent on the initial stock level, as introduced by Gupta and Vrat [3]. The demand rate is not known a priori, but is dependent on the order quantity and reorder point determined by the model. That is, the larger the initial stock level during the order cycle, the greater the demand rate during the cycle. Even for the basic, initial-stock-dependent demand model (before we consider the case with a different demand during the in-stock and stockout periods), we cannot find a closed-form solution for most functional forms of demand. The simple linear function, where D = a + bQ, is an exception; the solution to this model is presented in Appendix A. However, to find the optimal solution for most
25
functional forms, we must simply solve the following equation (a univariate search technique could be used as the order quantity is the only decision variable):
where fo is the demand rate as a function of the initial stock level, andfh is its first derivative. So, while we do not generally find a closed-form solution, the solution methodology is quite straightforward. Now consider the extension of this model to the situation in which there are different demand rates for the in-stock and stockout periods. There is now a constant demand rate during the in-stock period, D =f(S), and a constant demand rate during the stockout period, kD. The demand rate is now a function of two variables: (1) the initial stock level, as this determines the demand rate for the stockdependent demand items, and (2) the reorder point, as this dictates the amount of time that shortages are being incurred and demand is at the reduced rate. Since the demand rates are still constant, it can be shown that we will not end an inventory cycle with positive stock on hand; therefore, the analysis of this model will be conducted in the same manner as the basic in-stock/stockout model presented in Section 2 by simply replacing the demand parameter in Eq. (1) with the appropriate demand function. Unfortunately, differential calculus methods will generally not result in a system of simple quadratic equations that can provide a closed-form solution. However, the model can easily be solved by using a nonlinear optimization software package or a bivariate search technique if the second-order conditions are satisfied.
3.2. Demand as a function of the instantaneous stock level
Baker and Urban [6] investigated the situation in which the demand rate is a function of the instantaneous inventory level, not the order-up-to level. In this situation, the demand rate is not constant but
T.L. UrbanlInt. J. Production
26
changes as the stock level changes, D =f(i). where i is the inventory level. The demand rate is greater during the beginning of an order cycle due to the higher inventory level and decreases as the stock level decreases. In this situation, it may be desirable to end the inventory cycle with stock on hand in order to maintain higher inventory levels and, therefore, higher demand rates. While this may be a more realistic scenario, it will further complicate the analysis of the models. Many of the functional forms of the demand rate would not be appropriate with the in-stock/stockout model. These functional forms, such as the power function investigated by Baker and Urban [6], will result in zero demand when the inventory level reaches zero; thus, we are never in a ‘stockout’ condition. If, in fact, the demand rate is completely dependent on the current stock level, we would likely expect this to be the case. This may be the case with applications such as used-car dealers; a customer goes to the dealership and is more likely to find a car to purchase if there are more to choose from. If there are no cars available, no sale is made. On the other hand, if the level of demand is only partially due to the stock level, other kinds of functional forms may be appropriate. One possibility is utilizing a model formulation like that of Datta and Pal [13] in which there is a stock-leveldependent demand rate until a given stock level, So, is realized, after which the demand rate is constant. This model can then be extended to allow shortages, so the in-stock/stockout model will reflect a different constant demand rate during the stockout period. In this situation, the demand rate would be of the form
D =
ai”,
i 2 So,
c&,
0 < i d So,
1 kc&,
Economics
40 (1995) 21-28
where t1 = (Sle8 - SO’-s)/,(l - /I) is the length of the in-stock period in which the inventory level is greater than So with the demand rate decreasing as the inventory level decreases, t2 = So/D = SA-Bfu is the length of the in-stock period in which the inventory level is less than So resulting in a constant demand rate, and t3 = - sfD = - s/kc& is the stockout period with the reduced demand rate. This inventory function is illustrated in Fig. 4. To analyze this model, we must evaluate four separate cases concerning the relative size of S, So, and s: Case 1: Case2: Case 3: Case 4:
S < So. S>s>S,. S > So 2 s > 0. S > So 2 s and s < 0.
The first three cases have previously been investigated. In case 1, the demand rate is always constant; the inventory level is never large enough to realize the stock-dependent demand. The analysis discussed in Section 2 of this paper is appropriate for this situation to allow for the different demand rate during the stockout period. In the second case, the demand rate never becomes constant, the inventory is replenished before reaching So. The original results of Baker and Urban [6], using separable programming, are appropriate here. In case 3, we are never in a stockout position since the reorder point is nonnegative. The analysis of Urban
(5)
i < 0.
It then follows that the inventory function will be: [&Y-ai=
or(1 - P)CJ”“~~‘,
so -as@ - Cl), - kaS;(t - tl - c2),
0 < f < t,,
t1
Fig. 4. Inventory curve for the in-stock/stockout stock-dependent demand.
model
with
T.L. Urban/M.
Cl43 is appropriate in which separable programming was again suggested as a solution methodology. Case 4 has not yet been studied. In this scenario, all possible demand rates can be identified. As the inventory level drops from S to So, the inventorylevel-dependent demand rate is observed. As i decreases from Se to 0, there is a constant demand rate, and when we are in a stockout condition, the decreased demand rate is realized. The objective function for this case is as follows (the development of this profit function is given in Appendix B): zI,(S,s) = [(l - /?)[2a(m - l)C(S - s) - (2S2-8 - /3S;-9c,/(2
- /?)
- s2Sia C2jk - 2&,]]/ [S’-fl - /3$-l
- (1 - /3)sS;a/k] _ (7)
Nonlinear programming can then be used as a solution methodology for this case as well, in the same manner as case 2 (since the variables will be in polynomial form, separable programming can be utilized). The optimal solution of the model can then be determined from the solution of each case: ZI*(S,
S) = IllaX(l7j(s,S)}. j
27
J. Production Economics 40 (1995) 2/-28
(8)
4. Conclusion The current inventory research literature contains two types of models in which the demand rate of an item varies due to its stock level. The first type, stock-level-dependent demand models, assumes that the demand rate is a function of the stock level, either the initial stock level or the instantaneous stock level. The other type, instock/stockout models, assumes that the demand rate is different during stockout periods than it is when stock is on hand. This paper reevaluates the in-stock/stockout model and formulates it as a profit-maximization model. This model is then generalized to incorporate the effect of stock-level-dependent demand rates. Models in which the demand rate is dependent on the initial stock level (the order-up-to level)
are analyzed as are models in which the demand rate is dependent on the instantaneous stock level.
Appendix A
In the basic, no-shortage inventory model with the demand rate a function of the initial stock level, Q, we have the same assumptions as the classical EOQ model (known and constant holding and ordering costs, etc.) with the exception of the stock-dependent demand. In this situation, the demand rate will be constant once the order quantity is established. Silver and Peterson [ 151 note that in the case of deterministic, constant demand rate and instantaneous replenishment, it is optimal to let the inventory level reach zero before reordering. Thus, the only decision variable will be the order quantity. The profit function will be as follows (note, a profit-maximization objective function is appropriate since the demand rate is determined by the order quantity, D =fo): LZ(Q)=(m-l)Cfp-fC1-QC,.
SQ
(A.11
Simply taking the first derivative of this function results in the following equation which can easily be solved for Q* (assuming the second-order conditions are met) for any given demand function:
(A.3 Most functional forms of demand will require a simple univariate search procedure to find the optimal solution. However, consider the case of a linear demand function, D = a + bQ. A closedform solution can be found for this particular functional form; it is Q* =
J
2aC3
cr - 2b(m - 1)C’
(A-3)
which is very similar to the traditional EOQ formula, the only difference being the ‘adjustment’ to the holding cost (the second order conditions indicate that the objective function is strictly concave).
28
T.L. Urban/fnt. J. Production Economics 40 (1995) 21-28
Note, however, that this solution is valid only as long as b < (C,/2(m - l)C); as b + (C,/2(m - l)C), then Q*- 00.
Appendix B
To identify the average profit function for case 4, we consider each of the revenue/cost components individually: Average
net projit:
E = (m - l)C(S T
- s) ’
where T = tI + t2 + t3 =
kS’ -B - pkS;-a
- (1 - @S,p
a(1 - BP Average 11
inventory
[Sl-p
level:
- a(1 - /3)t]“(l-@)dt
+ +S;tl
I T= 0 T
Integrating, substituting for tl and tl, and rearranging the terms provides S2-8 - @$-8 ‘=
cr(2-fi)T
’
Average shortage 6-
-p3
Average
_
level:
;a:;.
number of orders per unit time:
1 fi=-_. T The objective function can then be expressed as n(s
s) = (m - l)C(S - s)
7
T
S2-@ - +flSi-’ -
42 - LOT
C1
References
Cl1 Schary,
P.B. and Becker, B.W., 1972. Distribution and final demand: The influence of availability. Mississippi Valley J. Bus. Econom., 8: 17-26. PI Wolfe, H.B., 1968. A model for control of style merchandise. Industrial Mgmt. Rev., 9: 69-82. c31Gupta, R. and Vrat, P., 1986. Inventory model for stock dependent consumption rate. Opsearch, 23: 19-24. G. and Vrat, P., 1990. An EOQ model c41 Padmanabhan, for items with stock dependent consumption rate and exponential decay. Eng. Costs Prod. Econom., 18: 241-246. [51 Gerchak, Y. and Wang, Y., 1994. Periodic review inventory models with inventory-level-dependent demand. Naval Res. Logist., 41: 99-116. [6] Baker, R.C. and Urban, T.L., 1988. A deterministic inventory model with an inventory-level-dependent demand rate. J. Oper. Res. Sot., 39: 823-831. [7] Mandal, B.N. and Phaujdar, S., 1989. An inventory model for deteriorating items and stock-dependent consumption rate. J. qper. Res. Sot., 40: 483488. PI Pal, S., Goswami, A. and Chaudhuri, KS., 1993. A deterministic inventory model for deteriorating items with stock-dependent demand rate. Int. J. Prod. Econom., 32: 29 l-299. fixed c91Baker, R.C. and Urban, T.L., 1991. Deterministic order-level inventory models: An application for replenishment of radioactive source material for irradiation sterilizers. Eur. J. Oper. Res., 50: 249-256. Cl01Goh, M., 1992. Some results for inventory models having inventory level dependent demand rate. Int. J. Prod. Econom., 27: 155-160. Cl11Goh, M., 1994. EOQ models with general demand and holding cost functions. Eur. J. Oper. Res., 73: S&54. Cl21Mandal, B.N. and Ghosh, A.K., 1991. A note on an inventory model with different demand rates during stock-in and stock-out period. Int. J. Mgmt. Systems, 7: 33-36. Cl31Datta, T.K. and Pal, A.K., 1990. A note on an inventory model with inventory-level-dependent demand rate. J. Oper. Res. Sot., 41: 971-975. Cl41Urban, T.L., 1992. An inventory model with an inventorylevel-dependent demand rate and relaxed terminal conditions. J. Oper. Res. Sot., 43: 721-724. Cl51Silver, E.A. and Peterson, R., 1985. Decision Systems for Inventory Management and Production Planning, 2nd ed. Wiley, New York, p. 176.
production economics
ELSEVIER
Int. J. Production
Economics
40 (1995) 21-28
Inventory models with the demand rate dependent on stock and shortage levels Timothy L. Urban Department
of Quantitative Methods and MIS, The University of Tulsa, 600 South College Avenue, Tulsa, OK 74104.3189, USA Received
20 June 1994; accepted
for publication
14 March 1995
Abstract This paper investigates an inventory system in which the demand rate during stockout periods differs from that during the in-stock period by a given amount. The previous analysis conducted in this area formulated the model as a cost-minimization model. This type of formulation will intentionally try to decrease demand, by increasing the stockout period, in order to decrease costs. This paper appropriately considers the model as a profit-maximization model and develops a closed-form solution. The model is then further generalized to incorporate the effect of stock-level-dependent demand rates and considers initial-stock-dependent demand as well as instantaneous-stock-dependent demand rates.
1. Introduction
were suggested as the solution methodology as the typical calculus methods do not provide closedform solutions. Since then, extensions of this model have been developed for deteriorating items [4] and stochastic demand [S]. Baker and Urban [6] investigated the situation in which the demand rate is a function of the instantaneous stock level. In this case, the demand rate is continuously changing as the inventory level de&eases. Nonlinear (separable) programming was suggested as the solution methodology as we can no longer guarantee global concavity of the profit function. A great deal of research has since been conducted on extensions of this model, including models incorporating deteriorating items with linear stock-dependent demand [7] and nonlinear stock-dependent demand [S], fixed order-level models [9], finite replenishment and backorder models [lo], and models where the holding cost also varies with the inventory level
Traditional inventory models assume that demand is constant, insensitive to either the stock level or the shortage level. However, in some situations (e.g., retail sales), the demand rate is expected to be dependent on the stock level. Schary and Becker [l] discuss the impact of product availability for stimulating demand and Wolfe [2] presented empirical evidence of the demand rate changing as the stock level changes. Particularly in situations in which a retailer runs out of on-hand stock, we would expect to see the demand rate change, ranging somewhere between 100% backorders to 100% lost sales. Several models have recently been developed to incorporate the effect of demand dependence on the stock level. Gupta and Vrat [3] developed a model in which the demand rate is a function of the order quantity (the initial stock level). Search techniques 0925-5273/95/$09.50 0 1995 Elsevier SSDI 0925-5273(95)00036-4
Science B.V. All rights
Cl 11. reserved
22
T.L. UrbanlInt. J. Production Economics 40 (1995) 21-28
In a recent article by Mandal and Ghosh [12], a model was developed in which a different demand rate is observed during the ‘in-stock’ period, when inventory is available for the customer, and the stockout period, when there is no stock on hand. This model has obvious application to retail sales in which some customers may want to see the product before deciding whether or not to make a purchase; thus, the demand rate may be different when there is no stock on hand. The model was developed as a cost-minimization model in which the objective is to minimize the total cost of holding, shortage, ordering, and purchase cost of the items. A practical drawback of this formulation is that increasing the time of the stockout period can effectively reduce the total demand which, in turn, will reduce the overall costs. That is, it will be desirable to reduce demand by increasing the time during which shortages are incurred, in order to reduce the purchase cost of the items, even if the item is profitable. Of course, the objective is not merely to minimize cost, we also want to increase sales and profit. Traditional inventory models (i.e., EOQ models) can ignore the effect of revenue, as it is not dependent on the decision variables; it is a fixed amount (price times demand). However, in the in-stock/ stockout model, the demand is dependent on the decision variables; thus, a sound formulation of the model would need to incorporate the effect of revenue. In fact, it can be shown that the cost-minimization model is equivalent to assuming that the price markup is zero; that is, the product will be given away. For inventory models in which the decision variables - in this case, the order quantity and the reorder point - affect the demand rate, the model should be formulated as a profit-maximization model. The purpose of this paper is then (1) to formulate the in-stock/stockout model as a profit-maximization model to account for the profit-generating potential of an item, and (2) to generalize the model to include the effect of a stock-dependent demand rate, considering both initial-stock-dependent demand and instantaneous-stock-dependent demand models.
2. The basic in-stock/stockout
model
2. I. Model development
The model presented is the continuous, deterin-stock/stockout ministic, profit-maximization, model. The assumptions of this model are the same as the typical EOQ models with backorders - with the exceptions that the demand rate is different during in-stock and stockout periods and there is a constant markup on the items-and are as follows: 1. Replenishments are instantaneous with a known, constant lead time. 2. The unit cost, C, and the price markup, m, are known and constant. 3. The holding cost per unit time, C1, the shortage cost per unit time, CZ, and the procurement cost, C3, are known and constant. 4. The demand rate is known with a constant rate, D, during the in-stock period which is different from the rate during the stockout period by a constant stockout factor, k. 5. The time horizon is infinite. 6. The inventory system involves only one item and one stocking point. In this situation, the demand rate is not a function of the inventory level (this assumption will be relaxed in Section 3) and will decrease at a constant rate from the order-up-to point, S, as shown in Fig. 1. The entire stock is then depleted at time tr = S/D, at which point the demand rate changes, and the shortages reach the reorder point, s, at time T = tl + t2 = S/D - s/kD (note that tl and t2 are the number of time periods of the in-stock and stockout periods, respectively). We can then easily determine the average inventory level, f kS’/(kS - s), the average shortage level, fs’/(kS - s), and the number of orders per unit time, kD/(kS - s). Given this, the objective will be to maximize the average net profit; the profit function would be as follows: average net profit
rz(S, s) =
=
average gross profit
-
kD(m - l)C(S
average holding cost
-
average shortage cost
- s) - jkSZCl
-
average procurement cost
- is2C2
- kDC3
kS - s (1)
23
T.L. UrbanlInt.J. ProductionEconomics40 (1995) 21-28
‘I---+---‘*------3 Fig. 1. Inventory
curve t’or the in-stock/stockout
model.
The typical differential calculus methods can be used to find the optimal solution for the profitmaximization model. The optimal order-up-topoint, S*, and the optimal reorder point, s*, are as follows:
(2) and kDv
s* =
kc2 - qC1’
where 1=
- 2kCIC2C3 + vJkC1C2D[2C3(CI
+ kCZ) - v*D]
c1 [VZD - 2C,C3]
Note, rupees (Rs.) and paise are units of currency for several countries; there are 100 paise per rupee. Holding cost Cr = 5 paise per unit per week. Shortage cost CZ = 20 paise per unit per week. Ordering cost C3 = Rs. 10 per order. Unit cost C = Rs. 5 per unit. Demand parameters D = 8 units per week and k = 0.9. The solution to this problem using the cost-minimization approach would be an order quantity of 45 units and a reorder point of - 25 units resulting in an average cost of Rs. 41.0 per week. More time is spent in a stockout condition (t2 = 3.5 weeks) than in an in-stock condition (tI = 2.5 weeks) resulting in decreased total demand. The average demand rate using this solution will be 7.54 units per week. Now suppose that there is a markup of 50% percent (m = 1.5) on the cost of the item for a selling price of Rs. 7.50. The solution to this problem using the profit-maximization approach would be q* = 60 units and s* = - 4 units. Here, the average weekly sales are 7.94 units, an increase of 0.4 units per week, due to the reduced stockout time (cl = 7.0 weeks, tz = 0.6 weeks). This results in average revenues of Rs. 59.6 per week and costs of Rs. 42.4 per week providing a weekly profit of Rs. 17.2. Note that the total cost increased by about 6% over the cost-minimization approach, but the additional revenues, due to the increased demand, increase
the total profit by approximately
11%.
and v = (1 - k)(l - m)C. The optimal order quantity (q* = S* - s*) can easily be calculated from Eqs. (2) and (3) and the optimal value of IZ*(S, s) from Eq. (1). Due to the complexity of the model, we have not been able to prove quasiconcavity for all values of the parameters; however, all of the examples that we considered display this condition.
2.2. Numeric example To illustrate the effect of this model, consider the numeric example from Mandal and Ghosh [12].
2.3. EfSect of stockout factor and price markup
To gain further insight into the consequences of using this model, we will evaluate the effect of two parameters on the solution: the stockout factor, k, and the price markup, m. Consider first the stockout factor. As previously stated, the effect of k on the cost-minimization model is to encourage stockouts. This is depicted in the numeric example above, as more time was spent in a stockout condition than in an in-stock condition. If there was no change in the demand rate during the stockout period (i.e., k = 1, the equivalent of the classical EOQ model with backorders), we would realize demand of 8 units per week during the entire cycle
T.L. UrbanlInt. J. Production
24
Economics
40 (1995) 21-28
and have inventory available for 6.3 weeks of the 7.9-week cycle time. Of course, the effect of the stockout factor would be greater if the value of k were smaller; in fact, reducing k to 0.868 drives t1 to zero resulting in no inventory being held at any time (q* = - s* = 26.4 units) and an average demand of kD = 6.95 units per week. The effect of changes in the value of the demand parameter k will be just the opposite in the profitmaximization formulation as it is in the cost-minimization model. Now, a decrease in k will further encourage limiting the stockout time as the effect on demand will be greater. For k = 0.859, there will be no stockout period (t2 = 0). Increasing k to a value greater than one - implying the demand rate increases during stockout periods, as may be the case for some status items - provides more demand when there are shortages and will result in a longer stockout period. When k = 1.326, there will be no stock on hand (ti = 0), all demand will be during the stockout period. Fig. 2 illustrates the sensitivity of the solution (the in-stock period, ti, and the stockout period, tz) to changes in the value of k for our example. While the effect of the stockout factor on the in-stock and stockout periods is what we would expect, another interesting consequence that was
observed was its effect on the optimal cycle time and optimal order quantity. It was found that the largest values of the cycle time and order quantity occur when k is very near one; in fact, the maximum value of the cycle time for this example, T = tl + t2 = 7.91 weeks, occurred at k = 0.990 and the maximum value of the order quantity, q = 63.3 units, occurred at k = 1.013. As the impact of the stockout factor becomes more pronounced (that is, as k moves from 1.0 in either direction), the optimal order quantity and cycle time decrease. The effect that the price markup, m, has on the time the item is in stock versus the time it is out of stock is illustrated in Fig. 3 for the numeric example. Again, this reflects the importance of increasing the demand for profit maximization. Note that as the price markup increases, the stockout time decreases. By increasing the relative amount of time that there is stock on hand, the demand increases which allows increased profits. If the item is marked down (na < l), then the model does encourage stockouts by extending the stockout time, so the number of units sold is minimized. In fact, for m = 0, we realize the same solution as the costminimization model. On the other hand, there are no shortages (t2 = 0) at a markup of m = 1.71 (assuming that we can sell all of the units at this price),
Fig. 2. Effect of the demand and stockout periods.
Fig. 3. Effect of price stockout periods.
parameter,
k, on length of in-stock
markup,
m, on length
of in-stock
and
T.L. UrbanlInt. J. Production Economics 40 (1995) 21-28
and the solution is simply the classical EOQ solution without backorders (4* = 56.6 units, s* = 0 units). Again, it is interesting to note which values of m result in the largest cycle time and order quantity. The maximum value of the cycle time (T = 7.99 weeks) occurs when m = 1; thus, as the unit price deviates from the unit cost of the item, the optimal cycle time decreases. The effect of the price markup on the optimal order quantity is somewhat different; the maximum value of the order quantity (4 = 56.6 units) occurs when m = 1.064. It is at this value of the price markup that the revenue on the unit covers all of the stated costs; that is, the unit cost, holding cost, shortage cost, and procurement cost. Any deviation from zero net profit results in decreasing the order quantity. An interesting extension to the model would be to include the price markup as a decision variable.
3. Incorporating the stock-dependent demand model In this section, we will briefly consider the situation in which the demand rate is a function of the stock level as well as having the demand pattern change with an in-stock versus a stockout condition.
3.1. Demand as a function of the initial stock level First, suppose that the demand of an item is dependent on the initial stock level, as introduced by Gupta and Vrat [3]. The demand rate is not known a priori, but is dependent on the order quantity and reorder point determined by the model. That is, the larger the initial stock level during the order cycle, the greater the demand rate during the cycle. Even for the basic, initial-stock-dependent demand model (before we consider the case with a different demand during the in-stock and stockout periods), we cannot find a closed-form solution for most functional forms of demand. The simple linear function, where D = a + bQ, is an exception; the solution to this model is presented in Appendix A. However, to find the optimal solution for most
25
functional forms, we must simply solve the following equation (a univariate search technique could be used as the order quantity is the only decision variable):
where fo is the demand rate as a function of the initial stock level, andfh is its first derivative. So, while we do not generally find a closed-form solution, the solution methodology is quite straightforward. Now consider the extension of this model to the situation in which there are different demand rates for the in-stock and stockout periods. There is now a constant demand rate during the in-stock period, D =f(S), and a constant demand rate during the stockout period, kD. The demand rate is now a function of two variables: (1) the initial stock level, as this determines the demand rate for the stockdependent demand items, and (2) the reorder point, as this dictates the amount of time that shortages are being incurred and demand is at the reduced rate. Since the demand rates are still constant, it can be shown that we will not end an inventory cycle with positive stock on hand; therefore, the analysis of this model will be conducted in the same manner as the basic in-stock/stockout model presented in Section 2 by simply replacing the demand parameter in Eq. (1) with the appropriate demand function. Unfortunately, differential calculus methods will generally not result in a system of simple quadratic equations that can provide a closed-form solution. However, the model can easily be solved by using a nonlinear optimization software package or a bivariate search technique if the second-order conditions are satisfied.
3.2. Demand as a function of the instantaneous stock level
Baker and Urban [6] investigated the situation in which the demand rate is a function of the instantaneous inventory level, not the order-up-to level. In this situation, the demand rate is not constant but
T.L. UrbanlInt. J. Production
26
changes as the stock level changes, D =f(i). where i is the inventory level. The demand rate is greater during the beginning of an order cycle due to the higher inventory level and decreases as the stock level decreases. In this situation, it may be desirable to end the inventory cycle with stock on hand in order to maintain higher inventory levels and, therefore, higher demand rates. While this may be a more realistic scenario, it will further complicate the analysis of the models. Many of the functional forms of the demand rate would not be appropriate with the in-stock/stockout model. These functional forms, such as the power function investigated by Baker and Urban [6], will result in zero demand when the inventory level reaches zero; thus, we are never in a ‘stockout’ condition. If, in fact, the demand rate is completely dependent on the current stock level, we would likely expect this to be the case. This may be the case with applications such as used-car dealers; a customer goes to the dealership and is more likely to find a car to purchase if there are more to choose from. If there are no cars available, no sale is made. On the other hand, if the level of demand is only partially due to the stock level, other kinds of functional forms may be appropriate. One possibility is utilizing a model formulation like that of Datta and Pal [13] in which there is a stock-leveldependent demand rate until a given stock level, So, is realized, after which the demand rate is constant. This model can then be extended to allow shortages, so the in-stock/stockout model will reflect a different constant demand rate during the stockout period. In this situation, the demand rate would be of the form
D =
ai”,
i 2 So,
c&,
0 < i d So,
1 kc&,
Economics
40 (1995) 21-28
where t1 = (Sle8 - SO’-s)/,(l - /I) is the length of the in-stock period in which the inventory level is greater than So with the demand rate decreasing as the inventory level decreases, t2 = So/D = SA-Bfu is the length of the in-stock period in which the inventory level is less than So resulting in a constant demand rate, and t3 = - sfD = - s/kc& is the stockout period with the reduced demand rate. This inventory function is illustrated in Fig. 4. To analyze this model, we must evaluate four separate cases concerning the relative size of S, So, and s: Case 1: Case2: Case 3: Case 4:
S < So. S>s>S,. S > So 2 s > 0. S > So 2 s and s < 0.
The first three cases have previously been investigated. In case 1, the demand rate is always constant; the inventory level is never large enough to realize the stock-dependent demand. The analysis discussed in Section 2 of this paper is appropriate for this situation to allow for the different demand rate during the stockout period. In the second case, the demand rate never becomes constant, the inventory is replenished before reaching So. The original results of Baker and Urban [6], using separable programming, are appropriate here. In case 3, we are never in a stockout position since the reorder point is nonnegative. The analysis of Urban
(5)
i < 0.
It then follows that the inventory function will be: [&Y-ai=
or(1 - P)CJ”“~~‘,
so -as@ - Cl), - kaS;(t - tl - c2),
0 < f < t,,
t1
Fig. 4. Inventory curve for the in-stock/stockout stock-dependent demand.
model
with
T.L. Urban/M.
Cl43 is appropriate in which separable programming was again suggested as a solution methodology. Case 4 has not yet been studied. In this scenario, all possible demand rates can be identified. As the inventory level drops from S to So, the inventorylevel-dependent demand rate is observed. As i decreases from Se to 0, there is a constant demand rate, and when we are in a stockout condition, the decreased demand rate is realized. The objective function for this case is as follows (the development of this profit function is given in Appendix B): zI,(S,s) = [(l - /?)[2a(m - l)C(S - s) - (2S2-8 - /3S;-9c,/(2
- /?)
- s2Sia C2jk - 2&,]]/ [S’-fl - /3$-l
- (1 - /3)sS;a/k] _ (7)
Nonlinear programming can then be used as a solution methodology for this case as well, in the same manner as case 2 (since the variables will be in polynomial form, separable programming can be utilized). The optimal solution of the model can then be determined from the solution of each case: ZI*(S,
S) = IllaX(l7j(s,S)}. j
27
J. Production Economics 40 (1995) 2/-28
(8)
4. Conclusion The current inventory research literature contains two types of models in which the demand rate of an item varies due to its stock level. The first type, stock-level-dependent demand models, assumes that the demand rate is a function of the stock level, either the initial stock level or the instantaneous stock level. The other type, instock/stockout models, assumes that the demand rate is different during stockout periods than it is when stock is on hand. This paper reevaluates the in-stock/stockout model and formulates it as a profit-maximization model. This model is then generalized to incorporate the effect of stock-level-dependent demand rates. Models in which the demand rate is dependent on the initial stock level (the order-up-to level)
are analyzed as are models in which the demand rate is dependent on the instantaneous stock level.
Appendix A
In the basic, no-shortage inventory model with the demand rate a function of the initial stock level, Q, we have the same assumptions as the classical EOQ model (known and constant holding and ordering costs, etc.) with the exception of the stock-dependent demand. In this situation, the demand rate will be constant once the order quantity is established. Silver and Peterson [ 151 note that in the case of deterministic, constant demand rate and instantaneous replenishment, it is optimal to let the inventory level reach zero before reordering. Thus, the only decision variable will be the order quantity. The profit function will be as follows (note, a profit-maximization objective function is appropriate since the demand rate is determined by the order quantity, D =fo): LZ(Q)=(m-l)Cfp-fC1-QC,.
SQ
(A.11
Simply taking the first derivative of this function results in the following equation which can easily be solved for Q* (assuming the second-order conditions are met) for any given demand function:
(A.3 Most functional forms of demand will require a simple univariate search procedure to find the optimal solution. However, consider the case of a linear demand function, D = a + bQ. A closedform solution can be found for this particular functional form; it is Q* =
J
2aC3
cr - 2b(m - 1)C’
(A-3)
which is very similar to the traditional EOQ formula, the only difference being the ‘adjustment’ to the holding cost (the second order conditions indicate that the objective function is strictly concave).
28
T.L. Urban/fnt. J. Production Economics 40 (1995) 21-28
Note, however, that this solution is valid only as long as b < (C,/2(m - l)C); as b + (C,/2(m - l)C), then Q*- 00.
Appendix B
To identify the average profit function for case 4, we consider each of the revenue/cost components individually: Average
net projit:
E = (m - l)C(S T
- s) ’
where T = tI + t2 + t3 =
kS’ -B - pkS;-a
- (1 - @S,p
a(1 - BP Average 11
inventory
[Sl-p
level:
- a(1 - /3)t]“(l-@)dt
+ +S;tl
I T= 0 T
Integrating, substituting for tl and tl, and rearranging the terms provides S2-8 - @$-8 ‘=
cr(2-fi)T
’
Average shortage 6-
-p3
Average
_
level:
;a:;.
number of orders per unit time:
1 fi=-_. T The objective function can then be expressed as n(s
s) = (m - l)C(S - s)
7
T
S2-@ - +flSi-’ -
42 - LOT
C1
References
Cl1 Schary,
P.B. and Becker, B.W., 1972. Distribution and final demand: The influence of availability. Mississippi Valley J. Bus. Econom., 8: 17-26. PI Wolfe, H.B., 1968. A model for control of style merchandise. Industrial Mgmt. Rev., 9: 69-82. c31Gupta, R. and Vrat, P., 1986. Inventory model for stock dependent consumption rate. Opsearch, 23: 19-24. G. and Vrat, P., 1990. An EOQ model c41 Padmanabhan, for items with stock dependent consumption rate and exponential decay. Eng. Costs Prod. Econom., 18: 241-246. [51 Gerchak, Y. and Wang, Y., 1994. Periodic review inventory models with inventory-level-dependent demand. Naval Res. Logist., 41: 99-116. [6] Baker, R.C. and Urban, T.L., 1988. A deterministic inventory model with an inventory-level-dependent demand rate. J. Oper. Res. Sot., 39: 823-831. [7] Mandal, B.N. and Phaujdar, S., 1989. An inventory model for deteriorating items and stock-dependent consumption rate. J. qper. Res. Sot., 40: 483488. PI Pal, S., Goswami, A. and Chaudhuri, KS., 1993. A deterministic inventory model for deteriorating items with stock-dependent demand rate. Int. J. Prod. Econom., 32: 29 l-299. fixed c91Baker, R.C. and Urban, T.L., 1991. Deterministic order-level inventory models: An application for replenishment of radioactive source material for irradiation sterilizers. Eur. J. Oper. Res., 50: 249-256. Cl01Goh, M., 1992. Some results for inventory models having inventory level dependent demand rate. Int. J. Prod. Econom., 27: 155-160. Cl11Goh, M., 1994. EOQ models with general demand and holding cost functions. Eur. J. Oper. Res., 73: S&54. Cl21Mandal, B.N. and Ghosh, A.K., 1991. A note on an inventory model with different demand rates during stock-in and stock-out period. Int. J. Mgmt. Systems, 7: 33-36. Cl31Datta, T.K. and Pal, A.K., 1990. A note on an inventory model with inventory-level-dependent demand rate. J. Oper. Res. Sot., 41: 971-975. Cl41Urban, T.L., 1992. An inventory model with an inventorylevel-dependent demand rate and relaxed terminal conditions. J. Oper. Res. Sot., 43: 721-724. Cl51Silver, E.A. and Peterson, R., 1985. Decision Systems for Inventory Management and Production Planning, 2nd ed. Wiley, New York, p. 176.