The sensivity of inventory models with demand trend

Computers ind. Engng Vol. 18, No. 2, pp. 153-161, 1990

0360-8352/90 $3.00+0.00 Copyright © 1990 Pergamon Press plc

Printed in Great Britain. All rights reserved

THE

SENSITIVITY OF INVENTORY WITH DEMAND TREND

MODELS

CHRISTIAN N . MADU Department of Management Science, Lubin Graduate School of Business, Pace University, 1 Pace Plaza, New York, NY 10038, U.S.A.

(Received for publication 27 July 1989) Abstract--This paper conducts sensitivity analysis on inventory models with demand trends. Four demand trend models are considered. These are linear, quadratic, exponential decay, and exponential growth trend models. The sensitivity analysis tests the effect of errors in estimating holding and setup costs on the ordering policy and total cost. The result shows that if either of these costs is changed by a factor of 2 in either direction of the actual cost, the total cost and ordering policy become sensitive. However, the total cost for the exponential growth model is insensitive to changes in setup cost. More importantly, our results show that approximately the same magnitude of change on the cycle time is obtained for any change on the setup cost or the holding cost irrespective of the demand trend. Thus, the magnitude of change on cycle time is not a function of the demand trend. However, the magnitude of change in the total cost is a function of the demand trend.

INTRODUCTION

The Wilson lot size model, more popularly known as the economic order quantity (EOQ) model has been well studied in the inventory literature. Several questions have been raised about some of its assumptions such as the linear holding cost assumption and the assumption of constant demand [1-4]. The assumption on constant demand has recently received increased attention. In the EOQ model, the effect of demand trend is ignored in determining order quantities. Donaldson [5] investigated a particular case by assuming a linear trend in demand. He established an optimal condition for this case. Since his paper, numerous papers have appeared in the literature focusing on alternative solution techniques to the linear demand trend [6-11]. Some of these papers additionally compared their models to existing models in terms of cost savings. Although each of these models tend to show that Donaldson's model gives the minimum total cost over a set of examples, Donaldson's model is more involving computationally, thus making it unattractive in terms of application. One of the simpler alternatives to the linear demand trend pattern is the model by Mitra et al. [8]. However, this model gives higher cost when compared to Donaldson's case. This increased cost does not seem to be significant, thus making it attractive for practical purposes. The most commonly used demand trend lines are straight lines. This is mainly because they are simple and they frequently appear as logical trend assumptions [12]. The quadratic trend, which is an extension of the linear trend is also often mentioned in trend analysis. Polynomials of higher degree than the cubic are not commonly used for trend lines. In this discussion, our consideration is limited to the linear, quadratic, and exponential trend cases. By consideration of these different trends, the model to determine the optimum order quantity becomes more difficult. Our aim is to test the sensitivity of the optimum solutions reached with these demand patterns to small changes in the holding cost per unit and the setup cost. With the EOQ model, it is generally accepted that the model is not sensitive to small changes in the holding and setup costs. The effect of small changes in these two cost parameters on the optimal policy is of significance in managerial decision making. This will provide management with a guide to the flexibility or error tolerance in determining costs. NOTATIONS

r = holding cost per unit per order. k = setup cost per order. T = cycle time or the time between orders. 153

154

Q n D D(t) H

CHRISTIAN N. MADU

= = = = =

the order size. the number of times an order is placed. annual demand rate (constant). demand as a function of time t. the planning horizon. METHODOLOGY

The models presented in this paper are based on the procedures outlined by Mitra et al. [8]. Mitra et al. applied this procedure to a case where there is linear trend in demand. This EOQ extension by Mitra et al. is then applied to the quadratic and exponential demand trends. These models are presented in the Appendix. However, the focus here is on the sensitivity of the ordering policies derived to changes in the holding and setup costs. A computer program written in PC BASIC is used to carry out the sensitivity analysis. SENSITIVITY ANALYSIS It is widely accepted that the EOQ model is insensitive to small variations or errors in the estimates of holding and ordering costs [13]. This property indicates that with reasonable ordering and inventory holding cost estimates, a good approximation of the actual minimum cost order quantity can be obtained. Cost estimates are derived using some statistical techniques or inferred by management based on past experience [14]. The EOQ model was developed on the assumption of constant demand. Recently, extensions of the EOQ include the consideration of trend factor [5-11]. These recent models did not evaluate the sensitivity of their minimum total cost models to changes in the setup cost and holding cost. Since the order obtained while using the demand trend models is not constant, the sensitivity is carried out on the cycle time (time between orders), the number of orders placed during the planning horizon, and on the total cost. The specification of order quantity Q is therefore equivalent to the specification of T [15]. The model for measuring the change in total cost due to changes in holding and setup costs is presented below as equations (1) and (2). This same approach is followed in measuring the changes in the cycle time and the number of times to place an order due to changes in the holding and setup costs. Let ATC(r *) =

TC(r) - TC(r*) x 100% TC(r*)

(1)

ATC(k*) =

TC(k) - TC(k*) × 100% TC(k *)

(2)

and

where TC(*) is the total cost with the actual value of either r or k and TC(.) is the total cost with the estimate. Furthermore, define the factor or ratio as r/r* when k is fixed and k / k * when r is fixed. Zimmermann and Soverign [16] noted that the EOQ model is insensitive to errors in ordering and inventory holding costs if the errors are in the same direction. Therefore there is no need to have strict accuracy in estimating the parameters involved in the calculation of order quantity. This characteristic makes the EOQ model a widely used one. Hadley and Whitin [17] hypothesized that if the order quantity varies from either direction by a factor of 2, the total costs are increased only by 25%. Brown et al. [1] and Madu [4] respectively showed that this hypothesis depends on the shape of the holding cost. Brown [18] noted that if the lot size is within the range 70-140% of the true optimum, the total annual costs rise by less than 6% above the true minimum. Eilon [2, 3] has suggested optimal ranges be used to define Q. The present models with consideration of demand trend differ significantly from the studies cited above. With demand trend, order quantity is not constant as in the EOQ model but varies from

Sensitivity o f i n v e n t o r y m o d e l s

155

order to order. By consideration of trend, a model to predict the order quantity or lot size for each order is determined. Thus these models forecast the order quantity for each order period. Each time an order is placed, a different order size has to be computed to accomodate the trend effect. In this paper, the same factor used by Hadley and Whitin to determine the sensitivity of Q* to total cost is applied. The absolute changes in total cost and the ordering policies due to changes in holding and setup costs is determined.

Linear demand trend The analysis is carried out by keeping one variable constant and varying the other. The actual costs for r and k are fixed at $2 and $5 respectively. It is observed from Table 1, that if the holding cost changes in either direction by a factor of 2 that is 0.5 ~
Quadratic demand trend The results of the sensitivity analysis for the quadratic demand trend are presented in Tables 3 and 4. They are almost identical to those obtained in Tables 1 and 2 for the linear demand trend. In the interval 0.5 ~
Exponential demand trend Two exponential demand trend cases are considered. The first case is expressed as the exponential growth trend model while the second case is referred to as an exponential decay trend model. The exponential growth model can be illustrated by introducing a new product that gets wide market acceptance. The exponential decay model represents a declining product that is being phased out. Table 1. Sensitivity table (% changes)

Table 2. Sensitivity table (% changes)

Linear trend

Linear trend

k is fixed and r is changing, r=2, k =5, a = l , b = 6 , D = 3 4 , H = I I

r is fixed and k is changing,

r=2, k=5, a = l , b = 6 , D=34, H = l l

Ratio

Cycle time

No. of orders

Total cost

Ratio

Cycle time

No. of orders

Total cost

3 2.5 2 1.5 1.25 1.125 I 0.875 0.75 0.667 0.5 0.4 0.33

42.32 36.75 29.22 18.37 10.54 5.72 0.00 6.93 15.51 22.44 41.42 58.13 73.19

70.59 58.82 41.18 23.53 11.76 5.88 0.00 5.88 11.76 17.65 29.41 35.29 4 I. 18

72.41 59.31 41.51 23.37 11.82 6.00 0.00 6.22 12.51 18.09 29.25 36.21 41.82

3 2.5 2 1.5 1.25 1.125 1 0.875 0.75 0.667 0.5 0.4 0.33

73.19 58.13 41.42 22.44 11.90 6.17 0.00 6.48 13.40 18.37 29.22 36.75 42.32

41.18 35.29 29.41 17.65 11.76 5.88 0.00 5.88 17.65 23.53 41.18 58.82 70.59

74.26 59.48 41.50 22.85 11.67 6.00 0.00 6.69 11.90 17.71 29.25 36.28 42.53

156

CHRISTIAN N, MADU

Table 3. Sensitivity table (% changes)

Table 4. Sensitivity table (% changes)

Quadratic trend

Quadratic trend

k is fixed and r is changing,

• is fixed and k is changing, r = 2 , k = 5 , a = 5 , b = 2 . 5 , c = 3 , H = 11

r=2, k = 5 , a=5, b=2.5, c=3, H = l l Ratio

Cycle time

No. of orders

Total cost

Ratio

Cycle time

No. of orders

Total cost

3 2.5 2 1.5 1.25 1,125

42.30 36.78 29.20 18.39 10.57 5.75 0.00 6.90 15.40 22.53 41.38 58.16 72.87

69.23 53.85 38.46 19.23 11.54 3.85 0.00 7.69 15.38 19.23 30.77 38.46 42.31

70.82 56.50 39.77 21.62 11.67 5.21 0.00 7.19 13.78 18.86 29.46 36.53 42.31

3 2.5 2 1.5 1.25 1.125

73.33 58.16 41.38 22.53 I 1.72 5.98 0.00 6.44 13.33 18.39 29.20 36.78 42.30

42.31 38.46 30.77 19.23 11.54 7.69 0.00 7.69 15.38 19.23 38.46 53.85 69.23

72.72 58.66 4 I. 07 21.68 11.00 5.37 0.00 5.32 13.30 18.9! 30. I 1 37.40 43.06

I 0.875 0.75 0.667 0.5 0.4 0.33

1 0.875 0.75 0.667 0.5 0.4 0.33

The results of Table 5 showed that when 0.5 ~< r/r*<~ 2, the change in total cost is given by 49.88% ~< ATC(r*) ~<49.83%. In this interval, ATC(r*) is symmetrical. However, when r/r* > 2, the ATC(r*) is much greater than changes in total cost when r/r*< 0.5. It is also observed that a change in r* by a factor of 1.25 in either direction will lead to a 25% change in total cost. The exponential growth trend model is therefore very sensitive to changes in the holding cost. A similar test using the setup cost k, shows that the total cost is insensitive to changes in the setup cost. For the interval 0.33 <<.k/k* ~< 3 considered in this paper, total cost changed by less than 1%. However, the ordering policy is sensitive to changes in the setup cost as seen in the Table 6 results on the cycle time and the number of orders. For example, a change in setup cost by a factor of 2 in the upward direction will lead to a change in the cycle time of 41.38 and 30% for the number of orders. In the downward direction, this change by a factor of 2 will lead to a change in the cycle time of 29.44 and 40% for the number of orders. The inference drawn from these results is that the total cost is insensitive to changes in the setup cost when the demand trend follows an exponential growth but the ordering policy remains sensitive. Tables 7 and 8 present the results for exponential decay trend. From the results of Table 7, the total cost is very sensitive to changes in r. A change in r in the downward direction by a factor of 2 will lead to a 60% change in total cost. On the other hand, a change in r by a factor of 2 in the upward direction will lead to a 170% change in total cost. The results obtained for the exponential growth model using the factor of 2 for the upward and downward directions were 100 and 60% respectively. Errors in estimating r for the exponential decay trend has more profound financial consequences when the error is in the upward direction. Contrary to the results obtained for the exponential growth trend, the total cost of the exponential decay trend is sensitive to changes in the setup cost when k/k* ~<0.5 and when

Table 6. Sensitivity table (% changes)

Table 5. Sensitivity table (% changes) Exponential growth trend

Exponential growth trend

k is fixed and r is changing, A =0.11, 2 =0.01, k = 15, r = 6 , H = 11

r is fixed and k is changing, A = 100, 2 = 0.01, r = 6 , k = 15. H = 11

Ratio

Cycle time

No. of orders

Total cost

Ratio

Cycle time

No. of orders

Total cost

3 2.5 2 1.5 1.25 1.125

42.44 36.87 29.44 18.30 10.61 5.84 0.00 6.90 15.38 22.28 41.38 58.09 72.68

70.00 56.67 40.00 20.00 10.00 3.33 0.00 6.67 13.33 20.00 30.00 36.67 43,33

199.24 149.45 99.65 49.83 24.92 12.46

3 2.5 2 1.5 1.25 I. 125

72.94 58.09 41.38 22.28 11.67 6.10 0.00 6.63 13.53 18.30 29.44 36.87 42.44

43.33 36.67 30.00 20.00 10.00 6.67 0.00 6.67 13.33 20.00 40.00 56,67 70,00

0.43 0.34 0,24 0.13 0.07 0.03 0.00 0.04 0.08 0.11 0.18 0.22 0.25

I 0.875 0.75 0.667 0.5 0.4 0.33

0.00 12.47 24.93 33.21 49.88 59.86 66.36

I 0,875 0,75 0,667 0.5 0.4 0.33

Sensitivity o f i n v e n t o r y m o d e l s

157

Table 8. Sensitivity table (% changes)

Table 7. Sensitivity table (% changes) Exponential decay trend

Exponential decay trend

k is fixed and r is changing, A = 100, 2 = 0.01, r = 6 , k = 15, H = 11

r is fixed and k is changing, A = 100, 2 = 0.01, r = 6 , k = 15, H = 11

Ratio

Cycle time

No. of orders

Total cost

Ratio

Cycle time

No. of orders

Total cost

3 2.5 2 1.5 1.25 I. 125

42.21 36.68 29.40 18.34 10.55 5.78 0.00 6.78 15.58 22.36 41.46 58.04 72.86

71.43 57.14 42.86 21.43 10.71 7.14 0.00 7.14 14.29 17.86 28.57 35.71 42.86

362.59 259.50 170.38 73.49 33.95 20.27 0.00 18.16 34.29 42.27 60.76 70.66 78.23

3 2.5 2 1.5 1.25 1.125

73.12 58.04 41.46 22.36 11.81 6.03 0.00 6.53 13.32 18.34 29.40 36.68 42.21

42.86 35.71 28.57 17.86 10.71 3.57 0.00 7.14 14.29 21.43 42.86 57.14 71.43

34.79 26.65 21.52 13.41 8.63 0.08 0.00 5.90 9.87 15.69 35.19 43.80 54.20

I

0.875 0.75 0.667 0.5 0.4 0.33

1

0.875 0.75 0.667 0.5 0.4 0.33

k/k* > 2 (Table 8). However, the magnitude of this effect is very low in comparison to the effect of change on the total cost due to changes in the holding cost. Figure 1 shows the percentage changes in T due to changes in r and k over the four different demand trends applied in this paper. For a fixed r(k), changes in k(r) have the same effect on the cycle time irrespective of the demand trend. This result further demonstrates that the EOQ does not react readily to major parameter changes. However, it is noted that when r/r* < 1 or k/k* < 1, changes in k tend to have more effect on T than changes in r. The opposite of this result is obtained when r/r* or k/k* > 1. As these ratios approach 1, the effects of r and k on T converge on a single curve. Thus r and k have equal magnitude of effects on T as k/k* or r/r* approaches 1. The results obtained for percent changes in the number of orders due to changes in r and k are directly opposite to that obtained for T. This result is expected due to the inverse relationship between the number of orders and the cycle time. This result is shown in Fig. 2. Figure 3 shows the sensitivity of the total cost to changes in the setup cost. It is noted that with the exponential growth model, changes in total cost are insignificant when the ordering cost is continually changing. However, the total cost becomes sensitive to the changes in the holding cost. In Fig. 4, it is shown that changes in the holding cost have a more pronounced effect on the exponential demand trend models. These changes have equal effects on both the linear and quadratic demand models. 80

70

6o 50 14O c 0

" 0

30

20

10

m

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0.40

0.50

0.667

0.t'5

o.e75

1

r/rN(k/k

1.t25

t.25

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Fig. 1. C h a n g e s in T d u e to c h a n g e s in r a n d k ( % ) . CAIE 18/2--D

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2.5

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CHRISTIAN N. MADU 60 7O

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2

2.5

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Fig. 2. C h a n g e s in the n u m b e r of orders due to changes in r and k (%). A MICROCOMPUTER

APPROACH

A program is written in BASIC language and implemented on an IBC PC. The use of the computer program facilitates the computation of the ordering policies. Computer programming was also necessary in order to carry out the sensitivity analysis. With the EOQ model, a constant value for the optimum Q is easily obtained. However, with demand trend models, the optimum order size becomes a function of the demand trend. Each time an order is placed, a different order size may have to be supplied throughout the planning horizon. The program is interactive. The user supplies information such as the demand trend, the setup and holding costs, the constant demand rate and the length of the planning horizon. When the demand trend is specified, the program requests for information that are pertinent to that particular demand pattern. For example, with the linear demand trend, the constant coefficients such as a and b, may be supplied. Determination of these coefficients is based on other appropriate techniques such as the least-square method commonly used to obtain the best-fit line. These constants are pre-determined and are not computed by the computer program. 80 --

70 -60

8

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,

o Linear/quad

50 ~

+

exp. growth

O

._c 4O @ o~ c 0

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0,40

I 0.50

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I

0.667

0.75

0.875

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'1.125

I. 2 5

1.5

k/k ~ Fig. 3. Changes in total cost due to changes in k (%).

2

2,5

3

Sensitivity of inventory models

400

159

-

550 300 250 -

/

[3 I.ineor/quad +

exp.growth

0

exp. decoy

/

/ /

c 150 + o

g

100 50 0

0.33 0.40 0.50 0.667 0.75 0.875

1 r/r

1.125 1.25 1.5

2

2.5

3

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Fig. 4. Changesin total cost due to changesin r (%).

Thus, the essence of the model presented in this paper is to solve inventory problems when a demand trend exists. In forecasting literature, it is known and accepted that demand patterns exist. Demand patterns such as linear demand trend have been applied in numerous examples using regression analysis. Yet, the EOQ model continues to be widely generalized for all cases. The sensitivity analysis carried out here show some unique characteristics of the demand trends studied. Thus, there is a need to understand an item's demand pattern before any of these inventory models can be applied. As a real world example, the exponential demand trend is applied to any product that faces either wide market acceptance and rapid demand growth, or rejection in the market place with a drastic drop in demand. These erratic conditions can not be adequately modeled by making the assumption of constant demand as in the EOQ. The use of the demand trend model further makes inventory models more difficult to solve. This is due to the fact that there does not exist one single order size. This requires that a different order size be computed each time an order is placed. The use of microcomputers in this respect, considerably reduces the computational time required. Once the program is developed, only the input information has to be supplied by the user. The program generates output information such as the optimal lot sizes, the cycle times, the number of times to place an order, and the minimum total cost. The program is easily written for the models presented in this paper. Since the models do not require large numbers of computations, the program runs efficiently on a microcomputer. CONCLUSION

This paper conducted sensitivity analysis on the models to determine order quantities when linear and nonlinear demand trends exist. The effect of changes in r and in k on the total cost was observed to be approximately identical for the linear and quadratic trend models, although, their ordering policies differ. It was further shown that the total cost for the exponential growth trend is insensitive to changes in k. The change in total cost for the exponential growth model when k is changing was found to be less than 1%. The exponential decay model tends to be more sensitive to changes in r or k than the exponential growth model. Another important result is that the models presented, with the exception of the exponential decay trend, tend to be most sensitive to changes in r and k in the upward direction. Furthermore, this paper shows that, unlike the EOQ model, when there is a demand trend, the total cost may be sensitive to the errors in the estimates for the setup and holding costs depending on the demand trend type.

160

CHRISTIAN N. MADU

It is further suggested that management determine the level of error tolerance to fully get the benefit of this analysis. Although the magnitude of change in the cycle time does not appear to be dependent on the demand trend, the change in total cost is a function of demand trend. Acknowledgement--This study was funded by a 1987 Summer research grant from Pace University, New York.

REFERENCES 1. R. M. Brown, T. E. Conine, Jr and M. Tamarkin. A note on holding costs and lot-size errors. Decis. Sci. 17, 603--608 (1986). 2. S. Eilon. Dragons in pursuit of the EBQ. Ops Res. Q. 15, 347-354 (1964). 3. S. Eilon. A note on the optimal range. Mgmt Sci. 7, 56-61 (1969). 4. C. N. Madu. The sensitivity of (Q, S) inventory model. Omega 16, 61--64 (1988). 5. W. A. Donaldson. Inventory replenishment policy for a linear trend in demand and analytical solution. Ops Res. Q. 28, 663-670 (1977). 6. S. K. Goyal, M. Kusy and R. Soni. A note on the economic replenishment interval for an item with a linear trend in demand. Engng Costs Prod. Econ. 10, 253-255 (1986). 7. R. J. Henry. Inventory replenishment policy for increasing demands. J. Ops Res. Soc. 30, 611-617 (1979). 8. A. Mitra, J. F. Cox and R. R. Jesse. A note on determining order quantities with a linear trend in demand. J. Ops Res. Soc. 35, 141-144 (1984). 9. R. I. Phelps. Optimal inventory rule for a linear trend in demand with a constant replenishment period. J. Ops Res. Soc. 31, 439-442 (1980). 10. E. Ritchie. Practical inventory replenishment policies for a linear trend in demand followed by a period of steady demand. J. Ops Res. Soc. 31, 605-613 (1980). 11. E. A. Silver. A simple inventory replenishment decision rule for a linear trend in demand. J. Ops Res. Soc. 30, 71--75 (1979). 12. N. N. Barish and S. Kaplan. Economic Analysis for Engineering and Managerial Decision Making, 2nd edn, p. 579. McGraw-Hill, New York (1978). 13. D. R. Anderson, D. J. Sweeney and T. A. Williams. Quantitative Methods for Business, 3rd edn, p. 597. West, St, Paul, Minn. (1986). 14. A. C. Hax and D. Candca. Production and Inventory Management. p. 135. Prentice-Hall, N.J. (1974). 15. L. A. Johnson and D. C. Montgomery. Operations Research in Production, Planning, Scheduling and Inventory Control. Wiley, New York (1974). 16. H. J. Zimmermann and M. G. Soverign. Quantitative Models for Production Management, p. 352. Prentice-Hall, N.J. (1974). 17. G. Hadley and T. M. Whitin. Analysis of Inventory Systems. Prentice-Hall, N.J. (1963). 18. R. G. Brown. Materials Management Systems. Wiley, New York (1977).

APPENDIX

The theory presented in this section for determining order quantity was initially applied by Mitra et al. [8] to linear demand trend. The linear trend is a first-degree polynomial of the general form Y = a + bt + ct2 + dt3 + . . . (A1) The second degree polynomial Y = a +bt +ct 2

(A2)

obtained from this general form is commonly called a quadratic. The quadratic trend model is a simple extension of the linear trend model and involves only the addition of a term in t 2. The constants a, b, and c are obtained in the same fashion as the linear trend model, by solving the normal equations resulting from the least square method. Suppose that Y = D(t) where D(t) is the demand rate at time t, a is the demand rate at time 0, and b and c are the trend values. If H is used to denote the planning horizon, then D *H is equal to the total demand where D* is an equivalent constant demand rate. The total demand over the planning horizon is given as, M =

j.H

D(t) dt = a H + (b/2)H 2 + (c/6)H 3. o An order interval T* is then calculated from the EOQ model with the constant demand rate as T* = [2k /rD*] v2

where k is the setup cost or cost per order in $ and r is the inventory carrying cost, in $ per unit of time. Let n = the number of times an order is placed, where n = [M/(D*T*)] +

(A3)

(A4)

(A5)

where this represents the smallest integer greater than or equal to the inner term. There is a constant time T* between orders except for the last order where the time interval between the last order and H may not equal T*.

Sensitivity of inventory models

161

Let ~tj=jT* and /~j = ( j - I)T* and ), = (n - 1)T*. The ordering policy is to order Qj units at time tj when the j t h order is placed. tj=f~

and

j=l

(A6)

. . . . . n.

Thus QJ = IaJ' D ( t ) dt = a(uj - fj) + 0.5b (u2 _ fl2) + (c/6) (~,~ - fl})

(A7)

and n-I j~l

The total cost is defined as TC(Q) = nk + ~ Rj

(A8)

j=l

where nk is the total ordering cost and the second term in equation (AS) is the total inventory cost. The inventory cost for each j t h order is computed as Rj=r

j=l

[t-(j-l)r*lD(t)dt

..... n-1

,J/b and R. = r

[t - (n - 1) T*] D ( t ) dt.

(A9)

7

Equation (A9) gives that Rj = r I0.5a(a 2 - fl2) + 0.333b(~t} _ fly) 3 + 0.125c(~tj4 - flj) , - flj{a(~tj - flj) + 0.5b(at~ - flY) + 0.1667c(~t~ - fl3)}]

(AI0)

R n = r [0.5a(H 2 - V2) + 0.333b(H 3 _ ),3) + 0.125c(H 4 - ),4) _ ), { a ( H - ~) + 0.5b(H 2 - ),2) + 0.1667(H 3 _ ),3)}]

(AI 1)

The major results for the linear trend and exponential trend (decay and growth) models are presented below without repeating the derivations done for the quadratic d e m a n d trend. The four models presented in this section were used to conduct the sensitivity analysis. Linear d e m a n d trend [8] M = a H + 0.5bH z

(AI2)

Qj = a T * + b [j - 0.5] (T*) 2

(AI3)

Ry = 0.5r[a(T*) 2 + b { j - (I/3)} (T*)3] j = 1. . . . . n - 1

(A14)

R, = r [(a / 2 ) H 2 + (b / 3 ) H 3 - ( a l l + (b / 2 ) H 2) (n - I)T* + (a/2) (n - 1)2(T*) 2 + (b/6) (n - I)3(T*) 3]

(A15)

Exponential demand trend

(1) Exponential decay M = A / 2 [1 - - e -~']

(AI6)

Qj = h / 2 [e -~-tj- Or' - e -~jr']

(AI7)

Rj = { ( - 1/2) e -~/r' [IT* + 1/2] + (1/2) e -~U-~)r" x [ ( / - (1/2)

(j -

1) e -~'r" + ( 1 / 2 ) ( j -

I)T* + 1/2]

1) e -~'(/- l)r.} rA

(AI8)

R n = rA { ( - l / 2 ) e - ~ n [ H + 1/2] + 1/2 e -~'(n-°r" [(n - I)T* + 1/2] + 1/2 (n - 1)T*e -~'n -

1/2 (n - I)T* e -:-~n-I)r'}

(A19)

(2) Exponential growth M = A/2 [e';'n- 1]

(A20)

Q j = A e ;Jr' [1 - e -;'r']

(A2I)

Rj = ( r A / 2 ) { T*e ~,jr" - ( j - 1) T* [e~Jr* - e ~-(j- l)r.]}

(A22)

R, = ( r A / 2 ) { H e :-n - 2(n - I)T* e :-n + (n - I)T* e ;-~- °r'}

(A23)

It should be noted that only M, Qj, Ry and R~ change for each d e m a n d trend considered. The other general equations presented for the quadratic trend can be applied in all the other trend models.