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1The William Paterson University of New Jersey, Wayne, NJ, USA; and 2Hung Kuang ... *Correspondence: J-T Teng, Cotsakos College of Business, The William.
Journal of the Operational Research Society (2004) 55, 495–503

r 2004 Operational Research Society Ltd. All rights reserved. 0160-5682/04 $25.00 www.palgrave-journals.com/jors

Deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time J-T Teng1* and H-L Yang2 1 The William Paterson University of New Jersey, Wayne, NJ, USA; and 2Hung Kuang University, Taichung, Taiwan ROC

In today’s time-based competition, the unit cost of a high-tech product declines significantly over its short life cycle while its demand increases. In this paper, we extend the classical economic order quantity model to allow for not only time-varying demand but also fluctuating unit cost. In addition, we also allow for shortages and partial backlogging. We then prove that the optimal replenishment schedule not only exists but also is unique. In addition, we also show that the total cost is a convex function of the number of replenishments, which simplifies the search for the optimal number of replenishments to find a local minimum. Moreover, we further simplify the search process by providing an intuitively good starting search point. Journal of the Operational Research Society (2004) 55, 495–503. doi:10.1057/palgrave.jors.2601678 Keywords: inventory; modeling; fluctuating cost; partial backlogging; deteriorating items

Introduction The classical economic order quantity (EOQ) model assumes not only a constant demand rate but also a fixed unit cost (ie, either purchase or production cost). However, as we know, the demand rate remains stable only in the maturity stage of a product life cycle. Moreover, in time-based competition today, the unit cost of a high-tech product declines significantly over its short product life cycle while its demand increases. For example, the cost of a personal computer drops constantly as shown in Lee et al.1 Therefore, using the EOQ formulation in stages other than the maturity stage or for a high-tech product with short life cycle will cause varying magnitudes of error. In addition, the cost of purchases as a percentage of sales is often substantial (52% for all industry) as shown in Heizer and Render.2 Consequently, adding the purchasing strategy into EOQ model is vital. In the growth stage of a product life cycle, the demand rate can be well approximated by a linear form. Consequently, Resh et al,3 and Donaldson4 established an algorithm to determine the optimal replenishment number and timing for a linearly increasing demand pattern. Barbosa and Friedman5 then generalized the solutions for power-form demand models. Furthermore, Henery6 extended the demand to any log-concave demand function. *Correspondence: J-T Teng, Cotsakos College of Business, The William Paterson University of New Jersey, 1600 Valley Road, Wayne, NJ 07470, USA. E-mail: [email protected]

Following the approach of Donaldson, Dave7 developed an exact replenishment policy for an inventory model with shortages. In contrast to the traditional replenishment policy that does not start with shortages, Goyal et al8 proposed an alternative that starts with shortages in every cycle, and suggested that their alternative outperforms the traditional approach. Lately, Teng et al9 investigated various inventory replenishment models with shortages, and mathematically proved that the alternative by Goyal et al is, indeed, less expensive to operate than the traditional policy. Hariga and Goyal10 then developed an iterative procedure that is simpler than that by Goyal et al. Later, Teng11 proposed a simple and computationally efficient optimal method in recursive fashion to solve the problem. In many real-life situations, products deteriorate continuously such as medicine, volatile liquids, blood banks, and others. Consequently, Dave and Patel12 discussed an inventory model for deteriorating items when shortages were not allowed. Sachan13 then extended their model to allow for shortages. Hariga14 developed optimal EOQ models with log-concave demand for deteriorating items under three replenishment policies. Recently, Yang et al15 provided various inventory models with time-varying demand patterns under inflation. Goyal and Giri16 reviewed the contributions on the literature in modelling of deteriorating inventory. The characteristic of all of the above articles is that the unsatisfied demand (due to shortages) is completely backlogged. However, for fashionable commodities and hightech products with short product life cycle, the willingness

496 Journal of the Operational Research Society Vol. 55, No. 5

for a customer to wait for backlogging during a shortage period is diminishing with the length of the waiting time. Hence, the longer the waiting time is, the smaller the backlogging rate would be. To reflect this phenomenon, Chang and Dye17 developed an inventory model in which the proportion of customers who would like to accept backlogging is the reciprocal of a linear function of the waiting time. Concurrently, Papachristos and Skouri18 established a partially backlogged inventory model in which the backlogging rate decreases exponentially as the waiting time increases. Recently, Teng et al19,20 extend the fraction of unsatisfied demand backordered to any decreasing function of the waiting time up to the next replenishment. In contrast to the traditional EOQ model, we assume here that not only the demand function but also the unit cost is positive and fluctuating with time (which is more general than increasing, decreasing, and log-concave functions). In addition, for generality, we assume that the backlogging rate of unsatisfied demand is a decreasing function of the waiting time. As a result, our proposed model is not only suitable for any given time horizon during a product life cycle, but also in a general framework that includes numerous previous models such as references mentioned as special cases. We then prove without loss of generality that the optimal replenishment schedule uniquely exists, and propose a onedimensional iterative method to find the optimal schedule. In addition, we show that the total relevant cost (ie, the sum of the holding, backlogging, lost sales, and purchase costs) in the system is a convex function of the number of replenishments. Consequently, the search for the optimal number of replenishments is reduced to find a local minimum. We further simplify the search process by establishing an intuitively good starting value for the optimal number of replenishments. Moreover, we use a couple of numerical examples to illustrate the algorithm. Finally, we make a summary and provide some suggestions for future research.

b(0) ¼ 1. To guarantee the existence of an optimal solution, we assume that b(t) þ tb 0 (t)X0, where b 0 (t) is the first derivative of b(t). Note that if b(t) ¼ 1 (or 0) for all t, then shortages are completely backlogged (or lost). 6. In the lost-sales case, the opportunity cost is the sum of the revenue loss and the cost of lost goodwill. Hence, the opportunity cost here is greater than the unit purchase cost. For details, see Teng et al.19 For convenience, the following notation is used throughout this paper: H= f(t)=

y= cf= cv(t)=

ch= cb= cl=

n= ti= Si=

C 0 (t)=

S(s, t)=

Assumptions and notation The mathematical model of the inventory replenishment problem is based on the following assumptions: 1. The planning horizon of the inventory problem here is finite and is taken as H time units. The initial and the final inventory levels are both zero during the time horizon H. 2. Replenishment is instantaneous. 3. Lead time is zero. 4. A constant fraction of the on-hand inventory deteriorates per unit of time and there is no repair or replacement of the deteriorated items. 5. Shortages are allowed. Unsatisfied demand is backlogged, and the fraction of shortages backordered is a decreasing function of time t, denoted by b(t), where t is the waiting time up to the next replenishment, and 0pb(t)p1 with

=

the time horizon under consideration. demand rate at time t, we assume that f(t) is greater than zero in (0, H), and continuous in the planning horizon [0, H]. the deterioration rate. the fixed purchase cost (or ordering cost) per order. the variable purchase cost per unit at time t, we assume that cv(t) is greater than zero, and differentiable in (0, H). the inventory holding cost per unit per unit time. the backlogging cost per unit per unit time, if the shortage is backlogged. the unit opportunity cost of lost sales, if the shortage is lost; we assume WLOG that cv(t)ocl for all t. the number of replenishments over [0, H] (a decision variable). the ith replenishment time (a decision variable), i ¼ 1,2,y,n. the time at which the inventory level reaches zero in the ith replenishment cycle (a decision variable), i ¼ 1,2,y,n. ch þ y cv(t) ¼ the marginal total inventory carrying cost (ie, the marginal sum of inventory and deterioration costs) per unit at time t. cb(ts)b(ts) þ [clcv(t)][1b(ts)], where s is the beginning of the shortage and t (with spt) is the replenishment time; the expected shortage cost (ie, the expected sum of backlogging and lost-sale costs) per unit at time t.

It is obvious that the longer the waiting time, the higher the expected shortage cost. As a result, we may assume WLOG that S(s, t) is a non-decreasing function of t (ie, the marginal expected shortage cost with respect to t is St(s, t)X0).

Mathematical model The ith replenishment is made at time ti. The quantity received at ti is used partly to meet the accumulated backorders in the previous cycle from time sil to ti

J-T Teng and H-L Yang—Deterministic economic order quantity models 497

Figure 1

Graphical representation of inventory model.

(si1oti). The inventory at ti gradually reduces to zero at si (si4ti). Consequently, based on whether the inventory is permitted to start and/or end with shortages, we have four possible cases, which were introduced in Teng et al.9,21 For simplicity, we use the inventory model depicted graphically in Figure 1. The objective of the inventory problem here is to determine the number of replenishments n, and the timing of the reorder points {ti} and the shortage points {si} in order to minimize the total relevant cost. Next, we formulate the level of inventory at time t as I(t), tiptpsi. Since the inventory is depleted by the combined effect of demand and deterioration, the inventory level at time t during the ith replenishment cycle is governed by the following differential equation: dIðtÞ ¼ f ðtÞ  yIðtÞ; ti ptpsi dt

ð1Þ

with the boundary condition I(si) ¼ 0. Solving the differential Equation (1), we have

and the cumulative number of lost sales at time t during [si1, ti) is LðtÞ ¼

Zt

½1  bðti  uÞf ðuÞdu; si1 ptpti ;

si1

ð5Þ

i ¼ 1; 2; . . . ; n Consequently, the time-weighted backorders due to shortages during the ith cycle is Bi ¼

Zti

BðtÞdt ¼

si1

Zti

ðti  tÞbðti  tÞf ðtÞdt

ð6Þ

si1

and the total number of lost sales due to shortages during the ith cycle is Li ¼ Lðti Þ ¼

Zti

½1  bðti  tÞ f ðtÞdt

ð7Þ

si1

IðtÞ ¼ eyt

Zsi

eyu f ðuÞdu ¼

t

Zsi e

yðutÞ

f ðuÞdu; ti ptpsi

ð2Þ

t

As a result, we obtain the time-weighted inventory during the ith cycle as Ii ¼

Zsi

IðtÞdt ¼

ti

¼

1 y

Zsi ti

Zsi

½eyt

Zsi

ð3Þ

½eyðtti Þ  1f ðtÞdt; i ¼ 1; 2; . . . ; n

ti

Similarly, the cumulative number of backorders at time t during [si1, ti) is BðtÞ ¼

Zt si1

Qi ¼ Bðti Þ þ Iðti Þ Zsi Zti bðti  tÞf ðtÞdt þ eyðtti Þ f ðtÞdt ¼ si1

eyu f ðuÞdudt

t

From (2) and (4), we have the order quantity at ti in the ith replenishment cycle as

ð8Þ

ti

Therefore, the purchase cost during the ith replenishment cycle is Pi ¼ cf þ cv ðti ÞQi 2t 3 Zi Zsi yðtt Þ i ¼ cf þ cv ðti Þ4 bðti  tÞf ðtÞdt þ e f ðtÞdt5 si1

ti

ð9Þ bðti  uÞf ðuÞdu; si1 ptpti ; i ¼ 1; 2; . . . ; n

ð4Þ

Hence, if n replenishment orders are placed in [0, H], then the total relevant cost of the inventory system during the

498 Journal of the Operational Research Society Vol. 55, No. 5

than buying it later. Likewise, if c0v ðtÞp  Sti ðt; ti Þo0, then the declining rate of unit cost cv 0 (t) is larger than or equal to the marginal expected shortage cost per unit Sti ðt; ti Þ. Consequently, delaying the purchase until the end of the planning horizon H is cheaper than buying it earlier. Note that neither of these two cases happens in the real life.

planning horizon H for the model is as follows: TCðn; fsi g; fti gÞ ¼

n X

ðPi þ ch Ii þ cb Bi þ c1 Li Þ

i¼1 n Z X

ti

¼ ncf þ

i¼1

f½cb ðti  tÞ þ cv ðti Þ  cl bðti  tÞ

si1

þ cl gf ðtÞdt þ

n X i¼1

Zsi

nhc

h

y

i o þ cv ðti Þ ðeyðtti Þ  1Þ þ cv ðti Þ f ðtÞdt

ti

ð10Þ with 0 ¼ s0ot1 and sn ¼ H. The problem is to determine n, {si} and {ti} such that TC(n, {si}, {ti}) in (10) is minimized.

Theoretical results and solution For a fixed value of n, the necessary conditions for TC(n, {si}, {ti}) to be minimized are: qTC(n, {si}, {ti})/qsi ¼ 0 and qTC(n, {si}, {ti})/qti ¼ 0, for i ¼ 1,2,y,n. Consequently, we obtain ½cb ðti þ 1  si Þ þ cv ðti þ 1 Þ  cl bðti þ 1  si Þ þ cl  cv ðti Þ hc i h ¼ þ cv ðti Þ ðeyðsi ti Þ  1Þ y ð11Þ and ½ch þ ycv ðti Þ 

c0v ðti Þ

Zsi

eyðtti Þ f ðtÞdt

ti

¼

Zti

Lemma 2 If C 0 ðti Þ4c0v ðti Þ4  Sti ðt; ti Þ for all ti, and t*1 is unique, then the solution to Equations (11) and (12) is also unique.

f½cb þ c0v ðti Þbðti  tÞ

ð12Þ

si1

þ ½cb ðti  tÞ þ cv ðti Þ  cl b 0 ðti  tÞgf ðtÞdt respectively. Equations (11)–(12) now imply the following results. Lemma 1 (a) If C 0 (ti)pcv 0 (ti) for all ti, then the optimal solution is n* ¼ 1 and t*1 ¼ 0. (b) If c0v ðti ÞpSti ðt; ti Þ for all ti, then the optimal solution is * n ¼ 1 and t*1 ¼ H. Proof See Appendix A. The results in (a) and (b) of Lemma 1 can be interpreted as follows. For (a), the condition 0oC 0 (t)pcv 0 (t) means that the increasing rate of the unit purchase cost cv 0 (t) is higher than or equal to the marginal inventory carrying cost per unit C 0 (t) (which includes inventory and deterioration costs). Therefore, buying and storing a unit now is less expensive

Proof See Appendix B. Theorem 1 If C 0 ðti Þ4c0v ðti Þ4  Sti ðt; ti Þ for all ti, then we have (a) t*1 is unique and the solution to Equations (11) and (12) is also unique (ie, the optimal values of {s*i } and {t*i } are uniquely determined). (b) Equations (11) and (12) are the necessary and sufficient conditions for finding the absolute minimum TC(n, {si}, {ti}). Proof See Appendix C. The result in (a) of Theorem 1 reduces the 2n-dimensional problem of finding {s*i } and {t*i } to a one-dimensional problem. Since s0 ¼ 0, we only need to find t*1 to generate s*1 by (12), t*2 by (11), and then the rest of {s*i } and {t*i } uniquely by repeatedly using (12) and (11). For any chosen t*1, if s*n ¼ H, then t*1 is chosen correctly. Otherwise, we can easily find the optimal t*1 by standard search techniques. For any given value of n, the solution procedure for finding {s*i } and {t*i } can be obtained by the algorithm in Yang et al15 with L ¼ 0 and U ¼ H/n. Next, we show that the total relevant cost TC(n, {s*i }, {t*i }) is a convex function of the number of replenishments. As a result, the search for the optimal replenishment number, n*, is reduced to find a local minimum. For simplicity, let TCðnÞ ¼ TCðn; fs i g; ft i gÞ:

ð13Þ

By applying Bellman’s principle of optimality,22 we have the following theorem: Theorem 2 If C 0 (t)4cv 0 (t)4St(s, t) for all t, then TC(n) is convex in n. Proof Use the same argument as in the proof of Friedman,23 or Theorem 2 of Teng et al.21 For details, see Appendix D.

Estimation of the replenishments number Now, we will establish an estimate of the optimal number of replenishments, n*. For computational simplicity, we assume

J-T Teng and H-L Yang—Deterministic economic order quantity models 499

here that the length of ti þ 1si approximately is 1. By using the average unit cost cv to replace cv ðti Þ, we obtain the expected shortage cost as Sðsi ; ti þ 1 Þ cb bð1Þ þ ðcl  cv Þ½1  bð1Þ

ð14Þ

and the marginal total inventory carrying cost as cv C 0 ðtÞ ¼ ch þ ycv ðtÞ ch þ y

ð15Þ

By using a similar estimate discussed in Teng,11 we have the estimate of the number of replenishments as n1 ¼rounded integer of ðch þ ycv Þfcb bð1Þ þ ðcl  cv Þ½1  bð1ÞgHQðHÞ 1=2 2cf fch þ ycv þ cb bð1Þ þ ðcl  cv Þ½1  bð1Þg ð16Þ RH

where QðHÞ ¼ 0 f ðtÞdt. It is obvious that searching for n* by starting with n in (16) will reduce the computational complexity significantly, comparing to starting with n ¼ 1 (such as in Chang and Dye,17 Dave7 or Papachristos and Skouri18). The algorithm for determining the optimal number of replenishments n* is similar to the algorithm in Yang et al15 with two initial trial values of n*, say n as in (16) and n1.

Numerical examples Example 1 Let f(t) ¼ 10e0.98t, H ¼ 4, cf ¼ 250, ch ¼ 40, cb ¼ 50, cl ¼ 500, cv(t) ¼ 200 þ 20e2t, y ¼ 0.08, b(t) ¼ e0.2t in appropriate units. After integration and calculation, we have cv ¼ 202.499, and b(1)E0.82. By (16), we obtain the estimate number of replenishments n1 ¼ 12. From computational results, we get TC(10) ¼ 106428.83, TC(11) ¼ 106413.58, and TC(12) ¼ 106442.40. Therefore, the optimal

i ti si cv(ti)

i ti si cv(ti)

number of replenishments is 11, and the optimal replenishment schedule is shown in Table 1. Example 2 Let f(t) ¼ 40 þ 3t, H ¼ 5, cf ¼ 250, ch ¼ 80, cb ¼ 120, cl ¼ 500, cv(t) ¼ 200 þ 20e2t, y ¼ 0.08, b(t) ¼ e0.02t in appropriate units. Since cv ¼ 202, and b(1)E0.98, we get from (16) that n1 ¼ 11. Judging from the computational results that TC(10) ¼ 53647.52, TC(11) ¼ 53604.13, and TC(12) ¼ 53609.59, we know that the optimal replenishment number is 11, and the optimal replenishment schedule is shown in Table 2.

Conclusions In this paper, we consider the inventory lot-size model for deteriorating items with deterministic varying demand and unit variable purchasing cost. The model allows for shortages and not completely backlogged. We show that the optimal replenishment schedule exists uniquely. In addition, we prove that the total cost associated with the inventory system is a convex function of the number of replenishments. Hence, the search for the optimal number of replenishments is simplified for finding a local minimum. Moreover, we characterize the influences of the demand patterns over the replenishment cycles and others. The model developed here can be further improved by employing more elaborate inventory models. A fruitful area of future research is to examine how other inventory model, such as price discount and dynamic lot sizing models, can be properly integrated into this inventory lot-size model.

Acknowledgements—The authors would like to thank Professor MawSheng Chern at National Tsing-Hua University in Taiwan for his constructive comments. This research was partially supported by the National Science Council of the Republic of China under Grant NSC92-2213-E-007-065, and a two-month research grant in 2003 by the Graduate Institute of Management Sciences in Tamkang University in Taiwan. In addition, the principal author’s research was supported by

Table 1

The optimal replenishment schedule for Example 1

1

2

3

4

5

6

7

8

9

10

11

0.5233 0.9309 207.023

1.2028 1.5416 201.804

1.7331 2.0103 200.625

2.1596 2.3917 200.266

2.5144 2.7134 200.131

2.8177 2.9917 200.071

3.0823 3.2367 200.042

3.3169 3.4557 200.026

3.5276 3.6536 200.017

3.7187 3.8341 200.012

3.8936 4.0000 200.008

Table 2

The optimal replenishment schedule for Example 2

1

2

3

4

5

6

7

8

9

10

11

0.2751 0.5027 211.537

0.7357 0.9854 204.592

1.2013 1.4580 201.810

1.6654 1.9227 200.715

2.1251 2.3803 200.285

2.5790 2.8314 200.115

3.0270 3.2763 200.047

3.4691 3.7153 200.019

3.9055 4.1487 200.008

4.3366 4.5769 200.003

4.7624 5.0000 200.002

500 Journal of the Operational Research Society Vol. 55, No. 5 the ART for Research and a Summer Research Funding from the William Paterson University of New Jersey.

References 1 Lee HL, Padmanabhan V, Taylor TA and Whang S (2000). Price protection in the personal computer industry. Mngt Sci 46(4): 467–482. 2 Heizer J and Render B (2000). Operations Management, 6th edn. Prentice-Hall: NJ. 3 Resh M, Friedman M and Barbosa LC (1976). On a general solution of the deterministic lot size problem with timeproportional demand. Opns Res 24: 718–725. 4 Donaldson WA (1977). Inventory replenishment policy for a linear trend in demand: an analytical solution. Opl Res Q 28: 663–670. 5 Barbosa LC and Friedman M (1978). Deterministic inventory lot size models-a general root law. Mngt Sci 24: 819–826. 6 Henery RJ (1979). Inventory replenishment policy for increasing demand. J Opl Res Soc 30(7): 611–617. 7 Dave U (1989). A deterministic lot-size inventory model with shortages and a linear trend in demand. Naval Res Logis 36: 507–514. 8 Goyal SK, Morin D and Nebebe F (1992). The finite horizon trended inventory replenishment problem with shortages. J Opl Res Soc 43(12): 1173–1178. 9 Teng JT, Chern MS and Yang HL (1997). An optimal recursive method for various inventory replenishment models with increasing demand and shortages. Naval Res Logist 44: 791–806. 10 Hariga MA and Goyal SK (1995). An alternative procedure for determining the optimal policy for an inventory item having linear trend in demand. J Opl Res Soc 46(4): 521–527. 11 Teng JT (1996). A deterministic replenishment model with linear trend in demand. Opns Res Lett 19: 33–41. 12 Dave U and Patel LK (1981). (T, Si) policy inventory model for deteriorating items with time proportional demand. J Opl Res Soc 32(1): 137–142. 13 Sachan RS (1984). On (T, Si) policy inventory model for deteriorating items with time proportional demand. J Opl Res Soc 35(11): 1013–1019. 14 Hariga MA (1996). Optimal EOQ models for deteriorating items with time-varying demand. J Opl Res Soc 47: 1228–1246. 15 Yang HL, Teng JT and Chern MS (2001). Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand. Naval Res Logist 48: 144–158. 16 Goyal SK and Giri BC (2001). Recent trends in modeling of deteriorating inventory. Eur J Opl Res 134: 1–16. 17 Chang HJ and Dye CY (1999). An EOQ model for deteriorating items with time varying demand and partial backlogging. J Opl Res Soc 50: 1176–1182. 18 Papachristos S and Skouri K (2000). An optimal replenishment policy for deteriorating items with time-varying demand and partial–exponential type–backlogging. Opns Res Lett 27: 175–184. 19 Teng JT, Chang HJ, Dye CY and Hung CH (2002). An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging. Opns Res Let 30: 387–393.

20 Teng JT, Yang HL and Ouyang LY (2003). On an EOQ Model for Deteriorating Items with Time-Varying Demand and Partial Backlogging. J Opl Res Soc 54(4): 432–436. 21 Teng JT, Chern MS, Yang HL and Wang YJ (1999). Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand. Opns Res Lett 24: 65–72. 22 Bellman RE (1957). Dynamic Programming. Princeton University Press: Princeton, NJ. 23 Friedman MF (1982). Inventory lot-size models with general time-dependent demand and carrying cost function. INFOR 20: 157–167.

Appendix A. Proof of Lemma 1 Let TCi ðsi1 ; ti ; si Þ Zti f½cb ðti  tÞ þ cv ðti Þ  cl bðti  tÞ þ cl gf ðtÞdt ¼ si1

Zsi nh i o ch þ cv ðti Þ ðeyðtti Þ  1Þ þ cv ðti Þ f ðtÞdt þ y ti

ðA1Þ We then have qTCi =qti ¼

Zti

½Sti ðt; ti Þ þ c0v ðti Þf ðtÞdt

si1

þ

Zsi

ðA2Þ ½c0v ðti Þ  C0 ðti Þeyðtti Þ f ðtÞdt

ti

If cv 0 (t)XC 0 (t)40, then we know from (A2) that qTCi/ qtiX0. Therefore, for any given i, TCi is increasing with ti. This implies that TCi(si1, ti, si)XTCi(si1, si1, si) for any fixed i. Consequently, we obtain s n Zi nh i X ch þ cv ðti Þ ðeyðtti Þ  1Þ TCðn; fsi g; fti gÞXncf þ y i¼1 si1 o þ cv ðti Þ f ðtÞdt s n Zi nh i X ch þ cv ð0Þ ðeyt  1Þ y i¼1 si1 o þ cv ð0Þ f ðtÞdt

Xncf þ

ZH h  ch þ cv ð0Þ ðeyt  1Þ y 0 i þ cv ð0Þ f ðtÞdt

Xcf þ

ðA3Þ This completes the proof of (a). Next, if c0v ðti Þp  Sti ðt; ti Þo0, with tpti, then we know from (A2) that qTCi/qtip0. Thus, for any given i, TCi

J-T Teng and H-L Yang—Deterministic economic order quantity models 501

is decreasing with ti. This implies TCi(si1, ti, si)X TCi(si1, si, si), for any i. Therefore, we know n Z X

GðxÞ ¼ ½ch þ ycv ðti Þ 

si

TCðn; fsi g; fti gÞXncf þ

i¼1

Xcf þ

ZH

uniquely exists. Similarly, from (12), we set Zx

c0v ðti Þ

eyðtti Þ f ðtÞdt

ti

½Sðt; HÞ þ cv ðHÞf ðtÞdt

si1

þ

Zti

f½cb þ c0v ðti Þbðti  tÞ

si1

f½cb ðH  tÞ þ cv ðHÞ  cl 

þ ½cb ðti  tÞ þ cv ðti Þ  cl b0 ðti  tÞgf ðtÞdt Zx ¼ ½c0v ðti Þ  C 0 ðti Þeyðtti Þ f ðtÞdt

0

bðH  tÞ þ cl gf ðtÞdt ðA4Þ

ðB4Þ

ti

which proves part (b). þ

½Sti ðt; ti Þ þ c0v ðti Þf ðtÞdt with xXti

si1

Appendix B. Proof of Lemma 2 From (11), we know that the optimal value of si (ie, s*i ) is the interior point between ti and ti þ 1 because if si ¼ ti or ti þ 1, then Equation (11) does not hold. TC(n, {si}, {ti}) is a continuous (and differentiable) function minimized over the compact set [0, H]2n. Hence, there exists an absolute minimum. The optimal value of ti (ie, t*i ) cannot be on the boundary since TC(n, {si}, {ti}) increases when any one of the ti’s is shifted to the end points 0 or H. Consequently, there exists at least an inner optimal solution satisfied (11) and (12). Since s0 ¼ 0 and t*1 is unique, if we can prove that both s*i generated by (12) and t*i þ 1 by (11) are uniquely determined, then we prove Lemma 2. For any given si and ti, from (11), we set FðxÞ ¼ ½cb ðx  si Þ þ cv ðxÞ  cl bðx  si Þ hc i h  ½cl  cv ðti Þ þ þ cv ðti Þ ðeyðsi ti Þ  1Þ y ¼ Sðsi ; xÞ  cv ðxÞ þ cv ðti Þ hc i h þ þ cv ðti Þ ðeyðst ti Þ  1Þ with xXsi y

Zti

Likewise, if C 0 (t)4cv 0 (t)4St(si, t) for all tXsi, then we have Gðti Þ ¼

Zti

½Sti ðt; ti Þ þ c0v ðti Þ f ðtÞdt40

si1

Taking the first derivative of G(x) with respect to x, we obtain G0 ðxÞ ¼ ½c0v ðti Þ  C 0 ðti Þeyðxti Þ f ðxÞo0

Appendix C. Proof of Theorem 1 For any fixed n, differentiating TC(n, {si}, {ti}) with respect to t1 and simplifying terms, we obtain

ðB1Þ qTCðn; fsi g; fti gÞ=qt1 ¼

n X

cl f ðti Þ

i¼1



n1 X

dti dt1

f½cb ðti þ 1  si Þ þ cv ðti þ 1 Þ  cl 

i¼1

hc i h þ cv ðti Þ Fðsi Þ ¼  cv ðsi Þ þ cv ðti Þ þ y

ðeyðsi ti Þ  1Þ

bðti þ 1  si Þ þ cl gf ðsi Þ n Z X

dsi dt1

ti

þ

i¼1

f½cb þ c0v ðti Þbðti  tÞ

si1

By the mean value theorem, we get þ ½cb ðti  tÞ þ cv ðti Þ  cl b0 ðti  tÞgf ðtÞdt ðB2Þ

for some tipkipsi. By taking the first derivative of F(x) with respect to x, we obtain F 0 ðxÞ ¼ ½Sx ðsi ; xÞ þ c0v ðxÞo0

ðB6Þ

As a result, we know that there exists a unique si (4ti) such that G(si) ¼ 0. Thus, the solution to Equation (12) uniquely exists. Therefore, we complete the proof of Lemma 2.

If C 0 (t)4cv 0 (t)4St(si, t) for all t, then we have

Fðsi Þ ¼ f½ch þ ycv ðti Þeyðki ti Þ  c0v ðki Þgðsi  ti Þ40

ðB5Þ

ðB3Þ

Consequently, there exists a unique ti þ 1 (4si) such that F(ti þ 1) ¼ 0, which implies that solution to Equation (11)

þ

n nh X ch i¼1

y

dti dt1

i þ cv ðti Þ ðeyðsi ti Þ  1Þ

n o dsi X dti þ cv ðti Þ f ðsi Þ  cv ðti Þf ðti Þ dt1 i¼1 dt1



n X i¼1

½ch þ ycv ðti Þ 

c0v ðti Þ

Zsi ti

eyðtti Þ f ðtÞdt

dti dt1

ðC1Þ

502 Journal of the Operational Research Society Vol. 55, No. 5

If we relax sn to be any number, then we know from (11) and (12) that n X

dti dt 1 i¼1 nhc i o dsn h þ cv ðtn Þ ðeyðsn tn Þ  1Þ þ cv ðtn Þ f ðsn Þ þ 40 y dt1 ðC2Þ

qTCðn; fsi g; fti gÞ=qt1 ¼

In order to prove that T*(n, 0, H) is strictly convex in n, we choose H1 and H2 such that s n ðn þ 1; 0; H1 Þ ¼ s n þ 1 ðn þ 2; 0; H2 Þ ¼ H

½cl  cv ðti Þ f ðti Þ

which implies that TC(n, {si}, {ti}) without the boundary condition of sn ¼ H is an increasing function of t1. From (11) and (12), we know that if t1 ¼ 0 (or H), then sn (t1) ¼ 0 (or 4H). As a result, for any given n, there exists a unique t1 such that TC(n, {si}, {ti}) in (10) is minimized with sn ¼ H. This proves (a). To prove (b), we know from (11) that the optimal value of si (ie, s*i ) is the interior point between ti and ti þ 1 because if si ¼ ti or ti þ 1, then Equation (11) does not hold. TC(n, {si},{ti}) is a continuous (and differentiable) function minimized over the compact set [0, H]2n. Hence, there exists an absolute minimum. The optimal value of ti (ie, t*i } cannot be on the boundary since TC(n, {si},{ti}) increases when any one of the tis is shifted to the end points 0 or H. Consequently, (11) and (12) are the necessary and sufficient conditions for the absolute minimum.

Appendix D. Proof of Theorem 2

and s 0 ðn þ 1; 0; H1 Þ ¼ s 0 ðn þ 2; 0; H2 Þ ¼ 0

Employing the principle of optimality on (D5) again, we have T ðn þ 1; 0; H1 Þ ¼ Minimize fT ðn; 0; tÞ þ Tð1; t; H1 Þg t2½0;H1 

¼ T ðn; 0; HÞ þ Tð1; H; H1 Þ ðD6Þ and T ðn þ 2; 0; H2 Þ ¼ Minimize fT ðn þ 1; 0; tÞ t2½0;H2 

¼ T ðn þ 1; 0; HÞ þ Tð1; H; H2 Þ respectively. Since H is an optimal interior point in T*(n þ 1, 0, H1) and T*(n þ 2, 0, H2), we know that  qT ðn; 0; tÞ qTð1; t; H1 Þ þ ðD8Þ  ¼0 qt qt t¼H and

ðD1Þ TCðn; fsi g; fti gÞ ¼ RðnÞ þ Tðn; 0; HÞ Pn where R(n) ¼ ncf and T (n, 0, H)= i¼1 ðPi þ ch Ii þ cb Bi þ cl Li Þ  ncf . Firstly, we know that R(n) is an increasing convex function of n. Next, by Bellman’s principle of optimality,22 we know that the minimum value of T(n, 0, H) is

 qT ðn þ 1; 0; tÞ qTð1; t; H2 Þ þ  ¼0 qt qt t¼H

2w 3 Z Zb yðtwÞ e f ðtÞdt5 Tð1; a; bÞ ¼ cv ðwÞ4 bðw  tÞf ðtÞdt þ a

þ

ðD2Þ

ch y

þ cb

Let t ¼ H and hence T*(n, 0, H) o T*(n1, 0, H). The strict inequality follows because the minimum in (D2) occurs at an interior point. Thus, T*(n, 0, H) is strictly decreasing in n. Recursive application of (D2) yields the following relations:

s i ðn; 0; HÞ ¼ s i ðn  j; 0; s nj ðn; 0; HÞÞ; i ¼ 1; 2; . . . ; n  j  1:

Zb

w

½eyðtwÞ  1f ðtÞdt

w

þ Tð1; t; HÞg:

i ¼ 1; 2; . . . ; n  j  1;

ðD9Þ

Utilizing the fact that

T ðn; 0; HÞ ¼ Minimize fT ðn  1; 0; tÞ

t i ðn; 0; HÞ ¼ t i ðn  j; 0; t nj ðn; 0; HÞÞ;

ðD7Þ

þ Tð1; t; H2 Þg

Let us set

t2½0;H

ðD5Þ

Zw

ðw  tÞbðw  tÞf ðtÞdt

a

þ cl

Zw

½1  bðw  tÞf ðtÞdt

ðD10Þ

a

and ðD3Þ qTð1; a; bÞ ¼ f½cb ðw  aÞ þ cv ðwÞ  cl bðw  aÞ qa þ cl gf ðaÞ ðD11Þ ðD4Þ

where w is the replenishment time between (a, b), and both a and b are the time at which the inventory level drops to zero

J-T Teng and H-L Yang—Deterministic economic order quantity models 503

since k(t) ¼ [(ch/y) þ cv(t)]ey(Ht) is a decreasing function for all toH. This implies that T*(n, 0, H)T*(n þ 1, 0, H) is a strictly increasing function of H. Thus,

in the cycle (a, b). We then obtain   qT ðn; 0; tÞ qTð1; t; H1 Þ ¼    qt qT t¼H t¼H

T ðn; 0; HÞ  T ðn þ 1; 0; HÞoT ðn; 0; H1 Þ  T ðn þ 1; 0; H1 Þ

¼ f½cb ðt n þ 1 ðn þ 1; 0; H1 Þ  HÞ þ cv ðt n þ 1 ðn þ 1; 0; H1 ÞÞ  cl 

bðt n þ 1 ðn þ 1; 0; H1 Þ  HÞ þ cl gf ðHÞ nh ch i yðHt n ðn;0;HÞÞ ¼ cv ðt n ðn; 0; HÞÞ þ 1 ½e y  þ cv ðt v ðn; 0; HÞÞ f ðHÞ; ðby ð11ÞÞ ðD12Þ where t*n(n, 0, H) and t*n þ 1(n þ 1, 0, H1) are the corresponding last replenishment time when n orders are placed in [0, H], and n þ 1 orders are placed in [0, H1], respectively. Similarly,   qT ðn þ 1; 0; tÞ qTð1; t; H2 Þ ¼    qt qt t¼H t¼H nh i ch yðHt ðn þ 1;0;HÞÞ nþ1 ½e ¼ cv ðt n þ 1 ðn þ 1; 0; HÞÞ þ  1 y þ cv ðt n þ 1 ðn þ 1; 0; HÞÞgf ðHÞ ðD13Þ

ðD15Þ

Again, by (D2) and the principle of optimality, we have T ðn; 0; H1 Þ  T ðn þ 1; 0; H1 Þ ¼ Minimize fT ðn  1; 0; tÞ þ Tð1; t; H1 Þ t2½0;H1 

ðD16Þ



 T ðn; 0; HÞ  Tð1; H; H1 Þ Let t ¼ H in (D16), we have T ðn; 0; H1 Þ  T ðn þ 1; 0; H1 ÞoT ðn  1; 0; HÞ ðD17Þ  T ðn; 0; HÞ By (D15) and (D17), we have T ðn; 0; HÞ  T ðn þ 1; 0; HÞoT ðn  1; 0; HÞ  T ðn; 0; HÞ

ðD18Þ

which implies T*(n, 0, H) is convex in n, and hence, TC(n) ¼ R(n) þ T*(n, 0, H) is also convex in n.

Subtracting (D12) from (D13), we have

Received June 2003; accepted October 2003

q ½T ðn; 0; tÞ  T ðn þ 1; 0; tÞjt¼H qt n ch yðHt n ðn;0;HÞÞ ½e  eyðHtn þ 1 ðn þ 1;0;HÞÞ  ¼ y þ cv ðt n ðn; 0; HÞÞeyðHtn ðn;0;HÞÞ

 cv ðt n þ 1 ðn þ 1; 0; HÞÞeyðHtn þ 1 ðn þ 1;0;HÞÞ gf ðHÞ40 ðD14Þ