A periodic review replenishment model with a refined delivery scenario

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A periodic review replenishment model with a reﬁned

delivery scenario

Chi Chiang

Department of Management Science, National Chiao Tung University, Hsinchu, Taiwan, ROC

a r t i c l e

i n f o

Available online 22 August 2008 Keywords:

Inventory Periodic review Stochastic model

a b s t r a c t

This paper considers a single-item replenishment problem where every n periods the buyer plans for the quantity delivered for each upcoming period. Assuming that demand not filled immediately is backlogged, we suggest that the quantity delivered in the immediately upcoming period be of any size and the quantity delivered in each of the subsequent (n1) periods be equal to or smaller than a fixed quantity Q. We derive the average cost per n periods and compute the optimal order-up-to level for the proposed replenishment policy. Next, assuming a fixed cost for reviewing inventory and planning deliveries, we obtain the optimal n by minimizing the expected cost per period. Finally, the optimal Q can also be determined. The proposed periodic review policy is easy to implement.

1. Introduction

One of the most widely used stochastic inventory models is an order-up-to periodic review policy. Unlike a continuous review model where the order quantity is ﬁxed, an order-up-to periodic policy places an order in every period that will raise inventory to a target level. Since demand ﬂuctuates period by period, the order quantity varies too. This may result in costs of adjustment in practice due to, for example, changes in capacity or production plans. These costs are incurred by the supplier but may be charged to the buyer (e.g., Urban, 2000). However, an ordinary order-up-to policy implicitly as-sumes that these costs do not exist (or simply ignores these costs).

It is possible that the order quantity is made (almost) ﬁxed in periodic review systems. Such a scenario is attractive and desirable to both the supplier and the buyer. It simpliﬁes the production, order picking, delivery, unloading (for the supplier), inspection process, and inventory record updating procedure (for the buyer).

It avoids any extra costs that might be incurred due to variations in the order quantity. It also agrees with the JIT management philosophy. However, since demand is stochastic in the real world, a certain mechanism is needed to absorb variations of demand. In general, there are several ways to achieve this. One is through a standing order inventory system where provision is made for procuring extra units in the case of an emergency and selling off excess inventory. Standing order systems were ﬁrst considered by Rosenshine and Obee (1976) and recently studied inChiang (2007). Another is through a two-supplier inventory system (Janssen and de Kok, 1999) in which while one supplier delivers a ﬁxed quantity in each period the other ships any quantity needed to raise inventory to a target level.

In this paper, we propose a third approach to facilitat-ing the scenario of a (almost) fixed order quantity in periodic review systems. We suppose that every n periods the buyer, who employs an order-up-to policy, plans for the quantity delivered for each upcoming period. Assum-ing that demand not filled immediately is backlogged, we suggest that a fixed quantity Q be delivered by the supplier for each of the upcoming n periods except perhaps the first several periods. To be more specific, the Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/ijpe

Int. J. Production Economics

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immediately upcoming period’s shipment size is adjusted first, and, if not sufficient, the upcoming second period’s shipment size is adjusted next, and so on, such that the quantity shipped in each of the subsequent periods is exactly Q. Thus, the quantity to be delivered in the immediately upcoming period is of any size and the quantity delivered in each of the subsequent (n1) periods is equal to or smaller than Q. For example, suppose that Q ¼ 5 and n ¼ 5. If demand of the previous 5 periods is 28 (resp. 22), the quantity delivered in each of the upcoming 5 periods is 8 (resp. 2), 5, 5, 5, 5, respectively. On the other hand, if demand of the previous 5 periods is 17 (resp. 13), the quantity delivered in each of the upcoming 5 periods is 0, 2 (resp. 0), 5 (resp. 3), 5, 5, respectively. For a certain n, the optimal order-up-to level for the delivery scenario described is computed by minimizing the average n-periods’ cost. Next, assuming that there is a fixed cost incurred every n periods for auditing the inventory level as well as planning and adjusting order quantities, the optimal n can be obtained by minimizing the average cost per period through a simple procedure. Finally, the optimal Q can also be determined. Hence, the proposed replenishment policy is easy to implement.

Note thatFlynn and Garstka (1990)consider a related but more complex problem where one schedules delivery quantities for the next n periods that are generally not equal to one another. Flynn and Garstka (1997) further extend the analysis to determine the optimal review period. Chiang (2001) also studies a delivery splitting periodic model where n shipments are scheduled in future time points that are evenly separated. However, three major shortcomings limit the applicability of Chiang’s model. First, these n shipments are of different sizes. Second, the costs of adjustment due to changes in the shipment size are not included. Third, the interval between delivery epochs may not be an integral multiple of a basic time unit (e.g., a day). Thus, the suggested model may not be easily implemented in practice. Other related periodic review models are investigated in Ehrhardt (1997) and Urban (2000). The former considers the problem of selecting a ﬁxed replenishment quantity to be delivered in each of n consecutive periods in the future, while the latter describes a multi-period ‘‘recurrent’’ newsvendor problem where changes in the order quantity result in an additional cost to the buyer.

The rest of this paper is organized as follows. In Section 2, we propose a periodic review replenishment model with the delivery scenario described above. In Section 3, we consider a simpliﬁed version of the proposed policy. Section 4 reports some computational results. Finally, Section 5 concludes this paper.

2. A periodic review replenishment model

Consider the following replenishment problem: every n periods (one period is 1 day, for example) the buyer reviews an item and plans its shipment size for each upcoming period to raise the inventory position (i.e., inventory on hand minus backorders plus inventory on

order) to a target level (i.e., an order-up-to policy is used). Assume that the demand of each period is independently and identically distributed. As explained in Section 1, it benefits the supplier as well as the buyer if a fixed quantity Q is shipped in each period. Such a situation occurs only when demand in the previous n periods is exactly nQ. If demand of the previous n periods is not nQ, the buyer would like to adjust first the immediately upcoming period’s shipment size so that the quantity shipped in each of the subsequent (n1) periods is exactly Q; if adjustment to the immediately upcoming period’s shipment size is not sufficient, the buyer would adjust next the upcoming second period’s shipment size so that the quantity shipped in each of the subsequent (n2) periods is Q; and so on. Notice that we have implicitly assumed in the above scenario that the immediately upcoming period’s shipment size can be adjusted if requested by the buyer. If it takes a positive lead time (an integral multiple of the period length) to adjust the shipment size, our replenishment model can be modified by appropriately defining Gi(Yjn, Q) introduced below, as

in an ordinary periodic review model (e.g.,Porteus, 1990). For the proposed replenishment policy to be clearer, let fk( ) be the probability density function of k periods’

demand, k ¼ 1, y, n, and D the demand during the previous n periods (its probability density function is thus fn( )). If DX(n1)Q, the quantity delivered for each of the

upcoming n periods is D(n1)Q, Q, y, Q, respectively; if (n1)Q4DX(n2)Q, the quantity delivered for each upcoming period is 0, D(n2)Q, Q,y, Q, respectively; if (n2)Q4DX(n3)Q, the quantity delivered for each upcoming period is 0, 0, D(n3)Q, Q,y, Q, respectively; and so on. Let Qibe the expected quantity shipped in the

upcoming ith period. Then

be the mean demand of a period, we have the following result.

Theorem 2. If Qo

m

, then Q14

m

.

Proof. The expected demand of n periods is n

m

, i.e., E[D] ¼ n

m

and thus Q1+Q2+?+Qn¼n

m

. Since QipQ for

2pipn, it follows that if Qo

m

, then Q14

m

.

Let Gi(Yjn, Q) be the average cost of the upcoming ith

period for the proposed delivery scenario given an order-up-to level YX0 and G(Yjn, Q) the average total n-periods’ cost, i.e.,

GðYjn; Q Þ G1ðYjn; Q Þ þ G2ðYjn; Q Þ þ þ GnðYjn; Q Þ. (3)

Only inventory holding and shortage costs (which are charged based on the ending inventory of a period) are included above. The procurement cost cE[D] (where c is

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the unit cost) for n periods is excluded since it is constant. Let h and p be the holding and shortage cost per unit per period, respectively. To derive Gi(Yjn, Q), for ip(n1), we

consider two cases: YX(ni)Q and Yo(ni)Q. In the former case, Gi(Yjn, Q) is evaluated according to whether

DX(ni)Q or Do(ni)Q. In the latter case, Gi(Yjn, Q) is

evaluated according to whether DX(ni)Q, (ni)Q4DXY, or Y4D. It follows that for ip(n1),

GiðYjn; Q Þ ¼ Z1 ðniÞQ fnðzÞdz ZYðniÞQ 0 hðY ðn iÞQ xÞ ( f_iðxÞdxþ Z1 YðniÞQ pðxY þ ðn iÞQ Þf_iðxÞdx ) þ ZðniÞQ 0 fnðzÞ ZYz 0 hðY zxÞ ( fiðxÞdx þ Z1 Yz pðxY þzÞf_iðxÞdx dz if YXðn iÞQ , (4)

GiðYjn; Q Þ ¼ pði

m

þ ðn iÞQ YÞ

The following lemma can be veriﬁed by examining the second derivative of Gi(Yjn, Q).

Lemma 3. Gi(Yjn, Q) for each i is convex on Y.

Hence, the optimal order-up-to level, denoted by Y*, can be obtained by equating DG(Yjn, Q) to zero and solving for Y. For discrete demand distribution, we ﬁnd the smallest

value of Y such that DG(Yjn, Q)X0. The minimum n-periods’ cost is G(Y*jn, Q).

Next, we investigate the issue of how often one should schedule for deliveries. Let K be the ﬁxed cost for reviewing the inventory level as well as planning and adjusting order quantities (particularly the order quantity of the immediately upcoming period, since it can be of any size). Part of this cost is incurred by the buyer (for reviewing inventory); the remaining part that is incurred by the supplier due to changes in the shipment size is charged to the buyer (as mentioned in Section 1). It is assumed that the entire ﬁxed cost K is incurred even if the quantity to be delivered in each of the upcoming n periods is exactly Q. The average cost per period, denoted by AC(Q, n), is given by

ACðQ ; nÞ ¼ ½GðY

jn; Q Þ þ K=n. (10) Minimizing AC(Q, n) requires a certain procedure of determining the optimal n, denoted by n*. Since the computation shows that AC(Q, n) is quasi-convex (but not necessarily convex) in n given Q (though we cannot prove this), we suggest that one simply tabulates AC(Q, n) as a function of n to obtain n*, as an approximate periodic review model determines the optimal review period (Hadley and Whitin, 1963, Section 5-2). Let AC*(Q)AC(Q, n*(Q)).

Finally, it may be desirable to determine the optimal Q, denoted by Q*, such that AC*(Q) is minimized. A simple method of ﬁnding Q* is to enumerate feasible (possibly integral) values of Q and choose the lowest AC*(Q).

3. A simpliﬁed policy

In the above delivery scenario, the quantity shipped in the immediately upcoming period is of any size and the quantity shipped in each of the subsequent (n1) periods is equal to or smaller than Q. Suppose that excess inventory in the immediately upcoming period can be salvaged or returned to the supplier at c per unit. Then, only the quantity shipped in this period is variable and the quantity shipped in each of the subsequent (n1) periods is exactly Q. This greatly simpliﬁes the replenishment policy. Referring to the second paragraph of Section 2, if Do(n1)Q, the quantity shipped for each upcoming period is 0, Q, Q,y, Q, respectively, and the excess units (n1)QD in the immediately upcoming period are salvaged or returned to the supplier. If Q ¼

m

and these excess units are regarded as a negative order quantity from the supplier, then Q1¼Q and thus Q1¼Q2¼ ? ¼Qn1¼Qn. For this

simpliﬁed policy, Gi(Yjn, Q), for ip(n1), in expressions (4)

if YXðn iÞQ , (11)

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and expression (6) remains unchanged for Gn(Yjn, Q). Also,

DGi(Yjn, Q), for ip(n1), reduce to

DGiðYjn; Q Þ ¼ p þ ðh þ pÞ

Z YðniÞQ 0

fið

x

Þd

x

if YXðn iÞQ , (13)

DGiðYjn; Q Þ ¼ p if Yoðn iÞQ. (14)

Let Ys _{be the optimal order-up-to level for this}

simpliﬁed model. By comparing (13) to (7) and (14) to (8), one can easily verify that YsXY*.

4. Computational results

+3, n ¼ 1, 2, y, 20, and p ¼ $10, $100, and $1000) and solve 900 problems. It is found that G(Y*jn, Q)/ n is increasing in n. This implies that if K ¼ 0, n* ¼ 1, i.e., the proposed model reduces to an ordinary periodic policy where an order is placed in every period.

the simpliﬁed policy yields almost the same one-period cost as the proposed model, since the probability of salvaging excess units is very low, and if Q is very large the simpliﬁed policy may give a higher one-period cost than

Table 1

AC*(Q) for various values of Q (data:m¼4, p ¼ $1000, and h ¼ $1)

Proposed model Simpliﬁed policy

K Q n* Y* AC*(Q) n* Y* AC*(Q) Savings (%)

$0 1 11 $8.29 1 11 $8.29 50 2 5 33 28.27 5 33 28.27 3 6 38 26.30 6 38 26.30 4 7 43 24.21 7 43 24.11 5 8 50 22.83 9 55 22.12 6 7 47 23.21 8 53 21.88 4.16 7 6 44 24.55 6 45 23.30 100 2 7 42 36.43 7 42 36.43 3 9 52 33.26 9 52 33.26 4 11 63 29.92 11 63 29.79 5 12 70 27.87 13 75 26.73 4.09 6 10 64 29.29 11 71 27.35 7 8 56 31.66 9 66 30.10 200 2 11 61 47.37 11 61 47.37 3 13 71 42.50 13 71 42.50 4 18 95 37.07 18 95 36.93 5 18 100 34.44 20 110 32.67 5.14 6 14 87 37.72 15 95 35.28 7 12 81 41.39 12 87 39.92

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the proposed model. For example, if K ¼ $200 and Q ¼ 7 in Table 2, then n* ¼ 13, Y* ¼ 82 and AC*(Q) ¼ $36.96 for the proposed model and n* ¼ 12, Y* ¼ 84 and AC*(Q) ¼ $37.31 for the simpliﬁed policy. This is because a large Q yields a relatively high Y* for the simpliﬁed policy, since the system needs some safety stock for the immediately upcoming period and still has Q to be delivered for each of the subsequent (n1) periods; in order for Y* not to be too

high, n* may be smaller and K is thus spread over less periods.

5. Conclusion

This paper considers a single-item replenishment problem where every n periods the buyer plans for the

Table 2

$0 1 9 $6.24 1 9 $6.24 50 2 6 33 24.12 6 33 24.12 3 7 38 21.98 7 38 21.97 4 8 43 19.77 8 43 19.65 5 9 51 18.83 10 56 18.05 4.14 6 7 43 19.93 8 50 18.89 7 6 40 21.46 6 43 20.80 100 2 8 41 31.38 8 41 31.38 3 10 51 28.00 10 51 28.00 4 13 66 24.46 13 66 24.32 5 14 75 23.32 15 80 22.18 4.89 6 10 60 25.58 11 68 24.32 7 9 58 28.01 9 63 27.63 200 2 12 59 41.31 12 59 41.31 3 15 73 36.14 15 73 36.14 4 20 97 30.43 20 97 30.27 5 19 100 29.36 21 110 27.70 5.65 6 15 88 33.46 15 91 32.19 7 13 82 36.96 12 84 37.31

Note that the lowest AC*(Q) values given K are underlined.

Table 3

$0 1 7 $3.93 1 7 $3.93 50 2 7 30 18.53 7 30 18.53 3 8 35 16.29 8 35 16.29 4 12 55 14.12 12 55 13.99 5 11 55 14.23 11 56 13.62 3.54 6 8 44 15.89 8 47 15.63 7 7 41 17.34 6 39 17.94 100 2 10 42 24.63 10 42 24.63 3 12 51 21.11 12 51 21.11 4 18 80 17.53 18 80 17.38 0.86 5 16 79 18.13 16 80 17.39 6 12 65 20.85 11 64 20.91 7 11 64 22.90 9 60 24.51 200 2 14 57 33.06 14 57 33.06 3 18 75 27.64 18 75 27.64 4 24 105 22.04 24 105 21.86 0.82 5 22 108 23.49 22 110 22.68 6 18 97 27.70 15 87 28.57 7 15 86 30.62 12 80 34.00

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quantity delivered for each upcoming period. Depending on the demand of the previous n periods, the quantity delivered in the immediately upcoming period may be of any size and the quantity delivered in each of the subsequent (n1) periods is equal to or smaller than a ﬁxed quantity Q. If excess inventory in the immediately upcoming period can be salvaged or retuned to the supplier at the original purchase cost, the quantity shipped in each of the subsequent (n1) periods is exactly Q.

Computation shows that the optimal Q is greater than or equal to the mean period demand. However, as the ratio p/h decreases, the optimal n increases and the optimal Q may approach the mean period demand. In addition, as K increases the optimal n increases, and if K ¼ 0 the optimal n is equal to 1. In this sense, the ordinary order-up-to policy where an order is placed in every period could be regarded as a special case of the proposed replenishment model. More importantly, as K is larger, the proposed

Table 4

$0 1 6 $4.60 1 6 $4.60 50 1 8 23 18.51 8 23 18.51 2 10 29 15.81 10 29 15.66 3 10 32 15.63 11 36 14.55 6.91 4 8 30 17.48 7 29 16.79 5 7 28 18.83 6 30 19.30 100 1 11 30 23.91 11 30 23.91 2 16 43 19.57 16 43 19.40 3 15 46 19.83 16 50 18.42 5.88 4 11 39 22.69 11 44 22.33 5 11 42 24.58 9 44 26.12 200 1 16 41 31.24 16 41 31.24 2 26 66 24.32 26 66 24.13 3 21 63 25.53 21 65 23.83 2.01 4 17 58 29.81 15 60 30.24 5 16 59 32.37 12 59 35.92

Note that the lowest AC*(Q) values given K are underlined.

Table 5

$0 1 12 $7.48 1 12 $7.48 50 4 5 40 26.37 5 40 26.37 5 6 47 24.46 6 47 24.45 6 7 55 22.54 7 55 22.45 7 8 64 21.32 9 71 20.82 2.35 8 7 60 21.81 8 68 20.86 9 6 55 23.10 6 56 22.24 100 4 8 60 34.10 8 60 34.10 5 9 67 31.04 9 67 31.03 6 11 82 27.93 11 82 27.81 7 13 99 26.22 13 99 25.39 3.17 8 10 83 27.64 11 91 26.33 9 9 79 29.97 9 82 29.01 200 4 11 79 44.53 11 79 44.53 5 14 99 39.78 14 99 39.78 6 18 128 34.69 18 128 34.55 7 19 141 32.54 20 148 31.30 3.81 8 15 121 35.72 15 123 34.03 9 12 104 39.42 12 109 38.59

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model becomes more attractive relative to the ordinary order-up-to policy.

References

Chiang, C., 2001. Order splitting under periodic review inventory systems. International Journal of Production Economics 70 (1), 67–76.

Chiang, C., 2007. Optimal control policy for a standing order inventory system. European Journal of Operational Research 182 (2), 695–703. Ehrhardt, R., 1997. A model of JIT make-to-stock inventory with stochastic demand. Journal of the Operational Research Society 48, 1013–1021.

Flynn, J., Garstka, S., 1990. A dynamic inventory model with periodic auditing. Operations Research 38 (6), 1089–1103.

Flynn, J., Garstka, S., 1997. The optimal review period in a dynamic inventory model. Operations Research 45 (5), 736–750.

Hadley, G., Whitin, T.M., 1963. Analysis of inventory systems. Prentice-Hall, Englewood Cliffs, NJ.

Janssen, F., de Kok, T., 1999. A two-supplier inventory model. Interna-tional Journal of Production Economies 59, 395–403.

Porteus, E.L., 1990. Stochastic inventory theory. In: Heyman, D.P., Sobel, M.J. (Eds.), Handbooks in OR & MS, Vol. 2. Elsevier Science Publishers, Amsterdam (Chapter 12).

Rosenshine, M., Obee, D., 1976. Analysis of a standing order inventory system with emergency orders. Operations Research 24, 1143–1155. Urban, T.L., 2000. Supply contracts with periodic, stationary