A note on periodic review inventory models with stochastic supplier's visit intervals and fixed ordering cost

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A note on periodic review inventory models with stochastic supplier

’s

visit intervals and

ﬁxed ordering cost

Chi Chiang

n

Department of Management Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC

a r t i c l e i n f o

Available online 23 August 2013 Keywords: Inventory model Periodic review Lost-sales Dynamic programming Compound Poisson

a b s t r a c t

Most periodic review models in the inventory literature have assumed afixed length of the review periods. In this note, we extend the work ofChiang (2008), and consider backlogged and lost-sales periodic review models where the review periods are of a variable length and there is afixed cost of ordering for replenishment. Assuming that period lengths are independently and identically distributed, we show (using an exact method of computing inventory holding costs) that an (s, S) policy is optimal for the infinite horizon problem. The periodic review policies developed are thus easy to implement. The computation shows that if thefixed cost of ordering is small, one needs to use the proposed periodic policies.

1. Introduction

Most periodic review systems in the inventory-control literature have assumed aﬁxed length of the review periods. It is possible in practice that periodic systems have the review periods of a variable length. Such systems arise mainly from supply uncertainties. For example, Chiang (2008) observed that many supermarkets have suppliers who come to visit regularly and replenish inventories for them. However, the supplier does not always come in constant time intervals. Depending on her visit plans or work schedules, she often arrives at a particular supermarket earlier or later than planned. The elapsed time between two consecutive visits varies in nature. See alsoErtogral and Rahim (2005)for supply chain settings where the replenishment epochs are not under the retailer′s control (i.e., under the supplier′s control), andTang and Musa (2011)for a variety of supply chain risks or uncertainties.

To the best of our knowledge, the issue of the period length variability or replenishment interval randomness is investigated only recently byErtogral and Rahim (2005) and Chiang (2008).

Ertogral and Rahim (2005) derived the expected profit per replenishment cycle by assuming constant demand; Chiang (2008) used dynamic programming to develop periodic review inventory models with stochastic demand. However, these studies assumed that thefixed cost of ordering for replenishment is zero. In this paper, we extend the work of Chiang (2008) and incorporate afixed cost of ordering. It is possible that the supplier visits a retailer and charges a service expense if the retailer′s

inventory is replenished. Moreover, instead of using an approx-imate method as inErtogral and Rahim (2005)andChiang (2008), we use an exact approach of computing inventory holding costs. We assume that period lengths are independently and identically distributed (iid), as in the above two studies, and examine both the backlogged and lost-sales periodic review inventory problems. We will show that the optimal policy is of the (s, S) type. Hence, existing algorithms (e.g.,Zheng and Federgruen, 1991) could be used to ﬁnd the optimal s and S. The periodic review models developed can be viewed as a generalization of ordinary periodic models where the period length isﬁxed.

The computation shows that when thefixed cost of ordering is small (but not small enough to be neglected, such that an order is always placed at a review epoch), ignoring the period length variability can incur unnecessary large losses, especially if lead-time is zero, shortage is costly, demand variability is small, and/or the period length is volatile. These results agree with Chiang (2008). Hence, one needs to use the proposed ordering policies, and the suggestions made inChiang (2008)apply here, e.g., if the replenishment epochs are under the supplier′s control, the retailer should somehow persuade the supplier to visit more regularly, or even cooperate or form a strategic alliance with the supplier in the long run;Prajogo et al. (2012)recently showed that strategic long-term relationship, one of the three supplier management practices suggested, has a positive relationship with afirm′s operational performance, andCheng et al. (2012)found that a purchasingfirm tends to form quanxi networks with its key supplier to improve communication and thus reduce supply risk.. However, when the fixed ordering cost is large, ignoring the period length variability causes small or virtually no losses to afirm, especially if lead-time is long or demand variability is large. The implication of this is that Contents lists available atScienceDirect

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it is alright to use the ordinary periodic review models in the case of largeﬁxed ordering costs.

2. Backlogged periodic review inventory models

Weﬁrst consider the case where demand not satisﬁed at once is backlogged. We use the same notation as in Chiang (2008). Demand is stochastic with mean rateμ per day, and is assumed to be non-negative and independently distributed in disjoint time intervals. Let T denote the period length and D the demand during T. Successive T′s are assumed to be iid random variables. Let ϕ( ) be the probability density function (pdf) of T and g( |T) be the conditional pdf of D. Also, let c be the unit purchase cost,α the discount rate, y the inventory position (i.e., inventory on hand minus backorder plus inventory on order) after a possible order is placed at a review epoch, and H the expected one-period inven-tory holding and shortage costs (H is a function of y).

We assume without loss of generality (also for simplicity) that replenishment is immediate. The case of a positive (constant) lead-time L can be handled by appropriately redeﬁning H, i.e., given time 0 at a review epoch, H is charged for the time interval [L, T þ L) (seeChiang, 2008 or Porteus, 1990). Let K denote theﬁxed cost of ordering and Vn(x) the expected discounted cost with n periods

remaining until the end of the planning horizon when the starting inventory position is x and an optimal ordering policy is used at every review epoch. Vn(x) satisﬁes the functional equation

VnðxÞ ¼ minxr yfKδðy–xÞþcyþHðyÞþET½eaTEDjT½Vn1ðy–DÞg–cx;

ð1Þ where δ( ) is the Dirac-delta function that is equal to 1 if the argument is positive and 0 otherwise, c(y– x) is the procurement cost, and ET[e-αTED|T[Vn-1(y– D)]] is the expected discounted cost

from the next review epoch to the end of the planning horizon. Since T is stochastic, the planning time horizon in (1) is of a random length, as opposed to theﬁxed time horizon of N periods commonly studied in the literature. Eq.(1)is basically the same as expression (1) ofChiang (2008), except that aﬁxed cost K is presented.

Letβ ¼ ET[e αT]. Deﬁne φ( ) ET[e αTg( |T)]/β, i.e., βφ( )¼

ET[e αTg( |T)] is the discount density of D and φ( ) is the

“normal-ized” pdf of D.Chiang (2008)showed that ET eaTEDjT Vn1ðy–DÞ ¼Z 1 0 eaTðZ 1 0 Vn1ðy–DÞgðDjTÞdDÞϕðTÞdT ¼ βZ 1 0 Vn1ðy–DÞ ðZ 1 0 eaTgðDjTÞϕðTÞdTÞ=βdD ¼ βED Vn1ðy–DÞ ð2Þ

where the expectation EDis taken over the pdfφ( ). Thus, Vn(x)

can be written by

VnðxÞ ¼ minxr yfKδðy–xÞþcyþHðyÞþβED½Vn1ðy–DÞg–cx: ð3Þ

There are a few approaches in the inventory literature of computing the one-period cost function H(y).Chiang (2008)used an approximate method based on Hadley and Whitin (1963, pp. 237–239). In this paper, we adopt an exact approach that charges the holding and shortage costs based on inventory on hand and backlogged demand, respectively at the end of each period. Let h′ be the holding cost per unit per period (irrespective of its length), and p the shortage cost per unit per period. If T is constant, H(y) is expressed by

HðyÞ ¼ Zy þ 0 h′ðy–DÞgðDjTÞ dDþZ 1 y þ pðD–yÞgðDjTÞ dD; ð4Þ

where ( )þ max{ , 0}. For the present model in which T is stochastic, H(y) is given by

HðyÞ ¼ Z y þ 0 h′ðy–DÞgn_{ðDÞ dDþ}Z 1 y þ pðD–yÞgn_{ðDÞ dD} _ð5Þ

where g*( ) ET[g( |T)]. However, since the length of a period is

not constant, the holding cost could be computed in proportion to it. Let h be the holding cost per day per unit held at the end of a period. Then H(y) is given by

HðyÞ ¼ Z y þ 0 hE½Tðy–DÞg′ðDÞdDþZ 1 y þ pðD–yÞgn_ðDÞdD; _ð6Þ

where g′( ) ET[Tg( |T)]/E[T]. Note that the two pdf′s g*( ) and

g′( ) do not differ much, especially if T′s variability is not large, and h′ should be equal to hE[T].

We can easily verify that H(y), given by either (5) or (6), is convex. Thus, we have the following theorem.

Theorem 1. The optimal policy for Vn(x) in (3) is of the (s, S) type.

Proof. H(y) is convex and Vn(x) is in the form of expression (1) of

Iglehart (1963).□

Hence, a stationary (s, S) policy is optimal for the inﬁnite horizon model; in other words, if x r s, order the amount S – x; otherwise if x4 s, do not order. Deﬁne

GðyÞ cyð1βÞþHðyÞ: ð7Þ To compute the two optimal operational parameters s* and S*, we use G(y) and the discount renewal density βφ( ) in a solution procedure (e.g., Veinott and Wagner, 1965). If α ¼ 0 (i.e., the undiscounted-cost criterion is used), we use H(y) and the density ET[g( |T)] instead. For discrete demand distributions, Zheng and

Federgruen (1991)had developed an efﬁcient algorithm to obtain s* and S*, along with the long-run average cost C(s, S). If T is constant, denote by s′ and S′ the two optimal parameters obtained. If K ¼ 0, an order-up-to policy is optimal and the optimal S* is found by minimizing G(y) (Veinott and Wagner, 1965), which is the expected cost of the upcoming period (not including the constant procurement cost cβED[D]). In other words, S* is the

solution to Z 1

S

gnðDÞdD ¼ ½cð1–βÞþh′=ðh′þpÞ ð8Þ if H(y) given by (5) is used, or the solution to

Z 1 S hE½Tg′ðDÞdDþZ 1 S pgnðDÞdD ¼ cð1–βÞþhE½T ð9Þ if H(y) given by (6) is used. Since p should be greater than c(1– β), S* is guaranteed to be obtained.

3. Lost-sales periodic review inventory models

Next, we consider the situation where demand not satisfied at once is lost. Assume that replenishment is immediate. Let z be the order quantity placed at a review epoch and redefine x as the starting on-hand inventory. Then Vn(x) satisfies the recursive

equation

VnðxÞ ¼ minzZ 0fKδðzÞþczþHðxþzÞþET½eαTEDjT½Vn1ððxþz–DÞþÞg

¼ minxr yfKδðy–xÞþcyþHðyÞþET½eαTEDjT½Vn1ððy–DÞþÞg–cx:

ð10Þ Eq.(10)is the same as expression (9) ofChiang (2008), except that aﬁxed ordering cost is presented. Through a transformation as in (2), we can rewrite (10) by

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Veinott and Wagner (1965)showed that (11) can be viewed as a backlog model in which a credit of βc is given to each unit of demand actually backlogged. Thus, the following result holds. Theorem 2. The optimal policy for Vn(x) in (11) is of the (s, S) type.

Note that G(y) in (7) is replaced by GðyÞ cyð1–βÞþHðyÞ–βcZ

1

y ðD–yÞφðDÞdD;

ð12Þ where the last term is the credit given to demand not satisﬁed. If α ¼ 0, (12) simpliﬁes to GðyÞ ¼ Z y 0 h′ðy–DÞgn_ðDÞdDþZ1 y ðp–cÞðD–yÞ g n_ðDÞdD; _ð13Þ

if the holding cost is charged irrespective to the length of a period, or simpliﬁes to GðyÞ ¼ Z y 0 hE½Tðy–DÞg′ðDÞdDþZ 1 y ðp–cÞðD–yÞ g n_ðDÞdD; _ð14Þ

if the holding cost is computed in proportion to the length of a period. Notice that p (the shortage cost per unit) has a different meaning in the lost-sales problem; it should be larger here, since it usually includes the sales price.

Suppose now that there is a positive (constant) lead-time L for replenishment which is less than or equal to the minimum T (i.e., there is at most one outstanding order at any time). The dynamic program can be formulated by adding Kδ(z) into expres-sion (8) of Chiang (2008). However, the resulting program becomes more difﬁcult to solve than expression (8) of Chiang (2008)(see references therein). If K is small or can be neglected, we suggest that one uses the heuristic policy proposed inChiang (2008). If K is large, Section 4 shows that ignoring the period length variability incurs insigniﬁcant or no losses for the case of L ¼ 0. We suspect that this is also true for L 4 0, given the result in

Table 6(explained below) and theﬁnding inChiang (2008)that a positive L would dilute the effect of the variable T on expected cost. Hence, one can safely use the dynamic programming for-mulation (17)–(19) ofChiang (2007)where T isﬁxed, and solve it directly (see Chiang, 2007 for more details and computational results).

4. Computational results

We investigate the effect of the variable T on expected cost, if a ﬁrm fails to incorporate it when developing inventory control policies. The common data used are α ¼ 0, h ¼ $0.1 (i.e., the holding cost is charged per day per unit held at the end of a period), and E[T] ¼ 5 days. We assume that the period length T is either triangularly or uniformly distributed. In the former case, Pr (T ¼ 4) ¼ Pr(T ¼ 6) ¼ 1/4 and Pr(T ¼ 5) ¼ 1/2; in the latter case, Pr(T ¼ 4) ¼ Pr(T ¼ 5) ¼ Pr(T ¼ 6) ¼ 1/3. Also, we assume that demand is Poisson or compound Poisson distributed. For Poisson demand,μ ¼ 5/day. Let pidenote the probability of the order size

of i units for each customer arrival. If demand is compound Poisson, thenμ ¼ 2/day and p1 ¼ p2 ¼ p3 ¼ p4 ¼ 0.25 and pi

4 0 for i 4 4 (hence, the mean of demand still equals 5 per day but the variance is larger than that of the simple Poisson case).

Consider the backlogged model. Supposeﬁrst that K ¼ $0 and L ¼ 0. As we see fromTables 1and2, ignoring T′s variability, i.e., using S′ of the ordinary periodic model when in fact T is random, can incur unnecessary large costs, especially if T is uniformly distributed (i.e., has a larger variability) or p is high. Moreover, it appears that the variable T has a less signi_{ﬁcant impact on cost if} demand variability is large (i.e., demand is compound Poisson). Now if L ¼ 4 (other things being equal as inTable 1), we see from

Table 3that a positive L will dilute the effect of the variable T on cost. These results agree with those found inChiang (2008).

Table 1

Effect of variable T on cost (K ¼ 0, L ¼ 0, and demand is Poisson).

p S′ (A) T is triangularly distributed (B) T is uniformly distributed

S* G(S*) G(S′) % off S* G(S*) G(S′) % off

5 32 33 5.60 5.74 2.5 34 5.79 6.09 5.2

20 35 38 7.45 8.32 11.7 38 7.68 9.02 17.4

80 38 41 9.12 10.97 20.3 42 9.30 12.08 29.9

Table 2

Effect of variable T on cost (K ¼ 0, L ¼ 0, and demand is compound Poisson).

5 37 38 8.97 9.02 0.6 38 9.14 9.22 0.9

20 44 45 12.22 12.31 0.7 46 12.43 12.62 1.5

80 49 52 15.18 15.69 3.4 52 15.44 16.24 5.2

Table 3

Effect of variable T on cost (K ¼ 0, L ¼ 4, and demand is Poisson).

5 54 55 6.87 6.97 1.5 56 7.03 7.22 2.7

20 59 61 9.14 9.33 2.1 61 9.33 9.70 4.0

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Next, consider the case of K 4 0. Assume L ¼ 0. We show in

Tables 4and5the effect of the variable T on the long-run average cost. As we see, T′s variability impacts less on the cost than when K ¼ 0 inTables 1and 2, respectively. This is because a positive K usually yields a higher S, thus reducing the effect of a possibly long T on shortage. However, if K is small enough, s* is not very lower than S* (which may equal its counterpart when K ¼ 0), indicating that one actually uses an order-up-to policy (i.e., K is always incurred). Nevertheless, if K is not small enough (such that an order may not be placed at a review epoch), ignoring T′s variability can still cause large losses, especially if demand variability is small, shortage is costly, and/or T′s variability is high. However, as K

becomes larger (K ¼ $80, for example), aﬁrm incurs smaller or virtually no losses if using the optimal s′ and S′ of the ordinary periodic model. Now if L ¼ 4 (other things being equal as in

Table 4), we observe fromTable 6that a positive L will dilute the effect of the variable T on the long-run average cost. In fact, as L becomes longer, the variable T has a less (and eventually no) impact on cost (more computational results are available from the author upon request).

Finally, consider the lost-sales model with zero lead-time. Computation (for α ¼ 0) will yield the same operational para-meters as inTables 1and2(if K ¼ 0) or inTables 4and5(if K4 0), if p is varied such that (p– c) is the same as p in the backlogged

Table 4

Effect of variable T on cost (K4 0, L ¼ 0, and demand is Poisson).

K p (A) T is triangularly distributed (B) T is uniformly distributed

(s′, S′) (s*, S*) C(s*, S*) C(s′, S′) % off (s*, S*) C(s*, S*) C(s′, S′) % off 10 5 (23, 32) (24, 34) 15.59 15.74 1.0 (24, 34) 15.79 16.09 1.9 20 (28, 35) (30, 38) 17.45 18.32 5.0 (32, 38) 17.68 19.02 7.6 80 (32, 38) (34, 41) 19.12 20.97 9.7 (35, 42) 19.30 22.08 14.4 20 5 (21, 56) (21, 57) 22.07 22.10 0.1 (21, 58) 22.29 22.34 0.2 20 (27, 60) (28, 63) 24.60 24.82 0.9 (29, 63) 24.92 25.27 1.4 80 (31, 64) (33, 67) 26.66 27.20 2.0 (34, 68) 27.02 27.89 3.2 50 5 (18, 80) (18, 81) 34.42 34.43 0.0 (18, 82) 34.52 34.54 0.1 20 (25, 85) (26, 87) 37.36 37.46 0.3 (26, 88) 37.60 37.76 0.4 80 (30, 89) (31, 92) 39.65 39.88 0.6 (32, 93) 39.90 40.35 1.1 80 5 (16, 102) (16, 101) 43.40 43.40 0.0 (16, 100) 43.34 43.35 0.0 20 (24, 108) (25, 108) 46.52 46.56 0.1 (25, 108) 46.69 46.76 0.1 80 (29, 112) (31, 113) 48.92 49.11 0.4 (31, 113) 49.14 49.49 0.7 Table 5

Effect of variable T on cost (K4 0, L ¼ 0, and demand is compound Poisson).

(s′, S′) (s*, S*) C(s*, S*) C(s′, S′) % off (s*, S*) C(s*, S*) C(s′, S′) % off 10 5 (25, 38) (25, 40) 18.59 18.68 0.5 (26, 41) 18.69 18.85 0.9 20 (33, 44) (34, 46) 22.00 22.22 1.0 (35, 47) 22.18 22.56 1.7 80 (40, 50) (42, 52) 25.03 25.36 1.3 (42, 53) 25.26 25.83 2.3 20 5 (22, 60) (23, 61) 24.37 24.39 0.1 (23, 61) 24.45 24.48 0.1 20 (31, 68) (32, 69) 28.32 28.40 0.3 (33, 70) 28.48 28.61 0.5 80 (38, 74) (40, 76) 31.72 31.97 0.8 (41, 77) 31.95 32.34 1.2 50 5 (19, 83) (19, 84) 35.99 35.99 0.0 (19, 84) 36.03 36.03 0.0 20 (29, 92) (30, 92) 40.31 40.32 0.0 (30, 93) 40.44 40.46 0.1 80 (36, 98) (38, 100) 43.91 44.06 0.3 (38, 100) 44.10 44.35 0.6 80 5 (16, 100) (17, 100) 44.53 44.53 0.0 (17, 100) 44.55 44.56 0.0 20 (27, 108) (28, 109) 49.07 49.11 0.1 (28, 110) 49.18 49.24 0.1 80 (35, 115) (37, 117) 52.78 52.89 0.2 (37, 117) 52.95 53.15 0.4 Table 6

Effect of variable T on cost (K4 0, L ¼ 4, and demand is Poisson).

(s′, S′) (s*, S*) C(s*, S*) C(s′, S′) % off (s*, S*) C(s*, S*) C(s′, S′) % off 10 5 (44, 54) (44, 55) 16.86 16.97 0.7 (45, 56) 17.02 17.22 1.2 20 (50, 59) (52, 61) 19.14 19.33 1.0 (52, 61) 19.33 19.70 1.9 80 (55, 63) (57, 65) 21.14 21.57 2.0 (58, 66) 21.36 22.08 3.4 20 5 (42, 77) (42, 78) 22.90 22.93 0.1 (42, 79) 23.07 23.13 0.3 20 (49, 83) (50, 85) 25.85 25.94 0.3 (50, 86) 26.11 26.29 0.7 80 (54, 87) (56, 90) 28.29 28.66 1.3 (56, 91) 28.60 29.18 2.0 50 5 (38, 101) (38, 102) 35.04 35.05 0.0 (38, 103) 35.13 35.15 0.1 20 (47, 107) (47, 109) 38.44 38.49 0.1 (48, 110) 38.62 38.71 0.2 80 (52, 112) (54, 115) 41.10 41.37 0.7 (54, 116) 41.34 41.77 1.0 80 5 (36, 122) (36, 122) 43.81 43.81 0.0 (36, 121) 43.85 43.85 0.0 20 (45, 130) (46, 130) 47.50 47.56 0.1 (46, 129) 47.64 47.72 0.2 80 (51, 135) (53, 136) 50.30 50.50 0.4 (53, 135) 50.49 50.81 0.6

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model. This result can be seen by comparing, for example, (14) to (6).

5. Conclusions

In this paper, we consider periodic inventory models where the review periods are of a variable length and there is aﬁxed cost of ordering for replenishment. Assuming that period lengths are independently and identically distributed, we show that an (s, S) policy is optimal for the inﬁnite horizon problem. Hence, existing algorithms could be used to obtain the optimal s and S. The periodic review inventory policies developed in this paper are thus easy to implement.

The computation shows that when theﬁxed ordering cost is small, ignoring the period length variability can incur large costs if lead-time is zero, shortage is costly, and/or demand variability is small. It also shows that aﬁrm is more vulnerable to the period length variability if the period length is volatile. These results agree with those inChiang (2008); hence, the suggestions made in

Chiang (2008)apply here.

The computation also shows that when thefixed ordering cost is large, ignoring the period length variability incurs insignificant or no loss, particularly if lead-time is long or demand variability is large. This means that one need not use the proposed periodic review models in the case of largefixed ordering costs.

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