Journal of Scientific & Industrial Research 180
J SCI IND RES VOL 69 MARCH 2010
Vol. 69, March 2010, pp. 180-187
An inventory model with temporary price discount when lead time links to order quantity Chih-Te Yang1*, Liang-Yuh Ouyang2, Kun-Shan Wu3 and Hsiu-Feng Yen4 1 2
Department of Industrial Engineering & Management, Ching Yun Unversity, Jung-Li, Taoyuan 320, Taiwan
Department of Management Sciences and Decision Making, 3Department of Business Administration, 4Graduate Institute of Management Sciences, Tamkang University, Tamsui, Taipei 251, Taiwan Received 23 June 2009; revised 31 December 2009; accepted 04 January 2010
This paper presents effects of a temporary price discount offered by supplier on a retailer’s replenishment policy, when lead time is linked to order quantity. A decision process for retailers is developed in deciding whether to adopt a regular or special order policy during a temporary sales period. Optimal special order quantity is determined by maximizing total cost saving between special and regular orders, and illustrated by several numerical examples along with sensitivity analysis of optimal solution. Keywords: Inventory, Lead time, Temporary price discount
Introduction Various inventory models have been proposed to gain deeper insight into relationship between price discounts and order policy. Naddor1 proposed inventory models of response to a temporary price discount where special sale coincides with replenishment time. Barker & Vilcassim 2 studied situation where special sale coincides with retailer ’s sales period. Ardalan 3 developed optimal order policies with possible combinations of replenishment time and sales period. Tersine & Schwarzkopf4 relaxed assumption for length of special sale by permitting more than one replenishment cycle before sale expiration. Goyal5 discussed temporary price discount model for calculating average inventory of a regular policy. Aull-Hyde 6 presented a temporary price discounts model for backlogs permissible. Various studies7-10 are available of response to a temporary sale price. However, in these studies, almost all the lead time (LT) was assumed to be constant and zero. In reality, LT is non-zero and variable due to time to fill order at source, shipping time, and time required to carry out paperwork. Several studies1115 are available on inventory problems with variable LT. *Author for correspondence Tel: +886 2 2502998#4617; Fax: +886 3 4683298 E-mail:
[email protected]
This study investigates effects of a temporary price discount offered by supplier on retailer’s replenishment policy with LT linked to order quantity. Model Formulation LT of a special order policy Ls is always longer than that of regular order policy L , when retailer orders large quantities, Qs . This may lead to an inventory shortage before special order quantity arrives, implying that a shortage cost is incurred. Therefore, retailer may place a large special order of quantity Qs (adopt Ls ) or alternatively adopt L . Temporary Price Discount Occurs at Retailer’s Reorder Time Point ( ts
= tr )
In this situation, retailer may place a large special order of quantity Qs , or alternatively may place a regular order of quantity Q* , at discounted price (1 − δ ) c at retailer’s replenishment time, followed by EOQ policy with original price c for next replenishment (Fig. 1). Due to length of LT of Ls being greater than length of LT for L , a shortage occur and backorder may be placed if retailer decides to order for Qs . Therefore, total cost ( TCS1 ) consisting of costs for ordering, purchasing, shortage and holding is expressed as
YANG et al: TEMPORARY PRICE DISCOUNT EFFECT WHEN LEAD TIME LINKS TO ORDER QUANTITY
181
..... .....
Fig. 1—Special vs regular order policies when temporary price discount occurs at retailer's reorder time point ( ts
2
bD Q − Q* (1− δ )hc ( P − D)Qs + DQ* TCS1 = A + (1− δ )cQs + s + P 2 P 2D
2
…(1) If retailer adopts a regular EOQ policy instead of placing a large special order quantity policy during the period Qs / D , then total cost ( TCN 1 ) is as * TCN 1 = A + (1 − δ )cQ cQ +
+
(1 − δ ) hcQ * 2D
= tr )
Temporary Price Discount Does Not Occur At Retailer’s Reorder Time Point ( ts
≠ tr )
When supplier offers a temporary price discount at time point ts , if retailer decides to place a special order quantity Qs , then retailer receives items after time length Ls . Consequently, inventory level q before special order quantity Qs arrives is given by
2
q = Q * − D ts −
(Qs − Q * ) AD hcQ * * + cD + D 2 Q
DQ s . P
With regard to total cost saving, two possible cases …(2)
From Eqs (1) and (2), total cost saving between special and regular orders during time period Qs / D can be obtained as
are found based on value of q : (1) q ≥ 0 and (2) q < 0 (Fig. 2).
q≥0 When q ≥ 0 , it means that inventory level either
Case 1:
+ or 0 before Q s arrives. Total cost of special order during time period (Qs + q ) / D is given as
g1 (Q s ) = TCN 1 − TCS 1 2
=
Qs − Q * AD hcQ * (1 − δ ) hcQ* bD Qs − Q* − * + δ cD + + D Q P 2 2D 2
−
( 1 − δ ) hc ( P − D ) Q s + DQ 2D P
*
2
TCS 21 = A + (1 − δ ) cQ s + +
2
…(3)
(1 − δ ) hcQ s 2D
hc * DQ s Q − D ts − 2D P
DQ s * Q s + 2 Q − D t s − P …(4)
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J SCI IND RES VOL 69 MARCH 2010
Fig. 2—Special vs regular order policies when temporary price discount doesn't occur at retailer's reorder time point ( ts
Total cost of time period (Qs + q) / D under EOQ policy is represented by TCN21 =
g21(Qs ) = TCN21 − TCS21 =
2 DQs Qs AD hc * hcQ* Q − D ts − + * + cD + 2D P 2 D Q
≠ tr )
Qs AD hcQ* (1−δ )hcQs DQ Q + 2Q* − Dts − s − A * + δ cD+ − 2 2D s D Q P
…(6) Case 2:
q 0.
Thus, when temporary price discount occurs at retailer ’s reorder point, optimal special quantity
Qs* = Qs1 . Temporary Price Discount Does Not Occur at Retailer’s Reorder
…(9)
Point ( ts
1 AD hcQ* * + δ cD + 2 2DP [bD + (1 − δ )hc( P − D) ] D Q DP 2
2
Time Point ( ts Case 1:
q≥0
≠ tr )
[ Q s ≤ P ( Q * − Dt s ) / D ]
Taking first order derivative of g 21 (Qs ) in Eq. (6) with respect to Qs and letting result equal zero gives dg 21 (Q s ) 1 AD hcQ * (1 − δ ) hcQ s P − 2 D = * + δ cD + − dQ s D Q 2 2D P −
(1 − δ ) hc DQ * Q s + 2 Q − D ts − P 2D
s
= 0
…(13) By solving Eq. (13), a unique solution of Qs (say
Qs 21 ) is obtained as Qs 21 =
P[( AD / Q * + δcD + hcQ * / 2 ) − (1 − δ ) hc (Q * − t s D )] (1 − δ ) hc ( P − 2 D )
…(14)
2
and second order derivative of g 21 (Qs ) in Eq. (6) with …(11)
respect to Qs is
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d 2 g 21 ( Q s ) dQ s2
=−
(1 − δ ) hc P − 2 D D P
…(15) Substituting Eq. (14) into Eq. (6), corresponding maximum total cost saving for Case 1 is
g 21 (Q s 21 ) =
(1 − δ ) hcQ s221 2D
Given that d 2 g 22 (Qs ) / dQs2 < 0 , g 22 (Qs ) is a concave function of Qs , a unique value of Qs (say Qs 22 ) which maximizes g 22 (Qs ) can be found. Solving Eq. dg 22 (Qs ) / dQs = 0 gives Q s 22 =
P − 2D − A P
AD P hcQ* + δ cD + − (1− δ )hc(Q* − ts D) − A * 2(1− δ )hcD(P − 2D) Q 2
…(16) It is worthwhile noting that, when P ≤ 2 D , then g 21 (Qs 21 ) < 0 , which means that retailer’s total cost saving is negative. In this situation, optimal decision for retailer is not to order until next replenishment time, at which point they should order regular quantity Q * . ∆1 = Qs21 For convenience, assume that * * − P Q − Dts / D , ∆ 2 = g 21 (Qs 21 ) , and Qs21 is optimal special order quantity. Then
(
)
bD
Substituting Eq. (19) into Eq. (9), corresponding maximum total cost saving for Case 2 is
(
=
DP 2
bD + (1 − δ ) hc ( P − D ) 2 [ bD 2 + (1 − δ ) hc ( P − D ) 2 ] 2
if
Similarly, to obtain optimal special order quantity * ) for Case 2, let ∆ 3 = Q s 22 − P (Q − Dt s ) / D and ∆ 4 = g 22 (Q s 22 ) , then * ( Qs 22
Optimal special order quantity for Case 2 as follows:
dg 22 ( Q s ) hcQ * b DQ s 1 AD = − Q * + Dt s * + δ cD + − dQ s D Q 2 P P
(1 − δ ) hc Q D P − D * Q s + Q − D ts − s D P P
…(17) and 2 s
)2 − A
Theorem 2
Taking first and second order derivatives of g 22 (Qs ) in Eq. (9) with respect to Qs , gives
dQ
(
…(20)
Qs*21
d 2 g 22 ( Q s )
2
)
b + (1 − δ ) hc Q * − Dt s 2D
otherwise.
q < 0 [ Q s > P ( Q * − Dt s ) / D ]
−
−
P > 2 D, ∆1 ≤ 0 and ∆ 2 > 0,
= 0 , it means that optimal policy for When retailer is not to order until next replenishment time and at that time order regular order quantity Q* . Case 2:
1 AD hcQ* b (1 − δ)hc(P − D) + Q* − Dts − * + δ cD + 2 DP P D Q
(
Q s , = 21 0,
)
b + (1−δ )hc(P − D)2 2 b + (1− δ )hc * 2 g22(Qs22 ) = Q − Dts − A Qs22 − 2 2D 2DP
Optimal special order quantity for Case 1 as follows:
Remark 1
AD hcQ * + δ cD + Q* 2
…(19)
Theorem 1
Qs*21
2
b (1 − δ ) hc ( P − D ) + (Q * − Dt s ) − DP P
2
=
1 + (1 − δ ) hc ( P − D ) D DP 2
2
Qs , Qs*22 = 22 0,
if
∆3 > 0 and ∆4 > 0,
otherwise.
Summarizing above results, an algorithm is developed to obtain optimal solution, Qs* , for two situations. Algorithm Step 1
b D (1 − δ ) hc ( P − D ) 2 = − − P P DP
…(18)
Qs*
If t s = t r , then optimal special order quantity is = Qs1 . Go to Step 6.
YANG et al: TEMPORARY PRICE DISCOUNT EFFECT WHEN LEAD TIME LINKS TO ORDER QUANTITY
Table 1—Optimal solutions of Example 1 under different values of
P
tr
ts
1500
0.1054
0.1054
2000
2500
3000
0.1581
0.1897
0.2108
0.2259
*
P
TCN *
TCS *
g*
8248.44
7988.78
259.66
0.01
Qs 22 = 858.06
9394.66
8978.07
416.58
0.1581
Qs1 = 824.79
8864.78
8610.99
253.79
0.01
Qs 22 = 905.92
9918.69
9631.44
287.24
0.1897
Qs1 = 813.45
8740.63
8514.90
225.72
0.01
Qs 22 = 841.99
9218.78
9089.56
129.21
0.2108
Qs1 = 781.51
8390.91
8193.66
197.25
8029.11
7855.14
173.96
0.2259
0 Qs1 = 748.47
0.01
0
0.01 3500
Qs* Qs1 = 768.50
185
*
and TCS denote minimum total cost for regular and special order policy, respectively; g * is corresponding maximum total cost saving; — indicates retailer will not order until next replenishment time and adopts regular EOQ policy.
TCN
Step 2
Step 6
Determine values of Qs 21 and Qs 22 from Eq. (14) and Eq. (19), respectively. Step 3
Calculate
values
of
∆ 1 = Q s 21 − P Q − D t s / D ,
and
(
)
*
∆ 2 = g 21 (Qs21 ) in Eq. (16). If P > 2 D , * ∆1 ≤ 0 and ∆ 2 > 0 , then set Qs 21 = Qs 21 and g 21 (Qs*21 ) = ∆ 2 ; otherwise, set Qs*21 = 0 and g 21 (Qs*21 ) = 0 . Step 4
Calculate
values
of
*
∆ 3 = Qs 22 − P(Q − Dt s ) / D , ∆ 4 = g 22 (Qs22 ) in Eq. (20). If ∆ 3 > 0 * ∆ 4 > 0 , then set Qs 22 = Qs 22 g 22 (Qs*22 ) = ∆ 4 ; otherwise, set Qs*22 * and g 22 (Qs 22 ) = 0 .
and
max { g 21 ( Q s*21 ), g 22 ( Q s*22 )} .
If
and and
=0
Step 5
Find
g 21 (Qs*21 ) ≥ otherwise,
g 22 (Qs*22 Qs* = Qs*22 .
) , then
Qs*
= Qs*21
;
Stop. Once optimal solution is obtained, corresponding minimum total costs can be found for regular and special order, and subsequently determining maximum cost saving. Numerical Examples To illustrate solution process for two situations, following examples are presented Example 1
An inventory system with following data is considered: c = $10 / unit , D = 1000 units/year , A = $150 / order , b = $5 /unit/year, δ = 0.05 and h = 0.3 . From these data, it can be found that Q* = 316.23 . Sets of values of P and t s are
P ∈{1500, 2000, 2500, 3000, 3500} and t s ∈ {t r , 0.01} , where t r = (Q * / D) − L = (Q * / D) − (Q * / P) . From algorithm, optimal order policies depending on different values of P , t r and t s can be obtained (Table 1). assumed as
From Table 1, it is observed that retailer will determine optimal special order quantity by trading off benefits of price discount against additional holding cost and shortage cost, if necessary. For example, for
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J SCI IND RES VOL 69 MARCH 2010
t s = 0.01 and P = 3000 or 3500 in Table 1, optimal value of Qs* = 0 , which implies that retailer should ignore temporary price discount offered by supplier and will not order until next replenishment time and then adopt regular EOQ policy. For other cases, optimal policy of retailer is to order a special quantity. In addition, when supplier’s production rate increases, special order quantity increases at the beginning and decreases later. However, total cost saving decreases.
Table 2—Effect of changes in various parameters of Example 2 Parameter
Qs*
TCN
*
TCS *
g*
c
-50 -25 +25 +50
17.0 5.5 -3.0 -4.7
-39.4 -19.8 20.2 40.7
-39.0 -19.7 20.0 40.4
-52.0 -25.4 24.8 49.2
δ
-50 -25 +25 +50
-15.5 -14.5 7.9 15.8
-15.5 -14.5 7.9 15.8
-13.6 -12.7 6.8 13.5
-79.6 -59.6 44.9 93.3
D
-50 -25 +25 +50
0 -21.6 11.9 12.3
-20.5 10.9 10.5
-18.4 9.5 8.4
-91.4 57.7 80.8
A
-50 -25 +25 +50
-21.2 -9.7 8.6 16.3
-23.2 -10.8 9.7 18.6
-23.5 -10.9 9.8 18.9
-14.5 -5.7 3.8 6.3
h
-50 -25 +25 +50
34.2 13.2 -9.4 -16.6
30.8 11.9 -8.5 -15.0
30.3 11.7 -8.4 -14.8
47.1 17.7 -12.2 -21.3
b
-50 -25 +25 +50
15.1 6.1 -4.4 -7.8
15.1 6.1 -4.4 -7.8
15.2 6.2 -4.5 -7.8
13.7 5.6 -4.0 -7.1
Example 2
In this example, a sensitivity analysis is conducted demonstrating robustness of model, wherein effects of changes in system parameters ( c , δ , D , A , h , and b ) on optimal special order quantity and optimal total cost saving are examined. Data used are same as those in Example 1. For convenience, considering the case when P = 2000 and t s = 0.01 . Sensitivity analysis is conducted by changing each of the parameters by − 50% , − 25% , + 25% and + 50% and simultaneously altering one parameter at a time while all other parameters remain unchanged (Table 2). Based on results in Table 2, following managerial insights are gained: 1) When purchase cost c increases, special order quantity Qs* decreases. However, total cost for regular order policy TCN * , total cost for special order policy TCS * , and total cost saving g * increases. Thus as purchase cost increases, it is more beneficial for retailer to adopt special order policy. Nevertheless, special order quantity will decrease; 2) As Qs* , TCN * , * TCS * and g increase, δ increases. This implies that greater the price discount, larger the special order quantity, total cost of regular order, total cost of special order and total cost saving for retailer. Results are consistent with basic common theory. Moreover, δ is sensitive to total cost saving. From supplier’s viewpoints, offering retailer a high temporary discount rate will lead to increased sales volume; 3) Qs* , TCN * , TCS * and g * increase when D and A increase. It implies that when either market demand rate or ordering cost or both increases, retailer will order a larger quantity. Moreover, when market demand rate D is sufficiently low (value of D reduces to 50% in Table 2) it is not worthwhile for retailer to place a special order; and 4) Accordingly, when either h or b or both increases, Qs* , TCN * , * TCS * , and g decrease. It implies that when either holding costs or shortage costs or both are higher, retailer will order a lower quantity and total cost saving will decrease. Additionally, holding cost is more sensitive to optimal solutions than shortage cost.
Change, %
Change % e, %
Conclusions This study presented effects of a temporary price discount offered by supplier on a retailer’s replenishment policy. An algorithm was established to determine optimal order polices. Numerical examples were presented to illustrate solution procedure. Acknowledgements Authors greatly appreciate referees for their valuable and helpful suggestions. Notation and Assumptions Notation production rate of the supplier. P,
D, c, δ,
market demand rate. unit purchasing price. price discounted fraction from the supplier (0 < δ
A, h,
< 1 ).
ordering cost per regular or special order. holding cost rate.
YANG et al: TEMPORARY PRICE DISCOUNT EFFECT WHEN LEAD TIME LINKS TO ORDER QUANTITY
b, q,
shortage cost per unit per unit time.
L, Ls , r,
the length of lead time of the regular order quantity.
tr ,
the reorder time at reorder point of regular order policy.
ts ,
the time at which the supplier offers a temporary price
2
inventory level before the special order quantity arrives.
the length of lead time of the special order quantity.
3 4
the reorder point of regular order policy (in terms of inventory level).
discount.
Q,
order quantity under regular order policy.
Q* ,
optimal order quantity under regular order policy
Qs ,
order quantity at discount price.
Qs* ,
optimal order quantity at discount price.
5
6 7
8
Assumptions (1) Deterioration of item is not considered in the model. (2) Supplier offers retailer a temporary price discount at discount rate δ . Temporary discount occurs at only one instant in time. (3) Inventory is continuously reviewed and it is replenished whenever inventory level falls to r under regular order policy. (4) Prior to and after temporary price discount, retailer adopts an economic order quantity (EOQ) policy and optimal order
9
10
11 12 13
*
quantity is Q = 2 AD ( hc ) . (5) Length of lead time depends on order quantity. (6) Shortages are allowed and items can be backordered under special order policy. (7) None of the price discount is passed on to customers.
14
References
15
1
Naddor E, Inventory Systems (Wiley, New York) 1966, 96102.
187
Barker R C & Vilcassim N, Continuous review price change inventory model, Production & Inventory Mgmt, 24 (1983) 67-72. Ardalan A, Optimal ordering policies in response to a sale, IIE Trans, 20 (1988) 292-294. Tersine R J & Schwarzkopf A B, Optimal stock replenishment strategies in response to a temporary price reduction, J Business Logistics, 10 (1989) 123-145. Goyal S K, A comment on Martin’s: Note on an EOQ model with a temporary sale price, Int J Prod Econ, 43 (1996) 283284. Aull-Hyde R L, A backlog inventory model during restricted sale periods, J Operat Res Soc, 47 (1996) 1192-1200B. Bhavin J S, EOQ model for time-dependent deterioration rate with a temporary price discount, Asia-Pacific J Operat Res, 22 (2005) 479-485. Dye C Y, Chang H J & Wu C H, Purchase-inventory decision models for deteriorating items with a temporary sale price, Int J Inform & Mgmt Sci, 18 (2007) 17-35. Bhaba R S & Mahmood A K, Optimal ordering policies in response to a discount offer, Int J Prod Econ, 100 (2006) 195211. Abad P L, Buyer’s response to a temporary price reduction incorporating freight costs, Eur J Operat Res, 182 (2007) 10731083. Bramson M J, The variable lead-time problem in inventory control, Operat Res Qual, 13 (1962) 41. Kaplan R S, A dynamic inventory model with stochastic leadtimes, Mgmt Soc, 16 (1970) 491-507. Ouyang L Y, Chen C K & Chang H C, Quality improvement, setup cost and lead-time reductions in lot size reorder point models with an imperfect production process, Compu & Operat Res, 29 (2002) 1701-1717. Pan J C-H, Lo M C & Hsiao Y C, Optimal reorder point inventory models with variable lead time and backorder discount consideration, Eur J Operat Res, 158 (2004) 488505. Lee W C, Wub J W & Lei C L, Computational algorithmic procedure for optimal inventory policy involving ordering cost reduction and back-order discounts when lead time demand is controllable, Appl Math & Compu,189 (2007) 186-200.
