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Journal of China University of Science and Technology Vol.50- 2011. 11
Abad 之供應商暫時降價存貨模式的最適封閉解
Closed-Form Optimal Solutions for Abad’s Inventory Models
When Supplier Offers Temporary Price Reduction
摘要
以一個零售商而言,對供應商的臨時調整價格必須立即做出採購與否的反應
不是一件容易的事。當供應商公告在某一時間點其供貨價格將有所調整時，零售
商會依自身的庫存量來決定下次訂貨量的多寡。一般當供應商突然採取暫時性的
降價促銷時，零售商會大量採購以節省採購成本， Abad (2003) 探討了供應商將
商品售價暫時性降價後零售商採購的兩種模式。本篇文章中將探討 Abad (2003) 所
討論的兩種模式，針對其尋找最適訂貨量及額外獲利提出一正確且快速簡明的修
正方法；文章中所提出的方法為一封閉解的型式，可廣泛運用到現有的存貨模型
中增加決策之效益。
Abstract
It’s not so easy for a retailer to make an instantaneously respond to the
procurement decision while supplier announces the sales price may change after a
decided time. Due to the replenishment is always occurring when the inventory is
depleted and the inventory level is zero. In general, retailers may depend on their own
inventory levels or the time of the last opportunities to purchase larger quantity before
some or all of the cost parameters may change and aim to save the costs. Abad (2003)
presented the inventory system when supplier offers temporary price reduction as two
models. In this article, we deal with alternative approaches to present simple solutions
in order to decide the two Abad (2003) models. We derive the closed-form solutions to
find the profit and the number of lots purchased by the reseller during the promotion.
The method is easy-to-apply and effective in decision making.
Keywords: Temporary price reduction; Discounted price, trade promotion, closed-form
solution
I. Introduction
The economic order quantity (EOQ) model is popular in supply chain management.
1
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
The traditional EOQ inventory model supposes that the inventory parameters are
constant during the sale period. Schwarz (1972) discussed the finite horizon EOQ model,
the costs of the model were static and the optimal ordering number can be found during
the finite horizon. In real life, there are many reasons for suppliers offer a temporarily
price discount to retailers. The retailers may engage in purchasing additional stock at
reduced price and sale at regular price later. Lev and Weiss (1990) considered the case
where the cost parameters may change, and the horizon may be finite as well as infinite.
However, the lower and upper bounds they used do not guarantee boundary conditions
are met. Tersine (1994) proposed a temporary price discount model, the optimal EOQ
policy is obtained by maximizing the difference between regular EOQ cost and special
ordering quantity cost during the sale period. Martin (1994) revealed that Tersine’s
(1994) representation of average inventory in the total cost is flawed, and suggested the
true representation of average inventory. But Martin (1994) sacrificed the closed-form
solution in solving objective function, instead of using search methods to find special
order quantity and maximum gain. Wee and Yu (1997) assumed that the items
deteriorated exponentially with time and temporary price discount purchase occurred at
the regular and non-regular replenishment time. Abad (1997) provided a procedure to
determine the optimal response of reseller when the supplier offers a temporary
reduction in price. he assumed that the trade promotion is offered by the supplier only at
one point in time and the market demand for the product is elastic with respect to the
selling price. The reseller may engage in purchasing additional stock at the reduced
price offered by the supplier for later sale at the regular selling price. Abad (2003)
extended the analysis to the case where the promotion lasted a finite time interval and
he considered two cases. Nevertheless, there exist some flaws on Abad (2003)
computational procedure. Sarker and Kindi (2006) proposed five different cases of the
discount sale scenarios in order to maximize the annual gain of the special ordering
quantity. Kovalev and Ng (2008) shown a discrete version of the classic EOQ problem,
they assumed that time and the product are continuously divisible and demand occurs at
a constant rate. Cárdenas-Barrón (2009b) pointed out there are some technical and
mathematical expression errors in Sarker and Kindi (2006) and presented the closed
form solutions for the optimal total gain cost. Li (2009) presented a solution method
which modified Kovalev and Ng’s (2008) search method to find the optimal number of
orders. Cárdenas-Barrón et al. (2010) proposed economic lot size model that the
supplier offers a temporary discount and specified a minimum quantity of additional
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Journal of China University of Science and Technology Vol.50- 2011. 11
units to purchase. García-Laguna et al. (2010) illustrated a method to obtain the solution
of the classic EOQ and economic production quantity models when the lot size must be
an integer quantity. Their approach obtained a rule to discriminate between the situation
in which the optimal solution is unique and when there are two optimal solutions.
Chang et al. (2011) used closed-form solutions to solve Martin (1994) and Wee and Yu
(1997) EOQ models with a temporary price discount. Other authors also considered
similar issues have been performed by Khouja and Park (2003), Wee et al. (2003),
Cárdenas-Barrón (2009a), etc.
The purpose of this article is to present two algorithms that are simpler than Abad’s
(2003) computing procedure. Either in any one of Abad’s (2003) two cases, we propose
closed-form solutions to represent the computing procedure. Only follow the steps we
suggested, you can quickly and correctly find the number of lots and the incremental
profit.
II. Notation
p0
p
We will follow the same notation used in Abad (2003).
the reseller’s regular selling price to customer
the reseller’s discounted selling price to customer, p  p0
D( p0 ) the annual customer demand associated with the selling price p 0 , D0  D( p0 )
D ( p ) the annual customer demand associated with the selling price p , D  D( p )
the fixed ordering receiving and placement cost for the reseller
C
the per unit price charged by the supplier to the reseller
v
r
the annual holding cost fraction for the reseller
t0
the cycle time in regular policy, t 0  2C / rvD0
the reseller’s profit rate under regular policy, W0  ( p0  v  2Crv / D0 ) D0
W0
the temporary price reduction offered by the manufacturer to the reseller ($/unit)
the duration over which the promotion is offered by the manufacture (years)
the number of lots purchased by the reseller during the promotion
m
Q
the lot size by the reseller for each cycle, Q  DT / m
 ( p, m) the incremental profit over time span T
d
T


the duration in which the selling price from reseller to customer is p
the duration in which the selling price from reseller to customer is p 0
 ( p, m, , ) the incremental profit over time span T    
3
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
III. Model formulation
3.1 Reseller’s response when the discount is applicable on units resold during the
supplier’s promotion
The inventory pattern for Abad’s (2003) first model is shown in Fig. 1. When
reseller sets regular selling price p 0 to customer, the reseller’s profit rate under regular
policy is W0 ,
p 0 can be solved by p  D / D   v  Crv / 2 D ,
W0  ( p0  v  2C r v/ D0 ) D0 and D  dD / dp . There are relations between annual
demand D ( p ) and the selling price p which the reseller sets. The higher selling
price p , the lower annual demand D ( p ) is. The supplier offered temporary price
reduction d to reseller during the interval [0, T ] . The supplier can monitor the
reselling price and insure that discount is applicable only on ‘sell-through’ units. During
the interval [0, T ] , the product discounted selling price is p . There are m identical
cycles with cycle time T / m . The last cycle ends at time T where inventory is zero.
The reseller’s problem is to maximize the additional profit during time interval [0, T ] .
The objective function is
max  ( p, m)  ( p  v  d ) DT 
hDT 2
 mC  TW0
2m
(1)
where h  r (v  d ) . We define Hessian matrix as below
  2
 2   (2 D  2  DD )T CD  



pm  
p 2
D
D 
H  2

2
CD 
2C 
   
 

 pm m 2  
D
m


Because Hessian matrix is a semi-negative definite matrix,  ( p, m) would exist
maximum value in Eq.(1). Let p (m ) be the solution of

 0 . i.e., p (m ) is the
p
solution of
p
D
hT
vd 
D
2m
(2)
Hence, Eq. (1) can be represented by
z (m)   [ p(m), m]  [ p(m)  v  d ]DT 
hDT 2
 mC  TW0
2m
(3)
dz (m) DhT 2
d 2 z (m)
DhT 2


C


 0 . We can say that z (m) is a
,
dm
2m 2
dm 2
4m 3
concave function of m . It means that there exists integer m such that z (m) has
Because
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Journal of China University of Science and Technology Vol.50- 2011. 11
maximum value. By z (m  1)  z (m)  0 , we can find
 1
1 DhT 2 
m   


2
4
2C 

(4)
Taking m into Eq. (2),
p
D
hT
vd 
D
 1
1 DhT 2 
2  


4
2C 
 2
(5)
We can find the value of p (m ) from the above equation. The complete algorithm for
solving  ( m, p ) is shown below:
Algorithm 1
Step 1. Compute reseller’s regular selling price p 0 by
p
where D 
D
Crv
v
D
2D
dD
, and D0  D( p0 )
dp
Step 2. Compute reseller’s profit rate W0 by
W0  ( p0  v 
2C r v
) D0
D0
Step 3. Compute m by
 1
1 DhT 2
m   

4
2C
 2



Step 4. Compute reseller’s discounted selling price p by
p
D
hT
vd 
D
 1
1 D h T2 
2  


4
2C 
 2
where D  D( p) , h  r (v  d ) .
Step 5. The additional profit  ( p, m) during interval [0, T ] is
 ( p, m)  ( p  v  d ) DT 
5
hDT 2
 mC  TW0
2m
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
3.2 Reseller’s response when the discount is applicable on units purchased during
the supplier’s promotion
The inventory pattern for Abad’s (2003) second model is shown in Fig. 2. The
supplier offered temporary price reduction d to reseller during the interval [0, T ] . The
supplier cannot monitor the reselling price. The reseller can purchase a large lot before
the end of the promotion and sell a portion of the large lot later at the regular price.
There are m identical cycles with cycle time T / m . The last cycle ends at time T
where inventory is zero. At T the reseller purchased a large lot. During the interval
[0, T   ] , the product discounted selling price is p . During the interval
(T   , T     ] , the product regular selling price is p 0 . The reseller’s problem is to
maximize the additional profit during the interval [0, T     ] . The objective function
is
max
 (m, p,  , )  ( p  v  d ) D(T   )  ( p 0  v  d ) D0  (m  1)C
DT 2 D 2 D0 2
 h(


 D0 )  (T     )W0
2m
2
2
(6)
subject to   0 ,   0 , where h  r (v  d ) .
Because the model during the interval [T , T     ] in Fig. 2 is like the model
during the interval [0, z   ] suggested by Abad (1997) , shown in Fig. 3. We can use
the procedure suggested by Abad (1997) to find p , D ,  ,  . And the value m in
Eq. (6) is obtained by Eq. (4). The complete algorithm for solving  (m, p, , ) is
shown below:
Algorithm 2
Step 1. Using the step 1 and step 2 in Algorithm 1 to find p 0 and W0 .
Step 2. Compute reseller’s discounted selling price p by
G ( p)  G ( p0 )
2G ( p) 
D  0
D  D0
where G ( p)  ( p  v  d ) D , G ( p) 
dG
and D  D( p) .
dp
Step 3. Compute  by

[ p  (v  d )]D  [ p0  (v  d )]D0
r (v  d )( D  D0 )
Step 4. Compute  by
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Journal of China University of Science and Technology Vol.50- 2011. 11
 
d  2C r v/ D0
r (v  d )

Step 5. Compute m by
 1
1 D h T2 
m   


4
2C 
 2
where h  r (v  d ) .
Step 6. The additional profit  (m, p, , ) during time interval [0, T     ] is
 (m, p,  , )  ( p  v  d ) D(T   )  ( p 0  v  d ) D0  (m  1)C
 h(
DT 2 D 2 D0 2


 D0 )  (T     )W0
2m
2
2
IV. Numerical example
We use the same data of Abad (2003) to show that we can quickly and correctly
find the additional profits  ( p, m) and  (m, p, , ) . Table 1 and Table 2 are the
results of comparing our method to Abad’s (2003) method. Using Abad’s (2003) data,
v  $8 / unit , d  $8 / unit , C  $80 / order , r  0.5$ / $ / yr , T  0.25yr and
demand D( p)  10000000 p 3 . For first case, the discount is applicable on units resold
during the supplier’s promotion, we can follow steps in Algorithm 1 and obtain
p0  12.26$ / unit , W0  21254$ / yr , p  11.025$ / unit , D  7462 unit/yr , m  3 and
the additional profit is  ( p, m)  6084.85$ . For second case, the discount is applicable
on units purchased during the supplier’s promotion, we can follow steps in Algorithm 2
and obtain p0  12.26$ / unit , W0  21254$ / yr , p  11.29$ / unit , D  6956 unit/yr ,
  0.177 yr ,   0.141 yr ,
and the additional profit is
m3
 (m, p, , )  2294.69$ . Either in case 1 or case2, we can quickly and correctly find
the number of lots m and the incremental profit. Our result in case 2, the incremental
profit  (m, p, , )  2294.69$ is larger than Abad’s incremental profit
 (m, p, , )  2289.6$ .
V. Conclusion
The main purpose of this paper is to propose easy-to-apply methods in order to
determine Abad’s (2003) temporary price reduction over an interval offered by the
supplier. The two algorithms we proposed not only solve the tediously numerical
7
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
iteration calculation but also find the exact incremental profit. It can also apply to the
cases when supplier temporarily reduce sale price during the sale period. Numerical
examples show that the two algorithms proposed in this paper is accurate and rapid.
Table 1. Comparing our method to Abad’s (2003) method for case 1
p
m  ( p, m)
p0
W0
D
Abad’s method 12.26
Our method
12.26
21254
11.025 7462
3
6084.85
21254
11.025 7462
3
6084.85
Table 2. Comparing our method to Abad’s (2003) method for case 2
p

m  (m, p, , )
p0
W0

D
Abad’s method 12.26
21254
11.10 7312 0.156 0.162
3
2289.6
Our method
21254
11.29 6956 0.177 0.141
3
2294.69
12.26
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Journal of China University of Science and Technology Vol.50- 2011. 11
Inventory level
Q
time
0
T
m
2T
m
…
T  t0
T
Fig. 1. Adad’s (2003) Case 1 model.
Inventory level
Q
0
T
m
2T
m
…
T
T 
T   
time
Fig. 2. Adad’s (2003) Case 2 model.
inventory level
zD  D0
zD
z 
z
Fig 3. Adad’s model suggested in 1997.
9
time
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
References
1.
2.
3.
Abad PL. (1997). Optimal policy for a reseller when the supplier offers a
temporary reduction in price. Decision Sciences, 28, 737-649.
Abad PL. (2003). Optimal price and lot size when the supplier offers a temporary
price reduction over an interval. Computers & Operations research, 30, 63-74.
Cárdenas-Barrón, L.E. (2009a). Optimal ordering policies in response to a discount
offer: Extensions. International Journal of Production Economics, 122(2),
774-782.
4.
Cárdenas-Barrón, L.E. (2009b). Optimal ordering policies in response to a discount
offer: Corrections. International Journal of Production Economics, 122(2),
783-789.
5.
Cárdenas-Barrón, L.E., Smith, N.R. and Goyal, S.K. (2010). Optimal order size to
take advantage of a one-time discount offer with allowed backorders. Applied
Mathematical Modelling, 34(6), 1642-1652.
Chang, H.J., Lin, W.F. and Ho, J.F. (2011) Closed-form solutions for Wee’s and
Martin’s EOQ models with a temporary price discount. International Journal of
Production Economics, 131(2), 528-534.
6.
7.
García-Laguna, J., San-José, L.A., Cárdenas-Barrón, L.E. and Sicilia, J. (2010).
The integrality of the lot size in the basic EOQ and EPQ models: Applications to
other production-inventory models. Applied Mathematics and Computation, 216(5),
1660-1672.
8. Khouja, M. and Park, S. (2003). Optimal lot sizing under continuous price decrease.
Omega, 31(6), 539–545.
9. Kovalev, A. and Ng, C.T. (2008). A discrete EOQ problem is solvable in O(log n)
time. European Journal of Operational Research, 189(3), 914-919.
10. Lev, B. and Weiss, H. J. (1990). Inventory models with cost changes. Operations
Research, 38(1), 53-63.
11. Li, C.L. (2009). A new solution method for the finite-horizon discrete-time EOQ
problem. European Journal of Operational Research, 197(1), 412–414.
12. Martin, G.E. (1994). Note on an EOQ model with a temporary sale price.
International Journal of Production Economics, 37(2), 241-243.
13. Schwarz, L.B. (1972). Economic order quantities for products with finite demand
horizon. AIIE Transactions, 4, 234-236.
10
Journal of China University of Science and Technology Vol.50- 2011. 11
14. Sarker, B.R. and Kindi, M.A. (2006). Optimal ordering policies in response to a
discount offer. International Journal of Production Economics, 100(2), 195–211.
15. Tersine, R.J. (1994). Principles of Inventory and Materials Management, 4th ed.
Prentice-Hall, Englewood Cliffs, NJ.
16. Wee, H.M. and Yu, J. (1997). A deteriorating inventory model with a temporary
price discount. International Journal of Production Economics, 53(1), 81-90.
17. Wee, H.M., Chung, S.L. and Yang, P.C. (2003). Technical Note - A modified EOQ
model with temporary sale price derived without derivatives. The Engineering
Economist, 48(2), 190-195.
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