Download Iuuuiu

Journal of China University of Science and Technology Vol.49- 2011. 08
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
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 [7] 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 [7] 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的修正方法；文章中所提出的方
法為一封閉解的型式，尤其當有整數運算子存在數學模型中，可免除數值搜尋時
的費時與產生的誤差，可廣泛運用到現有的存貨模型中。
Abstract
Supplier announces some or all of cost parameters may change after a decided time.
According to the inventory is depleted at the time of the last opportunity to purchase
before some or all of the cost parameters may change, Abad [7] presented the inventory
system when supplier offers temporary price reduction as two models. In this article,
author deals with alternative approaches to present simple solutions in order to decide
the two Abad’s [7] models: (1) the discount is applicable to units resold during the
promotion, and (2) the discount is applicable to units purchased during the promotion.
Author derives the closed-form solutions to find the additional profit and the number of
lots purchased by the reseller during the promotion. Also author uses the same data of
Abad [7] to demonstrate the closed-form solutions can obtain better results than Abad’s
iteration method.
Keywords: Temporary price reduction; Discounted price, trade promotion
I. Introduction
The economic order quantity (EOQ) model is popular in supply chain management.
The traditional EOQ inventory model supposes that the inventory parameters are
1
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
constant during the sale period. Schwarz [1] 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 [2] 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 [3] 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 [4] revealed that Tersine’s [3]
representation of average inventory in the total cost is flawed, and suggested the true
representation of average inventory. But Martin [4] 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 [5] assumed that the items deteriorated
exponentially with time and temporary price discount purchase occurred at the regular
and non-regular replenishment time. Abad [6] 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 [7] 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’s [7] computational procedure. Sarker and
Kindi [8] proposed five different cases of the discount sale scenarios in order to
maximize the annual gain of the special ordering quantity. Kovalev and Ng [9] 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 [10]
pointed out there are some technical and mathematical expression errors in Sarker and
Kindi [8] and presented the closed form solutions for the optimal total gain cost. Li [11]
presented a solution method which modified Kovalev and Ng’s [9] search method to
find the optimal number of orders. Cárdenas-Barrón et al. [12] proposed economic lot
size model that the supplier offers a temporary discount and specified a minimum
quantity of additional units to purchase. García-Laguna et al. [13] illustrated a method
to obtain the solution of the classic EOQ and economic production quantity models
2
Journal of China University of Science and Technology Vol.49- 2011. 08
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. [14] used closed-form solutions to solve
Martin [4] and Wee and Yu [5] EOQ models with a temporary price discount. Other
authors also considered similar issues have been performed by Khouja and Park [15],
Wee et al. [16], Cárdenas-Barrón [17], etc.
The purpose of this article is to present two algorithms that are simpler than Abad’s
[7] computing procedure. Either in any one of Abad’s [7] 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
We will follow the same notation used in Abad [7].
the reseller’s regular selling price to customer
p0
p
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 )
C
v
r
t0
W0
the fixed ordering receiving and placement cost for the reseller
the per unit price charged by the supplier to the reseller
the annual holding cost fraction for the reseller
the cycle time in regular policy, t 0  2C / rvD0
the reseller’s profit rate under regular policy, W0  ( p0  v  2CRv / D0 ) D0
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    
.
III. Model formulation
3
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
3.1 Reseller’s response when the discount is applicable on units resold during the
supplier’s promotion
The inventory pattern for Abad’s [7] first model is shown in figure 1. When reseller
sets regular selling price p 0 , the reseller’s profit rate under regular policy is W0 ,
p  D / D   v  Crv / 2 D
where
can
be
solved
by
and
p0
W0  ( p0  v  2CRv / D0 ) D0 . 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
hDT 2
max  ( p, m)  ( p  v  d ) DT 
 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 
   
 

2
 pm m  
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
where D 
D
hT
vd 
D
2m
(2)
dD
. Hence, Eq. (1) can be represented by
dp
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
maximum value. By z (m  1)  z (m)  0 , we can find
Because
4
Journal of China University of Science and Technology Vol.49- 2011. 08
 1
1 DhT 2
m   

4
2C
 2



(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 reseller’s discounted selling price p by
p
D
hT
vd 
D
 1
1 DhT 2 
2  


4
2C 
 2
where D  D( p) , h  r (v  d ) .
Step 4. Compute m by
 1
1 D h T2 
m   


2
4
2C 

Step 5. The additional profit  ( p, m) during interval [0, T ] is
hDT 2
 ( p, m)  ( p  v  d ) DT 
 mC  TW0
2m
3.2 Reseller’s response when the discount is applicable on units purchased during
5
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
the supplier’s promotion
The inventory pattern for Abad’s [7] second model is shown in figure 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
 h(
DT 2 D 2 D0 2


 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 [6] , shown in fig.3. We can use the
procedure suggested by Abad [6] 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. 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 
2CRv
) D0
D0
Step 3. Compute reseller’s discounted selling price p by
G ( p)  G ( p0 )
2G ( p) 
D  0
D  D0
6
Journal of China University of Science and Technology Vol.49- 2011. 08
where G ( p)  ( p  v  d ) D , G ( p) 
dG
and D  D( p) .
dp
Step 4. Compute  by

[ p  (v  d )]D  [ p0  (v  d )]D0
r (v  d )( D  D0 )
Step 5. Compute  by
 
d  2C r v/ D0
r (v  d )

Step 6. Compute m by
 1
1 D h T2 
m   


4
2C 
 2
where h  r (v  d ) .
Step 7. 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 [7] 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 [7] method. Using Abad’s [7] 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
7
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
 (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 [7] temporary price reduction over an interval offered by the supplier.
The two algorithms we proposed not only solve the tediously numerical 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 [7] 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 [7] 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
8
Journal of China University of Science and Technology Vol.49- 2011. 08
Inventory level
Q
time
0
T
m
2T
m
…
T  t0
T
Fig. 1. Inventory pattern over time.
Inventory level
Q
0
T
m
2T
m
…
T
T 
T   
time
Fig. 2. Inventory pattern over time.
inventory level
zD  D0
zD
z 
z
Fig 3. Adad’s [6] model suggested in 1997.
9
time
Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers
Temporary Price Reduction
References
[1] Schwarz, L.B. (1972). Economic order quantities for products with finite demand
horizon. AIIE Transactions, 4, 234-236.
[2] Lev, B. and Weiss, H. J. (1990). Inventory models with cost changes. Operations
Research, 38(1), 53-63.
[3] Tersine, R.J. (1994). Principles of Inventory and Materials Management, 4th ed.
Prentice-Hall, Englewood Cliffs, NJ.
[4] Martin, G.E. (1994). Note on an EOQ model with a temporary sale price.
International Journal of Production Economics, 37(2), 241-243.
[5] 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.
[6] Abad PL. (1997). Optimal policy for a reseller when the supplier offers a temporary
reduction in price. Decision Sciences, 28, 737-649.
[7] 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.
[8] 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.
[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]Cárdenas-Barrón, L.E. (2009b). Optimal ordering policies in response to a discount
offer: Corrections. International Journal of Production Economics, 122(2), 783-789.
[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]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.
[13]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.
[14]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.
10
Journal of China University of Science and Technology Vol.49- 2011. 08
[15]Khouja, M. and Park, S. (2003). Optimal lot sizing under continuous price decrease.
Omega, 31(6), 539–545.
[16]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.
[17]Cárdenas-Barrón, L.E. (2009a). Optimal ordering policies in response to a discount
offer: Extensions. International Journal of Production Economics, 122(2), 774-782.
11