Download Production Cost Disruption Management of Retailer-led Supply Chain

Production Cost Disruption Management of Retailer-led Supply
Chain
1
1,2
YU Mingnan , DI Junfeng
1
School of Management, Dalian University of Technology, P.R.China, 116024
2
School of Economics and Management, Henan Polytechnic University, P.R.China, 454000
[email protected]
：
Abstract This paper studies a two-level supply chain model including one dominant retailer and one
manufacturer. Firstly, one buy back contract is designed to coordinate the supply chain without
disruption; Yet this coordination may be impaired when production cost disrupts and the optimal
production quantity is given when production cost disrupts; Finally, an adjusted buy back contract with
anti-disruption-ability is proposed.
Key words retailer-led buy back contract production cost disruption supply chain coordination
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1 Introduction
Presently, study of supply chain disruption mainly concentrates on demand fluctuation and supply
[1]
interruption or delay . Regardless of demand disruption and production cost disruption, new plan
revised must consider some extra expenses caused by the deviation plan. The first paper about supply
chain disruption was offered by Qi, Bard and Yu. They showed how the quantity discount contract
coordinates a two-tied supply chain with linear determinant demand and demand disruption
[3]
[4]
[2]
; Yu Hui,
for coping with
Chen Jian and Yu Gang explored wholesale price contract and buy back contract
demand disruption under random demand.
In all the literatures above, the manufacturer is dominant. But in consumable industrial age, the status of
retailer and manufacturer has varied, for example, medicine industrial, computer hardware, clothing,
[5]
electronic products, furniture industrial etc . Thus, this paper will investigate a retailer-led supply
chain including a manufacturer and a retailer under cost disruption, and random demand is assumed. In
contrast to previous literatures, this paper has such traits below (1) The retailer is the Stackelberg leader;
(2) the retail price is exogenous variable; (3) the demand is stochastic.
This paper proceeds as follows. Section 2 designs a buy back contract to coordinate the supply chain
without production cost disruption; the cooperation is concerned when production cost disrupts in
Section 3; conclusions are in Section 4.
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2 Buy back contract of retailer-led supply chain without disruption
2.1 Model Assumptions
♦
A short-life-cycle product ,one manufacturer with infinite capacity and one dominant retailer ;
♦
Leading time is zero and the retailer has no ordering cost ;
♦
The information is symmetrical and members are rational in supply chain system.
2.2 Basical notations
p Retail price, exogenous and determined
c Manufacturing cost per unit
r Loss of goodwill cost per unit due to stock out at the retailer
v Salvage value per unit of the manufacturer or the retailer
w Wholesale price per unit
587
q
b
x
；
Order size of the retailer
Return price of unit residual product offered by the manufacturer
Stochastic customer demand with differentiable and monotony increasing cumulated distribution
function F ( x) , probability distribution function f ( x ) , mean µ and standard deviation σ .
More notations will be defined later when needed. Without loss of generality, p > w > c > v
，
w ≥ b ≥ v .These inequalities ensure that each firm makes positive profit and the chain will not produce
infinite products. We can easily get retailer’s expected inventory is I ( q) =
∫
q
0
F ( x )dx , so expected
sale quantity is q − I (q ) and expected shortage quantity is µ − q + I ( q ) .
[6]
Because retailer is dominant , Zhongsheng Hua and Sijie Li assumed that the retailer has the ability to
influence the manufacturer’s wholesale price by setting his/her potential maximal order quantity, and
announce the sensitivity of his/her order quantity to the manufacturer’s wholesale price, that is:
q = q0 − β w
(1)
In Eq. (1),
β > 0 is the sensitivity of retailer’s order quantity to the manufacturer’s wholesale price;
parameter q0 is the retailer’s maximal order quantity considering the retail-market conditions.
2.3 Buy back contract without disruption
Retailer and manufacturer forecast the market demand x first, then retailer ascertains maximal possible
order quantity q0 , based on q0 , the manufacturer declares wholesale price w (in fact, from Eq. (1),
when q0 and w are ascertained, q is also ascertained),at last, they negotiate b .
The retailer’s expected profit is:
Eπ R (q0 , w, b) = ( p − w + r )(q0 − β w) + (b − p − r ) I (q0 − β w) − r µ
(2)
The manufacturer’s expected profit is:
Eπ M (q0 , w, b) = ( w − c )(q0 − β w) + (v − b) I (q0 − β w)
(3)
The profit of the whole channel in our model only depends on the order quantity, expected profit of
supply chain is:
Eπ (q) = ( p − c + r )q − ( p − v + r ) I (q ) − r µ
(4)
dE 2π R
= −( p − v + r ) f (q) < 0 , there exits only one q* and the expected profit of the
Because
2
dq
supply chain is maximum.
F ( q* ) =
p−c +r
p−v +r
(5)
Cachon explored the common form of buy back contract, for a “ λ ” ( 0 < λ < 1 , λ is the
manufacturer’s share from the supply chain’s expected profit), according to Eq. (2), (3) and (5), buy
back contract ( q0 , w, b should satisfy:
[7]
）
 w −c
b * −v
 p−c +r = p−v +r = λ

 * *
*
q = q0 − β w

p −c+ r
F (q* ) =
p −v+ r

*
(6)
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From Eq. (3), we have:
Eπ M = ( w − c)q + (v − b) I (q )
(7)
By substituting Eq. (4) and (6) into (7), we can get:
Eπ M = λ ( p − c + r )q − λ ( p − v + r ) I (q) = λ Eπ + λ r µ
(8)
Therefore, the manufacturer’s expected profit is the linear affine function of supply chain’s expected
profit. When q = q , Eπ M increases with λ ’s increasing, so manufacturer’s expected profit is the
*
increasing function of λ .
Coordination of supply chain has two targets: one is to maximize supply chain’s profit and the other is
to improve each member’s profits. Therefore, λ is confined to a given range and negotiated between
manufacturer and retailer.
3 coordination of retailer-led supply chain with cost disruption
For the original supply chain, manufacturer has prepared for production according to the demand
forecast and the optimal ordering quantity. Before the sale season arrives, disruption happens and the
~
~
production cost varies from c to c , and ∆c = c − c .Therefore, the optimal production quantity will
change. Here two parameters are introduced: θ1 > 0 is extra expense per unit when the capacity
increases and θ 2 > 0 is extra loss per unit when the capacity decreases.
~
*
~
*
~
*
Theorem 1 If the optimal capacity is q , then: when ∆c > 0 , q ≤ q ; when ∆c < 0 , q ≥ q .
~
*
*
*
~
*
Proof we assume q > q when ∆c > 0 .so q satisfies:
~
*
~
~
~
~
（ 9）
max E π ( q ) = ( p − ( c + ∆ c ) + r ) q − ( p − v + r ) I ( q ) − r µ − θ 1 ( q − q )
*
~
~
~2
~
*
For d ( E π ( q)) / d q < 0 , we can get the only q from
2
~
F (q* ) = p − c + r − ∆c − θ1 / ( p − v + r ) .
~
*
Because F is monotony increasing with F ( q ) < F ( q ) , which is contradictive inequality. A similar
*
~
*
proof can confirm the other result of this theorem. If the cost is less, we may get q ≥ q .
 ~*
q∨
~
 *
*
Theorem 2 q =  q
~
q∧*
~
*
∧
satisfies F ( q ) =
∆c ≤ −θ1
~
~
−θ1 < ∆c < θ 2 , where q∨* satisfies F (q∨* ) =
*
~
p − c + r − ∆c − θ1
*
and q∧
p−v+r
∆c ≥ θ 2
p − c + r − ∆c + θ 2
.
p−v +r
~
*
Proof For ∆c > 0 , then q ≤ q , the optimal capacity satisfies:
~
~
*
~
~
~
max E π (q) = ( p − (c + ∆c ) + r ) q − ( p − v + r ) I (q ) − r µ − θ 2 (q* − q )
589
(10)
~
~
~
~
~
~
We can easily know that E π ( q ) is a concave function of q .When d ( E π ( q)) / d q = 0 , we may get:
p − c + r − ∆c + θ 2
p−v +r
~
F (q ) =
Now we discuss the above equation as follows:
( ) ∆c ≥ θ 2
ⅰ
p − c + r − ∆c + θ 2
~
F (q) =
p−v+r
ⅱ)
≤
p−c+r
p−v+r
~
~
~
= F ( q* ) and q ≤ q* ,so the optimal capacity q* = q∧* .
0 < ∆c < θ 2
~ ~
p − c + r − ∆c + θ 2 p − c + r
F (q ) =
>
= F (q* ) and concave E π (q ) ,so the optimal capacity
p−v +r
p −v+r
(
~
~
q* = q * .
When ∆c < 0 , we may take the similar method to prove. Concluded from theorem 1 and theorem 2,
when the cost disruption is small, quondam plan is of strong robustness and needn’t be altered; when the
change is large, the plan should be adjusted to maximize supply chain’s profit. When ∆c > 0 and ∆c is
too larger, the manufacturer’s production quantity may be zero, which will cause the profit of supply
chain is negative and the supply chain ruptures.
Proposition 1: when the production disrupted, if we adopt former buy back contract, supply chain is not
coordinated.
Proof when cost disrupted, expected profit of manufacturer and supply chain are:
~
~
~
~
~
~
E π M (q, w, b) = ( w − c − ∆c ) q + (v − b) I (q ) − θ1 (q − q* )+ − θ 2 (q* − q )+
~
~
~
~
(11)
~
~
E π (q ) = ( p − (c + ∆c ) + r ) q − ( p − v + r ) I (q) − r µ − θ1 (q − q* )+ − θ 2 (q* − q )+
(12)
+
Where y = max( y, 0) , for y is arbitrary .Substituting Eq.6 and Eq.12 into Eq.11 gives:
~
~
~
~
~
~
~
E π M (q ) = λ E π (q ) + µλ r − (1 − λ )[∆c q + θ1 (q − q* ) + + θ 2 (q* − q )+ ]
(13)
It is obvious that manufacturer’s expected profit no longer is the affine function of supply chain’s
expected profit, so supply chain is uncoordinated.
Therefore, when the cost disrupts, it is essential to modify the former buy back contract and gain
anti-disruption buy back contract for new supply chain coordination.
Proposition 2 the buy back contract below may coordinate the supply chain with product cost
disruption:
~
 * ~*
*
=
+
q
q
β
w
 0
 ∆c + θ1
 ~

*
*
w = w + (1 − λ )φ (∆c ) .Where φ (∆c) =  ∆c
−∆c + θ
~
2

b* = b*

∆c ≤ −θ1
~
−θ1 < ∆c < θ 2 , q* satisfies theorem 2.
∆c ≥ θ 2
Proof we discuss in three situations:
( ) ∆c ≤ −θ1
ⅰ
~
~
~
~
Eπ M (q, w, b) = ( w − c − ∆c − θ1 ) q + (v − b) I (q) + θ1q*
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（14）
~
~
~
~
E π (q) = ( p − c − r − ∆c − θ1 ) q − ( p − v + r ) I (q ) − r µ + θ1q*
（15）
w − c − ∆c − θ 1
b−v
= λ .We may obtain that profit of manufacturer is the affine
=
p − c + r − ∆c − θ 1 p − v + r
Let
~
*
~
*
function, so supply chain is coordinated when q = q∨ .
(ⅱ) −θ1 < ∆c < θ 2
~
~
~
（16）
（17）
~
Eπ M (q, w, b) = ( w − c − ∆c) q + (v − b) I (q)
~
~
~
~
E π ( q ) = ( p − c − r − ∆c ) q − ( p − v + r ) I ( q ) − r µ
w − c − ∆c
b−v
=
= λ .We may obtain that profit of manufacturer is the affine function,
Let
p − c + r − ∆c p − v + r
~
*
so supply chain is coordinated when q = q .
(ⅲ) ∆c ≥ θ
~
*
2
~
~
~
E π M (q, w, b) = ( w − c − ∆c + θ 2 ) q + (v − b) I (q ) − θ 2 q*
~
~
~
~
E π ( q ) = ( p − c − r − ∆c + θ 2 ) q − ( p − v + r ) I ( q ) − r µ − θ 2 q *
Let
（18）
（19）
w − c − ∆c + θ 2
b−v
= λ .We may obtain that profit of manufacturer is the affine
=
p − c + r − ∆c + θ 2 p − v + r
~
*
~
*
function, so supply chain is coordinated when q = q∧ .
 ∆c + θ1

Let φ ( ∆c) =  ∆c
−∆c + θ

2
∆c ≤ −θ1
−θ1 < ∆c < θ 2 .Integrating(
∆c ≥ θ 2
ⅰ)、(ⅱ )and (ⅲ ), we may achieve that
supply chain is newly coordinated.
4 Conclusions
After analyzing coordination of retailer-led supply chain without and with cost disruption, we may
draw conclusions as below: retailer-led supply chain may be coordinated by buy back contract without
disruption; former buy back contract can’t coordinate the supply chain when product cost disrupts;
revised buy back contract may deal with cost disruption and coordinate the supply chain.
References
[1]Yu Hui,Chen Jian,Yu Gang. How to coordinate supply chain under disruptions. Systems
Engineering-Theory Practice,2005(7): 9~16. (in Chinese)
[2]Qi X., Bard J.F and Yu G. Supply chain coordination with demand disruptions. Omega, 2004(32):
301~312.
[3]Yu Hui, Chen Jian, Yu Gang. Managing wholesale price contract in the supply chain under
disruptions. Systems Engineering-Theory Practice,2006(8):33~41. (in Chinese)
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[4]Yu Hui, Chen Jian, Yu Gang. Supply Chain Coordination under Disruptions with Buy Back Contract.
Systems Engineering-Theory Practice,2005(8):38~43. (in Chinese)
[5]Kumar N.The power of trust in manufacturer–retailer relationships. Harvard Business Review 1996
(6):92~109.
[6]Zhongsheng Hua, Sijie Li.Impacts of demand uncertainty on retailer’s dominance and
manufacturer-retailer supply chain cooperation, Omega, 2008(36) 697 714.
[7]Cachon G P. Supply chain coordination with contracts. working paper, Univerisity of Pennsylvania,
Philadelphia,PA.2002.
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