Download Backup Contracts under Supply Risks and Stochastic Demand

Backup Contracts under Supply
Risks and Stochastic Demand
Jing Hou
Outline
•
•
•
•
Literature Review
Model
Sensitivity Analysis
Comparison Study
Literature Review
• Supply disruption management
• Decision-making under supply and demand
uncertainty
• Backup Agreement
Contributions
• First, we study the contract-based coordination
mechanisms between a buyer and a backup supplier and
address the buyer’s decisions on the optimal order quantity
under two different contract types.
• Second, unlike most existing research efforts that study
recurrent supply uncertainties and disruptions risks
separately or treat the two risks indifferently, we take both
risk types into account and compare their impacts.
• Finally, we provide an in-depth analysis of the impact of the
supply uncertainties and compare the effects of two
contracts on mitigating supply and demand risks: a standby contract and a make-to-order contract, which helps the
buyer to choose a better way of using a backup supplier.
Models
• Consider a single period problem where the buyer faces stochastic
demand and two kinds of supply risks from the main supplier:
supply disruptions and recurrent supply uncertainties. During every
cycle, the buyer faces an actual demand of x units. Denote F(x) is
the distribution function of the demand at the market and its
average is uD. The buyer (the retailer) has two supply options: one
cheaper but unreliable (this supplier is referred to as the main
supplier), and the other is perfectly reliable and responsive, but is
more expensive (this one is referred to as the backup supplier). The
main supplier is prone to disruption with a probability of p, in which
case the main supplier becomes totally unavailable. If there is no
disruption, random y units are yielded with distribution function of
G(y) having a mean of S and standard deviation σ. The buyer sells
the goods to the market at the price of s and the wholesale price
from the main supplier is cm.
1 The 1st type of Backup Supply Contract: a Standby Contract
• In a stand-by contract, in the normal situations, the
buyer only orders from its main supplier, and the
backup supplier holds a certain inventory level of M
units (referred to as the reservation quantity) which is
paid at e per unit by the buyer.
• When supply disruption occurs, the buyer has to order
from the backup supplier to satisfy the demand. If the
actual demand x is smaller than M, the buyer only
orders x units at the wholesale price of cb; else if x is
larger than M, he has to order the extra units (x - M) at
the price of cp, the value of which is higher than cb.
1-期望利润
E (1b )  (1  p){eM  ( s  cm )[  ydF ( x)dG ( y ) 
x y
 xdF ( x)dG( y)]}
x y
M


0
M
M
 p[( s  cb )  xdF ( x)  (c p  cb ) M  dF ( x)  ( s  cp)  xdF ( x)]
• The first term is the revenue without supply
disruption but with recurrent supply uncertainty,
which includes the premium of the reservation at
the backup supplier and the profit of selling the
goods available from the main supplier.
• The second term is the revenue when there is
supply disruption at the main supplier.
1-最优决策
• Proposition 1: For the buyer, the optimal reservation
quantity M* at the backup supplier is given by
(1  p)eM *  p(c p  cb )(1  F (M * ))
Proposition 2: The optimal reservation quantity M*
increases with the disruption probability p, the
wholesale price difference (cp-cb), but decreases with
the premium e.
2 The 2nd type of Backup Supply
Contract: a Make-to-order Contract
• In a make-to-order contract, the buyer uses the
backup supplier as a regular producer during
normal situations, that is, has a proportion () of
the demand to be produced in a make-to-order
fashion by the backup supplier at regular times;
and the main supplier has an order of (1 - ) of
the demand. When supply disruption happens,
the buyer will simply order more to satisfy the
total demand. During the whole cycle, the
wholesale price from the backup supplier is cp.
2-期望利润
E ( b2 )  ( s  c p ) E ( D)[(1  p)  p]
(1  p)( s  cm )[
(1   ) xdF ( x)dG ( y )   y dF ( x)dG ( y )]



(1 ) x y
(1 ) x  y
The first term is the revenue of selling the goods
bought from the backup supplier. The second term is
the revenue of selling the goods bought from the main
supplier when there is no disruption.
2-最优决策
• Proposition 3: The optimal proportion * for
the buyer does not change with the disruption
probability p.
• Proposition 4: For the buyer, the optimal
proportion * is given by


0
(1 * ) x
 xf ( x)dx  g ( y)dy 
( s  c p ) E ( D)
s  cm
Sensitivity Analysis
• In this section we particularly consider the
different impacts of σ and p on the decisionmaking of the buyer under the two contracts.
• the demand is assumed to follow a uniform
distribution U[300,700]
• the main supplier’s delivery quantity is
assumed to follow a normal distribution with S
= 500.
• cm = 80, cb = 100, s = 200, cp = 120, e = 11.
1) The 1st type of Backup Supply Contract: a
Stand-by Contract
90
60000
80
50000
70
60
M*
σ=15
40000
σ=30
E * (1b )
50
σ=50
30000
σ=70
40
σ=90
σ=110
20000
30
σ=130
20
σ=150
10000
10
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
p
The buyer’s optimal reservation quantity
M* vs. the supply disruption p
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
p
The buyer’s optimal expected value
vs. the supply uncertainties
Impact
• In the following analysis, we define the impact
of σ (or p) on the buyer’s expected profit as:
max{ E * ( bi )}  min{ E * ( bi )}
, i  1,2.
*
b
min{ E ( i )}
So the impact of σ changes from 6.1% (p = 0) to 0.7445% (p = 0.9);
the impact of p changes from 28.4% (σ = 15) to 21.9% (σ = 150) in this case.
2) The 2nd type of Backup Supply Contract: a
Make-to-order Contract
0.25
60000
0.2
50000
σ=15
σ=30
σ=50
σ=70
σ=90
σ=110
σ=130
σ=150
40000
0.15
*
E * ( b2 )
30000
0.1
20000
0.05
10000
0
0
15
30
50
70
90
110
130
150
σ
The optimal proportion,  *
vs. the recurrent supply uncertainty, σ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
p
The buyer’s optimal expected value
vs. the supply uncertainties
The Comparison of the Two Contracts
• First, about the buyer’s decision-making under two contracts: M*
and , the supply uncertainties from the main supplier have different
impacts.
• (1) The optimal reservation quantity M* increases with the
disruption probability but has no relationship with the recurrent
uncertainty. That is because the buyer does not order from the
backup supplier during normal situations to deal with the recurrent
supply risk. The reservation is only used for mitigating supply
disruption risk.
• (2) The optimal proportion * does not change with the disruption
probability p, and it does not strictly increase with the recurrent
supply uncertainty. This is because the buyer can use the backup
supplier to mitigate the recurrent supply uncertainty, and during
disruption period, the backup supplier’s wholesale price does not
change with the order quantity.
• Second, the supply uncertainties have different
impacts on the expected profits of the buyer
under two contracts: (1) the impacts of σ are
similar. (2) the disruption probability p has more
impacts on the buyer’s expected profit under the
second contract, because when the buyer makes
the decision of *, it does not change with p; the
buyer’s expected profit is directly impacted by p.
But the buyer changes the value of M* with p,
which in some way mitigates the impact of p on
the buyer’s expected profit.
• Third, on the choice of the two contacts, we can see
from the numerical analysis that the buyer’s expected
profits are higher under the 2nd contract except when p
= 0.9, which is almost impossible to happen, which
means the make-to-order contract is better than the
stand-by contract for the buyer under such pricing
decisions of the backup supplier. One of the reasons,
we may say, is that under the make-to-order contract,
the backup supplier is used to mitigate not only the
supply disruptions, but also the recurrent supply
uncertainty. But under the stand-by contract, the buyer
has no way to deal with the recurrent supply
uncertainty.