List of Figures ................................................................................................................................. 1
List of Tables .................................................................................................................................. 1
Executive Summary ............................................................ Ошибка! Закладка не определена.
Introduction .............................................................. Ошибка! Закладка не определена.
1.0
1.1.
Literature Review ................................................. Ошибка! Закладка не определена.
2.0 Background ................................................................... Ошибка! Закладка не определена.
2.1. Definition of Variables ............................................. Ошибка! Закладка не определена.
2.2. Assumptions ............................................................. Ошибка! Закладка не определена.
2.3. Methodology ............................................................ Ошибка! Закладка не определена.
2.3. Non-coordinating Contract Cases ........................................................................................ 7
2.4. Coordinating Contract Cases ................................................................................................ 9
2.4.1. Revenue-Sharing Contract ............................................................................................. 9
2.4.2. Buy-Back Contract ...................................................................................................... 10
2.4.3. Quantity Flexibility Contract ....................................................................................... 12
3.0 Numerical Analysis ................................................................................................................. 16
3.1. Non-Coordinated Contract Cases ....................................................................................... 16
3.2. Coordinating Contract Cases .............................................................................................. 21
4.0 Extended Results and Discussion ........................................................................................... 33
5.0 Conclusion .............................................................................................................................. 33
6.0 Work Cited .............................................................................................................................. 33
7.0 Appendix ................................................................................................................................. 33
List of Figures
No table of figures entries found.
List of Tables
No table of figures entries found.
gₛ or 𝑔𝑣
cₛ or 𝑐𝑣
s
Buy-Back
Quantity
Flexibility
Online
Offline
Online
Offline
Online
140.65
142.28
140.65
142.28
140.65
9.9
9.9
-
9
-
0.185
0.402
0.66
-
As previously stated, the team was able to show the proof to calculate the optimal order
quantities in a non-coordinated case versus a coordinated case. Within a non-coordinated case to
obtain the optimal order quantity the profit of the retailer is differentiated and then expected sales
is isolated to obtain the F(Q*) equation. The proof starts with the expected equation the team
wanted and then illustrates how it was calculated. As this paper deals with both offline and
online prices there are two optimal order quantities depending on if the price is set as the offline
price or the online price.
Proof: Non-coordinated case, Optimal order quantity, Offline case:
𝐹 (𝑄 ∗ ) = 1 −
𝑐𝑟 + 𝑤 − 𝑠𝑟
𝑝 − 𝑠𝑟 + 𝑔𝑟
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑑𝑃
𝑑𝑃
𝑑𝑃
𝑑𝑃
𝑑𝑃
𝑑𝑃
𝑑𝑃
𝑑𝑃 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟 (𝑄 ∗ ) = 𝑝
𝑆(𝑄 ∗ ) + 𝑠𝑟 ( 𝑄 ∗ −
𝑆(𝑄 ∗ )) − 𝑔𝑟 ( 𝐷 −
𝑆(𝑄 ∗ )) − 𝑐𝑟 ( 𝑄 ∗ ) − 𝑤
𝑄
𝑑𝑞
𝑑𝑞
𝑑𝑞
𝑑𝑞
𝑑𝑞
𝑑𝑞
𝑑𝑞
𝑑𝑞
0 = 𝑝𝑆 ′ (𝑄 ∗ ) + 𝑠𝑟 − 𝑠𝑟 (𝑆 ′ (𝑄 ∗ )) + 𝑔𝑟 (𝑆 ′ (𝑄 ∗ )) − 𝑐𝑟 − 𝑤
𝑐𝑟 + 𝑤 − 𝑠𝑟 = 𝑝𝑆 ′ (𝑄 ∗ ) − 𝑠𝑟 (𝑆 ′ (𝑄 ∗ )) + 𝑔𝑟 (𝑆 ′ (𝑄 ∗ ))
𝑐𝑟 + 𝑤 − 𝑠𝑟
= 𝑆 ′ (𝑄 ∗ )
𝑝 − 𝑠𝑟 + 𝑔𝑟
𝑐𝑟 + 𝑤 − 𝑠𝑟
= 1 − 𝐹(𝑄 ∗ )
𝑝 − 𝑠𝑟 + 𝑔𝑟
𝐹(𝑄 ∗ ) = 1 −
𝑐𝑟 + 𝑤 − 𝑠𝑟
𝑝 − 𝑠𝑟 + 𝑔𝑟
𝐹(𝑄 ∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 1 −
3 + 18 − 9
53 − 9 + 12
𝐹(𝑄 ∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 0.7857
The 𝐹(𝑄∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 0.7857 from the calculation after subbing in the offline price and the
corresponding values from Table 2. After obtaining this value the team referenced the “Normal
Probability Distribution and Partial Expectations Table” that can be found in Appendix A. The
team then found the corresponding z-value at this F(Q*) value. It should be noted that since the
F(Q*) was between two values it was interpolated to get the exact value found in Table 4. After
getting the z-value the team utilized the Newsvendor model with the corresponding mean and
standard deviation found in Table 3 to find the Optimal Quantity for the offline demand.
𝑧 = 0.7917 𝑎𝑓𝑡𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝐹(𝑄 ∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 0.7857
(𝑄 ∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 𝜇 + 𝑧𝜎
(𝑄 ∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 100 + (0.7917)(30)
(𝑄 ∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 123.751
For non-coordinated cases, the derivation is the same for both the online and offline case
however the F(Q*) would change as the online price is not the same as the offline price. This
would then change the F(Q*), as well as the z-value and the overall optimal order quantity in an
online case. The proof to calculate the optimal order quantity for the online case can be seen
below.
Proof: Non-coordinated case, Optimal order quantity, Online case:
𝐹(𝑄 ∗ )𝑜𝑛𝑙𝑖𝑛𝑒 = 1 −
3 + 18 − 9
47 − 9 + 12
𝐹(𝑄 ∗ )𝑜𝑛𝑙𝑖𝑛𝑒 = 0.7600
𝑧 = 0.7064 𝑎𝑓𝑡𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝐹(𝑄 ∗ )𝑜𝑛𝑙𝑖𝑛𝑒 = 0.7600
(𝑄 ∗ )𝑜𝑛𝑙𝑖𝑛𝑒 = 𝜇 + 𝑧𝜎
(𝑄 ∗ )𝑜𝑛𝑙𝑖𝑛𝑒 = 100 + (0.7064)(30)
(𝑄 ∗ )𝑜𝑛𝑙𝑖𝑛𝑒 = 121.192
Within the coordinated cases the aim is to optimize the profit in the full supply chain. This is
achieved by considering both the retailer and the supplier. The Optimal order quantity of the
supply chain can be achieved by taking the derivative of the profit of the retailer and the profit of
the supplier. As previously noted, the equation that is wanted if shown first followed by how it is
achieved.
Proof: Coordinated case, optimal order quantity for the supply chain, Offline case.
∗
𝐹 (𝑄𝑠𝑐
)=1−
𝑐 − 𝑠𝑟
𝑝 − 𝑠𝑟 + 𝑔
𝑐𝑣 + 𝑐𝑟 −𝑠𝑟
𝑝 − 𝑠𝑟 + 𝑔𝑟 + 𝑔𝑠
∗ )=1−
𝐹(𝑄𝑠𝑐
∗ )
∗ )
∗
∗ ))
∗ ))
∗ )
∗
∗ ))
∗ ))
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑐 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝑠𝑟 (𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑟 (𝐷 − 𝑆(𝑄𝑠𝑐
− 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑞 + 𝑠𝑠 (𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑠 (𝐷 − 𝑆(𝑄𝑠𝑐
−
∗ ) + 𝑤𝑄 ∗
𝑐𝑠 (𝑄𝑠𝑐
𝑠𝑐
𝑑𝑃
𝑑𝑞
∗ )=𝑝
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑐 (𝑄𝑠𝑐
𝑑𝑃
𝑑𝑞
𝑑𝑃
𝑠𝑠 (
𝑑𝑃
𝑑𝑞
𝑑𝑃
∗ )+𝑠 (
𝑆(𝑄𝑠𝑐
𝑟
𝑑𝑞
∗
𝑄𝑠𝑐
−
𝑑𝑃
𝑑𝑞
𝑑𝑞
∗ −
𝑄𝑠𝑐
𝑑𝑃
𝑑𝑞
𝑑𝑃
∗ )) − 𝑔 (
𝑆(𝑄𝑠𝑐
𝑟
𝑑𝑃
∗ ))
𝑆(𝑄𝑠𝑐
− 𝑔𝑠 (
𝑑𝑞
𝐷−
𝑑𝑃
𝑑𝑞
𝑑𝑞
𝐷−
∗ ))
𝑆(𝑄𝑠𝑐
− 𝑐𝑠
𝑑𝑃
𝑑𝑞
𝑑𝑃
𝑑𝑞
∗ )) − 𝑐
𝑆(𝑄𝑠𝑐
𝑟
∗ )
(𝑄𝑠𝑐
+𝑤
𝑑𝑃
𝑑𝑞
𝑑𝑃
𝑑𝑞
∗ )−𝑤
(𝑄𝑠𝑐
𝑑𝑃
𝑑𝑞
∗ +
𝑄𝑠𝑐
∗
𝑄𝑠𝑐
∗ ) = 𝑝𝑆 ′ (𝑄 ∗ )+𝑠 + 𝑠 (−𝑆 ′ (𝑄 ∗ )) + 𝑔 (𝑆 ′ (𝑄 ∗ )) − 𝑐 − 𝑤 + s − 𝑠 (𝑆 ′ (𝑄 ∗ )) + 𝑔 (𝑆 ′ (𝑄 ∗ )) − 𝑐 + 𝑤
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑐 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑟
s
𝑠
𝑠𝑐
𝑟
𝑠𝑐
𝑠
∗
∗
∗
∗
∗
0 = 𝑝𝑆 ′ (𝑄𝑠𝑐 ) +𝑠𝑟 − 𝑠𝑟 (𝑆 ′ (𝑄𝑠𝑐 )) + 𝑔𝑟 (𝑆 ′ (𝑄𝑠𝑐 )) − 𝑐𝑟 + ss − 𝑠𝑠 (𝑆 ′ (𝑄𝑠𝑐 )) + 𝑔𝑟 (𝑆 ′ (𝑄𝑠𝑐 )) − 𝑐𝑠
∗
∗
∗
∗
∗
𝑐𝑠 − ss + 𝑐𝑟 −𝑠𝑟 = 𝑝𝑆 ′ (𝑄𝑠𝑐 ) −𝑠𝑟 (𝑆 ′ (𝑄𝑠𝑐 )) + 𝑔𝑟 (𝑆 ′ (𝑄𝑠𝑐 )) − 𝑠𝑠 (𝑆 ′ (𝑄𝑠𝑐 )) + 𝑔𝑟 (𝑆 ′ (𝑄𝑠𝑐 ))
Within coordinated contracts the aim is to have no salvage for the supplier as the supplier
and retailer would work together to get an optimal order quantity where they are both successful.
From this we assume that the salvage rate of the supplier = 0.
∗
∗
∗
∗
𝑐𝑠 + 𝑐𝑟 −𝑠𝑟 = 𝑝𝑆 ′ (𝑄𝑠𝑐 ) −𝑠𝑟 (𝑆 ′ (𝑄𝑠𝑐 )) + 𝑔𝑟 (𝑆 ′ (𝑄𝑠𝑐 )) + 𝑔𝑠 (𝑆 ′ (𝑄𝑠𝑐 ))
𝑐𝑠 + 𝑐𝑟 −𝑠𝑟
∗
= 𝑆 ′ (𝑄𝑠𝑐 )
𝑝 − 𝑠𝑟 + 𝑔𝑟 + 𝑔𝑠
𝑐𝑠 + 𝑐𝑟 −𝑠𝑟
∗ )
= 1 − 𝐹(𝑄𝑠𝑐
𝑝 − 𝑠𝑟 + 𝑔𝑟 + 𝑔𝑠
∗ )=1−
𝐹(𝑄𝑠𝑐
𝑐𝑠 + 𝑐𝑟 −𝑠𝑟
𝑝 − 𝑠𝑟 + 𝑔𝑟 + 𝑔𝑠
∗ )
𝐹(𝑄𝑠𝑐
𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 1 −
11 + 3 − 9
53 − 9 + 12 + 7
∗ )
𝐹(𝑄𝑠𝑐
𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 0.920
After finding the 𝐹(𝑄𝑠𝑐∗ )𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 0.920 the “Normal Probability Distribution and Partial
Expectations Table” in the Appendix A can be referenced to find the corresponding z-value.
After finding the z-value the Newsboy model is utilized to obtain the optimal order quantity of
the supply chain in the offline case.
∗ )
𝑧 = 1.41 𝑤ℎ𝑒𝑛 𝐹(𝑄𝑠𝑐
𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 0.920
∗ )
(𝑄𝑠𝑐
𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 𝜇 + 𝑧𝜎
∗ )
(𝑄𝑠𝑐
𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 100 + (1.41)(30)
∗ )
(𝑄𝑠𝑐
𝑜𝑓𝑓𝑙𝑖𝑛𝑒 = 142.30
Similarly, to the non-coordinated case, the derivation is the same for that of online and
offline in the coordinated case however the price would change. After referencing Appendix A
and interpolating the corresponding z-value can be found and used to solve for the optimal order
quantity of the supply chain.
Proof: Coordinated case, optimal order quantity for the supply chain, Online case.
∗ )
𝐹(𝑄𝑠𝑐
=1−
𝑐𝑠 + 𝑐𝑟 −𝑠𝑟
𝑝 − 𝑠𝑟 + 𝑔𝑟 + 𝑔𝑠
∗ )
𝐹(𝑄𝑠𝑐
𝑜𝑛𝑙𝑖𝑛𝑒 = 1 −
11 + 3 − 9
47 − 9 + 12 + 7
∗ )
𝐹(𝑄𝑠𝑐
𝑜𝑛𝑙𝑖𝑛𝑒 = 0.9123
∗ )
𝑧 = 1.354 𝑎𝑓𝑡𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝐹(𝑄𝑠𝑐
𝑜𝑛𝑙𝑖𝑛𝑒 = 0.9123
∗ )
(𝑄𝑠𝑐
𝑜𝑛𝑙𝑖𝑛𝑒 = 𝜇 + 𝑧𝜎
∗ )
(𝑄𝑠𝑐
𝑜𝑛𝑙𝑖𝑛𝑒 = 100 + (1.354)(30)
∗ )
(𝑄𝑠𝑐
𝑜𝑛𝑙𝑖𝑛𝑒 = 140.62
2.3. Non-coordinating Contract Cases
When determining the various contract equations that will be utilized to calculate the
profit for the retailer and supplier, the team took the basic retailer and supplier profit functions
and then applied it to the corresponding contract case to obtain the same equations found in the
paper. Where T is the transfer function for each contract case, I(q) is the expected leftover
inventory calculated by 𝐼(𝑞) = 𝑞 − 𝑆(𝑞) and lost sales is calculated by 𝐿(𝑞) = 𝜇 − 𝑆(𝑞). Within the
supplier profit depending on the contract a corresponding salvage rate is calculated; in other
cases it is assumed to be 0.
Basic Profit Equations for Retailer and Supplier
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝐼(𝑞)) − 𝑔𝑟 (𝐿(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = 𝑠𝑠 (𝐼(𝑞)) − 𝑔𝑠 (𝐿(𝑞)) − 𝑐𝑠 (𝑞) + 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = 𝑠𝑠 (𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑇
As previously stated within a non-coordinated contract case it is assumed that the retailer
has the optimal order quantity, and the supplier is trying to match that order quantity. Therefore,
it is found that the Q* is the order quantity used in the following proofs. The non-coordinated
cases illustrate a wholesale-price contract that contains a transfer function 𝑇(𝑄∗ , 𝑤) = 𝑤𝑄∗ . The
first equation is the equation that is presented in the paper followed by the proof on how to
obtain it. It is crucial to note that the demand will change so mean is substituted for the demand.
Proof: Non-coordinated, Stochastic Demand
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = 𝑠𝑠 (𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
Therefore, the final equations after taking the original profit equation for both supplier
and retailer can be found to match that of the equation found in the paper. Now applying this to
the deterministic case however as assumed applying the equation for demand will be substituted.
Proof: Non-Coordinated Deterministic Demand
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 )) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = 𝑠𝑠 (𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 )) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
Again, this illustrates that from the basic profit equations for both the retailer and the
supplier the deterministic equations match that of those in the paper.
2.4. Coordinating Contract Cases
The same basic profit equations for retailer and supplier can now be applied to the
coordinated cases. Within each case the transfer function (T) would be the main difference, once
the transfer function is applied to the basic profit equation, the proof equations would match the
ones found in the paper.
2.4.1. Revenue-Sharing Contract
It can be seen that for revenue-sharing contract the transfer function is as followed:
∗ , 𝑤 ′ , ϕ) = (𝑤 ′ + (1 − ϕ)s) 𝑄 ∗ + (1 − ϕ)(p − s) 𝑆(𝑄 ∗ )
𝑇(𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
The corresponding basic retailers profit function can be found to be:
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = (ϕ(p − sr ) + g r )𝑆(𝑞) − (𝑤𝑟 + 𝑐𝑟 − ϕsr )(q) − g r 𝜇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = ϕ(p𝑆(𝑞) − 𝑠𝑟 𝑆(𝑞) + 𝑔𝑟 𝑆(𝑞)) − (𝑤𝑟 𝑞 + 𝑐𝑟 𝑞 − ϕsr 𝑞) − g r 𝜇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = ϕ(p𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − (𝑤𝑟 𝑞 + 𝑐𝑟 𝑞)
The corresponding basic supplier profit function can be found to be:
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = 𝑠𝑠 (𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑇
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + (𝑤 ′ + (1 − ϕ)s − g ) 𝑄 ∗ + (1 − ϕ)(p − s
= 𝑠𝑠 (𝑄𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
s
𝑠𝑐
∗ )
+ g s ) 𝑆(𝑄𝑠𝑐
∗ ) = −𝑔 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤 ′ 𝑄 ∗ + (1 − ϕ)[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝜇 − 𝑆(𝑄 ∗ ))]
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
If we apply the calculated basic equations for the revenue sharing to the ones found in the
paper, we can obtain the proof in both the stochastic and deterministic demands for both retailer
and supplier. The first equation is the expected equation from the paper followed by how it was
obtained after using the basic equations derived above. Within the Deterministic case the demand
is substituted with the corresponding demand equation.
Proof: Coordinated, Revenue Sharing Stochastic Demand
∗ )
∗ )
∗
∗ ))
∗ ))]
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= ϕ[𝑝𝑆(𝑄𝑠𝑐
+ 𝑠𝑟 (𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑟 (𝐷 − 𝑆(𝑄𝑠𝑐
− 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤′𝑄𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = ϕ(p𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − (𝑤𝑟 𝑞 + 𝑐𝑟 𝑞)
∗ ) = ϕ(p𝑆(𝑄 ∗ ) + 𝑠 (𝑄∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − (𝑤 𝑄 ∗ + 𝑐 𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟 𝑠𝑐
𝑟 𝑠𝑐
∗ ) = ϕ[p𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ ))] − (𝑤 𝑄 ∗ ) − 𝑐 𝑄∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟 𝑠𝑐
𝑟 𝑠𝑐
∗ ) = −𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤′𝑄 ∗ + (1 − ϕ)[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ ))]
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
∗ ) = −𝑔 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤 ′ 𝑄 ∗ + (1 − ϕ)[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝜇 − 𝑆(𝑄 ∗ ))]
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
∗ ) = −𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤 ′ 𝑄 ∗ + (1 − ϕ)[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ ))]
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
Proof: Coordinated, Revenue Sharing, Deterministic Demand
∗ ) = ϕ[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ ))] − 𝑐 (𝑄 ∗ ) − 𝑤′𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
1
2
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = ϕ(p𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − (𝑤𝑟 𝑞 + 𝑐𝑟 𝑞)
∗ ) = ϕ[p𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ ))] − (𝑤 𝑄 ∗ ) − 𝑐 𝑄∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟 𝑠𝑐
𝑟 𝑠𝑐
∗ )
∗ )
∗
∗ ))
∗ ))]
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= ϕ[p𝑆(𝑄𝑠𝑐
+ 𝑠𝑟 (𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑟 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄𝑠𝑐
− (𝑤𝑟 𝑄𝑠𝑐
− 𝑐𝑟 𝑄𝑠𝑐
∗ ) = −𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤 ′ 𝑄 ∗ +
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
1
2
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ ))]
(1 − ϕ)[𝑝𝑆(𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
1
2
𝑠𝑐
∗ ) = −𝑔 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤 ′ 𝑄 ∗ + (1 − ϕ)[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝜇 − 𝑆(𝑄 ∗ ))]
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
∗ ) = −𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤 ′ 𝑄 ∗ +
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
1
2
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ ))]
(1 − ϕ)[𝑝𝑆(𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
1
2
𝑠𝑐
2.4.2. Buy-Back Contract
The same way that the team was able to find the revenue-sharing and non-coordinated
equations can be applied here to find the buyback contract equations. It is known that the transfer
function for the buy-back case is as followed:
𝑇 (𝑞, 𝑤, 𝛽) = 𝑤𝑞 − 𝛽𝐼(𝑞)
𝑇 (𝑞, 𝑤, 𝛽) = 𝑤𝑞 − 𝛽(𝑞 − 𝑆(𝑞))
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − (𝑤𝑞 − 𝛽(𝑞 − 𝑆(𝑞)))
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑤𝑞 + 𝛽(𝑞 − 𝑆(𝑞))
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = 𝑠𝑠 (𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = 𝑠𝑠 (𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑤𝑞 − 𝛽(𝑞 − 𝑆(𝑞))
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = (𝑠𝑠 − 𝛽)(𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑤𝑞
Based on the obtained equations above it can now be provided and compared to that of
the ones found in the paper. The proof below illustrates the proof for both demand types and for
both the supplier and retailer.
Proof: Coordinated, Buy-Back, Stochastic Demand
∗ )
∗ )
∗
∗ ))
∗ ))
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝛽(𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑟 (𝐷 − 𝑆(𝑄𝑠𝑐
− 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑄𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑤𝑞 + 𝛽(𝑞 − 𝑆(𝑞))
∗ ) = 𝑝𝑆(𝑄 ∗ ) − 𝑔 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗ + 𝛽(𝑄 ∗ − 𝑆(𝑄 ∗ ))
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝛽(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝛽(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = (𝑠 − 𝛽)(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = (𝑠𝑠 − 𝛽)(𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑤𝑞
∗ ) = (𝑠 − 𝛽)(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝜇 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
∗ ) = (𝑠 − 𝛽)(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
Proof: Coordinated, Buy-Back, Deterministic Demand
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝛽(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 ((𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑟
1
2
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑤𝑞 + 𝛽(𝑞 − 𝑆(𝑞))
∗ ) = 𝑝𝑆(𝑄 ∗ ) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗ + 𝛽(𝑄 ∗ − 𝑆(𝑄 ∗ ))
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠𝑐
∗ )
∗ )
∗
∗ ))
∗ ))
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝛽(𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑟 (𝐷 − 𝑆(𝑄𝑠𝑐
− 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑄𝑠𝑐
∗ )
∗ )
∗
∗ ))
∗ ))
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝛽(𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑟 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄𝑠𝑐
− 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑄𝑠𝑐
∗ )
∗
∗ ))
∗ ))
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
= (𝑠𝑠 − 𝛽)(𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑠 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄𝑠𝑐
− 𝑐𝑠 (𝑄𝑠𝑐
+ 𝑤𝑄𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑞) = (𝑠𝑠 − 𝛽)(𝑞 − 𝑆(𝑞)) − 𝑔𝑠 (𝜇 − 𝑆(𝑞)) − 𝑐𝑠 (𝑞) + 𝑤𝑞
∗ )
∗
∗
∗
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
= (𝑠𝑠 − 𝛽)(𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
)) − 𝑔𝑠 (𝐷 − 𝑆(𝑄𝑠𝑐
)) − 𝑐𝑠 (𝑄𝑠𝑐
+ 𝑤𝑄𝑠𝑐
∗ ) = (𝑠 − 𝛽)(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
1
2
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
2.4.3. Quantity Flexibility Contract
When looking at the quantity flexibility contract, the demand and order quantity and the
quantity flexibility fraction are the main factors that are being affected. The transfer function of
quantity flexibility considers the 3 variables stated 𝑇(𝑄𝑠𝑐∗ , 𝑤, 𝛿). The transfer function was found to
be as followed however it is expected that the fraction would affect how the q is calculated:
𝑇(𝑞, 𝑤, 𝛿) = 𝑤𝑞
With this there are three cases for both stochastic and deterministic where the fraction is
less than the demand, where the demand is between the optimal quantity and the fraction optimal
quantity value and when the demand is greater than the actual optimal quantity. The proofs for
all three equations can be seen found below. Depending on the case the fraction multiplied by the
optimal quantity was used instead of the optimal quantity.
Proof: Coordinated, Quantity Flexibility, Stochastic Demand
Case 1 - Demand level is higher than quantity.
Within this case the expected value is the demand as the maximum is Demand as the Optimal
quantity is less than the demand level.
∗
𝐷 > (1 − 𝛿)𝑄𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 ((1 − 𝛿)𝑄 ∗ − 𝐷) − 𝑐 ((1 − 𝛿)𝑄 ∗ ) − 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑤𝑞
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑞)) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ )
∗ )
∗
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝑠𝑟 ((1 − 𝛿)𝑄𝑠𝑐
− 𝐷) − 𝑐𝑟 ((1 − 𝛿)𝑄𝑠𝑐
− 𝑤((1 − 𝛿)𝑄𝑠𝑐
)
∗ ) = −𝑐 ((1 − 𝛿)𝑄 ∗ ) + 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = −𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
∗ ) = −𝑐 ((1 − 𝛿)𝑄 ∗ ) + (𝑤(1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
Case 2: Demand Level is higher than agreed upon quantity but smaller than Q*sc
Within this case the optimal quantity is the max, where the demand is below it. However, the
demand is greater than the fraction multiplied by the optimal quantity. Salvage for retailer is
assumed to be the wholesale price. As the retailer would not pay a salvage rate but would pay for
the quantity of product.
∗ < 𝐷 < 𝑄∗
(1 − 𝛿)𝑄𝑠𝑐
𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑤(𝑄 ∗ − 𝐷) − 𝑐 (𝑄 ∗ ) − 𝑤(𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝐷) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑤(𝑄 ∗ − 𝐷) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = (𝑠 − 𝑤)(𝑄 ∗ − 𝐷) − 𝑐 (𝑄 ∗ ) + 𝑤(𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = (𝑠𝑠 − 𝑠𝑟 )(𝑄 ∗ − 𝐷) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = (𝑠𝑠 − 𝑤)(𝑄 ∗ − 𝐷) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
∗ ) = (𝑠 − 𝑤)(𝑄 ∗ − 𝐷) − 𝑐 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
Case 3: Demand level is higher that Q*sc
Within this case there is no salvage for both the retailer and the supplier, as the demand is greater
than the Optimal quantity.
∗
𝐷 > 𝑄𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤(𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑤𝑞
∗ )
∗ )
∗
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
− 𝑔𝑟 (𝐷 − 𝑆(𝑄𝑠𝑐
)) − 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑄𝑠𝑐
∗ )
∗
∗
∗ )
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
= −𝑔𝑠 (𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
)) − 𝑐𝑠 (𝑄𝑠𝑐
+ 𝑤(𝑄𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
∗ ) = −𝑔 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
The same assumptions used for the stochastic demand can be used for the deterministic
demand however the demand would change to reflect the deterministic demand equation.
Proof: Coordinated, Quantity Flexibility, Deterministic Demand
Case 1: Demand level higher than agreed upon quantity
∗
∝ −(𝛽 − 𝛾)(𝑝1 + 𝑝2 ) > (1 − 𝛿)𝑄𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 ((1 − 𝛿)𝑄 ∗ − (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ))) − 𝑐 ((1 − 𝛿)𝑄 ∗ ) − 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
1
2
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑤𝑞
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑞)) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 ((1 − 𝛿)𝑄 ∗ − 𝐷) − 𝑐 ((1 − 𝛿)𝑄 ∗ ) − 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 ((1 − 𝛿)𝑄 ∗ − (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ))) − 𝑐 ((1 − 𝛿)𝑄 ∗ ) − 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
1
2
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = −𝑐 ((1 − 𝛿)𝑄 ∗ ) + 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = −𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
∗ ) = −𝑐 ((1 − 𝛿)𝑄 ∗ ) + (𝑤(1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
Case 2: Demand level higher than agreed upon quantity but smaller than optimal Q*sc
∗ < ∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ) < 𝑄 ∗
(1 − 𝛿)𝑄𝑠𝑐
1
2
𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑤(𝑄 ∗ − ∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 )) − 𝑐 (𝑄 ∗ ) − 𝑤(𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
1
2
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
∗ )
∗ )
∗
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝑠𝑟 (𝑄𝑠𝑐
− 𝐷) − 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑄𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑤(𝑄 ∗ − (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ))) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
1
2
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = (𝑠 − 𝑤)(𝑄 ∗ − ∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 )) − 𝑐 (𝑄 ∗ ) + 𝑤(𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
1
2
𝑠
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = (𝑠𝑠 − 𝑠𝑟 )(𝑄 ∗ − 𝐷) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = (𝑠𝑠 − 𝑤)(𝑄 ∗ − 𝐷) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
∗ ) = (𝑠 − 𝑤)(𝑄 ∗ − 𝐷) − 𝑐 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
∗ ) = (𝑠 − 𝑤)(𝑄 ∗ − (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ))) − 𝑐 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
1
2
𝑠
𝑠𝑐
𝑠𝑐
Case 3: Demand level higher than optimal quantity
∗
∝ −(𝛽 − 𝛾)(𝑝1 + 𝑝2 ) > 𝑄𝑠𝑐
∗ ) = 𝑝𝑆(𝑄 ∗ ) − 𝑔 (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤(𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
1
2
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) + 𝑠𝑟 (𝑞 − 𝑆(𝑞)) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑇
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑞) = 𝑝𝑆(𝑞) − 𝑔𝑟 (𝜇 − 𝑆(𝑞)) − 𝑐𝑟 (𝑞) − 𝑤𝑞
∗ ) = 𝑝𝑆(𝑄 ∗ ) − 𝑔 (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
1
2
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = −𝑔 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤(𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 ((𝐷) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
∗ ) = −𝑔 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
3.0 Numerical Analysis
3.1. Non-Coordinated Contract Cases
Proof for Expected sales
𝑞
𝑆(𝑄 ∗ ) = 𝑞 − 𝑞𝐹(𝑞) + 𝑞𝐹(𝑞) − ∫ 𝐹(𝑦)𝑑𝑦
0
𝑞
𝑆(𝑄 ∗ ) = 𝑞 − ∫ 𝐹(𝑦)𝑑𝑦
0
For the stochastic model,
The Demand Distribution: Normal, Normal Mean: 0.2323, Mean:100 and Standard Deviation: 30
𝐹(𝑦) is
the Cumulative Distribution Function and is calculated using the following equation.
𝐶𝐷𝐹 =
1
2
[1 + erf (
𝑥− µ
𝜎∗√2
)]
where, (erf) is the error function and is calculated using the following equation.
2 𝑥 −𝑡2
∫ 𝑒
𝜋 0
erf(𝑥) =
erf(0.2323) =
2 0.2323 −𝑡2
∫
𝑒 𝑑𝑡
𝜋 0
erf(0.2323) = 0.25748284
CDF = 0.6287
Therefore, the CDF can be calculated to be 0.6287. Then substituting the CDF value in the
equation for expected sales,
142.28
𝑆(𝑄 ∗ ) = 142.28 − ∫
𝐹(𝑦)𝑑𝑦
0
𝑆(𝑄 ∗ ) = 97.55
Sample Calculations to achieve profit outcomes within a Stochastic Non-Coordinated Case
similar to the table. The demand of 96.67 was chosen to achieve the optimal units short for the
table Govindan and Malomfalean.
Sample Calculation: Stochastic Profit of Retailer
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄 ∗ ) = 94.65, 𝑠𝑟 = 9, 𝑄 ∗ = 123.75, 𝑔𝑟 = 12, 𝐷 = 96.67, 𝑐𝑟 = 3, 𝑤 = 18
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (123.75) = (53)(94.65) + 9(123.75 − 94.65) − 12(96.67 − 94.65) − 3(123.75) − (18)(123.75)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (123.75) = 5016.45 + 261.9 − 24.24 − 371.25 − 2227.5
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (123.75) = 2655.36
The calculated profit for the retailer at the optimal Q was found to be 2655.36, when
compared to the results found in Govindan and Malomfalean paper the result differs by 0.02,
which is due to rounding within the calculations. For the supplier the optimal quantity must be
met for the non-coordinated condition, this means that the expected sales will be the optimal
order quantity. To achieve the optimal demand for the table the demand was found to be 125.77
Govindan and Malomfalean.
Sample Calculations: Stochastic Profit of Supplier
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄 ∗ ) = 123.75, 𝑠𝑠 = 8, 𝑄 ∗ = 123.75, 𝑔𝑠 = 7, 𝐷 = 125.77, 𝑐𝑠 = 11, 𝑤 = 18
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (123.75) = 8(123.75 − 123.75) − 7(125.77 − 123.75) − 11(123.75) + 18(123.75)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (123.75) = 0 − 14.14 − 1361.25 + 2227.50
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (123.75) = 852.11
The calculated profit for the supplier at the optimal Q was found to be 852.11, when
compared to the results from in the Govindan and Malomfalean the result differs by 0.02, which
is due to rounding within the calculations.
Sample Calculation: Deterministic Profit of Retailer
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 )) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
Online
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 47, 𝑆(𝑄 ∗ ) = 63.16, 𝑠𝑟 = 9, 𝑄 ∗ = 121.19, 𝑔𝑟 = 12, 𝛼 = 120, 𝛽𝑟 = 9.9, 𝛾 = 10, 𝑐𝑟 = 3, 𝑤 = 18
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 )) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = (47 ∗ 63.16) + 9(121.19 − 47) − 12((120 − (9.9 − 10)(100) − 63.16) − (3 ∗ 121.19) − (18
∗ 121.19)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = (47 ∗ 63.16) + 9(121.19 − 47) − 12((120 − (9.9 − 10)(100) − 63.16) − (3 ∗ 121.19) − (18
∗ 121.19)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 2968.15 + 522.22 − 802.08 − 363.57 − 2181.42
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 945.16
Offline
𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄 ∗ ) = 32.12, 𝑠𝑟 = 9, 𝑄 ∗ = 123.75, 𝑔𝑟 = 12, 𝛼 = 120, 𝛽𝑟 = 9.9, 𝛾 = 10, 𝑐𝑟 = 3, 𝑤 = 18
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 )) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = (53 ∗ 32.12) + 9(123.75 − 32.12) − 12((120 − (9.9 − 10)(100) − 32.12) − (3 ∗ 123.75) − (18
∗ 123.75)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 1702.42 + 824.67 − 1174.56 − 371.25 − 2227.5
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = −71.66
Online + Offline
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠𝑟 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑟 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 )) − 𝑆(𝑄 ∗ )) − 𝑐𝑟 (𝑄 ∗ ) − (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 873.5
Sample Calculation: Deterministic Profit of Supplier
Online
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 47, 𝑆(𝑄 ∗ ) = 191.19, 𝑠𝑠 = 8, 𝑄 ∗ = 121.19, 𝑔𝑠 = 7, 𝛼 = 120, 𝛽𝑟 = 9.9, 𝛾 = 10, 𝑐𝑠 = 11, 𝑤 = 18
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 8(121.19 − 121.19) − 7((120 − (9.9 − 10)(100) − 121.19)) − (11 ∗121.19) + (18*121.19)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 0 − 0 − 1333.09 + 2181.42
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 848.33
Offline
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄 ∗ ) = 123.75, 𝑠𝑠 = 8, 𝑄 ∗ = 123.75, 𝑔𝑠 = 7, 𝛼 = 120, 𝛽𝑠 = 9.9, 𝛾 = 10, 𝑐𝑠 = 11, 𝑤 = 18
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 𝑠𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 8(123.75 − 123.75) − 7(120 − (9.9 − 10)(100) − 123.75) − (11 ∗ 123.75) + (18 ∗ 123.75)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 0 − 0 − 1361.25 + 2227.5
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 866.25
Online + Offline
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄 ∗ ) = 1714.58
Non-Coordinated
Retailers Profit
Supplier Profit
Total Profit Supply Chain
Deterministic Demand
873.5
1714.58
2588.08
Stochastic Demand
2655.36
852.11
3507.47
INSERT FIGURE OF PROFIT OF RETAILER (BOTH) (NON-COORDINATED)
Online
Demand
200
180
160
140
120
100
80
60
40
20
0
INSERT FIGURE OF PROFIT OF SUPPLIER (BOTH, NON-COORDINATED)
Profit of the Supplier (Online)
Profit of the Supplier
(Online)
900
800
700
600
500
400
300
200
100
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Profit
3.2. Coordinating Contract Cases
Proof for Expected sales
𝑞
𝑆(𝑄 ∗ ) = 𝑞 − 𝑞𝐹(𝑞) + 𝑞𝐹(𝑞) − ∫ 𝐹(𝑦)𝑑𝑦
0
𝑞
𝑆(𝑄 ∗ ) = 𝑞 − ∫ 𝐹(𝑦)𝑑𝑦
0
For the stochastic model,
The Demand Distribution: Normal, Normal Mean: 0.2323, Mean:100 and Standard Deviation: 30
𝐹(𝑦) is
the Cumulative Distribution Function and is calculated using the following equation.
𝐶𝐷𝐹 =
1
2
[1 + erf (
𝑥− µ
𝜎∗√2
)]
where, (erf) is the error function and is calculated using the following equation.
erf(𝑥) =
erf(0.2323) =
2 𝑥 −𝑡2
∫ 𝑒
𝜋 0
2 0.2323 −𝑡2
∫
𝑒 𝑑𝑡
𝜋 0
erf(0.2323) = 0.25748284
CDF = 0.6287
Therefore, the CDF can be calculated to be 0.6287. Then substituting the CDF value in the
equation for expected sales,
142.28
𝑆(𝑄 ∗ ) = 142.28 − ∫
𝐹(𝑦)𝑑𝑦
0
𝑆(𝑄 ∗ ) = 97.55
The sample calculations utilize this expected sales value. When looking at the sample
calculations of revenue sharing of the supplier the expected sales value is the optimal quantity
for the demand during offline 𝑄𝑠𝑐∗ = 142.3. The demand to achieve the maximum profit with this
corresponding expected sales and optimal order quantity was found to be 143.12 from Govindan
and Malomfalean. This is the addition of the expected sales and the units short. Within revenue
sharing the supplier will share the revenue with the retailer this is considered through the revenue
sharing fraction.
Sample Calculation: Revenue Sharing Stochastic Demand for Retailer
∗ )
∗
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄𝑠𝑐
= 97.55, 𝑠𝑟 = 9, 𝑄𝑠𝑐
= 142.3, 𝑔𝑟 = 12, 𝐷 = 98.4, 𝑐𝑟 = 3, 𝑤 ′ = 9, ϕ = 0.66
∗ ) = ϕ[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ ))] − 𝑐 (𝑄 ∗ ) − 𝑤′𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = 0.66[53(97.55) + 9(142.3 − 97.55) − 12(98.4 − 97.55)] − 3(142.3) − 9(142.3)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 0.66[5170.15 + 402.75 − 10.2] − 426.9 − 1280.7
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 3671.38 − 426.9 − 1280.7
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 1963.78
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
Sample Calculation: Revenue Sharing Stochastic Demand for Supplier
∗ ) = 142.30, 𝑠 = 8, 𝑄 ∗ = 142.3, 𝑔 = 7, 𝐷 = 143.12, 𝑐 = 11, 𝑤 ′ = 9, ϕ = 0.66
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠
∗ ) = −𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤′𝑄 ∗ + (1 − ϕ)[𝑝𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ ))]
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
∗ ) = −7(143.12 − 142.3) − 11(142.3) + 9(142.3)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠 (𝑄𝑠𝑐
+ (0.34)[53(142.3) + 8(142.3 − 142.3) − 7(143.12 − 142.3)]
∗ ) = −5.74 − 1565.3 + 1280.7 + 2564.25 − 1.952
𝑃𝑟𝑜𝑓𝑖𝑡𝑠 (𝑄𝑠𝑐
∗ ) = 2271.95
𝑃𝑟𝑜𝑓𝑖𝑡𝑠 (𝑄𝑠𝑐
Sample Calculation: Revenue Sharing Deterministic Demand for Retailer
Online
∗ ) = ϕ[p𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ ))] − 𝑐 𝑄 ∗ − (𝑤 ′𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑟
1
2
𝑠𝑐
𝑟 𝑠𝑐
𝑟 𝑠𝑐
∗ ) = 0.6[3016.93 + 0 − 2330] − (421.95) − (1265.85)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 303.37,
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
Offline
∗ ) = ϕ[p𝑆(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ ))] − 𝑐 𝑄 ∗ − (𝑤 ′𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
𝑟
1
2
𝑠𝑐
𝑟 𝑠𝑐
𝑟 𝑠𝑐
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 0.4[(53 ∗ 33.19) + 9(142.28 − 142.28) − 12(120 − (9.9 − 10)(100) − 33.19)] − (3 ∗ 142.28)
− (9 ∗ 142.28)
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 0.66[(1759.07) + 0 − 1161.72] − (426.84) − (1280.53)
∗ ) = 201.6042
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
Online + Offline
∗ ) = 505.97
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
Sample Calculation: Revenue Sharing Deterministic Demand for Supplier
Online
∗
∗ ) = −𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤 ′𝑄𝑠𝑐 +
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
1
2
𝑠𝑐
𝑠
𝑠𝑐
∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ ))]
(1 − ϕ)[𝑝𝑆(𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
1
2
𝑠𝑐
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
= −12(120 − (9.9 − 10)(100) − 123.75) − (11 ∗ 140.65) + (9 ∗ 140.65) + 0.6[(47 ∗ 140.65)
+ 8(140.65 − 140.65) − 12(120 − (9.9 − 10)(100) − 140.65)]
∗ ) = 0 − 1547.15 + 1265.83 + 0.6[0 +0 -0]
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 281.32
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
Offline
∗
∗ )
∗ ))
∗ )
𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
= −𝑔𝑠 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄𝑠𝑐
− 𝑐𝑠 (𝑄𝑠𝑐
+ 𝑤 ′𝑄𝑠𝑐 +
∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ ))]
(1 − ϕ)[𝑝𝑆(𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠
1
2
𝑠𝑐
∗ ) = -12(120 – (9.9 - 10)(100) – 142.28) – (3 *142.28) + (9 *142.28) + (0.4)[(47*142.28) + 9(142.28 – 142.28)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
– 12(120 – (9.9 - 10)(100) – 142.28 )]
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
= 0 − 426.84 + 1280.52 + 0
∗ ) = 853.68
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
Online + Offline
∗ ) = 1135
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
To optimal order quantity as previously shown is 142.3. The buy-back value has also
been provided as 9.9. The buy-back value must be within the range of the equation below. The
expected sales are the same being 97.55. The demand was found to be 98.4. Applying all these
the calculations for buy-back stochastic demand for the retailer can be found. For the supplier the
demand utilized to get the results was 98.91, which was the addition of the expected sales 97.55
and units short of satisfying order 1.36.
Proof of buy-back value provided:
∗ − 𝑠 𝑆(𝑄 ∗ ) + (𝑐 + 𝑤 − 𝑔 )𝑄 ∗
∗ − (𝑔 − 𝑐 + 𝑤)(𝑄 ∗ ) + 𝑠 (𝑄 ∗ − 𝑆(𝑄 ∗ ))
(𝑠𝑟 + 𝑔𝑟 − 𝑐𝑟 − 𝑤)𝑄𝑠𝑐
(𝑔𝑠 − 𝑐𝑠 + 𝑤)𝑄𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑟
𝑠𝑐
𝑠
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
≤𝛽≤
∗
∗
∗ − 𝑆(𝑄 ∗ )
𝑄𝑠𝑐 − 𝑆(𝑄𝑠𝑐 )
𝑄𝑠𝑐
𝑠𝑐
∗ − 9𝑆(97.55) + (3 + 18 − 12)142.3
(9 + 12 − 3 − 18)𝑄𝑠𝑐
≤𝛽
142.3 − 97.55
(7 − 11 + 18)142.3 − (7 − 11 + 18)(142.3) + 8(142.3 − 97.55)
≤
142.3 − 97.55
402.75
965.46
≤𝛽≤
44.75
44.75
9 ≤ 𝛽 ≤ 21.57
Therefore the buy-back value provided is correct and is within the range therefore it can be used.
Sample Calculation: Buy-Back Stochastic Demand for Retailer:
∗ )
∗
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄𝑠𝑐
= 97.55, 𝑠𝑟 = 9, 𝑄𝑠𝑐
= 142.3, 𝑔𝑟 = 12, 𝐷 = 98.4, 𝑐𝑟 = 3, 𝑤 = 18, 𝛽 = 9.9
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝛽(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = (53)(97.55) + (9.9)(142.28 − 97.55) − 12(98.4 − 97.55) − 3(142.3) − 18(142.3)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 5170.15 + 442.83 − 10.2 − 426.9 − 2561.4
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 2614.48
Sample Calculation: Buy-Back Stochastic Demand for Supplier
∗ )
∗
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄𝑠𝑐
= 97.55, 𝑠𝑠 = 8, 𝑄𝑠𝑐
= 142.3, 𝑔𝑠 = 7, 𝐷 = 98.91, 𝑐𝑠 = 11, 𝑤 = 18, 𝛽 = 9.9
∗ ) = (𝑠 − 𝛽)(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 (𝐷 − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
= (8 − 9.9)(142.3 − 97.55) − 7(98.91 − 97.55) − 11(142.3) + 18(142.3)
∗ ) = (8 − 9.9)(142.3 − 97.55) − 7(98.91 − 97.55) − 11(142.3) + 18(142.3)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
∗ ) = (−85.025) − 9.52 − 1565.3 + 2561.4
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 901.55
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
Sample Calculation: Buy-Back Deterministic Demand for Retailer
Online
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝛽(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔 ((𝛼 − (𝛽 − 𝛾)(𝑝 + 𝑝 ) − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) − 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑠𝑐
𝑟
1
2
𝑠𝑐
𝑟
𝑠𝑐
𝑠𝑐
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= (47 ∗ 64.19) + 9.9(140.65 − 64.19) − 12((120 − (9.9 − 10)(100) − 64.19) − (3 ∗ 140.65) − (18
∗ 140.65)
∗ ) = (3017) + 756.9 − 789.72 − 421.95 − (2531.7)
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ ) = 720.25
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
Offline
∗ )
∗ )
∗
∗ ))
∗ ))
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝛽(𝑄𝑠𝑐
− 𝑆(𝑄𝑠𝑐
− 𝑔𝑟 ((𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄𝑠𝑐
− 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑄𝑠𝑐
∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= (53 ∗ 33.19) + 9.9(142.28 − 33.19) − 12(120 − (9.9 − 10)(100) − 33.19) − (3 ∗ 142.28) − (18
∗ 142.28)
∗ ) = 1759.07 + 1079.99 − 1161.72 − 426.84 − 2561.04
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
∗ ) = −148.83
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
Online + Offline
∗ ) = 571.42
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
Sample Calculation: Buy-Back Deterministic Demand for Supplier
Online
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = (𝑠𝑠 − 𝛽)(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ )
= (8 − 0.0006539)(140.65 − 64.19) − 7(120 − (9.9 − 10)(100) − 64.19) − (11 ∗ 140.65) + (18
∗ 140.65)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 611.63 − 460.67 – 1547.15 +2531.7
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 1135.51
Offline
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = (𝑠𝑠 − 𝛽)(𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑔𝑠 (𝛼 − (𝛽 − 𝛾)(𝑝1 + 𝑝2 ) − 𝑆(𝑄 ∗ )) − 𝑐𝑠 (𝑄 ∗ ) + 𝑤𝑄 ∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ )
= (8 − 0.0006539)(142.28 − 33.19) − 7(120 − (9.9 − 10)(100) − 33.19) − (11 ∗ 142.28) + (18
∗ 142.28)
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 872.64 − 1565.08 + 2561.04
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 1190.93
Online + Offline
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄 ∗ ) = 2326.44
Sample Calculation: Quantity Flexibility Stochastic Demand for Retailer
∗ = 142.3, 𝑔 = 7, 𝐷 = 98.4, 𝑐 = 11, 𝑤 ′ = 9, ϕ = 0.66
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄 ∗ ) = 97.55, 𝑠𝑠 = 8, 𝑄𝑠𝑐
𝑠
𝑠
∗
𝐷 > 𝑄𝑠𝑐
∗ )
∗ )
∗
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
− 𝑔𝑟 (𝐷 − 𝑆(𝑄𝑠𝑐
)) − 𝑐𝑟 (𝑄𝑠𝑐
− 𝑤𝑄𝑠𝑐
∗
𝐷 > (1 − 𝛿)𝑄𝑠𝑐
∗ )
∗ )
∗
∗ )
∗
𝑃𝑟𝑜𝑓𝑖𝑡𝑟𝑒𝑡𝑎𝑖𝑙𝑒𝑟 (𝑄𝑠𝑐
= 𝑝𝑆(𝑄𝑠𝑐
+ 𝑠𝑟 ((1 − 𝛿)𝑄𝑠𝑐
− 𝐷) − 𝑐𝑟 ((1 − 𝛿)𝑄𝑠𝑐
− 𝑤((1 − 𝛿)𝑄𝑠𝑐
)
Sample Calculation: Quantity Flexibility Stochastic Demand for Supplier
∗ = 142.3, 𝑔 = 7, 𝐷 = 98.4, 𝑐 = 11, 𝑤 ′ = 9, ϕ = 0.66
𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 𝑝 = 53, 𝑆(𝑄 ∗ ) = 97.55, 𝑠𝑠 = 8, 𝑄𝑠𝑐
𝑠
𝑠
∗
𝐷 > 𝑄𝑠𝑐
∗ ) = −𝑔 (𝑄 ∗ − 𝑆(𝑄 ∗ )) − 𝑐 (𝑄 ∗ ) + (𝑤𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
∗
𝐷 > (1 − 𝛿)𝑄𝑠𝑐
∗ ) = −𝑐 ((1 − 𝛿)𝑄 ∗ ) + (𝑤(1 − 𝛿)𝑄 ∗ )
𝑃𝑟𝑜𝑓𝑖𝑡𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 (𝑄𝑠𝑐
𝑠
𝑠𝑐
𝑠𝑐
Sample Calculation: Quantity Flexibility Deterministic Demand for Retailer
Online
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 ((1 − 𝛿)𝑄 ∗ − (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ))) − 𝑐 ((1 − 𝛿)𝑄 ∗ ) − 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
1
2
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = (47 ∗ 64.19) + 9(1 − 0.402)140.65 − (120 − (9.9 − 0.402)(100)) − 3((1 − 0.402)140.65
𝑃 (𝑄𝑠𝑐
− 18((1 − 0.402)140.65
∗ ) = 3388 + 189.8775 – 252 – 1534
𝑃 (𝑄𝑠𝑐
∗ ) = 1791.8775
𝑃 (𝑄𝑠𝑐
Offline
∗ ) = 𝑝𝑆(𝑄 ∗ ) + 𝑠 ((1 − 𝛿)𝑄 ∗ − (∝ −(𝛽 − 𝛾)(𝑝 + 𝑝 ))) − 𝑐 ((1 − 𝛿)𝑄 ∗ ) − 𝑤((1 − 𝛿)𝑄 ∗ )
𝑃 (𝑄𝑠𝑐
𝑠𝑐
𝑟
𝑠𝑐
1
2
𝑟
𝑠𝑐
𝑠𝑐
∗ ) = 53 ∗ 37.99 + 9((1 − 0.185)142.28 − (120 − (9.9 − 10)(100) − 3((1 − 0.402). 28 − 18(1 − 0.185)142.26
𝑃 (𝑄𝑠𝑐
∗ )
𝑃 (𝑄𝑠𝑐
= 2013.81 + 701.66 - 347.88 – 2087.26
∗ )
𝑃 (𝑄𝑠𝑐
= 280.33
∗ ) = 1791.8775 + 280.33 = 2072.2
𝑃 (𝑄𝑠𝑐
Sample Calculation: Quantity Flexibility Deterministic Demand for Supplier
Online
Offline
Coordinated –
Deterministic Demand
Retailers Profit
Supplier Profit
Total Profit Supply Chain
Revenue-Sharing
Contract
Buy-Back Contract
Quantity Flexibility
Contract
Coordinated – Stochastic
Demand
Retailers Profit
Supplier Profit
Total Profit Supply Chain
Revenue-Sharing
Contract
1963.78
2271.95
4235.73
Buy-Back Contract
Quantity Flexibility
Contract
2614.48
901.55
3516.03
INSERT FIGURE OF PROFIT OF RETAILER (3 CASES) (DETERMINISTIC)
INSERT FIGURE OF PROFIT OF SUPPLIER (3 CASES) (DETERMINISTIC)
INSERT FIGURE OF PROFIT OF RETAILER (3 CASES) (STOCHASTIC)
INSERT FIGURE OF PROFIT OF SUPPLIER (3 CASES) (STOCHASTIC)
Stochastic Demand (offline + online)
Retailer
Non-coordinated
Revenue Sharing
Buy-back
Quantity
Flexibility
Revenue Sharing
Buy-back
Quantity
Flexibility
Generated Revenue
Salvage Realized
Goodwill incurred
Marginal Cost
incurred
Wholesale value
incurred
Leftover inventory
Units short
Average sale
Supplier
Generated Revenue
Salvage Realized
Goodwill incurred
Marginal Cost
incurred
Wholesale value
incurred
Return Cost
Inventory level
Units short
Average sale
Deterministic Demand (offline + online)
Retailer
Non-coordinated
Generated Revenue
Salvage Realized
Goodwill incurred
Marginal Cost
incurred
Wholesale value
incurred
Leftover inventory
Units short
Average sale
Supplier
Generated Revenue
Salvage Realized
Goodwill incurred
4670.57
1346.89
0.0
734.82
4408.92
0.0
95.29
0.0
0.0
2694.215
Marginal Cost
incurred
Wholesale value
incurred
Return Cost
Inventory level
Units short
Average sale
Deterministic Demand (online)
Retailer
Non-coordinated
Revenue Sharing
Buy-back
Quantity
Flexibility
Revenue Sharing
Buy-back
Quantity
Flexibility
Generated Revenue
Salvage Realized
Goodwill incurred
Marginal Cost
incurred
Wholesale value
incurred
Leftover inventory
Units short
Average sale
Supplier
Generated Revenue
Salvage Realized
Goodwill incurred
Marginal Cost
incurred
Wholesale value
incurred
Return Cost
Inventory level
Units short
Average sale
Deterministic Demand (offline)
Retailer
Non-coordinated
Generated Revenue
Salvage Realized
Goodwill incurred
Marginal Cost
incurred
Wholesale value
incurred
Leftover inventory
Units short
Average sale
Supplier
Generated Revenue
Salvage Realized
Goodwill incurred
Marginal Cost
incurred
Wholesale value
incurred
Return Cost
Inventory level
Units short
Average sale
Deterministic Demand (Total Performance online + offline)
Retailer
RevenueContribution
BuyContribution Quantity Contribution
Sharing
(%)
Back
(%)
Flexibility
(%)
Generated
Revenue
Salvage
Realized
Goodwill
incurred
Marginal
Cost incurred
Wholesale
value
incurred
Total Gain
Total Loss
Total
Supplier
Salvage
Realized
Goodwill
incurred
Marginal
Cost incurred
Wholesale
value
incurred
Return Cost
Total Gain
Total Loss
Total
Stochastic Demand (Total Performance online + offline)
Retailer
RevenueContribution
BuyContribution Quantity Contribution
Sharing
(%)
Back
(%)
Flexibility
(%)
Generated
Revenue
Salvage
Realized
Goodwill
incurred
Marginal
Cost incurred
Wholesale
value
incurred
Total Gain
Total Loss
Total
Supplier
Salvage
Realized
Goodwill
incurred
Marginal
Cost incurred
Wholesale
value
incurred
Return Cost
Total Gain
Total Loss
Total
INSERT FIGURE OF PERFORMANCE OF RETAILER (3 CASES) (DETERMINISTIC)
INSERT FIGURE OF PERFORMANCE OF SUPPLIER (3 CASES) (DETERMINISTIC)
INSERT FIGURE OF PERFORMANCE OF RETAILER (3 CASES) (STOCHASTIC)
INSERT FIGURE OF PERFORMANCE OF SUPPLIER (3 CASES) (STOCHASTIC)
4.0 Extended Results and Discussion
-
If ours results matched the paper any errors
-
What other papers have done may be different and similar
-
Additional examples
5.0 Conclusion
6.0 Work Cited
7.0 Appendix
7.1 Appendix A
The Figures below were referenced from the course content.