American Institute of Mathematical Sciences

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July  2017, 13(3): 1553-1586. doi: 10.3934/jimo.2017007

Optimum pricing strategy for complementary products with reservation price in a supply chain model

Mitali Sarkar , and  Young Hae Lee

Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea

* Corresponding author: mitalisarkar.ms@gmail.com (Mitali Sarkar), Phone: +82-1074901981, Fax: +82-31-436-8146

Received  November 2015 Published  December 2016

This paper describes a two-echelon supply chain model with two manufacturers and one common retailer. Two types of complementary products are produced by two manufacturers, and the common retailer buys products separately using a reservation price and bundles them for sale. The demands of manufacturers and retailer are assumed to be stochastic in nature. When the retailer orders for products, any one of manufacturers agrees to allow those products, and the rest of the manufacturers have to provide the same amount. The profits of two manufacturers and the retailer are maximized by using Stackelberg game policy. By applying a game theoretical approach, several analytical solutions are obtained. For some cases, this model obtains quasi-closed-form solutions, for others, it finds closed-form solutions. Some numerical examples, sensitivity analysis, managerial insights, and graphical illustrations are given to illustrate the model.

Keywords: Supply chain management, complementary products, reservation price, Stackelberg approach.
Mathematics Subject Classification: Primary: 90B05, 90B50; Secondary: 90B30.
Citation: Mitali Sarkar, Young Hae Lee. Optimum pricing strategy for complementary products with reservation price in a supply chain model. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1553-1586. doi: 10.3934/jimo.2017007
References:
[1]

A. Banerjee, A joint economic-lot-size model for purchaser and vendor, Decision Sciences, 17 (1986), 292-311.  doi: 10.1111/j.1540-5915.1986.tb00228.x.

[2]

L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258. 

[3]

C. S. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296.  doi: 10.1287/mksc.10.4.271.

[4]

A. L. EI-Ansary and L. W. Stern, Power measurement in the distribution channel, Journal of Marketing Research, 9 (1972), 47-52. 

[5]

G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700.  doi: 10.1080/07408170208928905.

[6]

J. Gabszewicz, N. Sonnac and X. Wauthy, On price competition with complementary goods, Economics Letters, 70 (2001), 431-437.  doi: 10.1016/S0165-1765(00)00383-9.

[7]

S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, International Journal of Production Research, 15 (1977), 107-111.  doi: 10.1080/00207547708943107.

[8]

S. K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241.  doi: 10.1111/j.1540-5915.1988.tb00264.x.

[9]

C. C. Hsieh and C. H. Wu, Coordinated decisions for substitutable products in a common retailer supply chain, European Journal of Operational Research, 196 (2009), 273-288.  doi: 10.1016/j.ejor.2008.02.019.

[10]

K. F. McCardle, K. Rajaram and C. S. Tang, Bundling retail products: Models and analysis, European Journal of Operational Research, 177 (2007), 1197-1217.  doi: 10.1016/j.ejor.2005.11.009.

[11]

N. M. Modak, S. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International Journal of Production Economics, 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.

[12]

S. Mukhopadhyay, X. Yue and X. Zhu, A Stackelberg model of pricing of complementary goods under information asymmetry, International Journal of Production Economics, 134 (2011), 424-433.  doi: 10.1016/j.ijpe.2009.11.015.

[13]

K. Pan, K. K. Lai, S. C. H. Leung and D. Xiao, Revenue-sharing versus wholesale price mechanisms under different channel power structures, European Journal of Operational Research, 203 (2010), 532-538.  doi: 10.1016/j.ejor.2009.08.010.

[14]

B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering 2016 (2016), Article ID 6318737, 14 pages. doi: 10.1155/2016/6318737.

[15]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151.  doi: 10.1016/j.apm.2012.07.026.

[16]

B. Sarkar and A. Majumder, Integrated vendor-buyer supply chain model with vendors setup cost reduction, Applied Mathematics and Computation, 224 (2013), 362-371.  doi: 10.1016/j.amc.2013.08.072.

[17]

B. Sarkar, S. Saren, D. Sinha and S. Hur, Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model Mathematical Problems in Engineering 2015 (2015), Article ID 469486, 13 pages. doi: 10.1155/2015/469486.

[18]

J. Wei, J. Zhao and Y. Li, Pricing decisions for complementary products with firms' different market powers, European Journal of Operational Research, 224 (2013), 507-519.  doi: 10.1016/j.ejor.2012.09.011.

[19]

J. Wei, J. Zhao and Y. Li, Price and warranty period decisions for complementary products with horizontal firms' cooperation/noncooperation strategies, Journal of Cleaner Production, 105 (2015), 86-102.  doi: 10.1016/j.jclepro.2014.09.059.

[20]

C. H. Wu, C. W. Chen and C. C. Hsieh, Competitive pricing decisions in a two echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2012), 265-274.  doi: 10.1016/j.ijpe.2011.07.020.

[21]

Z. Yao, S. C. H. Leung and K. K. Lai, Manufacturer's revenue-sharing contract and retail competition, European Journal of Operational Research, 186 (2008), 637-651.  doi: 10.1016/j.ejor.2007.01.049.

[22]

X. Yue, S. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192.  doi: 10.1016/j.jbusres.2005.06.005.

[23]

J. Zhao, W. Tang, R. Zhao and J. Wei, Pricing decisions for substitutable products with a common retailer in fuzzy environments, European Journal of Operational Research, 216 (2012), 409-419.  doi: 10.1016/j.ejor.2011.07.026.

show all references

References:
[1]

A. Banerjee, A joint economic-lot-size model for purchaser and vendor, Decision Sciences, 17 (1986), 292-311.  doi: 10.1111/j.1540-5915.1986.tb00228.x.

[2]

L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258. 

[3]

C. S. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296.  doi: 10.1287/mksc.10.4.271.

[4]

A. L. EI-Ansary and L. W. Stern, Power measurement in the distribution channel, Journal of Marketing Research, 9 (1972), 47-52. 

[5]

G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700.  doi: 10.1080/07408170208928905.

[6]

J. Gabszewicz, N. Sonnac and X. Wauthy, On price competition with complementary goods, Economics Letters, 70 (2001), 431-437.  doi: 10.1016/S0165-1765(00)00383-9.

[7]

S. K. Goyal, An integrated inventory model for a single supplier-single customer problem, International Journal of Production Research, 15 (1977), 107-111.  doi: 10.1080/00207547708943107.

[8]

S. K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment, Decision Sciences, 19 (1988), 236-241.  doi: 10.1111/j.1540-5915.1988.tb00264.x.

[9]

C. C. Hsieh and C. H. Wu, Coordinated decisions for substitutable products in a common retailer supply chain, European Journal of Operational Research, 196 (2009), 273-288.  doi: 10.1016/j.ejor.2008.02.019.

[10]

K. F. McCardle, K. Rajaram and C. S. Tang, Bundling retail products: Models and analysis, European Journal of Operational Research, 177 (2007), 1197-1217.  doi: 10.1016/j.ejor.2005.11.009.

[11]

N. M. Modak, S. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International Journal of Production Economics, 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.

[12]

S. Mukhopadhyay, X. Yue and X. Zhu, A Stackelberg model of pricing of complementary goods under information asymmetry, International Journal of Production Economics, 134 (2011), 424-433.  doi: 10.1016/j.ijpe.2009.11.015.

[13]

K. Pan, K. K. Lai, S. C. H. Leung and D. Xiao, Revenue-sharing versus wholesale price mechanisms under different channel power structures, European Journal of Operational Research, 203 (2010), 532-538.  doi: 10.1016/j.ejor.2009.08.010.

[14]

B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering 2016 (2016), Article ID 6318737, 14 pages. doi: 10.1155/2016/6318737.

[15]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2013), 3138-3151.  doi: 10.1016/j.apm.2012.07.026.

[16]

B. Sarkar and A. Majumder, Integrated vendor-buyer supply chain model with vendors setup cost reduction, Applied Mathematics and Computation, 224 (2013), 362-371.  doi: 10.1016/j.amc.2013.08.072.

[17]

B. Sarkar, S. Saren, D. Sinha and S. Hur, Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model Mathematical Problems in Engineering 2015 (2015), Article ID 469486, 13 pages. doi: 10.1155/2015/469486.

[18]

J. Wei, J. Zhao and Y. Li, Pricing decisions for complementary products with firms' different market powers, European Journal of Operational Research, 224 (2013), 507-519.  doi: 10.1016/j.ejor.2012.09.011.

[19]

J. Wei, J. Zhao and Y. Li, Price and warranty period decisions for complementary products with horizontal firms' cooperation/noncooperation strategies, Journal of Cleaner Production, 105 (2015), 86-102.  doi: 10.1016/j.jclepro.2014.09.059.

[20]

C. H. Wu, C. W. Chen and C. C. Hsieh, Competitive pricing decisions in a two echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2012), 265-274.  doi: 10.1016/j.ijpe.2011.07.020.

[21]

Z. Yao, S. C. H. Leung and K. K. Lai, Manufacturer's revenue-sharing contract and retail competition, European Journal of Operational Research, 186 (2008), 637-651.  doi: 10.1016/j.ejor.2007.01.049.

[22]

X. Yue, S. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192.  doi: 10.1016/j.jbusres.2005.06.005.

[23]

J. Zhao, W. Tang, R. Zhao and J. Wei, Pricing decisions for substitutable products with a common retailer in fuzzy environments, European Journal of Operational Research, 216 (2012), 409-419.  doi: 10.1016/j.ejor.2011.07.026.

Figure 1.  Graphical representation for Case 1.1, total profit of manufacturer 1 versus selling-price and lot size
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Figure 2.  Graphical representation for Case 1.1, total profit of manufacturer 2 versus selling-price
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Figure 3.  Graphical representation for Case 1.1, total profit of retailer versus selling-price of bundle product
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Figure 4.  Graphical representation for Case 1.2, total profit of manufacturer 1 versus selling-price and lot size
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Figure 5.  Graphical representation for Case 1.2, total profit of manufacturer 2 and retailer versus selling-price and selling-price of bundle product
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Figure 6.  Graphical representation for Case 2.1, total profit of manufacturer 2 versus selling-price and lot size
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Figure 7.  Graphical representation for Case 2.1, total profit of manufacturer 1 versus sellingprice
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Figure 8.  Graphical representation for Case 2.1, total profit of retailer versus selling-price of bundle product
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Figure 9.  Graphical representation for Case 2.2, total profit of manufacturer 2 versus selling price and lot size
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Figure 10.  Graphical representation for Case 2.2, total profit of manufacturer 1 and retailer versus selling-price and selling-price of bundle product
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Figure 11.  Comparative studies of cooperation and non-cooperation for the selling-price of product 2 of manufacturer 2 in Case 1. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
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Figure 12.  Comparative studies of cooperation and non-cooperation for the selling-price of bundle product of retailer in Case 1. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
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Figure 13.  Comparative studies of cooperation and non-cooperation for the selling-price of product 1 of manufacturer 1 in Case 2. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
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Figure 14.  Comparative studies of cooperation and non-cooperation for the selling-price of bundle product of retailer in Case 2. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy
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Table 1.  Comparison between the contributions of different authors
Author (s)SCMCompetitive price studyReservation priceGame approachStochastic demand
Choi [3] √ √
Yue et al. [22] √ √
Mukhopadhyay
et al. [12] √ √
Wei et al. [18] √ √ √
Cárdenas-Barrón and Sana [2] √ √
Sarkar [14] √ √
McCardle et al. [10] √ √ √
This Model √ √ √ √ √
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Author (s)SCMCompetitive price studyReservation priceGame approachStochastic demand
Choi [3] √ √
Yue et al. [22] √ √
Mukhopadhyay
et al. [12] √ √
Wei et al. [18] √ √ √
Cárdenas-Barrón and Sana [2] √ √
Sarkar [14] √ √
McCardle et al. [10] √ √ √
This Model √ √ √ √ √
Table  . 
Decision variables
$Q$ order quantity (units)
$P_{i}$ selling-price of product j, j=1, 2 (＄/unit)
$P_{r}$ selling-price of the bundle product (＄/unit)
Random variables
$D_{m_{i}}$ demand for product j, j=1, 2 (units)
$D_{r}$ demand for the bundle product (units)
Parameters
$C_{i}$ manufacturing cost of product j, j=1, 2 (＄/unit)
$h_{m_{i}}$ holding cost of product j per unit per unit time, j=1, 2 (＄/unit/unit time)
$h_{r}$ holding cost of the bundle product per unit per unit time (＄/unit/unit time)
$S_{m_{i}}$ setup cost per setup of product j, j=1, 2 (＄/unit)
$K_{m_{i}}$ production rate of product j, j=1, 2 (units)
$M$ known market size (units)
$A$ ordering cost per order of the retailer (＄/order)
$I_{m_{i1}}$ inventory of manufacturer i at $ t \in[0, t_{m_{i}}]$, i=1, 2
$I_{m_{i2}}$ inventory of manufacturer i at $t \in [t_{m_{i}}, T_{m_{i}}]$, i=1, 2
$AP_{m_{i}}$ expected average profit of manufacturer i, i=1, 2
$AP_{r}$ expected average profit of the retailer
$t_{m_{i}}$ time required for maximum inventory of manufacturer i, i=1, 2
$T_{m_{i}}$ cycle time of manufacturer i, i=1, 2
$R_{i}^{a}$ lower limit of reservation price of manufacturer i, i=1, 2
$R_{i}^{b}$ upper limit of reservation price of manufacturer i, i=1, 2
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Decision variables
$Q$ order quantity (units)
$P_{i}$ selling-price of product j, j=1, 2 (＄/unit)
$P_{r}$ selling-price of the bundle product (＄/unit)
Random variables
$D_{m_{i}}$ demand for product j, j=1, 2 (units)
$D_{r}$ demand for the bundle product (units)
Parameters
$C_{i}$ manufacturing cost of product j, j=1, 2 (＄/unit)
$h_{m_{i}}$ holding cost of product j per unit per unit time, j=1, 2 (＄/unit/unit time)
$h_{r}$ holding cost of the bundle product per unit per unit time (＄/unit/unit time)
$S_{m_{i}}$ setup cost per setup of product j, j=1, 2 (＄/unit)
$K_{m_{i}}$ production rate of product j, j=1, 2 (units)
$M$ known market size (units)
$A$ ordering cost per order of the retailer (＄/order)
$I_{m_{i1}}$ inventory of manufacturer i at $ t \in[0, t_{m_{i}}]$, i=1, 2
$I_{m_{i2}}$ inventory of manufacturer i at $t \in [t_{m_{i}}, T_{m_{i}}]$, i=1, 2
$AP_{m_{i}}$ expected average profit of manufacturer i, i=1, 2
$AP_{r}$ expected average profit of the retailer
$t_{m_{i}}$ time required for maximum inventory of manufacturer i, i=1, 2
$T_{m_{i}}$ cycle time of manufacturer i, i=1, 2
$R_{i}^{a}$ lower limit of reservation price of manufacturer i, i=1, 2
$R_{i}^{b}$ upper limit of reservation price of manufacturer i, i=1, 2
Table 2.  Input data
Player Market size (units) Manufacturer 1
$M=1500$		 Manufacturer 2
$M=1500$		 Retailer
$M=1500$
Setup cost
(＄/setup)		 $S_{m_{1}}=20$ $S_{m_{2}}=20$ $A=1$
Holding cost
(＄/unit/year)		 $h_{m_{1}}=0.015$ $h_{m_{2}}=0.015$ $h_r=0.01$
Production rate
(units/year)		 $K_{m_{1}}=2000$ $K_{m_{2}}=2000$ -
Purchasing cost
(＄/unit)		 $C_{1}=0.25$ $C_{2}=0.15$ -
Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9]
  -indicates that the parameter is not available for this case.
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Player Market size (units) Manufacturer 1
$M=1500$		 Manufacturer 2
$M=1500$		 Retailer
$M=1500$
Setup cost
(＄/setup)		 $S_{m_{1}}=20$ $S_{m_{2}}=20$ $A=1$
Holding cost
(＄/unit/year)		 $h_{m_{1}}=0.015$ $h_{m_{2}}=0.015$ $h_r=0.01$
Production rate
(units/year)		 $K_{m_{1}}=2000$ $K_{m_{2}}=2000$ -
Purchasing cost
(＄/unit)		 $C_{1}=0.25$ $C_{2}=0.15$ -
Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9]
  -indicates that the parameter is not available for this case.
Table 3.  Optimum results of Example 1
Case $Q^*$ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year
1.1 1433.14 0.63 0.53 1.33 195.39 246.92 36.37 - -
1.2 1433.14 0.63 0.44 1.29 195.39 - - - 294.18
2.1 1690.88 0.63 0.53 1.33 195.18 247.15 35.90 - -
2.2 1690.88 0.51 0.53 1.27 - 247.15 - 245.87 -
  -indicates that the average profit is not available for this case.
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Case $Q^*$ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year
1.1 1433.14 0.63 0.53 1.33 195.39 246.92 36.37 - -
1.2 1433.14 0.63 0.44 1.29 195.39 - - - 294.18
2.1 1690.88 0.63 0.53 1.33 195.18 247.15 35.90 - -
2.2 1690.88 0.51 0.53 1.27 - 247.15 - 245.87 -
  -indicates that the average profit is not available for this case.
Table 4.  Input data from McCardle et al. [10]
Player Market size (units) Manufacturer 1
$M=100$		 Manufacturer 2
$M=100$		 Retailer
$M=100$
Setup cost
(＄/setup)		 $S_{m_{1}}=0$ $S_{m_{2}}=0$ $A=0$
Holding cost
(＄/unit/year)		 $h_{m_{1}}=0$ $h_{m_{2}}=0$ $h_r=0$
Production rate
(units/year)		 $K_{m_{1}}=0$ $K_{m_{2}}=0$ -
Purchasing cost
(＄/unit)		 $C_{1}=0.25$ $C_{2}=0.25$ -
Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9]
  -indicates that the parameter is not available for this case.
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Player Market size (units) Manufacturer 1
$M=100$		 Manufacturer 2
$M=100$		 Retailer
$M=100$
Setup cost
(＄/setup)		 $S_{m_{1}}=0$ $S_{m_{2}}=0$ $A=0$
Holding cost
(＄/unit/year)		 $h_{m_{1}}=0$ $h_{m_{2}}=0$ $h_r=0$
Production rate
(units/year)		 $K_{m_{1}}=0$ $K_{m_{2}}=0$ -
Purchasing cost
(＄/unit)		 $C_{1}=0.25$ $C_{2}=0.25$ -
Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9]
  -indicates that the parameter is not available for this case.
Table 5.  Optimum results of Example 2
Case $Q^* $ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year
1.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - -
1.2 $300$ $0.625$ $0.541667$ $1.33$ $14.0625$ - - - $16.1458$
2.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - -
2.2 $300$ $0.625$ $0.541667$ $1.33333$ - $14.0625$ - 16.1458 -
  -indicates that the average profit is not available for this case.
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Case $Q^* $ units $P_{1}^{*}$ ＄/unit $P_{2}^{*}$ ＄/unit $P_{r}^{*}$ ＄/unit $AP_{m_{1}}^{*}$ ＄/year $AP_{m_{2}}^{*}$ ＄/year $AP_{{r}}^{*}$ ＄/year $AP_{m_{1}r}^{*}$ ＄/year $AP_{m_{2}r}^{*}$ ＄/year
1.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - -
1.2 $300$ $0.625$ $0.541667$ $1.33$ $14.0625$ - - - $16.1458$
2.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - -
2.2 $300$ $0.625$ $0.541667$ $1.33333$ - $14.0625$ - 16.1458 -
  -indicates that the average profit is not available for this case.
Table 6.  Sensitivity analysis for Case 1.1
Parameter change(in %) $AP_{m_{1}}$
(in %)		 $AP_{m_{2}}$
(in %)		 $AP_{r}$
(in %)
$M$ -50% -52.76 -52.18 -59.85
-25% -26.38 -26.09 -29.93
+25% +26.38 +26.09 +29.93
+50% +52.75 +52.18 +59.85
Parameter change(in %) $AP_{m_{1}}$
(in %)		 Parameter change(in %) $AP_{m_{2}}$
(in %)
$S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +1.97
-25% +0.99 -25% +0.98
+25% -0.99 +25% -0.98
+50% -1.98 +50% -1.95
$h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.42
-25% +1.06 -25% +1.42
+25% -0.94 +25% -0.71
+50% -1.78 +50% -1.42
$C_{1}$ -50% +38.63 $C_{2}$ -50% +22.18
-25% +18.54 -25% +10.82
+25% -17.04 +25% -10.29
+50% -32.58 +50% -20.04
Parameter change(in %) $AP_{r}$
(in %)		 Parameter change(in %) $AP_{r}$
(in %)
$A$ -50% +0.25 $h_{r}$ -50% +9.85
-25% +0.12 -25% +4.93
+25% -0.12 +25% -4.93
+50% -0.25 +50% -9.85
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Parameter change(in %) $AP_{m_{1}}$
(in %)		 $AP_{m_{2}}$
(in %)		 $AP_{r}$
(in %)
$M$ -50% -52.76 -52.18 -59.85
-25% -26.38 -26.09 -29.93
+25% +26.38 +26.09 +29.93
+50% +52.75 +52.18 +59.85
Parameter change(in %) $AP_{m_{1}}$
(in %)		 Parameter change(in %) $AP_{m_{2}}$
(in %)
$S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +1.97
-25% +0.99 -25% +0.98
+25% -0.99 +25% -0.98
+50% -1.98 +50% -1.95
$h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.42
-25% +1.06 -25% +1.42
+25% -0.94 +25% -0.71
+50% -1.78 +50% -1.42
$C_{1}$ -50% +38.63 $C_{2}$ -50% +22.18
-25% +18.54 -25% +10.82
+25% -17.04 +25% -10.29
+50% -32.58 +50% -20.04
Parameter change(in %) $AP_{r}$
(in %)		 Parameter change(in %) $AP_{r}$
(in %)
$A$ -50% +0.25 $h_{r}$ -50% +9.85
-25% +0.12 -25% +4.93
+25% -0.12 +25% -4.93
+50% -0.25 +50% -9.85
Table 7.  Sensitivity analysis for Case 1.2
Parameter change(in %) $AP_{m_{1}}$
(in %)		 $AP_{m_{2}r}$
(in %)
$M$ -50% -52.15 -53.04
-25% -26.24 -26.52
+25% +26.49 +26.52
+50% +53.18 +53.04
Parameter change(in %) $AP_{m_{1}}$
(in %)		 Parameter change(in %) $AP_{m_{2}r}$
(in %)
$S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +2.04
-25% +0.99 -25% +1.02
+25% -0.99 +25% -1.01
+50% -1.98 +50% -2.02
$h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.05
-25% +1.06 -25% +0.52
+25% -0.94 +25% -0.52
+50% -1.78 +50% -1.04
$C_{1}$ -50% +38.63 $C_{2}$ -50% +22.91
-25% +18.55 -25% +11.18
+25% -17.02 +25% -10.62
+50% -32.52 +50% -20.67
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Parameter change(in %) $AP_{m_{1}}$
(in %)		 $AP_{m_{2}r}$
(in %)
$M$ -50% -52.15 -53.04
-25% -26.24 -26.52
+25% +26.49 +26.52
+50% +53.18 +53.04
Parameter change(in %) $AP_{m_{1}}$
(in %)		 Parameter change(in %) $AP_{m_{2}r}$
(in %)
$S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +2.04
-25% +0.99 -25% +1.02
+25% -0.99 +25% -1.01
+50% -1.98 +50% -2.02
$h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.05
-25% +1.06 -25% +0.52
+25% -0.94 +25% -0.52
+50% -1.78 +50% -1.04
$C_{1}$ -50% +38.63 $C_{2}$ -50% +22.91
-25% +18.55 -25% +11.18
+25% -17.02 +25% -10.62
+50% -32.52 +50% -20.67
Table 8.  Sensitivity analysis for Case 2.1
Parameters change(in %) $AP_{m_{1}}$
(in %)		 $AP_{m_{2}}$
(in %)		 $AP_{r}$
(in %)
$M$ -50% -53.25 -52.57 -61.77
-25% -26.62 -26.28 -30.89
+25% +26.62 +26.28 +30.89
+50% +53.25 +52.57 +61.77
Parameter change(in %) $AP_{r}$
(in %)		 Parameter change(in %) $AP_{r}$
(in %)
$A$ -50% +0.21 $h_{r}$ -50% +11.77
-25% +0.11 -25% +5.89
+25% -0.11 +25% -5.89
+50% -0.21 +50% -11.77
Parameter change(in %) $AP_{m_{1}}$
(in %)		 Parameter change(in %) $AP_{m_{2}}$
(in %)
$S_{m_{1}}$ -50% +1.70 $S_{m_{2}}$ -50% +1.96
-25% +0.85 -25% +0.90
+25% -0.84 +25% -0.79
+50% -1.69 +50% -1.50
$h_{m_{1}}$ -50% +2.34 $h_{m_{2}}$ -50% +1.96
-25% +1.17 -25% +0.90
+25% -1.17 +25% -0.79
+50% -2.34 +50% -1.50
$C_{1}$ -50% +38.76 $C_{2}$ -50% +22.27
-25% +18.63 -25% +10.86
+25% -17.13 +25% -10.32
+50% -32.76 +50% -20.09
Table Options

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Parameters change(in %) $AP_{m_{1}}$
(in %)		 $AP_{m_{2}}$
(in %)		 $AP_{r}$
(in %)
$M$ -50% -53.25 -52.57 -61.77
-25% -26.62 -26.28 -30.89
+25% +26.62 +26.28 +30.89
+50% +53.25 +52.57 +61.77
Parameter change(in %) $AP_{r}$
(in %)		 Parameter change(in %) $AP_{r}$
(in %)
$A$ -50% +0.21 $h_{r}$ -50% +11.77
-25% +0.11 -25% +5.89
+25% -0.11 +25% -5.89
+50% -0.21 +50% -11.77
Parameter change(in %) $AP_{m_{1}}$
(in %)		 Parameter change(in %) $AP_{m_{2}}$
(in %)
$S_{m_{1}}$ -50% +1.70 $S_{m_{2}}$ -50% +1.96
-25% +0.85 -25% +0.90
+25% -0.84 +25% -0.79
+50% -1.69 +50% -1.50
$h_{m_{1}}$ -50% +2.34 $h_{m_{2}}$ -50% +1.96
-25% +1.17 -25% +0.90
+25% -1.17 +25% -0.79
+50% -2.34 +50% -1.50
$C_{1}$ -50% +38.76 $C_{2}$ -50% +22.27
-25% +18.63 -25% +10.86
+25% -17.13 +25% -10.32
+50% -32.76 +50% -20.09
Table 9.  Sensitivity analysis for Case 2.2
Parameter change(in %) $AP_{m_{2}}$
(in %)		 $AP_{m_{1}r}$
(in %)
$M$ -50% -52.57 -54.30
-25% -26.28 -27.15
+25% +26.28 +27.15
+50% +52.57 +54.30
Parameter change(in %) $AP_{m_{1r}}$
(in %)		 Parameter change(in %) $AP_{m_{2}}$
(in %)
$S_{m_{1}}$ -50% +1.76 $S_{m_{2}}$ -50% +1.96
-25% +0.88 -25% +0.90
+25% -0.88 +25% -0.79
+50% -1.75 +50% -1.50
$h_{m_{1}}$ -50% +1.64 $h_{m_{2}}$ -50% +1.96
-25% +0.82 -25% +0.90
+25% -0.82 +25% -0.79
+50% -1.64 +50% -1.50
$C_{1}$ -50% +40.31 $C_{2}$ -50% +22.27
-25% +19.36 -25% +10.86
+25% -17.77 +25% -10.32
+50% -33.95 +50% -20.09
Parameter change(in %) $AP_{m_{1r}}$
(in %)		 Parameter change(in %) $AP_{r}$
(in %)
$A$ -50% +0.04 $h_{r}$ -50% +1.72
-25% +0.02 -25% +0.86
+25% -0.02 +25% -0.86
+50% -0.04 +50% -1.72
Table Options

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Parameter change(in %) $AP_{m_{2}}$
(in %)		 $AP_{m_{1}r}$
(in %)
$M$ -50% -52.57 -54.30
-25% -26.28 -27.15
+25% +26.28 +27.15
+50% +52.57 +54.30
Parameter change(in %) $AP_{m_{1r}}$
(in %)		 Parameter change(in %) $AP_{m_{2}}$
(in %)
$S_{m_{1}}$ -50% +1.76 $S_{m_{2}}$ -50% +1.96
-25% +0.88 -25% +0.90
+25% -0.88 +25% -0.79
+50% -1.75 +50% -1.50
$h_{m_{1}}$ -50% +1.64 $h_{m_{2}}$ -50% +1.96
-25% +0.82 -25% +0.90
+25% -0.82 +25% -0.79
+50% -1.64 +50% -1.50
$C_{1}$ -50% +40.31 $C_{2}$ -50% +22.27
-25% +19.36 -25% +10.86
+25% -17.77 +25% -10.32
+50% -33.95 +50% -20.09
Parameter change(in %) $AP_{m_{1r}}$
(in %)		 Parameter change(in %) $AP_{r}$
(in %)
$A$ -50% +0.04 $h_{r}$ -50% +1.72
-25% +0.02 -25% +0.86
+25% -0.02 +25% -0.86
+50% -0.04 +50% -1.72
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