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October  2019, 15(4): 1701-1727. doi: 10.3934/jimo.2018118

Coordinating the supplier-retailer supply chain under noise effect with bundling and inventory strategies

Ata Allah Taleizadeh 1, , Leopoldo Eduardo Cárdenas-Barrón 2,, and  Roya Sohani 3,

1. 

School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran
2. 

School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México
3. 

Department of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran

* Corresponding author: Tel. +52 81 83284235, Fax +52 81 83284153. E-mail address:lecarden@itesm.mx (L.E. Cárdenas-Barrón)

Received  May 2017 Revised  January 2018 Published  October 2019 Early access  August 2018

In current competitive market, the products and their demand's uncertainty are high. In order to reduce these uncertainties the coordination of supply chain is necessary. Supply chain can be managed under two viewpoints typically: 1) centralized supply chain and 2) decentralized supply chain, and the coordination can be done in both types of chains. In the centralized supply chain there exists a global decision maker who takes all the best decisions in order to maximize the profit of the whole supply chain. Here, the useful information required to make the best decisions is open to all members of the chain. On the other hand, in the decentralized supply chain all members decide in a separate and sequential way, how to maximize their profits. In order to coordinate efficiently the supply chain, both supplier and retailer are involved in a coordination contract that makes it possible for the decentralized decisions to maximize the profit of the entire supply chain. In this context, the situation that the supplier-retailer chain faces is a two-stage decision model. In the first stage the supplier, based on former knowledge about the market, decides the production capacity to reserve for the retailer. In the second stage, after that demand information is updated, the retailer determines the bundle price and the quantity of bundles to order. This paper considers a supply chain comprised of one supplier and one retailer in which two complementary fashion products are manufactured and sold as a bundle. The bundle has a short selling season and a stochastic price dependent on demand with a high level of uncertainty. Therefore, this research considers that the demand rates are uncertain and are dependent on selling prices and on a random noise effect on the market. Profit maximization models are developed for centralized and decentralized supply chains to determine decisions on production capacity reservation, order quantity of bundled products and the bundle-selling price. The applicability of the developed models and solution method are illustrated with a numerical example.

Keywords: Pricing, Inventory; supply chain coordination, risk and profit sharing.
Mathematics Subject Classification: 90B05.
Citation: Ata Allah Taleizadeh, Leopoldo Eduardo Cárdenas-Barrón, Roya Sohani. Coordinating the supplier-retailer supply chain under noise effect with bundling and inventory strategies. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1701-1727. doi: 10.3934/jimo.2018118
References:
[1]

M. Armstrong and J. Vickers, Competitive non-linear pricing and bundling, The Review of Economic Studies, 77 (2010), 30-60.  doi: 10.1111/j.1467-937X.2009.00562.x.

[2]

R. Arora, Price bundling and framing strategies for complementary products, Journal of Product and Brand Management, 17 (2008), 475-484. 

[3]

M. Banciu and F. ∅degaard, Optimal product bundling with dependent valuations: The price of independence, European Journal of Operational Research, 255 (2016), 481-495.  doi: 10.1016/j.ejor.2016.05.022.

[4]

D. Barnes-Schuster, Y. Bassok and R. Anupindi, Coordination and flexibility in supply contracts with options, Manufacturing and Service Operations Management, 4 (2002), 171-207. 

[5]

R. J. Bennett and P. J. Robson, Exploring the market potential and bundling of business association services, Journal of Services Marketing, 15 (2001), 222-239. 

[6]

H. K. Bhargava, Retailer-driven product bundling in a distribution channel, Marketing Science, 31 (2012), 1014-1021. 

[7]

G. R. Bitran and J. C. Ferrer, On pricing and composition of bundles, Production and Operations Management, 16 (2007), 93-108. 

[8]

D. Brito and H. Vasconcelos, Interfirm bundling and vertical product differentiation, The Scandinavian Journal of Economics, 117 (2015), 1-27. 

[9]

Z. Bulut, Ü. Gürler and A. Sen, Bundle pricing of inventories with stochastic demand, European Journal of Operational Research, 197 (2009), 897-911.  doi: 10.1016/j.ejor.2006.09.106.

[10]

P. G. Cachon, Supply chain coordination with contracts. In: Graves, S., de Kok, T. (Eds.), Handbooks in Operations Research and Management Science. North Holland Press, 11 (2003), 229-340. 

[11]

G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: strengths and limitations, Management Science(1), 51 (2005), 30-44. 

[12]

A. Chakravarty, A. Mild and A. Taudes, Bundling decisions in supply chains, European Journal of Operational Research, 231 (2013), 617-630. 

[13]

H. Chen, Y. F. Chen, C. H. Chiu, T. M. Choi and S. Sethi, Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand, European Journal of Operational Research, 203 (2010), 70-80. 

[14]

J. Chen and P. C. Bell, Coordinating a decentralized supply chain with customer returns and price-dependent stochastic demand using a buyback policy, European Journal of Operational Research, 212 (2011), 293-300.  doi: 10.1016/j.ejor.2011.01.036.

[15]

K. L. Donohue, Efficient supply contracts for fashion goods with forecast updating and two production modes, Management Science, 46 (2000), 1397-1411. 

[16]

J. C. Eckalbar, Closed-form solutions to bundling problems, Journal of Economics and Management Strategy, 19 (2010), 513-544. 

[17]

H. Estelami, Consumer savings in complementary product bundles, Journal of Marketing Theory and Practice, 7 (1999), 107-114. 

[18]

J. C. Ferrer, H. Mora and F. Olivares, On pricing of multiple bundles of products and services, European Journal of Operational Research, 206 (2010), 197-208. 

[19]

J. S. Gans and S. P. King, Paying for loyalty: Product bundling in oligopoly, The Journal of Industrial Economics, 54 (2006), 43-62. 

[20]

R. N. Giri, S. K. Mondal and M. Maiti, Bundle pricing strategies for two complementary products with different channel powers, Annals of Operations Research, (2017), 1-25.  doi: 10.1007/s10479-017-2632-y.

[21]

M. Girju, A. Prasad and B. T. Ratchford, Pure components versus pure bundling in a marketing channel, Journal of Retailing, 89 (2013), 423-437. 

[22]

J. P. Guiltinan, The price bundling of services: A normative framework, The Journal of Marketing, (1987), 74-85. 

[23]

Ü. Gürler, S. Öztop and A. Şen, Optimal bundle formation and pricing of two products with limited stock, International Journal of Production Economics, 118 (2009), 442-462. 

[24]

R. Glenn Hubbard, A. Saha and J. Lee, To bundle or not to bundle: Firms' choices under pure bundling, International Journal of the Economics of Business, 14 (2007), 59-83. 

[25]

M. Li, H. Feng, F. Chen and J. Kou, Numerical investigation on mixed bundling and pricing of information products, International Journal of Production Economics, 144 (2013), 560-571. 

[26]

P. P. Mathur and J. Shah, Supply chain contracts with capacity investment decision: Two-way penalties for coordination, International Journal of Production Economics, 114 (2008), 56-70. 

[27]

C. Matutes and P. Regibeau, Compatibility and bundling of complementary goods in a duopoly, The Journal of Industrial Economics, 40 (1992), 37-54. 

[28]

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

[29]

S. K. 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. 

[30]

B. Nalebuff, Bundling as an entry barrier, The Quarterly Journal of Economics, 119 (2004), 159-187. 

[31]

H. Oppewal and B. Holyoake, Bundling and retail agglomeration effects on shopping behavior, Journal of Retailing and Consumer Services, 11 (2004), 61-74. 

[32]

E. C. Rosenthal, J. L. Zydiak and S. S. Chaudhry, Vendor selection with bundling, Decision Sciences, 26 (1995), 35-48. 

[33]

M. Sheikhzadeh and E. Elahi, Product bundling: Impacts of product heterogeneity and risk considerations, International Journal of Production Economics, 144 (2013), 209-222. 

[34]

S. Sheng, A. M. Parker and K. Nakamoto, The effects of price discount and product complementarity on consumer evaluations of bundle components, Journal of Marketing Theory and Practice, 15 (2007), 53-64. 

[35]

B. L. Simonin and J. A. Ruth, Bundling as a strategy for new product introduction: Effects on consumers' reservation prices for the bundle, the new product, and its tie-in, Journal of Business Research, 33 (1995), 219-230. 

[36]

A. A. Taleizadeh, S. T. A. Niaki, M. B. Aryanezhad and A. F. Tafti, A genetic algorithm to optimize multiproduct multiconstraint inventory control systems with stochastic replenishment intervals and discount, The International Journal of Advanced Manufacturing Technology, 51 (2010), 311-323. 

[37]

A. A. Taleizadeh and M. Noori-daryan, Pricing, manufacturing and inventory policies for raw material in a three-level supply chain, International Journal of Systems Science, 47 (2016), 919-931.  doi: 10.1080/00207721.2014.909544.

[38]

A. A. Taleizadeh, M. Noori-Daryan and K. Govindan, Pricing and ordering decisions of two competing supply chains with different composite policies: A Stackelberg game-theoretic approach, International Journal of Production Research, 54 (2016), 2807-2836. 

[39]

A. A. Taleizadeh, M. Noori-daryan and R. Tavakkoli-Moghaddam, Pricing and ordering decisions in a supply chain with imperfect quality items and inspection under buyback of defective items, International Journal of Production Research, 53 (2015), 4553-4582. 

[40]

A. A. Taleizadeh and D. W. Pentico, An economic order quantity model with a known price increase and partial backordering, European Journal of Operational Research, 228 (2013), 516-525.  doi: 10.1016/j.ejor.2013.02.014.

[41]

A. A. Taleizadeh, D. W. Pentico, M. S. Jabalameli and M. Aryanezhad, An economic order quantity model with multiple partial prepayments and partial backordering, Mathematical and Computer Modelling, 57 (2013), 311-323.  doi: 10.1016/j.mcm.2012.07.002.

[42]

T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects, Management Science, 48 (2002), 992-1007. 

[43]

A. Vamosiu, Optimal bundling under imperfect competition, International Journal of Production Economics, 195 (2018), 45-53. 

[44]

A. G. Vaubourg, Differentiation and discrimination in a duopoly with two bundles, International Journal of Industrial Organization, 24 (2006), 753-762. 

[45]

R. Venkatesh and W. Kamakura, Optimal bundling and pricing under a monopoly: Contrasting complements and substitutes from independently valued products, Journal of Business, 76 (2003), 211-231. 

[46]

Q. Wang, Discount pricing policies and the coordination of decentralized distribution systems, Decision Sciences, 36 (2005), 627-646. 

[47]

Y. Wang, L. Sun, R. Qu and G. Li, Price and service competition with maintenance service bundling, Journal of Systems Science and Systems Engineering, 24 (2015), 168-189. 

[48]

A. Wäppling, C. Strugnell and H. Farley, Product bundling strategies in Swedish markets: links to business orientation and perceived effects on consumer influence, International Journal of Consumer Studies, 34 (2010), 19-27. 

[49]

R. Yan, Managing channel coordination in a multi-channel manufacturer-retailer supply chain, Industrial Marketing Management, 40 (2011), 636-642. 

[50]

R. Yan and S. Bandyopadhyay, The profit benefits of bundle pricing of complementary products, Journal of Retailing and Consumer Services, 18 (2011), 355-361. 

[51]

R. Yan, C. Myers, J. Wang and S. Ghose, Bundling products to success: The influence of complementarity and advertising, Journal of Retailing and Consumer Services, 21 (2014), 48-53. 

[52]

R. Yan and Z. Pei, Retail services and firm profit in a dual-channel market, Journal of Retailing and Consumer Services, 16 (2009), 306-314. 

[53]

X. Yue, S. K. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192. 

show all references

References:
[1]

M. Armstrong and J. Vickers, Competitive non-linear pricing and bundling, The Review of Economic Studies, 77 (2010), 30-60.  doi: 10.1111/j.1467-937X.2009.00562.x.

[2]

R. Arora, Price bundling and framing strategies for complementary products, Journal of Product and Brand Management, 17 (2008), 475-484. 

[3]

M. Banciu and F. ∅degaard, Optimal product bundling with dependent valuations: The price of independence, European Journal of Operational Research, 255 (2016), 481-495.  doi: 10.1016/j.ejor.2016.05.022.

[4]

D. Barnes-Schuster, Y. Bassok and R. Anupindi, Coordination and flexibility in supply contracts with options, Manufacturing and Service Operations Management, 4 (2002), 171-207. 

[5]

R. J. Bennett and P. J. Robson, Exploring the market potential and bundling of business association services, Journal of Services Marketing, 15 (2001), 222-239. 

[6]

H. K. Bhargava, Retailer-driven product bundling in a distribution channel, Marketing Science, 31 (2012), 1014-1021. 

[7]

G. R. Bitran and J. C. Ferrer, On pricing and composition of bundles, Production and Operations Management, 16 (2007), 93-108. 

[8]

D. Brito and H. Vasconcelos, Interfirm bundling and vertical product differentiation, The Scandinavian Journal of Economics, 117 (2015), 1-27. 

[9]

Z. Bulut, Ü. Gürler and A. Sen, Bundle pricing of inventories with stochastic demand, European Journal of Operational Research, 197 (2009), 897-911.  doi: 10.1016/j.ejor.2006.09.106.

[10]

P. G. Cachon, Supply chain coordination with contracts. In: Graves, S., de Kok, T. (Eds.), Handbooks in Operations Research and Management Science. North Holland Press, 11 (2003), 229-340. 

[11]

G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: strengths and limitations, Management Science(1), 51 (2005), 30-44. 

[12]

A. Chakravarty, A. Mild and A. Taudes, Bundling decisions in supply chains, European Journal of Operational Research, 231 (2013), 617-630. 

[13]

H. Chen, Y. F. Chen, C. H. Chiu, T. M. Choi and S. Sethi, Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand, European Journal of Operational Research, 203 (2010), 70-80. 

[14]

J. Chen and P. C. Bell, Coordinating a decentralized supply chain with customer returns and price-dependent stochastic demand using a buyback policy, European Journal of Operational Research, 212 (2011), 293-300.  doi: 10.1016/j.ejor.2011.01.036.

[15]

K. L. Donohue, Efficient supply contracts for fashion goods with forecast updating and two production modes, Management Science, 46 (2000), 1397-1411. 

[16]

J. C. Eckalbar, Closed-form solutions to bundling problems, Journal of Economics and Management Strategy, 19 (2010), 513-544. 

[17]

H. Estelami, Consumer savings in complementary product bundles, Journal of Marketing Theory and Practice, 7 (1999), 107-114. 

[18]

J. C. Ferrer, H. Mora and F. Olivares, On pricing of multiple bundles of products and services, European Journal of Operational Research, 206 (2010), 197-208. 

[19]

J. S. Gans and S. P. King, Paying for loyalty: Product bundling in oligopoly, The Journal of Industrial Economics, 54 (2006), 43-62. 

[20]

R. N. Giri, S. K. Mondal and M. Maiti, Bundle pricing strategies for two complementary products with different channel powers, Annals of Operations Research, (2017), 1-25.  doi: 10.1007/s10479-017-2632-y.

[21]

M. Girju, A. Prasad and B. T. Ratchford, Pure components versus pure bundling in a marketing channel, Journal of Retailing, 89 (2013), 423-437. 

[22]

J. P. Guiltinan, The price bundling of services: A normative framework, The Journal of Marketing, (1987), 74-85. 

[23]

Ü. Gürler, S. Öztop and A. Şen, Optimal bundle formation and pricing of two products with limited stock, International Journal of Production Economics, 118 (2009), 442-462. 

[24]

R. Glenn Hubbard, A. Saha and J. Lee, To bundle or not to bundle: Firms' choices under pure bundling, International Journal of the Economics of Business, 14 (2007), 59-83. 

[25]

M. Li, H. Feng, F. Chen and J. Kou, Numerical investigation on mixed bundling and pricing of information products, International Journal of Production Economics, 144 (2013), 560-571. 

[26]

P. P. Mathur and J. Shah, Supply chain contracts with capacity investment decision: Two-way penalties for coordination, International Journal of Production Economics, 114 (2008), 56-70. 

[27]

C. Matutes and P. Regibeau, Compatibility and bundling of complementary goods in a duopoly, The Journal of Industrial Economics, 40 (1992), 37-54. 

[28]

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

[29]

S. K. 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. 

[30]

B. Nalebuff, Bundling as an entry barrier, The Quarterly Journal of Economics, 119 (2004), 159-187. 

[31]

H. Oppewal and B. Holyoake, Bundling and retail agglomeration effects on shopping behavior, Journal of Retailing and Consumer Services, 11 (2004), 61-74. 

[32]

E. C. Rosenthal, J. L. Zydiak and S. S. Chaudhry, Vendor selection with bundling, Decision Sciences, 26 (1995), 35-48. 

[33]

M. Sheikhzadeh and E. Elahi, Product bundling: Impacts of product heterogeneity and risk considerations, International Journal of Production Economics, 144 (2013), 209-222. 

[34]

S. Sheng, A. M. Parker and K. Nakamoto, The effects of price discount and product complementarity on consumer evaluations of bundle components, Journal of Marketing Theory and Practice, 15 (2007), 53-64. 

[35]

B. L. Simonin and J. A. Ruth, Bundling as a strategy for new product introduction: Effects on consumers' reservation prices for the bundle, the new product, and its tie-in, Journal of Business Research, 33 (1995), 219-230. 

[36]

A. A. Taleizadeh, S. T. A. Niaki, M. B. Aryanezhad and A. F. Tafti, A genetic algorithm to optimize multiproduct multiconstraint inventory control systems with stochastic replenishment intervals and discount, The International Journal of Advanced Manufacturing Technology, 51 (2010), 311-323. 

[37]

A. A. Taleizadeh and M. Noori-daryan, Pricing, manufacturing and inventory policies for raw material in a three-level supply chain, International Journal of Systems Science, 47 (2016), 919-931.  doi: 10.1080/00207721.2014.909544.

[38]

A. A. Taleizadeh, M. Noori-Daryan and K. Govindan, Pricing and ordering decisions of two competing supply chains with different composite policies: A Stackelberg game-theoretic approach, International Journal of Production Research, 54 (2016), 2807-2836. 

[39]

A. A. Taleizadeh, M. Noori-daryan and R. Tavakkoli-Moghaddam, Pricing and ordering decisions in a supply chain with imperfect quality items and inspection under buyback of defective items, International Journal of Production Research, 53 (2015), 4553-4582. 

[40]

A. A. Taleizadeh and D. W. Pentico, An economic order quantity model with a known price increase and partial backordering, European Journal of Operational Research, 228 (2013), 516-525.  doi: 10.1016/j.ejor.2013.02.014.

[41]

A. A. Taleizadeh, D. W. Pentico, M. S. Jabalameli and M. Aryanezhad, An economic order quantity model with multiple partial prepayments and partial backordering, Mathematical and Computer Modelling, 57 (2013), 311-323.  doi: 10.1016/j.mcm.2012.07.002.

[42]

T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects, Management Science, 48 (2002), 992-1007. 

[43]

A. Vamosiu, Optimal bundling under imperfect competition, International Journal of Production Economics, 195 (2018), 45-53. 

[44]

A. G. Vaubourg, Differentiation and discrimination in a duopoly with two bundles, International Journal of Industrial Organization, 24 (2006), 753-762. 

[45]

R. Venkatesh and W. Kamakura, Optimal bundling and pricing under a monopoly: Contrasting complements and substitutes from independently valued products, Journal of Business, 76 (2003), 211-231. 

[46]

Q. Wang, Discount pricing policies and the coordination of decentralized distribution systems, Decision Sciences, 36 (2005), 627-646. 

[47]

Y. Wang, L. Sun, R. Qu and G. Li, Price and service competition with maintenance service bundling, Journal of Systems Science and Systems Engineering, 24 (2015), 168-189. 

[48]

A. Wäppling, C. Strugnell and H. Farley, Product bundling strategies in Swedish markets: links to business orientation and perceived effects on consumer influence, International Journal of Consumer Studies, 34 (2010), 19-27. 

[49]

R. Yan, Managing channel coordination in a multi-channel manufacturer-retailer supply chain, Industrial Marketing Management, 40 (2011), 636-642. 

[50]

R. Yan and S. Bandyopadhyay, The profit benefits of bundle pricing of complementary products, Journal of Retailing and Consumer Services, 18 (2011), 355-361. 

[51]

R. Yan, C. Myers, J. Wang and S. Ghose, Bundling products to success: The influence of complementarity and advertising, Journal of Retailing and Consumer Services, 21 (2014), 48-53. 

[52]

R. Yan and Z. Pei, Retail services and firm profit in a dual-channel market, Journal of Retailing and Consumer Services, 16 (2009), 306-314. 

[53]

X. Yue, S. K. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192. 

Figure 1.  Impact $a_1$ between two products on the retailer's pricing strategy
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Figure 2.  Impact $a_1$ between two products on the wholesale pricing strategy
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Table 1.  Some recent works related to bundling strategy
Literature Strategies Selling price Demand rate Situation
Chakravarti et al. [12]BundlingBundle priceSelling priceDecentralized supply chains
Li et al. [25]Mix bundlingBundle priceSelling priceBi-level programming
Yan et al. [51]Bundle pricing and advertisingBundle priceSelling priceProduct complementary and advertisement of bundle product
Wang et al. [47]Service bundling—Service and Price bundlingDuopoly competitive environment
Banciu and ∅degaard [3]Different bundling——Simulation technique
Giri et al. [20]PricingBundling priceLinearly dependent on priceDuopoly market
Vamosiu [43]Imperfect CompetitionMixed bundling—Pure bundling
This paperBundlingBundle selling priceUncertain, selling price and random noise effect on marketCentralized and decentralized supply chains
Table Options

Download as excel

Literature Strategies Selling price Demand rate Situation
Chakravarti et al. [12]BundlingBundle priceSelling priceDecentralized supply chains
Li et al. [25]Mix bundlingBundle priceSelling priceBi-level programming
Yan et al. [51]Bundle pricing and advertisingBundle priceSelling priceProduct complementary and advertisement of bundle product
Wang et al. [47]Service bundling—Service and Price bundlingDuopoly competitive environment
Banciu and ∅degaard [3]Different bundling——Simulation technique
Giri et al. [20]PricingBundling priceLinearly dependent on priceDuopoly market
Vamosiu [43]Imperfect CompetitionMixed bundling—Pure bundling
This paperBundlingBundle selling priceUncertain, selling price and random noise effect on marketCentralized and decentralized supply chains
Table 2.  Effects of basic demand size $a_1$ to the contract for product 1 when $Q_{1}^{c} <M_{1}^{c} = 487$
$a_1$ $p_{1} $ $w_{1} $ $d_{1} $ $\alpha _{1} $ $Q_{1}^{c} $ $F(s_{1} )$
500237151133.502.643160.887
550259161144.501.853430.896
600282172156.001.083690.903
650304183167.000.293970.910
700326192178.00-0.504240.915
750348203189.00-1.294500.920
800371213200.50-2.504770.925
819.9380218205.00-2.364870.927
Table Options

Download as excel

$a_1$ $p_{1} $ $w_{1} $ $d_{1} $ $\alpha _{1} $ $Q_{1}^{c} $ $F(s_{1} )$
500237151133.502.643160.887
550259161144.501.853430.896
600282172156.001.083690.903
650304183167.000.293970.910
700326192178.00-0.504240.915
750348203189.00-1.294500.920
800371213200.50-2.504770.925
819.9380218205.00-2.364870.927
Table 3.  Effects of basic demand size $a_1$ to the contract for product 1, $ Q_{1}^{c} = M_{1}^{c} = 487$
$a_1$ $p_{1} $ $w_{1} $ $d_{1} $ $\alpha _{1} $ $F(s_{1} )$
820405231217.50-1.500.975
850427242228.50-1.680.965
880449253239.50-1.860.950
910472264251.00-2.040.940
940495275262.50-2.160.905
970518287274.00-2.310.880
1000541298285.50-2.460.855
Table Options

Download as excel

$a_1$ $p_{1} $ $w_{1} $ $d_{1} $ $\alpha _{1} $ $F(s_{1} )$
820405231217.50-1.500.975
850427242228.50-1.680.965
880449253239.50-1.860.950
910472264251.00-2.040.940
940495275262.50-2.160.905
970518287274.00-2.310.880
1000541298285.50-2.460.855
Table 4.  Comparison between coordination contract vs price-only contract for profit of product 1, Coordination contract: $M_{1}^{c} $ = 487 and total profit = 279270
$w_{1} $CapacitySupplier profitRetailer profitTotal profit
163449142000123210265210
199426125720105810231530
2504011095490740101694
2903789580175437171238
3203568191862549144467
Table Options

Download as excel

$w_{1} $CapacitySupplier profitRetailer profitTotal profit
163449142000123210265210
199426125720105810231530
2504011095490740101694
2903789580175437171238
3203568191862549144467
Table 5.  Effects of basic demand size to the contract under bundling policy, $ Q_{B}^{c} <M_{B}^{c} = 867 $
$a_1$ $p_{1B} $ $w_{B} $ $d_{B} $ $\alpha _{B} $ $Q_{B}^{c} $ $F(s_{B} )$
500279216174.508.005640.799
550303225186.506.105920.812
600327234198.504.206440.823
650350244210.002.276960.833
700374255222.000.417460.842
750398264234.00-1.487960.850
800421273245.50-3.468470.857
819.9431278250.50-4.188670.860
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$a_1$ $p_{1B} $ $w_{B} $ $d_{B} $ $\alpha _{B} $ $Q_{B}^{c} $ $F(s_{B} )$
500279216174.508.005640.799
550303225186.506.105920.812
600327234198.504.206440.823
650350244210.002.276960.833
700374255222.000.417460.842
750398264234.00-1.487960.850
800421273245.50-3.468470.857
819.9431278250.50-4.188670.860
Table 6.  Effects of basic demand size on the contract under bundling policy when, $ Q_{B}^{c} = M_{B}^{c} = 867 $
$a_1$ $p_{2B} $ $w_{B} $ $d_{B} $ $\alpha _{B} $ $F(s_{B} )$
820437279253.50-5.920.752
850474298272.0-5.900.750
880510315290.00-5.860.747
910546333308.00-5.600.742
940585353327.00-5.960.740
970621371345.50-6.250.739
1000658389364.00-6.350.736
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$a_1$ $p_{2B} $ $w_{B} $ $d_{B} $ $\alpha _{B} $ $F(s_{B} )$
820437279253.50-5.920.752
850474298272.0-5.900.750
880510315290.00-5.860.747
910546333308.00-5.600.742
940585353327.00-5.960.740
970621371345.50-6.250.739
1000658389364.00-6.350.736
Table 7.  Profit with bundling policy, Proposed contract: $M_{B}^{c} $ = 867 and total profit = 260621
$w_{B} $CapacitySupplier profitRetailer profitTotal profit
234838142490104540247030
25979412673087871214601
29974511241072037184447
3526689645360367156820
3866308579245607131399
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$w_{B} $CapacitySupplier profitRetailer profitTotal profit
234838142490104540247030
25979412673087871214601
29974511241072037184447
3526689645360367156820
3866308579245607131399
Table 8.  The results in numerical analysis
Percent change $p_{1} $ $p_{2} $ $p_{B} $ $F(s_{B} )$ $Q_{B}^{c} $ $w_{B} $ $\alpha _{B} $ $d_{B} $Retailer profitSupplier profit
$a_{2} =0.5$+5052.852.4838.446.02-29.9733.09-279.4433-7.07-4.66
+2525.8326.0318.633.12-12.8215.83-121.3915.994.003.47
+1515.4215.7011.082.04-7.059.35-68.069.513.993.13
-15-15.42-15.29-10.61-2.286.09-8.9951.94-9.11-7.31-5.91
-25-25.42-25.62-17.22-4.089.29-14.0390.83-14.98-14.60-9.97
-50Infeasible
$\theta=0.25$+5010.4210.331.650.362.081.0817.781.424.893.53
+255.004.960.940.240.960.728.330.812.291.86
+152.923.310.470.120.640.366.390.401.341.55
-15-3.33-3.31-0.710.00-0.32-0.36-3.06-0.61-1.91-0.08
-25-5.42-4.96-0.94-0.12-0.96-0.72-6.39-0.81-2.32-1.05
-50-10-10.33-1.65-0.24-1.06-1.08-14.72-1.42-4.72-3.42
$\lambda =0.35$+50Infeasible
+250.000.0013.212.40-7.0511.15-66.3911.346.835.81
+150.000.008.251.56-4.016.83-39.446.885.114.10
-150.000.00-8.25-1.684.17-6.4736.67-7.29-7.63-4.22
-250.000.00-13.44-3.005.61-11.1555.83-11.74-12.06-8.73
-500.000.00-25.71-6.838.33-20.8691.67-22.06-27.45-19.74
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Percent change $p_{1} $ $p_{2} $ $p_{B} $ $F(s_{B} )$ $Q_{B}^{c} $ $w_{B} $ $\alpha _{B} $ $d_{B} $Retailer profitSupplier profit
$a_{2} =0.5$+5052.852.4838.446.02-29.9733.09-279.4433-7.07-4.66
+2525.8326.0318.633.12-12.8215.83-121.3915.994.003.47
+1515.4215.7011.082.04-7.059.35-68.069.513.993.13
-15-15.42-15.29-10.61-2.286.09-8.9951.94-9.11-7.31-5.91
-25-25.42-25.62-17.22-4.089.29-14.0390.83-14.98-14.60-9.97
-50Infeasible
$\theta=0.25$+5010.4210.331.650.362.081.0817.781.424.893.53
+255.004.960.940.240.960.728.330.812.291.86
+152.923.310.470.120.640.366.390.401.341.55
-15-3.33-3.31-0.710.00-0.32-0.36-3.06-0.61-1.91-0.08
-25-5.42-4.96-0.94-0.12-0.96-0.72-6.39-0.81-2.32-1.05
-50-10-10.33-1.65-0.24-1.06-1.08-14.72-1.42-4.72-3.42
$\lambda =0.35$+50Infeasible
+250.000.0013.212.40-7.0511.15-66.3911.346.835.81
+150.000.008.251.56-4.016.83-39.446.885.114.10
-150.000.00-8.25-1.684.17-6.4736.67-7.29-7.63-4.22
-250.000.00-13.44-3.005.61-11.1555.83-11.74-12.06-8.73
-500.000.00-25.71-6.838.33-20.8691.67-22.06-27.45-19.74
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