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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

 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.

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:

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##### References:
Impact $a_1$ between two products on the retailer's pricing strategy
Impact $a_1$ between two products on the wholesale pricing strategy
Some recent works related to bundling strategy
 Literature Strategies Selling price Demand rate Situation Chakravarti et al. [12] Bundling Bundle price Selling price Decentralized supply chains Li et al. [25] Mix bundling Bundle price Selling price Bi-level programming Yan et al. [51] Bundle pricing and advertising Bundle price Selling price Product complementary and advertisement of bundle product Wang et al. [47] Service bundling — Service and Price bundling Duopoly competitive environment Banciu and ∅degaard [3] Different bundling — — Simulation technique Giri et al. [20] Pricing Bundling price Linearly dependent on price Duopoly market Vamosiu [43] Imperfect Competition Mixed bundling — Pure bundling This paper Bundling Bundle selling price Uncertain, selling price and random noise effect on market Centralized and decentralized supply chains
 Literature Strategies Selling price Demand rate Situation Chakravarti et al. [12] Bundling Bundle price Selling price Decentralized supply chains Li et al. [25] Mix bundling Bundle price Selling price Bi-level programming Yan et al. [51] Bundle pricing and advertising Bundle price Selling price Product complementary and advertisement of bundle product Wang et al. [47] Service bundling — Service and Price bundling Duopoly competitive environment Banciu and ∅degaard [3] Different bundling — — Simulation technique Giri et al. [20] Pricing Bundling price Linearly dependent on price Duopoly market Vamosiu [43] Imperfect Competition Mixed bundling — Pure bundling This paper Bundling Bundle selling price Uncertain, selling price and random noise effect on market Centralized and decentralized supply chains
Effects of basic demand size $a_1$ to the contract for product 1 when $Q_{1}^{c} <M_{1}^{c} = 487$
 $a_1$ $p_{1}$ $w_{1}$ $d_{1}$ $\alpha _{1}$ $Q_{1}^{c}$ $F(s_{1} )$ 500 237 151 133.50 2.64 316 0.887 550 259 161 144.50 1.85 343 0.896 600 282 172 156.00 1.08 369 0.903 650 304 183 167.00 0.29 397 0.910 700 326 192 178.00 -0.50 424 0.915 750 348 203 189.00 -1.29 450 0.920 800 371 213 200.50 -2.50 477 0.925 819.9 380 218 205.00 -2.36 487 0.927
 $a_1$ $p_{1}$ $w_{1}$ $d_{1}$ $\alpha _{1}$ $Q_{1}^{c}$ $F(s_{1} )$ 500 237 151 133.50 2.64 316 0.887 550 259 161 144.50 1.85 343 0.896 600 282 172 156.00 1.08 369 0.903 650 304 183 167.00 0.29 397 0.910 700 326 192 178.00 -0.50 424 0.915 750 348 203 189.00 -1.29 450 0.920 800 371 213 200.50 -2.50 477 0.925 819.9 380 218 205.00 -2.36 487 0.927
Effects of basic demand size $a_1$ to the contract for product 1, $Q_{1}^{c} = M_{1}^{c} = 487$
 $a_1$ $p_{1}$ $w_{1}$ $d_{1}$ $\alpha _{1}$ $F(s_{1} )$ 820 405 231 217.50 -1.50 0.975 850 427 242 228.50 -1.68 0.965 880 449 253 239.50 -1.86 0.950 910 472 264 251.00 -2.04 0.940 940 495 275 262.50 -2.16 0.905 970 518 287 274.00 -2.31 0.880 1000 541 298 285.50 -2.46 0.855
 $a_1$ $p_{1}$ $w_{1}$ $d_{1}$ $\alpha _{1}$ $F(s_{1} )$ 820 405 231 217.50 -1.50 0.975 850 427 242 228.50 -1.68 0.965 880 449 253 239.50 -1.86 0.950 910 472 264 251.00 -2.04 0.940 940 495 275 262.50 -2.16 0.905 970 518 287 274.00 -2.31 0.880 1000 541 298 285.50 -2.46 0.855
Comparison between coordination contract vs price-only contract for profit of product 1, Coordination contract: $M_{1}^{c}$ = 487 and total profit = 279270
 $w_{1}$ Capacity Supplier profit Retailer profit Total profit 163 449 142000 123210 265210 199 426 125720 105810 231530 250 401 10954 90740 101694 290 378 95801 75437 171238 320 356 81918 62549 144467
 $w_{1}$ Capacity Supplier profit Retailer profit Total profit 163 449 142000 123210 265210 199 426 125720 105810 231530 250 401 10954 90740 101694 290 378 95801 75437 171238 320 356 81918 62549 144467
Effects of basic demand size to the contract under bundling policy, $Q_{B}^{c} <M_{B}^{c} = 867$
 $a_1$ $p_{1B}$ $w_{B}$ $d_{B}$ $\alpha _{B}$ $Q_{B}^{c}$ $F(s_{B} )$ 500 279 216 174.50 8.00 564 0.799 550 303 225 186.50 6.10 592 0.812 600 327 234 198.50 4.20 644 0.823 650 350 244 210.00 2.27 696 0.833 700 374 255 222.00 0.41 746 0.842 750 398 264 234.00 -1.48 796 0.850 800 421 273 245.50 -3.46 847 0.857 819.9 431 278 250.50 -4.18 867 0.860
 $a_1$ $p_{1B}$ $w_{B}$ $d_{B}$ $\alpha _{B}$ $Q_{B}^{c}$ $F(s_{B} )$ 500 279 216 174.50 8.00 564 0.799 550 303 225 186.50 6.10 592 0.812 600 327 234 198.50 4.20 644 0.823 650 350 244 210.00 2.27 696 0.833 700 374 255 222.00 0.41 746 0.842 750 398 264 234.00 -1.48 796 0.850 800 421 273 245.50 -3.46 847 0.857 819.9 431 278 250.50 -4.18 867 0.860
Effects of basic demand size on the contract under bundling policy when, $Q_{B}^{c} = M_{B}^{c} = 867$
 $a_1$ $p_{2B}$ $w_{B}$ $d_{B}$ $\alpha _{B}$ $F(s_{B} )$ 820 437 279 253.50 -5.92 0.752 850 474 298 272.0 -5.90 0.750 880 510 315 290.00 -5.86 0.747 910 546 333 308.00 -5.60 0.742 940 585 353 327.00 -5.96 0.740 970 621 371 345.50 -6.25 0.739 1000 658 389 364.00 -6.35 0.736
 $a_1$ $p_{2B}$ $w_{B}$ $d_{B}$ $\alpha _{B}$ $F(s_{B} )$ 820 437 279 253.50 -5.92 0.752 850 474 298 272.0 -5.90 0.750 880 510 315 290.00 -5.86 0.747 910 546 333 308.00 -5.60 0.742 940 585 353 327.00 -5.96 0.740 970 621 371 345.50 -6.25 0.739 1000 658 389 364.00 -6.35 0.736
Profit with bundling policy, Proposed contract: $M_{B}^{c}$ = 867 and total profit = 260621
 $w_{B}$ Capacity Supplier profit Retailer profit Total profit 234 838 142490 104540 247030 259 794 126730 87871 214601 299 745 112410 72037 184447 352 668 96453 60367 156820 386 630 85792 45607 131399
 $w_{B}$ Capacity Supplier profit Retailer profit Total profit 234 838 142490 104540 247030 259 794 126730 87871 214601 299 745 112410 72037 184447 352 668 96453 60367 156820 386 630 85792 45607 131399
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 profit Supplier profit $a_{2} =0.5$ +50 52.8 52.48 38.44 6.02 -29.97 33.09 -279.44 33 -7.07 -4.66 +25 25.83 26.03 18.63 3.12 -12.82 15.83 -121.39 15.99 4.00 3.47 +15 15.42 15.70 11.08 2.04 -7.05 9.35 -68.06 9.51 3.99 3.13 -15 -15.42 -15.29 -10.61 -2.28 6.09 -8.99 51.94 -9.11 -7.31 -5.91 -25 -25.42 -25.62 -17.22 -4.08 9.29 -14.03 90.83 -14.98 -14.60 -9.97 -50 Infeasible $\theta=0.25$ +50 10.42 10.33 1.65 0.36 2.08 1.08 17.78 1.42 4.89 3.53 +25 5.00 4.96 0.94 0.24 0.96 0.72 8.33 0.81 2.29 1.86 +15 2.92 3.31 0.47 0.12 0.64 0.36 6.39 0.40 1.34 1.55 -15 -3.33 -3.31 -0.71 0.00 -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 $\lambda =0.35$ +50 Infeasible +25 0.00 0.00 13.21 2.40 -7.05 11.15 -66.39 11.34 6.83 5.81 +15 0.00 0.00 8.25 1.56 -4.01 6.83 -39.44 6.88 5.11 4.10 -15 0.00 0.00 -8.25 -1.68 4.17 -6.47 36.67 -7.29 -7.63 -4.22 -25 0.00 0.00 -13.44 -3.00 5.61 -11.15 55.83 -11.74 -12.06 -8.73 -50 0.00 0.00 -25.71 -6.83 8.33 -20.86 91.67 -22.06 -27.45 -19.74
 Percent change $p_{1}$ $p_{2}$ $p_{B}$ $F(s_{B} )$ $Q_{B}^{c}$ $w_{B}$ $\alpha _{B}$ $d_{B}$ Retailer profit Supplier profit $a_{2} =0.5$ +50 52.8 52.48 38.44 6.02 -29.97 33.09 -279.44 33 -7.07 -4.66 +25 25.83 26.03 18.63 3.12 -12.82 15.83 -121.39 15.99 4.00 3.47 +15 15.42 15.70 11.08 2.04 -7.05 9.35 -68.06 9.51 3.99 3.13 -15 -15.42 -15.29 -10.61 -2.28 6.09 -8.99 51.94 -9.11 -7.31 -5.91 -25 -25.42 -25.62 -17.22 -4.08 9.29 -14.03 90.83 -14.98 -14.60 -9.97 -50 Infeasible $\theta=0.25$ +50 10.42 10.33 1.65 0.36 2.08 1.08 17.78 1.42 4.89 3.53 +25 5.00 4.96 0.94 0.24 0.96 0.72 8.33 0.81 2.29 1.86 +15 2.92 3.31 0.47 0.12 0.64 0.36 6.39 0.40 1.34 1.55 -15 -3.33 -3.31 -0.71 0.00 -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 $\lambda =0.35$ +50 Infeasible +25 0.00 0.00 13.21 2.40 -7.05 11.15 -66.39 11.34 6.83 5.81 +15 0.00 0.00 8.25 1.56 -4.01 6.83 -39.44 6.88 5.11 4.10 -15 0.00 0.00 -8.25 -1.68 4.17 -6.47 36.67 -7.29 -7.63 -4.22 -25 0.00 0.00 -13.44 -3.00 5.61 -11.15 55.83 -11.74 -12.06 -8.73 -50 0.00 0.00 -25.71 -6.83 8.33 -20.86 91.67 -22.06 -27.45 -19.74
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