A Bi-Level Robust Optimization Model in Production Planning by Consideration of Pricing Decisions for Satisfying the Demand in a Competitive Environment: a Case Study

Document Type : Research Paper

Authors

1 Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran.

2 PhD Candidate, Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran.

Abstract

Bi-level programming is a mathematical programming, which there is another optimization problem at its constraints. According to the current fierce competition between large production companies to obtain a greater share of the market, this study develops a bi-level robust optimization model as the leader and the follower using Stackelberg game in the field of production planning. The leader company with higher leverage has decided to produce some new products that can be replaced with the existing products. The follower companies as a competitor, similar to the leader company, are looking to sell more. The follower companies do not have any intent and ability to produce such new products. Prices of the new products are determined using the tensile relations, which presented between the uncertain demand and price, creating the game between two levels of the model. After the linearization, the bi-level robust model is transformed to standard single-level model using conditions of Karush–Kuhn–Tucker (KKT). Finally, the accuracy and efficiency of the developed model have been verified by using the real data of Sarvestan Sepahan Company in Isfahan as the leader in the competitive market.

Keywords

Main Subjects


[1]      Xiao, T., Yang, D., (2008). “Price and service competition of supply chains with risk-averse retailers under demand uncertainty”, International Journal of Production Economics, 114(1): 187-200.
[2]      Rezapour, S., Farahani, R.Z., Ghodsipour, S.H., Abdollahzadeh, S., (2011). “Strategic design of competing supply chain networks with foresight”, Advances in Engineering Software, 42(4): 130-141.
[3]      Zhang, D., (2006). “A network economic model for supply chain versus supply chain competition”, Omega, 34(3): 283-295.
[4]      Boyaci, T., Gallego, G., (2004). “Supply Chain Coordination in a Market with Customer Service Competition”, Production and Operations Management, 13(1): 3-22.
[5]      Anderson, E.J., Bao, Y., (2010). “Price competition with integrated and decentralized supply chains”, European Journal of Operational Research, 200(1): 227-234.
[6]      Fallah, H., Eskandari, H., Pishvaee, M.S., (2015). “Competitive closed-loop supply chain network design under uncertainty”, Journal of Manufacturing Systems, 37: 649-661.
[7]      Makui, A., Heydari, M., Aazami, A., Dehghani, E., (2016). “Accelerating Benders decomposition approach for robust aggregate production planning of products with a very limited expiration date”, Computers & Industrial Engineering, 100: 34-51.
[8]      Saharidis, G.K., Ierapetritou, M.G., (2009). “Resolution method for mixed integer bi-level linear problems based on decomposition technique”, Journal of Global Optimization, 44(1): 29-51.
[9]      Mulvey, J.M., Vanderbei, R.J., Zenios, S.A., (1995). “Robust Optimization of Large-Scale Systems”, Operations Research, 43(2): 264-281.
[10]   Bernstein, F., Federgruen, A., (2004). “A General Equilibrium Model for Industries with Price and Service Competition”, Operations Research, 52(6): 868-886.
[11]   Candler, W., Norton, R., (1977). “Multi-level programming and development policy”, The World Bank.
[12]   Zhang, L., Rushton, G., (2008). “Optimizing the size and locations of facilities in competitive multi-site service systems”, Computers & Operations Research, 35(2): 327-338.
[13]   Bracken, J., McGill, J.T., (1973). “Mathematical Programs with Optimization Problems in the Constraints”, Operations Research, 21(1): 37-44.
[14]   Stackelberg, H.V., (1952). “The theory of the market economy”, Oxford University Press.
[15]   Vicente, L.N., Calamai, P.H., (1994). “Bilevel and multilevel programming: A bibliography review”, Journal of Global Optimization, 5(3): 291-306.
[16]   Candler, W., Fortuny-Amat, J., McCarl, B., (1981). “The Potential Role of Multilevel Programming in Agricultural Economics”, American Journal of Agricultural Economics, 63(3): 521-531.
[17]   Fortuny-Amat, J., McCarl, B., (1981). “A representation and economic interpretation of a two-level programming problem”, The Journal of the Operational Research Society, 32(9): 783-792.
[18]   Shimizu, K., Aiyoshi, E., (1981). “A new computational method for Stackelberg and min-max problems by use of a penalty method”, IEEE Transactions on Automatic Control, 26(2): 460-466.
[19]   Bard, J.F., Falk, J.E. (1982). “An explicit solution to the multi-level programming problem”, Computers and Operations Research, 9(1): 77-100.
[20]   Colson, B., Marcotte, P., Savard, G., (2005). “Bilevel programming: A survey”, 4OR, 3(2): 87-107.
[21]   Colson, B., Marcotte, P., Savard, G., (2007). “An overview of bilevel optimization”, Annals of Operations Research, 153(1): 235-256.
[22]   Bard, J.F. (1998). “Practical bilevel optimization: algorithms and applications”. Springer Science & Business Media, 30.
[23]   Farahani, R.Z., Rezapour, S., Drezner, T., Fallah, S., (2014). “Competitive supply chain network design: An overview of classifications, models, solution techniques and applications”, Omega (United Kingdom), 45: 92-118.
[24]   Ben-Ayed, O., Boyce, D.E., Blair, C.E., (1988). “A general bilevel linear programming formulation of the network design problem”, Transportation Research Part B: Methodological, 22(4): 311-318.
[25]   Bard, J., Moore, J., (1990). “A branch and bound algorithm for the bilevel programming problem”, SIAM Journal on Scientific and Statistical Computing, 11(2): 281-292.
[26]   Bard, J.F., Moore, J.T., (1992). “An algorithm for the discrete bilevel programming problem”, Naval Research Logistics, 39(3): 419-435.
[27]   Edmunds, T.A., Bard, J.F., (1992). “An algorithm for the mixed-integer nonlinear bilevel programming problem”, Annals of Operations Research, 34(1): 149-162.
[28]   Yang, H., (1995). “Heuristic algorithms for the bilevel origin-destination matrix estimation problem”, Transportation Research Part B, 29(4): 231-242
[29] Maher, M.J., Zhang, X., Vliet, Van. D., (2001). “A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows”, Transportation Research Part B: Methodological, 35(1): 23-40.
[30]   Burgard, A.P., Pharkya, P., Maranas, C.D., (2003). “OptKnock: A Bilevel Programming Framework for Identifying Gene Knockout Strategies for Microbial Strain Optimization”, Biotechnology and Bioengineering, 84(6): 647-657.
[31]   Gao, Z., Wu, J., Sun, H., (2005). “Solution algorithm for the bi-level discrete network design problem”, Transportation Research Part B: Methodological, 39(6): 479-495.
[32]   Shi, C., Lu, J., Zhang, G., Zhou, H., (2006). “An extended branch and bound algorithm for linear bilevel programming”, Applied Mathematics and Computation, 180(2): 529-537.
[33]   Sun, H., Gao, Z., Wu, J., (2008). “A bi-level programming model and solution algorithm for the location of logistics distribution centers”, Applied Mathematical Modelling, 32(4): 610-616.
[34]   Zhang, T., Zhao, Q., Wu, W., (2009). “Bi-level programming model of container port game in the container transport supernetwork”, Journal of Applied Mathematics and Computing, 31(1): 13-32.
[35]   Gelareh, S., Nickel, S., Pisinger, D., (2010). “Liner shipping hub network design in a competitive environment”, Transportation Research Part E: Logistics and Transportation Review, 46(6): 991-1004.
[36]   Küükaydin, H., Aras, N., Kuban Altinel, I., (2011). “Competitive facility location problem with attractiveness adjustment of the follower: A bilevel programming model and its solution”, European Journal of Operational Research, 208(3): 206-220.
[37]   Naimi Sadigh, A., Mozafari, M., Karimi, B., (2012). Manufacturer-retailer supply chain coordination: A bi-level programming approach. Advances in Engineering Software, 45(1): 144-152.
[38] Kristianto, Y., Helo, P., Jiao, R.J., (2013). “Mass customization design of engineer-to-order products using Benders’ decomposition and bi-level stochastic programming”, Journal of Intelligent Manufacturing, 24(5): 961-975.
[39] Rezapour, S., Zanjirani Farahani, R., (2014). “Supply chain network design under oligopolistic price and service level competition with foresight”, Computers & Industrial Engineering, 72: 129-142.
[40] Rezapour, S., Zanjirani Farahani, R., (2014). “Supply chain network design under oligopolistic price and service level competition with foresight”, Computers & Industrial Engineering, 72: 129-142.
[41] Rezapour, S., Farahani, R.Z., Fahimnia, B., Govindan, K., Mansouri, Y., (2015). “Competitive closed-loop supply chain network design with price-dependent demands”, Journal of Cleaner Production, 93: 251-272.
[42] Rashidi, E., Parsafard, M., Medal, H., Li, X., (2016). “Optimal traffic calming: A mixed-integer bi-level programming model for locating sidewalks and crosswalks in a multimodal transportation network to maximize pedestrians’ safety and network usability”, Transportation Research Part E: Logistics and Transportation Review, 91: 33-50.
[43] Han, J., Zhang, G., Hu, Y., Lu, J., (2016). “A solution to bi/tri-level programming problems using particle swarm optimization”, Information Sciences, 370: 519-537
[44] Mula, J., Poler, R., Garcia-Sabater, J.P., Lario, F. C., (2006). “Models for production planning under uncertainty: A review”, International Journal of Production Economics, 103(1): 271-285.
[45] Leung, S.C. H., Ng, W., (2007). “A goal programming model for production planning of perishable products with postponement”, Computers & Industrial Engineering, 53(3): 531-541.
[46] Leung, S.C.H., Chan, S.S.W., (2009). “A goal programming model for aggregate production planning with resource utilization constraint”, Computers & Industrial Engineering, 56(3): 1053-1064.
[47]   Zhang, J., Liu, X., Tu, Y.L., (2011). “A capacitated production planning problem for closed-loop supply chain with remanufacturing”, The International Journal of Advanced Manufacturing Technology, 54(5): 757-766.
[48]   Mirzapour Al-E-Hashem, S.M.J., Malekly, H., Aryanezhad, M.B., (2011). “A multi-objective robust optimization model for multi-product multi-site aggregate production planning in a supply chain under uncertainty”, International Journal of Production Economics, 134(1): 28-42.
[49]   Zhang, G., Shang, J., Li, W., (2011). “Collaborative production planning of supply chain under price and demand uncertainty”, European Journal of Operational Research, 215(3): 590-603.
[50]   Yaghin, R.G., Torabi, S.A., Ghomi, S.M.T.F., (2012). “Integrated markdown pricing and aggregate production planning in a two echelon supply chain: A hybrid fuzzy multiple objective approach”, Applied Mathematical Modelling, 36(12): 6011-6030.
[51]   Ramezanian, R., Rahmani, D., Barzinpour, F., (2012). “An aggregate production planning model for two phase production systems: Solving with genetic algorithm and tabu search”, Expert Systems with Applications, 39(1): 1256-1263.
[52]   Awudu, I., Zhang, J., (2013). “Stochastic production planning for a biofuel supply chain under demand and price uncertainties”, Applied Energy, 103: 189-196.
[53]   Da-Silva, A.F., Marins, F.A.S., (2014). “A Fuzzy Goal Programming model for solving aggregate production-planning problems under uncertainty: A case study in a Brazilian sugar mill”, Energy Economics, 45: 196-204.
[54]   Chakrabortty, R.K., Hasin, M.A.A., Sarker, R.A., Essam, D.L., (2015). “A possibilistic environment based particle swarm optimization for aggregate production planning”, Computers & Industrial Engineering, 88: 366-377.
[55]   Jabbarzadeh, A., Fahimnia, B., Sheu, J.B., (2015). “An enhanced robustness approach for managing supply and demand uncertainties”, International Journal of Production Economics, 183: 620-631.
[56] خیرخواه، امیرسامان، نوبری، آرش، حاجی‏پور، وحید، (1395). «ارایه الگوریتم رقابت استعماری چندهدفه جهت بهینه‏سازی مسئله‏ی برنامه‏‏ریزی تولید ادغامی پایا»، پژوهش‏های مهندسی صنایع در سیستم‏های تولید، 4(7): 1-15.
[57]   Entezaminia, A., Heidari, M., Rahmani, D., (2017). “Robust aggregate production planning in a green supply chain under uncertainty considering reverse logistics: a case study”, The International Journal of Advanced Manufacturing Technology, 90(5-8): 1507-1528.
[58]    ترکمن، سمیه، فاطمی قمی، سید محمد تقی، (1395). «برنامه‏ریزی تولید چندمرحله‏ای در زنجیره تأمین حلقه بسته همراه با راه‏اندازی‏های وابسته به توالی و انتقال راه‌اندازی»، پژوهش‏های مهندسی صنایع در سیستم‏های تولید، 4(9): 239-255.
[59]   Mokhtari, H., Hasani, A. (2017), “A multi-objective model for cleaner production-transportation planning in manufacturing plants via fuzzy goal programming”, Journal of Manufacturing Systems, 44: 230-242.
[60]   Makui, A., Ghavamifar, A., (2016). “Benders Decomposition Algorithm for Competitive Supply Chain Network Design under Risk of Disruption and Uncertainty”, Journal of Industrial and Systems Engineering, (special issue on supply chain): 30-50.
[61]   Vidal, C.J., Goetschalckx, M., (2001). “A global supply chain model with transfer pricing and transportation cost allocation”, European Journal of Operational Research, 129: 134-158.