A Novel Perspective on Resolving the Discounted Transportation Problem

Year 2025, Volume: 25 Issue: 4, 798 - 803, 04.08.2025

Nurdan Kara

https://doi.org/10.35414/akufemubid.1514404

Abstract

The transportation problem, a fundamental optimization problem, aims to optimize profits or reduce costs by transporting products from suppliers to destinations. Nevertheless, a new level of complexity is introduced when the cost per unit of transportation is based on the quantity of goods that are transported. This is the point at which quantity discounts are implemented. A segmented pricing structure is frequently established by suppliers to encourage the purchase of larger quantities by offering reduced per-unit costs. Traditional methods are unable to effectively address the distinctive challenge of incorporating quantity reductions, despite the extensive research conducted on the transportation problem. Our research explores this exact problem, acknowledging the constraints of current approaches to managing the complexities of quantity reductions. In response, we suggest a novel methodology that functions as a potent alternative. This method exceeds simply making use of a fixed cost per unit. Rather, it represents the fluctuating character of costs in relation to the volume of shipments. Image a staircase, with each step representing a distinct cost division that is initiated by a specific quantity range. This divided structure is easily incorporated into the optimization process by our methodology.

Keywords

Transportation problem, Separable Programming, Piecewise cost function, Weighting.

İndirimli Taşıma Problemine Yeni Bir Bakış

Abstract

Taşıma problemi, tedarikçilerden alıcı noktalara ürünlerin taşınmasıyla kar elini optimize etmeyi veya maliyetleri azaltmayı amaçlayan temel bir optimizasyon problemidir. Ancak, taşıma birimi başına maliyetin taşınan mal miktarına bağlı olduğu durumlarda yeni bir karmaşıklık seviyesi ortaya çıkar. İşte bu noktada miktar indirimleri ortaya çıkar. Tedarikçiler, daha büyük miktarların satın alınmasını teşvik etmek için birim başına maliyetleri düşürerek segmente edilmiş bir fiyatlandırma yapısı oluştururlar. Taşıma problemi üzerine yapılan kapsamlı araştırmalara rağmen, geleneksel yöntemler miktar indirimlerini etkin bir şekilde ele almada zorlanmaktadır. Araştırmamız, miktar indirimlerinin karmaşıklıklarını yönetme konusundaki yaklaşımların sınırlarını göz önünde bulundurarak bu özel problemi ele alır. Buna karşılık olarak, etkili bir alternatif olarak işlev gören yeni bir metodoloji öneriyoruz. Bu yöntem, sadece sabit bir birim maliyet kullanımını aşmanın yanısıra, maliyetlerin sevkiyat hacmiyle ilişkili dalgalanan karakterini temsil etmektedir. Her bir adımın belirli bir miktar aralığı tarafından tetiklenen farklı bir maliyet bölümünü temsil ettiği bir merdiven şeklindeki bölünmüş yapı, metodolojimiz tarafından optimizasyon sürecine kolayca entegre edilir.

Keywords

Taşıma problemi, Ayrılabilir programlama, Parçalı maliyet fonksiyonları, Ağırlıklandırma

References

• Altassan, K. M., El-Sherbiny, M. M., & Sasidhar, B. (2013). Near optimal solution for the step fixed charge transportation problem. Applied Mathematics & Information Sciences, 7(2), 661-669.
• Balachandran, V. and Perry, A.,1976. Transportation type problems with quantity discounts. Naval Research Logistics Quarterly, 23(2), 195-209.
• Christensen, T. R., & Labbé, M. (2015). A branch-cut-and-price algorithm for the piecewise linear transportation problem. European journal of operational research, 245(3), 645-655.
• Das, S.C., Manna, A.K., Rahman, M.S., Shaikh, A.A. and Bhunia, A.K., 2021a. An inventory model for non-instantaneous deteriorating items with preservation technology and multiple credit periods-based trade credit financing via particle swarm optimization. Soft Computing. 25 (7), 5365–5384. http://dx.doi.org/10.1007/s00500-020-05535-x.
• Falk, J. E., & Soland, R. M. (1969). An algorithm for separable nonconvex programming problems. Management Science, 15(9), 550-569.
• Hadley, G. and T.M. Whitin.,1963. Analysis of Znventory Systems (Prentice-Hall, Inc., Englewood Cliffs, N.J.
• Hochbaum, D. S., & Shanthikumar, J. G. (1990). Convex separable optimization is not much harder than linear optimization. Journal of the ACM (JACM), 37(4), 843-862.
• Huang, C. H. (2009). An effective linear approximation method for separable programming problems. Applied mathematics and computation, 215(4), 1496-1506.
• Khan, M.A.A., Shaikh, A.A. and Cárdenas-Barrón, L.E., 2021. An inventory model under linked-to-order hybrid partial advance payment, partial credit policy, all-units discount and partial backlogging with capacity constraint. Omega, 103, 102418. http://dx.doi.org/10.1016/j.omega.2021.102418.
• Kumar, R., Edalatpanah, S.A., Jha, S. and Singh, R., 2019. A Pythagorean fuzzy approach to the transportation problem. Complex Intell. Syst., 5 (2), 255–263. http://dx.doi.org/10.1007/s40747-019-0108-1.
• Li, H. L., and Yu, C. S., 1999. A global optimization method for nonconvex separable programming problems. European Journal of Operational Research, 117(2), 275-292.
• Manerba, D., Mansini, R. and Perboli, G., 2018. The capacitated supplier selection problem with total quantity discount policy and activation costs under uncertainty. Int. J. Prod. Econ., 198, 119–132. http://dx.doi.org/10.1016/j.ijpe.2018.01.035.
• Mashud, A.H.M., Wee, H.M., Sarkar, B. and Li, Y.H.C., 2020. A sustainable inventory system with the advanced payment policy and trade-credit strategy for a two-warehouse inventory system. Kybernetes, 50 (5), 1321–1348. http://dx.doi.org/10.1108/K-01-2020-0052
• McKeown PG. 1980. Solving incremental quantity discounted transportation problems by vertex ranking. Naval Research Logistics Quarterly, 27(3), 437-445.
• Mitlif, R.J., Rasheed, M., Shihab, S., 2020. An optimal algorithm for a fuzzy transportation problem. J. Southwest Jiaotong Univ. 55 (3), 1-10 http://dx.doi.org/10.35741/issn.0258-2724.55.3.7.
• Panja, S., Mondal, S.K., 2020. Exploring a two-layer green supply chain game theoretic model with credit linked demand and mark-up under revenue sharing contract. J. Clean. Prod. 250, 119491. http://dx.doi.org/10.1016/j.jclepro.2019.119491.
• Singh, G., Singh, A., 2021. Extension of particle swarm optimization algorithm for solving transportation problem in fuzzy environment. Appl. Soft Comput. 110, 107619. http://dx.doi.org/10.1016/j.asoc.2021.107619.
• Vogt L, Even J. “Piecewise linear programming solutions of transportation costs as obtained from rate tariffs”. AIIE Transactions, 4(2), 148-153,1972.
• Zhang, J., & Xu, C. (2010). Inverse optimization for linearly constrained convex separable programming problems. European Journal of Operational Research, 200(3), 671-679.