Multi-item inventory model with probabilistic demand function under permissible delay in payment and fuzzy-stochastic budget constraint: A signomial geometric programming method

Document Type : Research Paper


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


This study proposes a new multi-item inventory model with hybrid cost parameters under a fuzzy-stochastic constraint and permissible delay in payment. The price and marketing expenditure dependent stochastic demand and the demand dependent the unit production cost are considered. Shortages are allowed and partially backordered. The main objective of this paper is to determine selling price, marketing expenditure, credit period, and variables of inventory control simultaneously for maximizing the total profit. To solve the problem, first some transformations are applied to convert the original problem into a multi-objective nonlinear programming problem, of which each objective has signomial terms. Then, the multi-objective nonlinear programming problem is solved by first converting it into a single objective problem and then by using global optimization of signomial geometric programming problems. At the end, several numerical examples and sensitivity analysis are done to test model and solution procedure and also obtain managerial insights.


Main Subjects

Aggarwal, S. and Jaggi, C. (1995) 'Ordering policies of deteriorating items under permissible delay in payments', Journal of the operational research society, 658-662.
Cheng, T. (1991) 'EPQ with process capability and quality assurance considerations', Journal of the Operational Research Society, 42(8), 713-720.
Chung, K.-J. and Huang, Y.-F. (2003) 'The optimal cycle time for EPQ inventory model under permissible delay in payments', International Journal of Production Economics, 84(3), 307-318.
Chung, K.-J., Liao, J.-J., Ting, P.-S., Lin, S.-D. and Srivastava, H.M. (2015) 'The algorithm for the optimal cycle time and pricing decisions for an integrated inventory system with order-size dependent trade credit in supply chain management', Applied Mathematics and Computation, 268, 322-333.
Das, K., Roy, T.K. and Maiti, M. (2004) 'Multi-item stochastic and fuzzy-stochastic inventory models under two restrictions', Computers & Operations Research, 31(11), 1793-1806.
De, S.K. and Sana, S.S. (2015) 'Backlogging EOQ model for promotional effort and selling price sensitive demand-an intuitionistic fuzzy approach', Annals of Operations Research, 233(1), 57-76.
Goyal, S.K. (1985) 'Economic order quantity under conditions of permissible delay in payments', Journal of the operational research society, 335-338.
He, Y., Zhao, X., Zhao, L. and He, J. (2009) 'Coordinating a supply chain with effort and price dependent stochastic demand', Applied Mathematical Modelling, 33(6), 2777-2790.
Ho, C.-H., Ouyang, L.-Y. and Su, C.-H. (2008) 'Optimal pricing, shipment and payment policy for an integrated supplier–buyer inventory model with two-part trade credit', European Journal of Operational Research, 187(2), 496-510.
Huang, Y.-F. (2007) 'Economic order quantity under conditionally permissible delay in payments', European Journal of Operational Research, 176(2), 911-924.
Islam, S. and Roy, T.K. (2006) 'A fuzzy EPQ model with flexibility and reliability consideration and demand dependent unit production cost under a space constraint: A fuzzy geometric programming approach', Applied Mathematics and computation, 176(2), 531-544.
Jamal, A., Sarker, B. and Wang, S. (1997) 'An ordering policy for deteriorating items with allowable shortage and permissible delay in payment', Journal of the operational research society, 48(8), 826-833.
Kauffman, A. and Gupta, M.M. (1991) 'Introduction to Fuzzy Arithmetic, Theory and Application'.
Liang, Y. and Zhou, F. (2011) 'A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment', Applied Mathematical Modelling, 35(5), 2221-2231.
Luhandjula, M.K. (1983) 'Linear programming under randomness and fuzziness', Fuzzy Sets and Systems, 10(1-3), 45-55.
Maihami, R. and Abadi, I.N.K. (2012) 'Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging', Mathematical and Computer Modelling, 55(5), 1722-1733.
Maihami, R. and Karimi, B. (2014) 'Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts', Computers & Operations Research, 51, 302-312.
Maihami, R., Karimi, B. and Ghomi, S.M.T.F. (2017) 'Effect of two-echelon trade credit on pricing-inventory policy of non-instantaneous deteriorating products with probabilistic demand and deterioration functions', Annals of Operations Research, 257(1-2), 237-273.
Mandal, N.K., Roy, T.K. and Maiti, M. (2006) 'Inventory model of deteriorated items with a constraint: A geometric programming approach', European Journal of Operational Research, 173(1), 199-210.
Mutapcic, A., Koh, K., Kim, S. and Boyd, S. (2006) 'GGPLAB version 1.00: a Matlab toolbox for geometric programming'.
Panda, D., Kar, S. and Maiti, M. (2008) 'Multi-item EOQ model with hybrid cost parameters under fuzzy/fuzzy-stochastic resource constraints: a geometric programming approach', Computers & Mathematics with Applications, 56(11), 2970-2985.
Pang, Q., Chen, Y. and Hu, Y. (2014) 'Coordinating three-level Supply Chain by revenue-sharing contract with sales effort dependent demand', Discrete Dynamics in Nature and Society, 2014.
Pramanik, P., Maiti, M.K. and Maiti, M. (2017) 'A supply chain with variable demand under three level trade credit policy', Computers & Industrial Engineering, 106, 205-221.
Sadjadi, S.J., Hesarsorkh, A.H., Mohammadi, M. and Naeini, A.B. (2015) 'Joint pricing and production management: a geometric programming approach with consideration of cubic production cost function', Journal of Industrial Engineering International, 11(2), 209-223.
Samadi, F., Mirzazadeh, A. and Pedram, M.M. (2013) 'Fuzzy pricing, marketing and service planning in a fuzzy inventory model: a geometric programming approach', Applied Mathematical Modelling, 37(10), 6683-6694.
Sarkar, B., Saren, S. and Cárdenas-Barrón, L.E. (2015) 'An inventory model with trade-credit policy and variable deterioration for fixed lifetime products', Annals of Operations Research, 229(1), 677-702.
Soni, H. and Patel, K. (2012) 'Optimal pricing and inventory policies for non-instantaneous deteriorating items with permissible delay in payment: Fuzzy expected value model', International journal of industrial engineering computations, 3(3), 281-300.
Soni, H.N. (2013) 'Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment', International journal of production Economics, 146(1), 259-268.
Tabatabaei, S.R.M., Sadjadi, S.J. and Makui, A. (2017) 'Optimal production and marketing planning with geometric programming approach'.
Taleizadeh, A.A., Pentico, D.W., Jabalameli, M.S. and Aryanezhad, M. (2013) 'An EOQ model with partial delayed payment and partial backordering', Omega, 41(2), 354-368.
Tiwari, R., Dharmar, S. and Rao, J. (1987) 'Fuzzy goal programming—an additive model', Fuzzy sets and systems, 24(1), 27-34.
Xu, G. (2014) 'Global optimization of signomial geometric programming problems', European Journal of Operational Research, 233(3), 500-510.
Volume 11, Issue 2
April 2018
Pages 207-227
  • Receive Date: 12 January 2018
  • Revise Date: 17 March 2018
  • Accept Date: 17 March 2018
  • First Publish Date: 16 October 2018