{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T23:29:32Z","timestamp":1782430172393,"version":"3.54.5"},"reference-count":53,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T00:00:00Z","timestamp":1683504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Prince Sattam Bin Abdulaziz University","award":["PSAU\/2023\/R\/1444"],"award-info":[{"award-number":["PSAU\/2023\/R\/1444"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems"],"abstract":"<jats:p>This paper aims to design an inventory model for a retail enterprise with a profit maximization objective using the opportunity for a price discount facility given by a supplier. In the profit maximization objective, the demand should be increased. The demand can be boosted by lowering the selling price. However, lowering the selling price may not always give the best profit. Impreciseness plays a vital role during such decision-making. The decision-making and managerial activities may be imprecise due to some decision variables. For instance, the selling price may not be deterministic. A vague selling price will make the retail decision imprecise. To achieve this goal, the retailer must minimize impreciseness as much as possible. Learning through repetition may be a practical approach in this regard. This paper investigates the impact of fuzzy impreciseness and triangular dense fuzzy setting, which dilutes the impreciseness involved with managerial decisions. Based on the mentioned objectives, this article considers an inventory model with price-dependent demand and time and a purchasing cost-dependent holding cost in an uncertain phenomenon. This paper incorporates the all-units discount policy into the unit purchase cost according to the order quantity. In this paper, the sense of learning is accounted for using a dense fuzzy set by considering the unit selling price as a triangular dense fuzzy number to lessen the impreciseness in the model. Four fuzzy optimization methods are used to obtain the usual extreme profit when searching for the optimal purchasing cost and sale price. It is perceived from the numerical outcomes that a dense fuzzy environment contributes the best results compared to a crisp and general fuzzy environment. Managerial insights from this paper are that learning from repeated dealing activities contributes to enhancing profitability by diluting impreciseness about the selling price and demand rate and taking the best opportunity from the discount facility while purchasing.<\/jats:p>","DOI":"10.3390\/systems11050235","type":"journal-article","created":{"date-parts":[[2023,5,9]],"date-time":"2023-05-09T01:06:28Z","timestamp":1683594388000},"page":"235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["A Learning-Based Optimal Decision Scenario for an Inventory Problem under a Price Discount Policy"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7393-2568","authenticated-orcid":false,"given":"Alaa Fouad","family":"Momena","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, College of Engineering, Prince Sattam Bin Abdulaziz University, Al Kharj 11942, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5513-4095","authenticated-orcid":false,"given":"Mostafijur","family":"Rahaman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4160-0702","authenticated-orcid":false,"given":"Rakibul","family":"Haque","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, Nadia, West Bengal, Haringhata 741249, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8263-117X","authenticated-orcid":false,"given":"Shariful","family":"Alam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4690-2598","authenticated-orcid":false,"given":"Sankar Prasad","family":"Mondal","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, Nadia, West Bengal, Haringhata 741249, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,8]]},"reference":[{"key":"ref_1","first-page":"135","article-title":"How many parts to make at once","volume":"10","author":"Harris","year":"1913","journal-title":"Fact.-Mag. Manag."},{"key":"ref_2","unstructured":"Hadley, G., and Whitin, T.M. (1963). Analysis of Inventory Systems, Prentice-Hall."},{"key":"ref_3","unstructured":"Naddor, E. (1966). Inventory Systems, John Wiley."},{"key":"ref_4","first-page":"64","article-title":"A heuristic for selecting lot size quantities for the case of a deterministic time varying demand rate and discrete opportunities for replenishment","volume":"14","author":"Silver","year":"1973","journal-title":"Prod. Inventory Manag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1016\/0377-2217(95)00309-6","article-title":"An inventory model for deteriorating items with stock-dependent demand rate","volume":"95","author":"Giri","year":"1996","journal-title":"Eur. J. Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1080\/09537289508930252","article-title":"An optimal credit policy to increase supplier\u2019s profits with price-dependent demand functions","volume":"6","author":"Kim","year":"1995","journal-title":"Prod. Plan. Control"},{"key":"ref_7","first-page":"25","article-title":"An inventory model for deteriorating items with price dependent demand and time varying holding cost","volume":"10","author":"Roy","year":"2008","journal-title":"Adv. Model. Optim."},{"key":"ref_8","first-page":"2171","article-title":"An inventory model for Weibull deteriorating items with price dependent demand and time-varying holding cost","volume":"4","author":"Tripathy","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1007\/s13369-018-3593-4","article-title":"EOQ and EPQ production inventory models with variable holding cost: State-of-the-art review","volume":"44","author":"Alfares","year":"2019","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_10","first-page":"497","article-title":"A deterministic inventory model for deteriorating items with selling price dependent demand and three-parameter Weibull distributed deterioration","volume":"5","author":"Bhunia","year":"2014","journal-title":"Int. J. Ind. Eng. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1007\/s13198-013-0209-y","article-title":"An inventory model of price and stock dependent demand rate with deterioration under inflation and delay in payment","volume":"5","author":"Pal","year":"2014","journal-title":"Int. J. Syst. Assur. Eng. Manag."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s10479-014-1739-7","article-title":"An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation- and selling price-dependent demand and customer returns","volume":"226","author":"Ghoreishi","year":"2015","journal-title":"Ann. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2807","DOI":"10.1080\/00207543.2016.1154621","article-title":"Pricing and ordering decisions of two competing supply chains with different composite policies: A Stackelberg game-theoretic approach","volume":"54","author":"Taleizadeh","year":"2016","journal-title":"Int. J. Prod. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10479-017-2419-1","article-title":"An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment","volume":"254","author":"Mishra","year":"2017","journal-title":"Ann. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s40092-018-0269-3","article-title":"A credit policy approach in a two-warehouse inventory model for deteriorating items with price- and stock-dependent demand under partial backlogging","volume":"15","author":"Panda","year":"2019","journal-title":"J. Ind. Eng. Int."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"105557","DOI":"10.1016\/j.cie.2018.12.004","article-title":"An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit","volume":"139","author":"Barron","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3963","DOI":"10.1007\/s13369-021-06516-4","article-title":"A Generalized Production-Inventory Model with Variable Production, Demand, and Cost Rates","volume":"47","author":"Alfares","year":"2022","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4137","DOI":"10.1007\/s10462-022-10268-4","article-title":"An application of tournament differential evolution algorithm in production inventory model with green level and expiry time dependent demand","volume":"56","author":"Akhtar","year":"2022","journal-title":"Artif. Intell. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Hakim, M.A., Hezam, I.M., Alrasheedi, A.F., and Gwak, J. (2022). Pricing Policy in an Inventory Model with Green Level Dependent Demand for a Deteriorating Item. Sustainability, 14.","DOI":"10.3390\/su14084646"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1080\/02331934.2011.590485","article-title":"Optimal ordering and pricing policy with supplier quantity discounts and price-dependent stochastic demand","volume":"61","author":"Shi","year":"2012","journal-title":"Optimization"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.ins.2012.12.015","article-title":"Optimal inventory policy for the fuzzy newsboy problem with quantity discounts","volume":"228","author":"Chen","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.ijpe.2014.01.012","article-title":"An economic order quantity model with partial backordering and all-units discount","volume":"155","author":"Taleizadeh","year":"2014","journal-title":"Int. J. Prod. Econ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.cie.2015.01.005","article-title":"An economic order quantity model with partial backordering and incremental discount","volume":"82","author":"Taleizadeh","year":"2015","journal-title":"Comput. Ind. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"715","DOI":"10.3846\/13926292.2015.1108936","article-title":"Maximum-profit inventory model with stock-dependent demand, time dependent holding cost, and all-units quantity discounts","volume":"20","author":"Alfares","year":"2015","journal-title":"Math. Model. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.cie.2016.02.009","article-title":"Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts","volume":"94","author":"Alfares","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_26","unstructured":"Huang, T.S., Yang, M.F., Chao, Y.S., Kei, E.S.Y., and Chung, W.H. (2018, January 14\u201316). Fuzzy Supply Chain Integrated Inventory Model with Quantity Discounts and Unreliable Process in Uncertain Environments. Proceedings of the International MultiConference of Engineers and Computer Scientists, Hong Kong, China."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1007\/s40092-019-0311-0","article-title":"Economic order quantity model for growing items with incremental quantity discounts","volume":"15","author":"Sebatjane","year":"2019","journal-title":"J. Ind. Eng. Int."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1111\/itor.12632","article-title":"Price discount facility in an EOQ model for deteriorating items with stock-dependent demand and partial backlogging","volume":"26","author":"Shaikh","year":"2019","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_29","first-page":"61","article-title":"Optimal lot-size decision for deteriorating items with price-sensitive demand, linearly time-dependent holding cost under all-units discount environment","volume":"9","author":"Khan","year":"2020","journal-title":"Int. J. Syst. Sci. Oper. Logist."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"S1069","DOI":"10.1051\/ro\/2020106","article-title":"Joint pricing deteriorating inventory model considering product life cycle and advance payment with a discount facility","volume":"55","author":"Mashud","year":"2021","journal-title":"RAIRO-Oper. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2455","DOI":"10.1007\/s10462-021-10069-1","article-title":"Interval valued demand related inventory model under all-units discount facility and deterioration via parametric approach","volume":"55","author":"Rahman","year":"2022","journal-title":"Artif. Intell. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1246257","DOI":"10.1155\/2023\/1246257","article-title":"Fuzzy Inventory Model for Imperfect Items with Price Discount and Penalty Maintenance Cost","volume":"2023","author":"Kuppulakshmi","year":"2023","journal-title":"Math. Probl. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0165-0114(87)90029-7","article-title":"Fuzzy differential equations","volume":"24","author":"Kaleva","year":"1987","journal-title":"Fuzzy Sets Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1082","DOI":"10.1109\/TSMC.1987.6499320","article-title":"Fuzzy-set theoretic interpretation of economic order quantity","volume":"17","author":"Park","year":"1987","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_36","first-page":"1","article-title":"Inventory model with demand as type-2 fuzzy number: A fuzzy differential equation approach","volume":"15","author":"Debnath","year":"2018","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1016\/j.engappai.2012.10.017","article-title":"A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach","volume":"26","author":"Guchhait","year":"2013","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_38","first-page":"1962427","article-title":"Three-echelon supply chain model in an imperfect production system with inspection error, learning effect, and return policy under fuzzy environment","volume":"10","author":"Mahata","year":"2021","journal-title":"Int. J. Syst. Sci. Oper. Logist."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Manna, A.K., Barron, L.E.C., Dey, J.K., Mondal, S.K., Shaikh, A.A., Mota, A.C., and Garza, G.T. (2022). A fuzzy imperfect production inventory model based on fuzzy differential and fuzzy integral method. J. Risk Financ. Manag., 15.","DOI":"10.3390\/jrfm15060239"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"122","DOI":"10.2514\/8.155","article-title":"Factors affecting the cost of airplanes","volume":"3","author":"Wright","year":"1936","journal-title":"J. Aeronaut. Sci."},{"key":"ref_41","first-page":"90","article-title":"An EOQ model with fuzzy demand and learning in fuzziness","volume":"12","author":"Glock","year":"2012","journal-title":"Int. J. Serv. Oper. Manag."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1007\/s10845-014-1024-2","article-title":"A production-inventory model with imperfect production process and partial backlogging under learning considerations in fuzzy random environments","volume":"28","author":"Mahata","year":"2014","journal-title":"J. Intell. Manuf."},{"key":"ref_43","first-page":"1486","article-title":"EPQ model with learning consideration, imperfect production and partial backlogging in fuzzy random environment","volume":"46","author":"Kumar","year":"2015","journal-title":"Int. J. Syst. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1016\/j.cie.2015.05.014","article-title":"Incorporating human learning into a fuzzy EOQ inventory model with backorders","volume":"87","author":"Kazemi","year":"2015","journal-title":"Comput. Ind. Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2985","DOI":"10.3233\/IFS-151907","article-title":"An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning","volume":"30","author":"Shekarian","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5010","DOI":"10.1080\/00207543.2016.1165874","article-title":"A fuzzy lot-sizing problem with two stage composite human learning","volume":"54","author":"Kazemi","year":"2016","journal-title":"Int. J. Prod. Res."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2595","DOI":"10.3233\/JIFS-16372","article-title":"Demand uncertainty and learning in fuzziness in a continuous review inventory model","volume":"33","author":"Soni","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"469","DOI":"10.3233\/IFS-162160","article-title":"Triangular dense fuzzy sets and new defuzzification methods","volume":"31","author":"De","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"7243","DOI":"10.1007\/s00500-017-2726-0","article-title":"Triangular dense fuzzy lock sets","volume":"22","author":"De","year":"2018","journal-title":"Soft Comput."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Maity, S., De, S.K., and Mondal, S.P. (2019). A Study of an EOQ Model under Lock Fuzzy Environment. Mathematics, 7.","DOI":"10.3390\/math7010075"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2991","DOI":"10.1051\/ro\/2021127","article-title":"An estimation of effects of memory and learning experience on the EOQ model with price dependent demand","volume":"55","author":"Rahaman","year":"2021","journal-title":"RAIRO-Oper. Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s12046-021-01562-y","article-title":"Synergetic study of inventory management problem in uncertain environment based on memory and learning effects","volume":"46","author":"Rahaman","year":"2021","journal-title":"Sadhana"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/0020-0255(81)90017-7","article-title":"A procedure for ordering fuzzy subsets of the unit interval","volume":"24","author":"Yager","year":"1981","journal-title":"Inf. Sci."}],"container-title":["Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-8954\/11\/5\/235\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:31:29Z","timestamp":1760124689000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-8954\/11\/5\/235"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,8]]},"references-count":53,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["systems11050235"],"URL":"https:\/\/doi.org\/10.3390\/systems11050235","relation":{},"ISSN":["2079-8954"],"issn-type":[{"value":"2079-8954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,8]]}}}