{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T00:08:10Z","timestamp":1768867690883,"version":"3.49.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,9,5]],"date-time":"2018-09-05T00:00:00Z","timestamp":1536105600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2018,9,5]],"date-time":"2018-09-05T00:00:00Z","timestamp":1536105600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann Oper Res"],"published-print":{"date-parts":[[2020,9]]},"DOI":"10.1007\/s10479-018-3033-6","type":"journal-article","created":{"date-parts":[[2018,9,5]],"date-time":"2018-09-05T17:32:26Z","timestamp":1536168746000},"page":"341-369","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Mixture inventory model of lost sale and back-order with stochastic lead time demand on permissible delay in payments"],"prefix":"10.1007","volume":"292","author":[{"given":"Rabin Kumar","family":"Mallick","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kartik","family":"Patra","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shyamal Kumar","family":"Mondal","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,9,5]]},"reference":[{"key":"3033_CR1","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1057\/jors.1995.90","volume":"46","author":"SP Aggarwal","year":"1995","unstructured":"Aggarwal, S. P., & Jaggi, C. K. (1995). Ordering policies of deteriorating items under permissible delay in payments. Journal of the Operational Research Society, 46, 658\u2013662.","journal-title":"Journal of the Operational Research Society"},{"issue":"1","key":"3033_CR2","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1186\/s40064-016-2567-9","volume":"5","author":"A Banu","year":"2016","unstructured":"Banu, A., & Mondal, S. K. (2016). Analysis of credit linked demand in an inventory model with varying ordering cost. Springerplus, 5(1), 926\u2013939.","journal-title":"Springerplus"},{"key":"3033_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijpe.2014.07.002","volume":"155","author":"LE Cardenas-Barron","year":"2014","unstructured":"Cardenas-Barron, L. E., Chung, K. J., & Trevino-Garza, G. (2014). Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris. International Journal of Production Economics, 155, 1\u20137.","journal-title":"International Journal of Production Economics"},{"key":"3033_CR4","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1080\/00207720110091460","volume":"33","author":"HJ Chang","year":"2002","unstructured":"Chang, H. J., Hung, C. H., & Dye, C. Y. (2002). A finite time horizon inventory model with deterioration and time-value of money under the conditions of permissible delay in payments. International Journal of Systems Science, 33, 141\u2013151.","journal-title":"International Journal of Systems Science"},{"issue":"1","key":"3033_CR5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.1540-5414.2004.02332.x","volume":"35","author":"S Chopra","year":"2004","unstructured":"Chopra, S., Reinhardt, G., & Dada, M. (2004). The effect of lead time uncertainty on safety stocks. Decision Sciences, 35(1), 1\u201324.","journal-title":"Decision Sciences"},{"key":"3033_CR6","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/S0305-0548(98)80007-5","volume":"25","author":"KJ Chung","year":"1998","unstructured":"Chung, K. J. (1998). A theorem on the determination of economic order quantity under conditions of permissible delay in payments. Computers and Operations Research, 25, 49\u201352.","journal-title":"Computers and Operations Research"},{"key":"3033_CR7","doi-asserted-by":"crossref","first-page":"4653","DOI":"10.1016\/j.apm.2012.10.018","volume":"37","author":"KJ Chung","year":"2013","unstructured":"Chung, K. J., & Cardenas-Barron, L. E. (2013). The simplified solution procedure for deteriorating items under stock-dependent demand and two-level trade credit in the supply chain management. Applied Mathematical Modelling, 37, 4653\u20134660.","journal-title":"Applied Mathematical Modelling"},{"issue":"3","key":"3033_CR8","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1016\/j.cie.2012.12.020","volume":"64","author":"BC Das","year":"2013","unstructured":"Das, B. C., Das, B., & Mondal, S. K. (2013). Integrated supply chain model for a deteriorating item with procurement cost dependent credit period. Computer & Industrial Engineering, 64(3), 788\u2013796.","journal-title":"Computer & Industrial Engineering"},{"key":"3033_CR9","doi-asserted-by":"crossref","unstructured":"Das, B. C., Das, B., & Mondal, S. K. (2014a). An integrated inventory model with delay in payment for deteriorating item under Weibull distribution and advertisement cum price-dependent demand. International Journal of Operational Research, 20(3), 341\u2013368.","DOI":"10.1504\/IJOR.2014.062456"},{"key":"3033_CR10","doi-asserted-by":"crossref","unstructured":"Das, B. C., Das, B., & Mondal, S. K. (2014b). Optimal transportation and business cycles in an integrated production\u2013inventory model with a discrete credit period. Transportation Research Part E, 68, 1\u201313.","DOI":"10.1016\/j.tre.2014.04.007"},{"key":"3033_CR11","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.asoc.2014.11.057","volume":"28","author":"BC Das","year":"2015","unstructured":"Das, B. C., Das, B., & Mondal, S. K. (2015). An integrated production inventory model under interactive fuzzy credit period for deteriorating item with several markets. Applied Soft Computing, 28, 453\u2013465.","journal-title":"Applied Soft Computing"},{"issue":"1","key":"3033_CR12","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.ijpe.2011.09.007","volume":"136","author":"CH Glock","year":"2012","unstructured":"Glock, C. H. (2012). Lead time reduction strategies in a single-vendor\u2013single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. International Journal of Production Economics, 136(1), 37\u201344.","journal-title":"International Journal of Production Economics"},{"issue":"4","key":"3033_CR13","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1057\/jors.1985.56","volume":"36","author":"SK Goyal","year":"1985","unstructured":"Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research, 36(4), 335\u2013338.","journal-title":"Journal of the Operational Research"},{"issue":"1\u20132","key":"3033_CR14","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.apm.2012.02.010","volume":"37","author":"A Guria","year":"2013","unstructured":"Guria, A., Das, B., Mondal, S. K., & Maiti, M. (2013). Inventory policy for an item with inflation induced purchasing price, selling price and demand with immediate part payment. Applied Mathematical Modelling, 37(1\u20132), 240\u2013257.","journal-title":"Applied Mathematical Modelling"},{"issue":"2","key":"3033_CR15","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1504\/IJOR.2012.048866","volume":"15","author":"A Guria","year":"2012","unstructured":"Guria, A., Mondal, S. K., & Maiti, M. (2012). A two-ware house EOQ model with two-level delay in payment. International Journal of Operations Research, 15(2), 170\u2013194.","journal-title":"International Journal of Operations Research"},{"key":"3033_CR16","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/j.ejor.2010.11.025","volume":"211","author":"JC Hayya","year":"2011","unstructured":"Hayya, J. C., Harrison, T. P., & He, X. J. (2011). The impact of stochastic lead time reduction on inventory cost under order crossover. European Journal of Operational Research, 211, 274\u2013281.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"3033_CR17","first-page":"113","volume":"97","author":"XJ He","year":"2005","unstructured":"He, X. J., Kim, J. G., & Hayya, J. C. (2005). The cost of lead-time variability: The case of the exponential distribution. International Journal of Production Eco-nomics, 97(2), 113\u2013122.","journal-title":"International Journal of Production Eco-nomics"},{"key":"3033_CR18","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.ijpe.2013.02.019","volume":"144","author":"MA Hoque","year":"2013","unstructured":"Hoque, M. A. (2013). A vendor\u2013buyer integrated production\u2013inventory model with normal distribution of lead time. International Journal of Production Economics, 144, 409\u2013417.","journal-title":"International Journal of Production Economics"},{"issue":"1","key":"3033_CR19","doi-asserted-by":"crossref","first-page":"390","DOI":"10.24200\/sci.2017.4042","volume":"24","author":"CK Jaggi","year":"2017","unstructured":"Jaggi, C. K., Cardenas-Barron, L. E., Tiwaria, S., & Shafi, A. (2017). Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments. Scientia Iranica, Transactions E, 24(1), 390\u2013412.","journal-title":"Scientia Iranica, Transactions E"},{"key":"3033_CR20","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1016\/j.apm.2012.04.042","volume":"37","author":"JK Jha","year":"2013","unstructured":"Jha, J. K., & Shanker, K. (2013). Single-vendor multi-buyer integrated production\u2013inventory model with controllable lead time and service level constraints. Applied Mathematical Modelling, 37, 1753\u20131767.","journal-title":"Applied Mathematical Modelling"},{"issue":"2","key":"3033_CR21","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s10479-014-1602-x","volume":"240","author":"Y Li","year":"2016","unstructured":"Li, Y., Zhen, X., & Cai, X. (2016). Trade credit insurance, capital constraint, and the behavior of manufacturers and banks. Annals of Operations Research, 240(2), 395\u2013414.","journal-title":"Annals of Operations Research"},{"key":"3033_CR22","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0925-5273(99)00015-8","volume":"63","author":"HC Liao","year":"2000","unstructured":"Liao, H. C., Tsai, C. H., & Su, C. T. (2000). An inventory model with deteriorating items under inflation when a delay in payment is permissible. International Journal of Production Economics, 63, 207\u2013214.","journal-title":"International Journal of Production Economics"},{"issue":"3","key":"3033_CR23","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.apm.2005.11.016","volume":"31","author":"JJ Liao","year":"2007","unstructured":"Liao, J. J. (2007a). On an EPQ model for deteriorating items under permissible delay in payments. Applied Mathematical Modelling, 31(3), 393\u2013403.","journal-title":"Applied Mathematical Modelling"},{"issue":"6","key":"3033_CR24","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1287\/opre.25.6.977","volume":"25","author":"MJ Liberatore","year":"1977","unstructured":"Liberatore, M. J. (1977). Planning horizons for a stochastic lead-time inventory model. Operations Research, 25(6), 977\u2013988.","journal-title":"Operations Research"},{"key":"3033_CR25","doi-asserted-by":"crossref","first-page":"2433","DOI":"10.1016\/j.apm.2008.07.024","volume":"33","author":"AK Maiti","year":"2009","unstructured":"Maiti, A. K., Maiti, M. K., & Maiti, M. (2009). Inventory model with stochastic leadtime and price dependent demand incorporating advance payment. Applied Mathematical Modelling, 33, 2433\u20132443.","journal-title":"Applied Mathematical Modelling"},{"issue":"1\u20132","key":"3033_CR26","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10479-017-2419-1","volume":"254","author":"U Mishra","year":"2017","unstructured":"Mishra, U., Cardenas-Barron, L. E., Tiwari, S., Shaikh, A. A., & Trevino-Garza, G. (2017). An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment. Annals of Operations Research, 254(1\u20132), 165\u2013190.","journal-title":"Annals of Operations Research"},{"issue":"2","key":"3033_CR27","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1002\/nav.3800200205","volume":"20","author":"DC Montgomery","year":"1973","unstructured":"Montgomery, D. C., Bazaraa, M. S., & Keswani, A. K. (1973). Inventory models with a mixture of backorders and lost sales. Naval Research Logistics (NRL), 20(2), 255\u2013263.","journal-title":"Naval Research Logistics (NRL)"},{"issue":"1","key":"3033_CR28","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.ijpe.2011.09.013","volume":"136","author":"A Musa","year":"2012","unstructured":"Musa, A., & Sani, B. (2012). Inventory ordering policies of delayed deteriorating items under permissible delay in payments. International Journal of Production Economics, 136(1), 75\u201383.","journal-title":"International Journal of Production Economics"},{"key":"3033_CR29","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.amc.2013.08.062","volume":"224","author":"LY Ouyang","year":"2013","unstructured":"Ouyang, L. Y., Yang, C. T., Chan, Y. L., & Cardenas-Barron, L. E. (2013). A comprehensive extension of the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity. Applied Mathematics and Computation, 224, 268\u2013277.","journal-title":"Applied Mathematics and Computation"},{"issue":"1","key":"3033_CR30","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s10479-013-1510-5","volume":"233","author":"SS Sana","year":"2015","unstructured":"Sana, S. S. (2015). An EOQ model for stochastic demand for limited capacity of own warehouse. Annals of Operations Research, 233(1), 383\u2013399.","journal-title":"Annals of Operations Research"},{"key":"3033_CR31","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.amc.2015.03.010","volume":"259","author":"NH Shah","year":"2015","unstructured":"Shah, N. H., & Cardenas-Barron, L. E. (2015). Retailer\u2019s decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount. Applied Mathematics and Computation, 259, 569\u2013578.","journal-title":"Applied Mathematics and Computation"},{"key":"3033_CR32","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s10479-011-0944-x","volume":"199","author":"J Sicilia","year":"2012","unstructured":"Sicilia, J., San-Jose, L. A., & Garcia-Laguna, J. (2012). An inventory model where backordered demand ratio is exponentially decreasing with the waiting time. Annals of Operations Research, 199, 137\u2013155.","journal-title":"Annals of Operations Research"},{"key":"3033_CR33","doi-asserted-by":"crossref","first-page":"9234","DOI":"10.1016\/j.amc.2011.04.001","volume":"217","author":"AA Taleizadeh","year":"2011","unstructured":"Taleizadeh, A. A., Niaki, S. T. A., & Barzinpour, F. (2011). Multiple-buyer multiple-vendor multi-product multi-constraint supply chain problem with stochastic demand and variable lead-time: A harmony search algorithm. Applied Mathematics and Computation, 217, 9234\u20139253.","journal-title":"Applied Mathematics and Computation"},{"key":"3033_CR34","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.ijpe.2016.03.016","volume":"176","author":"S Tiwari","year":"2016","unstructured":"Tiwari, S., Cardenas-Barron, L. E., Khanna, A., & Jaggi, C. K. (2016). Impact of trade credit and inflation on retailer\u2019s ordering policies for non-instantaneous deteriorating items in a two-warehouse environment. International Journal of Production Economics, 176, 154\u2013169.","journal-title":"International Journal of Production Economics"},{"issue":"Part\u20131","key":"3033_CR35","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.ijpe.2015.10.020","volume":"171","author":"J Wu","year":"2016","unstructured":"Wu, J., Al-khateeb, F. B., Teng, J. T., & Cardenas-Barron, L. E. (2016). Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis. International Journal of Production Economics, 171(Part\u20131), 105\u2013115.","journal-title":"International Journal of Production Economics"},{"issue":"3","key":"3033_CR36","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1016\/j.ejor.2014.03.009","volume":"237","author":"J Wu","year":"2014","unstructured":"Wu, J., Ouyang, L. Y., Cardenas-Barron, L. E., & Goyal, S. K. (2014). Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing. European Journal of Operational Research, 237(3), 898\u2013908.","journal-title":"European Journal of Operational Research"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-018-3033-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10479-018-3033-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-018-3033-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,31]],"date-time":"2020-07-31T14:42:36Z","timestamp":1596206556000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10479-018-3033-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,5]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["3033"],"URL":"https:\/\/doi.org\/10.1007\/s10479-018-3033-6","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,9,5]]},"assertion":[{"value":"5 September 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}