Kareye Tamamlama Yöntemi ve Aritmetik-Geometrik Ortalama Eşitsizliğiyle İki Aşamalı Tedarik Zinciri Modelinin Çözümü ve Analizi

Year 2023, , 650 - 674, 24.12.2023

Harun Öztürk

https://doi.org/10.17065/huniibf.1281343

Abstract

Bu çalışmada, farklı ülkelerde bulunan tek bir üretici ve tek bir perakendeciden oluşan iki aşamalı tedarik zinciri problemi için bir bütünleşik stok kontrol modeli geliştirilmiştir. Bu çalışmanın amacı, bütünleşik toplam maliyeti minimum yapacak şekilde üreticinin parti sayısının ve perakendecinin parti büyüklüğünün, yani bütünleşik üretim-stok kontrol politikası parametrelerinin birlikte hesaplanmasıdır. Tek kalem ürünün siparişi, eşit büyüklükte partiler halinde teslim alınmaktadır. Perakendecinin teslim aldığı her parti iyi kaliteli ürünlerle birlikte kusurlu ürünler de içermektedir. Kusurlu ürünler, kalite kontrol işleminin ardından indirimli fiyattan satılmak üzere tek parti halinde stoktan çıkarılmaktadır; kusurlu ürün sayısı kadar iyi kaliteli fakat daha yüksek fiyatlı ürünler yerel bir tedarikçiden satın alınmaktadır. Üretici ve perakendecinin toplam stok maliyeti fonksiyonları elde edilmiş ve bütünleşik toplam stok maliyeti fonksiyonu türetilmiştir. Optimum çözüm diferensiyel hesabı kullanmadan aritmetik-geometrik ortalama eşitsizliği ve kareye tamamlama yöntemiyle elde edilmiştir. Sayısal bir örnek yardımıyla teorik sonuçlar elde edilmiş ve duyarlılık analizleri verilmiştir.

Keywords

Tedarik zinciri, Stok kontrolü, Matematik model, Kareye tamamlama tekniği, Aritmetik-geometirk ortalama eşitsizliği

The Complete Squares Method and the Arithmetic-Geometric Mean Inequality to Solve and Analyze A Two-Level Supply Chain Problem

Abstract

In this study, an integrated inventory model is developed for a two-level supply chain problem consisting of a single manufacturer and a single retailer located in different countries. The aim of this study is to jointly calculate the number of batches of the manufacturer and the batch size of the retailer, i.e. the parameters of the integrated production-inventory control policy, in a way that minimizes the integrated total cost. The order of a single type of product is delivered in batches of equal size. Each batch received by the retailer contains both good quality products and defective products. Defective products are removed from inventory as a single batch at a discounted price after a quality control process; good quality products, equal to the number of defective products, are purchased from a local supplier, but at a higher price. The total cost functions for the manufacturer and the retailer are obtained and the integrated total cost function is derived. The optimal solution is obtained by using the complete square method and the arithmetic-geometric mean inequality without using differential calculus. With the help of a numerical example, theoretical results were obtained and sensitivity analyses were given.

Keywords

Supply Chain, Inventory Control, Mathematical Model, Complete Squares Method, Arithmetic-Geometric Mean Inequality

References

• Banerjee, A. (1986). A joint economic‐lot‐size model for purchaser and vendor. Decision Sciences, 17(3), 292-311. https://doi.org/10.1111/j.1540-5915.1986.tb00228.x
• Banerjee, A., & Burton, J. S. (1994). Coordinated vs. independent inventory replenishment policies for a vendor and multiple buyers. International Journal of Production Economics, 35(1-3), 215-222. https://doi.org/10.1016/0925-5273(94)90084-1
• Barnett, M. A., Ziegler, M. R., & Byleen, K. E. (2017). Genel Matematik. (Çev. A. Sabuncuoğlu (Ed.)). Ankara, Nobel Akademik Yayıncılık.
• Buzacott, J. A. (1975). Economic order quantities with inflation. Journal of the Operational Research Society, 26(3), 553-558. https://doi.org/10.1057/jors.1975.113
• Can, T. (2015). Yöneylem Araştırması: Nedensellik Üzerine Diyaloglar I. Beta Basım Yayın.
• Cárdenas-Barrón, L. E. (2001). The economic production quantity (EPQ) with shortage derived algebraically. International Journal of Production Economics, 70(3), 289-292. https://doi.org/10.1016/S0925-5273(00)00068-2
• Chang, C. T., & Ouyang, L. Y. (2017). An arithmetic-geometric mean inequality approach for determining the optimal production lot size with backlogging and imperfect rework process. Journal of Applied Analysis & Computation, 7(1), 224-235. https://doi.org/10.1080/00207543.2019.1696491
• Chang, H. C., & Ho, C. H. (2011). A note on solving the EOQ model with imperfect quality subject to in-house inspection. IMA Journal of Management Mathematics, 22(3), 301-306. https://doi.org/10.1093/imaman/dpq002
• Chen, S. C., Cárdenas-Barrón, L. E., & Teng, J. T. (2014). Retailer’s economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity. International Journal of Production Economics, 155, 284-291. https://doi.org/10.1016/j.ijpe.2013.05.032
• Chiu, S. W. (2008). Production lot size problem with failure in repair and backlogging derived without derivatives. European Journal of Operational Research, 188(2), 610-615. https://doi.org/10.1016/j.ejor.2007.04.049 Chung, K. J. (2009). A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives: A comment. European Journal of Operational Research, 198(3), 979-982. https://doi.org/10.1016/j.ejor.2008.11.014
• Chung, K. J., Liao, J. J., Lin, S. D., Chuang, S. T., & Srivastava, H. M. (2020). Mathematical analytic techniques and the complete squares method for solving an inventory modelling problem with a mixture of backorders and lost sales. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 114, 1-10. https://doi.org/10.1007/s13398-019-00764-8
• Erkekoğlu, H. (2007). AB’ye tam üyelik sürecinde Türkiye’nin üye ülkeler karşısındaki göreli gelişme düzeyi: Çok değişkenli istatistiksel bir analiz. Kocaeli Üniversitesi Sosyal Bilimler Dergisi, 14, 28-50.
• Gautam, P., Maheshwari, S., Kausar, A., & Jaggi, C. K. (2021). Inventory models for imperfect quality items: A two-decade review. In P. K. Kapur, G. Singh, S. Panwar (eds.), Advances in Interdisciplinary Research in Engineering and Business Management (pp. 185-215). Springer.
• Goyal, S. K. (1976). An integrated inventory model for a single supplier-single customer problem. International Journal of Production Research, 15(1), 107-111. https://doi.org/10.1080/00207547708943107
• Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36, 35-38. https://doi.org/10.2307/2582421
• Goyal, S. K., & Nebebe, F. (2000). Determination of economic production—shipment policy for a single-vendor—single-buyer system. European Journal of Operational Research, 121, 175-178. https://doi.org/10.1016/S0377-2217(99)00013-2
• Grubbström, R. W. (1995). Modelling production opportunities—an historical overview. International Journal of Production Economics, 41(1-3), 1-14. https://doi.org/10.1016/0925-5273(95)00109-3
• Grubbström, R. W., & Erdem, A. (1999). The EOQ with backlogging derived without derivatives. International Journal of Production Economics, 59(1-3), 529-530. https://doi.org/10.1016/S0925-5273(98)00015-2
• Harris, F. W. (1913). How many parts to make at once. Factory, The Magazine of Management, 10(2), 135-136, 152. Hsieh, T. P., Chang, H. J., Weng, M. W., & Dye, C. Y. (2008). A simple approach to an integrated single-vendor single-buyer inventory system with shortage. Production Planning and Control, 19(6), 601-604. https://doi.org/10.1080/09537280802462789
• Hoque, M., & Goyal, S. K. (2005). An algebraically derived minimal cost solution technique of the ıntegrated vendor-buyer problem. International Journal of Operations Research, 2(1), 43-48.
• Hovelaque, V., & Bironneau, L. (2015). The carbon-constrained EOQ model with carbon emission dependent demand. International Journal of Production Economics, 164, 285-291. https://doi.org/10.1016/j.ijpe.2014.11.022
• Huang, C. K. (2002). An integrated vendor-buyer cooperative inventory model for items with imperfect quality. Production Planning & Control, 13(4), 355-361. https://doi.org/10.1080/09537280110112424
• Huang, Y. F. (2003). The deterministic inventory models with shortage and defective items derived without derivatives. Journal of Statistics and Management Systems, 6(2), 171-180. https://doi.org/10.1080/09720510.2003.10701076
• Huang, Y. F. (2006). An inventory model under two levels of trade credit and limited storage space derived without derivatives. Applied Mathematical Modelling, 30(5), 418-436. https://doi.org/10.1016/j.apm.2005.05.009
• Jaber, M. Y., Zanoni, S., & Zavanella, L. E. (2014). Economic order quantity models for imperfect items with buy and repair options. International Journal of Production Economics, 155, 126-131. https://doi.org/10.1016/j.ijpe.2013.10.014
• Jayaswal, M. K., Mittal, M., Sangal, I., & Yadav, R. (2021). EPQ model with learning effect for imperfect quality items under trade-credit financing. Yugoslav Journal of Operations Research, 31(2), 235-247. https://doi.org/10.2298/YJOR2002
• Khan, M., Jaber, M. Y., Guiffrida, A. L., & Zolfaghari, S. (2011). A review of the extensions of a modified EOQ model for imperfect quality items. International Journal of Production Economics, 132(1), 1-12. https://doi.org/10.1016/j.ijpe.2011.03.009
• Kobu, B. (1993). Üretim Yönetimi. Avcıol Basım-Yayın
• Leung, K. N. F. (2008). A use of the complete squares method to solve and analyze a quadratic objective function with two decision variables exemplified via a deterministic inventory model with a mixture of backorders and lost sales. International Journal of Production Economics, 113(1), 275-281. https://doi.org/10.1016/j.ijpe.2007.08.007
• Mahato, C., & Mahata, G. C. (2023). Optimal ordering policy under order-size dependent trade credit and complete backlogging derived algebraically. OPSEARCH, 60(1), 420-444. https://doi.org/10.1007/s12597-022-00614-z
• Nesin, A. (2019). Analiz I. Nesin Yayıncılık.
• Özdemir, A. İ. (2004). Tedarik zinciri yönetiminin gelişimi, süreçleri ve yararları. Erciyes Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi, 23, 87-96.
• Öztürk, H. (2022). Optimal manufacturer-buyer cooperative inventory models under unequal shipment policy with emergency replacement of sub-standard items. International Journal of Integrated Supply Management, 15(1), 49-73. https://doi.org/10.1504/IJISM.2022.119586
• Rahman, Md S., & Khatun, R. (2023). Generalised arithmetic mean-geometric mean inequality and its application to find the optimal policy of the classical EOQ model under interval uncertainty. Applied Mathematics E-Notes, 23, 90-99.
• Salameh, M. K., Abdul-Malak, M. U., & Jaber, M, Y. (1993). Mathematical modelling of the effect of human learning in the finite production ınventory model. Applied Mathematical Modelling, 17, 613-615. https://doi.org/10.1016/0307-904X(93)90070-W
• Salameh, M. K., & Jaber, M. Y. (2000). Economic production quantity model for ıtems with ımperfect quality. International Journal of Production Economics, 64, 59-64. https://doi.org/10.1016/S0925-5273(99)00044-4
• Schwaller, R. L. (1988). EOQ under ınspection costs. Production and Inventory Management, 29, 22-35. Seliaman, M. E., Cárdenas-Barrón, L. E., & Rushd, S. (2020). An algebraic decision support model for inventory coordination in the generalized n-stage non-serial supply chain with fixed and linear backorders costs. Symmetry, 12(12), 1998. https://doi.org/10.3390/sym12121998
• Seliaman, M. E., Khan, M., & Cárdenas-Barrón, L. E. (2018). Algebraic modelling of a two level supply chain with defective items. RAIRO-Operations Research, 52(2), 415-427. https://doi.org/10.1051/ro/2017063
• Taha, H. A. (2004). Yöneylem Araştırması. (Çev. Ş. A. Baray, Ş. Esnaf). Literatür Yayıncılık.
• Teerapabolarn, K., & Khamrod, S. (2014). The inventory models with backorders and defective items derived algebraically and AGM. International Journal of Pure and Applied Mathematics, 97(2), 225-230. http://dx.doi.org/10.12732/ijpam.v97i2.11
• Teng, H. M., & Hsu, P. H. (2015). Optimal production lots for items with imperfect production and screening processes without using derivatives. International Journal of Management and Enterprise Development, 14(2), 172-185. https://doi.org/10.1504/IJMED.2015.070100
• Teng, J. T. (2009). A simple method to compute economic order quantities. European Journal of Operational Research, 198(1), 351-353. https://doi.org/10.1016/j.ejor.2008.05.019
• Teng, J. T., Cárdenas-Barrón, L. E., & Lou, K. R. (2011). The economic lot size of the integrated vendor–buyer inventory system derived without derivatives: A simple derivation. Applied Mathematics and Computation, 217(12), 5972-5977. https://doi.org/10.1016/j.amc.2010.12.018
• Tu, Y. C., Huang, Y. F., Chen, Y. C., & Chen, H. F. (2011). Using simple methods to derive EOQ and EPQ models with shortage and imperfect quality. Journal of Information and Optimization Sciences, 32(6), 1333-1340. https://doi.org/10.1080/02522667.2011.10700122
• Öztürk, H. (2019). The derivation of production lot sizing with imperfect quality, inspection and rework using an algebraic approach. Journal of Research in Business, 4(2), 93-110. https://doi.org/10.23892/JRB.2019.56
• Wee, H. M., & Chung, C. J. (2007). A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives. European Journal of Operational Research, 177(2), 1289-1293. https://doi.org/10.1016/j.ejor.2005.11.035
• Wu, K. S., & Ouyang, L. Y. (2003). An integrated single-vendor single-buyer inventory system with shortage derived algebraically. Production Planning & Control, 14(6), 555-561. https://doi.org/10.1080/09537280310001613722
• Yang, P. C., & Wee, H. M. (2002). The economic lot size of the integrated vendor‐buyer inventory system derived without derivatives. Optimal Control Applications and Methods, 23(3), 163-169. https://doi.org/10.1002/oca.706
• Zhang, X., & Gerchak, Y. (1990). Joint lot sizing and inspection policy in an EOQ model with random yield. IIE Transactions, 22, 41-47. https://doi.org/10.1080/07408179008964156