Document Type : Article

**Authors**

Department of Mechanical Engineering, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir, 182320, India.

**Abstract**

One of the major challenges being faced by manufacturing companies today is the issue of addressing various aspects of perishable products in a supply chain environment. To address this issue, the integrated lot size problem for a perishable product has been investigated in the present work. The problem is modeled as a single vendor multiple buyer system. A variant of the truckload discount scheme is applied and the proposed model is formulated as a mixed integer program (MIP). The traditional warehouses are replaced by ‘cross-docks’ and situations in which, cross-docking would be more beneficial are highlighted. The problem of fleet selection is also addressed and various strategies to minimize the vendor cost are also highlighted for centralized and decentralized supply chains. Sensitivity analysis is then carried out on various input parameters such as setup cost at plant, variable transportation cost, fixed transportation cost, setup cost per order, holding cost and, lost cost, which underscores the significant impact of economies of scale in transportation on the total supply chain cost. Analysis of lead time-cost trade-off reveals that alternate modes of transportation could be explored, which significantly reduces the lead time of transportation, thereby minimizing the total supply chain cost.

**Keywords**

**Main Subjects**

References:

1. Nobari, A. and Kheirkhah, A. "Integrated and dynamic design of sustainable closed-loop supply chain network considering pricing", Scientia Iranica, 25(1), pp. 410-430 (2018).

2. Hajiaghaei-Keshteli, M. and Fathollahi-Fard, A.M.F. "Sustainable closed-loop supply chain network design with discount supposition", Neural Computing and Applications, pp. 1-35. (2018). DOI: https://doi.org/10.1007/s00521-018-3369-5.

3. Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E., and Shankar, R., Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies, Tata McGraw-Hill Education, Third Edn. (2008).

4. Farahani, R.Z., Rezapour, S., Drezner, T., and Fallah, S. "Competitive supply chain network design: An overview of classifications, models, solution techniques and applications", Omega, 45, pp. 92-118 (2014).

5. Koc, C . "An evolutionary algorithm for supply chain network designs with assembly line balancing", Neural Computing and Applications, 28(11), pp. 3183-3195 (2017).

6. Schmidt, G. and Wilhelm, W.E. "Strategic, tactical and operational decisions in multi-national logistics networks: a review and discussion of modelling issues", International Journal of Production Research, 38(7), pp. 1501-1523 (2000).

7. Fathollahi-Fard, A.M. and Hajiaghaei-Keshteli, M. "A stochastic multi-objective model for a closed-loop supply chain with environmental considerations", Applied Soft Computing, 69, pp. 232-249 (2018).

8. Eydi, A., Fazayeli, S., and Ghafouri, H. "Multi- period configuration of forward and recerse integrated supply chain networks with the choice of transportation mode", Scientia Iranica, 27(2), pp. 935-955 (2020). DOI: 10.24200/SCI.2018.5261.1175.

9. Mokhtari, H., Naimi-Sadigh, A., and Salmasnia, A. "A computational approach to economic production quantity model for perishable products with backordering shortage and stock-dependent demand", Scientia Iranica, 24(4), pp. 2138-2151 (2017).

10. Devika, K., Jafarian, A., and Nourbakhsh, V. "Designing a sustainable closed-loop supply chain network based on triple bottom line approach: A comparison of meta-heuristics hybridization techniques", European Journal of Operational Research, 235(3), pp. 594-615 (2014).

11. De La Vega, D.S., Vieira, J.G.V., Toso, E.A.V., and De Faria, R.N. "A decision on the truckload and less-thantruckload problem: An approach based on MCDA", International Journal of Production Economics, 195, pp. 132-145 (2018).

12. Bruno, G., Genovese, A., and Piccolo, C. "The capacitated lot sizing model: A powerful tool for logistics decision making", International Journal of Production Economics, 155, pp. 380-390 (2014).

13. Chandra, P. and Fisher, M.L. "Coordination of production and distribution planning", European Journal of Operational Research, 72(3), pp. 503-517 (1994).

14. Erenguc, S.S., Simpson, N.C., and Vakharia, A.J. "Integrated production/distribution planning in supply chains: An invited review", European Journal of Operational Research, 115(2), pp. 219-236 (1999).

15. Li, H. and Meissner, J. "Capacitated dynamic lot sizing with capacity acquisition", International Journal of Production Research, 49(16), pp. 4945-4963 (2011).

16. Rezaei, J. and Davoodi, M. "A joint pricing, lot-sizing, and supplier selection model", International Journal of Production Research, 50(16), pp. 4524-4542 (2012).

17. Amorim, P., Belo-Filho, M., Toledo, F., Almeder, C., and Almada-Lobo, B. "Lot sizing versus batching in the production and distribution planning of perishable goods", International Journal of Production Economics, 146, pp. 208-218 (2013).

18. Govindan, K., Jafarian, A., Khodaverdi, R., and Devika, K. "Two-echelon multiple-vehicle locationrouting problem with time windows for optimization of sustainable supply chain network of perishable food", International Journal of Production Economics, 152, pp. 9-28 (2014).

19. Ma, Z.J., Wu, Y., and Dai, Y. "A combined order selection and time-dependent vehicle routing problem with time widows for perishable product delivery", Computers & Industrial engineering, 114, pp. 101- 113 (2017).

20. Azadeh, A., Elahi, S., Farahani, M.H., and Nasirian, B. "A genetic algorithm-Taguchi based approach to inventory routing problem of a single perishable product with transshipment", Computers & Industrial Engineering, 104, pp. 124-133 (2017).

21. Dolgui, A., Tiwari, M.K., Sinjana, Y., Kumar, S.K., and Son, Y.J. "Optimizing integrated inventory policy for perishable items in a multi-stage supply chain", International Journal of Production Research, 56(1- 2), pp. 902-925 (2018).

22. Gitinavard, H., Ghodsypour, S.H., and Shirazi, M.A. "A bi-objective multi-echelon supply chain model with Pareto optimal points evaluation for perishable products under uncertainty", Scientia Iranica, 26(5), pp. 2952-2970 (2019). DOI: 10.24200/SCI.2018.5047.1060.

23. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M., and Mirjalili, S. "Multi-objective stochastic closed-loop supply chain network design with social considerations", Applied Soft Computing, 71, pp. 505-525 (2018).

24. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M., and Mirjalili, S. "Hybrid optimizers to solve a tri-level programming model for a tire closed-loop supply chain network design problem", Applied Soft Computing, 70, pp. 701-722 (2018).

25. Musavi, M. and Bozorgi-Amiri, A. "A multi-objective sustainable hub location-scheduling problem for perishable food supply chain", Computers & Industrial Engineering, 113, pp. 766-778 (2017).

26. Fathollahi-Fard, A.M. and Hajiaghaei-Keshteli, M. "A bi-objective partial interdiction problem considering different defensive systems with capacity expansion of facilities under imminent attacks", Applied Soft Computing, 68, pp. 343-359 (2018).

27. Hajiaghaei-Keshteli, M. and Fathollahi-Fard, A.M. "A set of efficient heuristics and meta-heuristics to solve a two-stage stochastic bi-level decision-making model for the distribution network problem", Computers & Industrial Engineering, 123, pp. 378-395 (2018).

28. Samadi, A., Mehranfar, N., Fathollahi Fard, A.M., and Hajiaghaei-Keshteli, M. "Heuristic-based metaheuristic to address a sustainable supply chain network design problem", Journal of Industrial and Production Engineering, 35(2), pp. 102-117 (2018).

29. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M., and Tavakkoli-Moghaddam, R. "The Social Engineering Optimizer (SEO)", Engineering Applications of Artificial Intelligence, 72, pp. 267-293 (2018c).

30. Stenmarck, A., Jensen, C., Quested, T., and Moates, G. "Estimation of European food waste levels", FUSIONS: EU Project under European Community's Seventh Framework Programme Under Grant Agreement No 311972 (2016). (Retrieved at http://www.eu-fusions.org/phocadownload/Publications/ Estimates%20of%20European%20food%20waste %20levels.pdf on 22 Dec 2018).

31. Goyal, S.K. and Giri, B.C. "Recent trends in modeling of deteriorating inventory", European Journal of Operational Research, 134(1), pp. 1-16 (2001).

32. Ghare, P. and Schrader, G. "A model for exponentially decaying inventory", Journal of Industrial Engineering, 14(5), pp. 238-243 (1963).

33. Nahmias, S. "Perishable inventory theory: A review", Operations Research, 30(4), pp. 680-708 (1982).

34. Hsu, V.N. "Dynamic economic lot size model with perishable inventory", Management Science, 46(8), pp. 1159-1169 (2000).

35. Abad, P. "Optimal lot size for a perishable good under conditions of finite production and partial backordering and lost sale", Computers & Industrial Engineering, 38, pp. 457-465 (2000).

36. Abad, P.L. "Optimal pricing and lot-sizing under conditions of perishability, finite production and partial backordering and lost sale", European Journal of Operational Research, 144, pp. 677-685 (2003).

37. Teng, J.T., Cardenas-Barron, L.E., Chang, H.J., Wu, J., and Hu, Y. "Inventory lot-size policies for deteriorating items with expiration dates and advance payments", Applied Mathematical Modelling, 40(19- 20), pp. 8605-8616 (2016).

38. Hiassat, A., Diabat, A., and Rahwan, I. "A genetic algorithm approaches for location-inventory-routing problem with perishable products", Journal of Manufacturing Systems, 42, pp. 93-103 (2017).

39. Sahebjamnia, N., Fathollahi-Fard, A.M., and Hajiaghaei-Keshteli, M. "Sustainable tire closed-loop supply chain network design: Hybrid metaheuristic algorithms for large-scale networks", Journal of Cleaner Production, 196, pp. 273-296 (2018).

40. Karimi, B., Fatemi Ghomi, S., and Wilson, J. "The capacitated lot sizing problem: A review of models and algorithms", Omega, 31(5), pp. 365-378 (2003).

41. Yano, C.A. and Lee, H.L. "Lot sizing with random yields: A review", Operations Research, 43(2), pp. 311-334 (1995).

42. Ben-Daya, M., Darwish, M., and Ertogral, K. "The joint economic lot-sizing problem: Review and extensions", European Journal of Operational Research, 185(2), pp. 726-742 (2008).

43. Robinson, P., Narayanan, A., and Sahin, F. "Coordinated deterministic dynamic demand lot-sizing problem: A review of models and algorithms", Omega, 37(1), pp. 3-15 (2009).

44. Onal, M., Romeijn, H.E., Sapra, A., and Van den Heuvel, W. "The economic lot-sizing problem with perishable items and consumption order preference", European Journal of Operational Research, 244, pp. 881-891 (2015).

45. Lee, W. "A joint economic lot size model for raw material ordering, manufacturing setup, and finished goods delivering", Omega, 33, pp. 163-174 (2005).

46. Pal, A., Chan, F., Mahanty, B., and Tiwari, M. "Aggregate procurement, production, and shipment planning decision problem for a three-echelon supply chain using swarm-based heuristics", International Journal of Production Research, 49, pp. 2873-2905 (2011).

47. Onal, M. "The two-level economic lot-sizing problem with perishable items", Operations Research Letters, 44, pp. 403-408 (2016).

48. Benton, W.C. and Park, S. "A classification of literature on determining the lot size under quantity discounts", European Journal of Operational Research, 92(2), pp. 219-238 (1996).

49. Nahmias, S. and Cheng, Y., Production and Operations Analysis, New York: McGraw-hill, 6 (2009).

50. Ke, G.Y. and Bookbinder, J.H. "Coordinating the discount policies for retailer, wholesaler, and less-thantruckload carrier under price-sensitive demand: A trilevel optimization approach", International Journal of Production Economics, 196, pp. 82-100 (2018).

51. Marandi, F. and Zegordi, S.H. "Integrated production and distribution scheduling for perishable products", Scientia Iranica, 24(4), pp. 2105-2118 (2017).

52. Chung, K.J., Ting, P.S., and Cardenas-Barron, L.E. "A simple solution procedure for solving the multidelivery policy into economic production lot size problem with partial rework", Scientia Iranica, Transaction E, Industrial Engineering, 24(5), pp. 2640-2644 (2017).

53. Optimization, D. "Xpress-mp manual" (2002).

54. Gueret, C., Prins, C., and Sevaux, M., Applications of Optimization with Xpress-mp, Dash Optimization Ltd (1999).

55. Chen, P., Guo, Y., Lim, A., and Rodrigues, B. "Multiple cross-docks with inventory and time windows", Computers & Operations Research, 33(1), pp. 43-63 (2006).

2. Hajiaghaei-Keshteli, M. and Fathollahi-Fard, A.M.F. "Sustainable closed-loop supply chain network design with discount supposition", Neural Computing and Applications, pp. 1-35. (2018). DOI: https://doi.org/10.1007/s00521-018-3369-5.

3. Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E., and Shankar, R., Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies, Tata McGraw-Hill Education, Third Edn. (2008).

4. Farahani, R.Z., Rezapour, S., Drezner, T., and Fallah, S. "Competitive supply chain network design: An overview of classifications, models, solution techniques and applications", Omega, 45, pp. 92-118 (2014).

5. Koc, C . "An evolutionary algorithm for supply chain network designs with assembly line balancing", Neural Computing and Applications, 28(11), pp. 3183-3195 (2017).

6. Schmidt, G. and Wilhelm, W.E. "Strategic, tactical and operational decisions in multi-national logistics networks: a review and discussion of modelling issues", International Journal of Production Research, 38(7), pp. 1501-1523 (2000).

7. Fathollahi-Fard, A.M. and Hajiaghaei-Keshteli, M. "A stochastic multi-objective model for a closed-loop supply chain with environmental considerations", Applied Soft Computing, 69, pp. 232-249 (2018).

8. Eydi, A., Fazayeli, S., and Ghafouri, H. "Multi- period configuration of forward and recerse integrated supply chain networks with the choice of transportation mode", Scientia Iranica, 27(2), pp. 935-955 (2020). DOI: 10.24200/SCI.2018.5261.1175.

9. Mokhtari, H., Naimi-Sadigh, A., and Salmasnia, A. "A computational approach to economic production quantity model for perishable products with backordering shortage and stock-dependent demand", Scientia Iranica, 24(4), pp. 2138-2151 (2017).

10. Devika, K., Jafarian, A., and Nourbakhsh, V. "Designing a sustainable closed-loop supply chain network based on triple bottom line approach: A comparison of meta-heuristics hybridization techniques", European Journal of Operational Research, 235(3), pp. 594-615 (2014).

11. De La Vega, D.S., Vieira, J.G.V., Toso, E.A.V., and De Faria, R.N. "A decision on the truckload and less-thantruckload problem: An approach based on MCDA", International Journal of Production Economics, 195, pp. 132-145 (2018).

12. Bruno, G., Genovese, A., and Piccolo, C. "The capacitated lot sizing model: A powerful tool for logistics decision making", International Journal of Production Economics, 155, pp. 380-390 (2014).

13. Chandra, P. and Fisher, M.L. "Coordination of production and distribution planning", European Journal of Operational Research, 72(3), pp. 503-517 (1994).

14. Erenguc, S.S., Simpson, N.C., and Vakharia, A.J. "Integrated production/distribution planning in supply chains: An invited review", European Journal of Operational Research, 115(2), pp. 219-236 (1999).

15. Li, H. and Meissner, J. "Capacitated dynamic lot sizing with capacity acquisition", International Journal of Production Research, 49(16), pp. 4945-4963 (2011).

16. Rezaei, J. and Davoodi, M. "A joint pricing, lot-sizing, and supplier selection model", International Journal of Production Research, 50(16), pp. 4524-4542 (2012).

17. Amorim, P., Belo-Filho, M., Toledo, F., Almeder, C., and Almada-Lobo, B. "Lot sizing versus batching in the production and distribution planning of perishable goods", International Journal of Production Economics, 146, pp. 208-218 (2013).

18. Govindan, K., Jafarian, A., Khodaverdi, R., and Devika, K. "Two-echelon multiple-vehicle locationrouting problem with time windows for optimization of sustainable supply chain network of perishable food", International Journal of Production Economics, 152, pp. 9-28 (2014).

19. Ma, Z.J., Wu, Y., and Dai, Y. "A combined order selection and time-dependent vehicle routing problem with time widows for perishable product delivery", Computers & Industrial engineering, 114, pp. 101- 113 (2017).

20. Azadeh, A., Elahi, S., Farahani, M.H., and Nasirian, B. "A genetic algorithm-Taguchi based approach to inventory routing problem of a single perishable product with transshipment", Computers & Industrial Engineering, 104, pp. 124-133 (2017).

21. Dolgui, A., Tiwari, M.K., Sinjana, Y., Kumar, S.K., and Son, Y.J. "Optimizing integrated inventory policy for perishable items in a multi-stage supply chain", International Journal of Production Research, 56(1- 2), pp. 902-925 (2018).

22. Gitinavard, H., Ghodsypour, S.H., and Shirazi, M.A. "A bi-objective multi-echelon supply chain model with Pareto optimal points evaluation for perishable products under uncertainty", Scientia Iranica, 26(5), pp. 2952-2970 (2019). DOI: 10.24200/SCI.2018.5047.1060.

23. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M., and Mirjalili, S. "Multi-objective stochastic closed-loop supply chain network design with social considerations", Applied Soft Computing, 71, pp. 505-525 (2018).

24. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M., and Mirjalili, S. "Hybrid optimizers to solve a tri-level programming model for a tire closed-loop supply chain network design problem", Applied Soft Computing, 70, pp. 701-722 (2018).

25. Musavi, M. and Bozorgi-Amiri, A. "A multi-objective sustainable hub location-scheduling problem for perishable food supply chain", Computers & Industrial Engineering, 113, pp. 766-778 (2017).

26. Fathollahi-Fard, A.M. and Hajiaghaei-Keshteli, M. "A bi-objective partial interdiction problem considering different defensive systems with capacity expansion of facilities under imminent attacks", Applied Soft Computing, 68, pp. 343-359 (2018).

27. Hajiaghaei-Keshteli, M. and Fathollahi-Fard, A.M. "A set of efficient heuristics and meta-heuristics to solve a two-stage stochastic bi-level decision-making model for the distribution network problem", Computers & Industrial Engineering, 123, pp. 378-395 (2018).

28. Samadi, A., Mehranfar, N., Fathollahi Fard, A.M., and Hajiaghaei-Keshteli, M. "Heuristic-based metaheuristic to address a sustainable supply chain network design problem", Journal of Industrial and Production Engineering, 35(2), pp. 102-117 (2018).

29. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M., and Tavakkoli-Moghaddam, R. "The Social Engineering Optimizer (SEO)", Engineering Applications of Artificial Intelligence, 72, pp. 267-293 (2018c).

30. Stenmarck, A., Jensen, C., Quested, T., and Moates, G. "Estimation of European food waste levels", FUSIONS: EU Project under European Community's Seventh Framework Programme Under Grant Agreement No 311972 (2016). (Retrieved at http://www.eu-fusions.org/phocadownload/Publications/ Estimates%20of%20European%20food%20waste %20levels.pdf on 22 Dec 2018).

31. Goyal, S.K. and Giri, B.C. "Recent trends in modeling of deteriorating inventory", European Journal of Operational Research, 134(1), pp. 1-16 (2001).

32. Ghare, P. and Schrader, G. "A model for exponentially decaying inventory", Journal of Industrial Engineering, 14(5), pp. 238-243 (1963).

33. Nahmias, S. "Perishable inventory theory: A review", Operations Research, 30(4), pp. 680-708 (1982).

34. Hsu, V.N. "Dynamic economic lot size model with perishable inventory", Management Science, 46(8), pp. 1159-1169 (2000).

35. Abad, P. "Optimal lot size for a perishable good under conditions of finite production and partial backordering and lost sale", Computers & Industrial Engineering, 38, pp. 457-465 (2000).

36. Abad, P.L. "Optimal pricing and lot-sizing under conditions of perishability, finite production and partial backordering and lost sale", European Journal of Operational Research, 144, pp. 677-685 (2003).

37. Teng, J.T., Cardenas-Barron, L.E., Chang, H.J., Wu, J., and Hu, Y. "Inventory lot-size policies for deteriorating items with expiration dates and advance payments", Applied Mathematical Modelling, 40(19- 20), pp. 8605-8616 (2016).

38. Hiassat, A., Diabat, A., and Rahwan, I. "A genetic algorithm approaches for location-inventory-routing problem with perishable products", Journal of Manufacturing Systems, 42, pp. 93-103 (2017).

39. Sahebjamnia, N., Fathollahi-Fard, A.M., and Hajiaghaei-Keshteli, M. "Sustainable tire closed-loop supply chain network design: Hybrid metaheuristic algorithms for large-scale networks", Journal of Cleaner Production, 196, pp. 273-296 (2018).

40. Karimi, B., Fatemi Ghomi, S., and Wilson, J. "The capacitated lot sizing problem: A review of models and algorithms", Omega, 31(5), pp. 365-378 (2003).

41. Yano, C.A. and Lee, H.L. "Lot sizing with random yields: A review", Operations Research, 43(2), pp. 311-334 (1995).

42. Ben-Daya, M., Darwish, M., and Ertogral, K. "The joint economic lot-sizing problem: Review and extensions", European Journal of Operational Research, 185(2), pp. 726-742 (2008).

43. Robinson, P., Narayanan, A., and Sahin, F. "Coordinated deterministic dynamic demand lot-sizing problem: A review of models and algorithms", Omega, 37(1), pp. 3-15 (2009).

44. Onal, M., Romeijn, H.E., Sapra, A., and Van den Heuvel, W. "The economic lot-sizing problem with perishable items and consumption order preference", European Journal of Operational Research, 244, pp. 881-891 (2015).

45. Lee, W. "A joint economic lot size model for raw material ordering, manufacturing setup, and finished goods delivering", Omega, 33, pp. 163-174 (2005).

46. Pal, A., Chan, F., Mahanty, B., and Tiwari, M. "Aggregate procurement, production, and shipment planning decision problem for a three-echelon supply chain using swarm-based heuristics", International Journal of Production Research, 49, pp. 2873-2905 (2011).

47. Onal, M. "The two-level economic lot-sizing problem with perishable items", Operations Research Letters, 44, pp. 403-408 (2016).

48. Benton, W.C. and Park, S. "A classification of literature on determining the lot size under quantity discounts", European Journal of Operational Research, 92(2), pp. 219-238 (1996).

49. Nahmias, S. and Cheng, Y., Production and Operations Analysis, New York: McGraw-hill, 6 (2009).

50. Ke, G.Y. and Bookbinder, J.H. "Coordinating the discount policies for retailer, wholesaler, and less-thantruckload carrier under price-sensitive demand: A trilevel optimization approach", International Journal of Production Economics, 196, pp. 82-100 (2018).

51. Marandi, F. and Zegordi, S.H. "Integrated production and distribution scheduling for perishable products", Scientia Iranica, 24(4), pp. 2105-2118 (2017).

52. Chung, K.J., Ting, P.S., and Cardenas-Barron, L.E. "A simple solution procedure for solving the multidelivery policy into economic production lot size problem with partial rework", Scientia Iranica, Transaction E, Industrial Engineering, 24(5), pp. 2640-2644 (2017).

53. Optimization, D. "Xpress-mp manual" (2002).

54. Gueret, C., Prins, C., and Sevaux, M., Applications of Optimization with Xpress-mp, Dash Optimization Ltd (1999).

55. Chen, P., Guo, Y., Lim, A., and Rodrigues, B. "Multiple cross-docks with inventory and time windows", Computers & Operations Research, 33(1), pp. 43-63 (2006).

Transactions on Industrial Engineering (E)

July and August 2020Pages 2021-2039