A Hybrid Fix-and-Optimize Heuristic for Integrated Inventory-Transportation Problem in a Multi-region Multi-facility Supply Chain
In this work, we study an integrated inventory-transportation problem in a supply chain consisting of region-bound warehouses located in different regions. The supply chain deals with multiple items that compete for storage space and transportation capacity with multi-modal transportation considering regional capacity constraint for each mode of transportation. The objective is to determine an optimal storage and transportation plan to satisfy the demand of all regions without shortages for
... t shortages for known procurement plan for all items. The problem is formulated as a mixed integer programming (MIP) model for minimizing the total costs over a finite planning horizon. An MIP-based fix-and-optimize (F&O) heuristic with several decomposition schemes is proposed to solve the problem efficiently. The performance of the decomposition schemes is investigated against the structure of the sub-problems obtained. To enhance the performance, F&O is crossbred with two metaheuristics -genetic algorithm (GA) and iterated local search (ILS) separately, which lead to hybrid heuristic approach. Extensive numerical experiments are carried out to analyze the performance of the proposed solution methodology by randomly generating several problem instances built using data collected from the Indian Public Distribution System. The proposed solution approach is found to be computationally efficient and effective, and outperforming state of the art MIP solver Cplex for practical size problem instances. Also, the hybridization of F&O heuristic with GA and ILS boosts its performance although with a justified increase in the computational time. Mathematics Subject Classification. 90B05, 90B06, 90B90, 90C11. aspects are practiced independently which may result in suboptimal performance of a supply chain. Therefore, the need for close coordination between inventory control and transportation planning is highlighted in the literature  . Along with the traditional supply chain, the challenge of competitive performance through efficient management of inventory and transportation is equally applicable to a non-profit supply chain such as public distribution system (PDS). PDS is practiced in most of the countries for distribution of essential commodities, particularly foodgrains, at subsidized prices to the needy people of society to ensure the food security. In the Indian PDS, foodgrains, mainly wheat and rice, are procured from farmers and distributed to the beneficiaries at subsidized prices to ensure availability and affordability of foodgrains. Serving to more than 813.5 million beneficiaries, the Indian PDS is the largest in the world. The supply chain of the Indian PDS has some special characteristics. It consists of multiple storage facilities (warehouses) located across the country to store the foodgrains to meet the demand of different regions. Each region has a procurement and demand strategy that is aggregated over all the warehouses located in that region. Consequently, it is necessary to decide which warehouse to store the procured items and when to withdraw the items to meet the demand of each region. Because of the limited available storage capacity, the major challenge includes the storage of a large amount of foodgrains in different warehouses. Also, due to the mismatch in procurement, demand, and the storage capacity across different regions, the other challenge is the transportation of foodgrains from regions with a surplus to regions with a deficit over a notably large geographic area by selecting an optimal mode of transportation. At present, these aspects are practiced in a decoupled fashion, which results in a higher cost. Therefore, in the present study, we propose an integrated inventory-transportation problem to jointly optimize the storage and transportation related costs over a finite planning horizon. This problem of storage and inter-state transportation of foodgrains is first studied by Tanksale and Jha  for the Indian public distribution system. Later, Tanksale and Jha  envisioned a multi-region multi-facility supply chain structure and analyzed an integrated inventory-transportation problem over a finite planning horizon. However, they have not considered the detailed characteristics of transportation of items. Therefore, in this research, we aim to generalize and extend the problem presented in  using multiple modes of transportation under regional capacity constraint for each mode of transportation. A capacitated multi-item inventory-transportation problem in a multi-period setting is formulated as mixed-integer programming (MIP) model to determine an optimal storage and transportation plan for all items by minimizing the total transportation and inventory related costs. Further, to deal with the increased complexity of the problem in the present study, we implement MIP-based fix-and-optimize (F&O) heuristic originally proposed by Helber and Sahling  . The idea behind F&O heuristic is to decompose the original problem into sub-problems such that each sub-problem has a fewer number of free binary variables to be optimized while keeping other binary variables fixed, and iteratively solving the series of subproblems that will provide the solution to the original problem. This approach has been found efficient and effective in case of many complex production planning, lot-sizing, and scheduling problems. In this work, we demonstrate the application of F&O heuristic to the problem under consideration. We propose several decomposition schemes obtained from the structure of the problem and an efficient greedy heuristic to construct an initial feasible solution. We demonstrated that the performance of a decomposition scheme depends on the subproblem size as well as the number of sub-problems formed. We further enhance F&O heuristic by hybridizing it with two metaheuristics -genetic algorithm (GA) and iterated local search (ILS) separately, as diversification tools. The contributions of the present study to the body of research are multi-fold, as illustrated below. -We have presented a capacitated inventory-transportation problem in a multi-region multi-facility supply chain over a finite planning horizon, which is stemmed from the Indian PDS by fully characterizing the transportation of items in the form of multi-modal transportation and region-wise transportation capacity constraints. The problem can be seen as capacitated multi-period inventory transportation problem. In this way, we extend and generalize the previous works of Tanksale and Jha [56, 57] .