Exact Algorithm

In this subsection, we present a literature review emphasizing those studies that implemented the exact algorithms to solve their proposed model. Several exact algorithms such as Branch-and Cut, Branch-and-Price and Lagrangian Relaxation algorithms have been implemented.

Archetti et al. (2007) were the first proposed branch-and-cut algorithms for a single product and single vehicle IRP. They have introduced a special formulation for maximum order policy. The instances solved are 30 customers and 6-period horizon and 50 customers with 3-period horizon. The formulation was improved by Solyali and Sural (2008, 2011) who introduced customer replenishment strategy by incorporating shortest path network and uses a heuristic approach to get initial bound to branch-and- cut algorithm. This new formulation enables the authors to solve 15 customers with 12 periods, 25 customers with 9 periods and 60 customers with 3 periods.

Recently, Coelho and Laporte (2013c) extended the formulation proposed by Archetti et al. (2007) by including the multiple vehicles known as Multi-vehicle IRP (MIRP). The

13

authors proposed a branch-and-cut algorithm for the exact solution of several classes of IRP. Several cases have been considered in their computational experiment namely MIRP solutions under the maximum level (ML) replenishment policy, MIRP solution with a homogeneous and heterogeneous fleet of vehicles, IRP with transshipment options and MIRP with additional consistency features. The computational experiments done on the benchmark instances and the computational results confirm the success of the proposed algorithm.

Coelho and Laporte (2013b) extended their work to propose branch-and-cut algorithm for solving multi product multi vehicle IRP (MMIRP) with deterministic demand and stockout cost is not allowed. In this paper, Coelho and Laporte (2013b) have implemented a solution of improvement algorithm after branch-and-cut identifies a new best solution. The purpose of solution improvement algorithm is to approximate the cost of a new solution resulting from the vertex removal and reinsertions. In this paper, the authors considered additional of two features namely the driver partial consistency and visiting space consistency. The driver partial consistency plays the role of increasing the quality of the solution provided by the IRP both to customers and suppliers in a multi- product environment. The results show that the visiting space helps in reducing the search space while providing meaningful solution. The computational experiments to test the efficiency of the algorithm for their proposed MMIRP model and MMIRP with the additional two consistency features are presented in this paper. The authors have proposed larger instances where the number of customers has increased to 50 and up to seven time periods.

In the most recent work, Desaulniers et al. (2015) works on a single supplier who produces a single product at each period over a finite horizon to fulfill the demand of a

14

set of customers by using a fleet of homogeneous capacitated vehicles. Each customer has their inventory capacity and initial inventory. The authors introduced an innovative formulation for the IRP and developed a state-of-the-art branch-price-and-cut algorithm for solving their proposed IRP model. The developed algorithm integrated known and new families of valid inequalities, appending an adaptation of the well-known capacity inequalities, as well as an ad hoc labeling algorithm in order to solve the column generation subproblems. The computational results showed that their algorithm outperforms existing exact algorithms for instances with more than three vehicles. The authors proved that the proposed valid inequalities, branching decisions, and other speed up strategies are effective.

Chien et al. (1989) is amongst the first to simulate a multiple period planning model where the model is based on a single period approach. This is achieved by passing some information from one period to the next through inter-period inventory flow. The authors have formulated their problem as a mixed integer program and developed a Lagrangian based procedure to generate both good upper bounds and heuristic solutions. Since then many researchers have focused their modeling on a finite planning horizon.

Yu et al. (2008) solved a large-scale IRP that delivers a single product with split delivery and vehicle fleet size constraint. The problem is solved by using a Lagrangian relaxation method and it combines with the surrogate subgradient method. The solution of the model obtained by the Lagrangian relaxation method is used to construct a near- optimal solution of the IRP by solving a series of assignment problems. Numerical experiments show that the proposed hybrid approach can find a high quality near- optimal solution for the IRP with up to 200 customers and 10 periods in a reasonable computation time.

15

Bard and Nananukul (2010) proposed a branch-and-price (B&P) algorithm for solving the production, inventory, distribution, routing problem (PIDRP), a variant of IRP. The model of this problem had included a single production facility, a set of customers with time varying demand, a finite planning horizon, and a fleet of homogeneous vehicles. The aim of this study is to construct a production plan and delivery schedule that minimizes the total cost while ensuring that each customer’s demand is met over the planning horizon. In this study, a new branching rule for dealing with an unstudied form of master problem degeneracy is introduced, while reducing the effects of symmetry and obtaining feasible solutions by combining a rounding heuristics and tabu search within B&P, and the use of column generation heuristics. The computational results indicated that the PIDRP instances with up to 50 customers and 8 time periods can be solved within 1 hour. The hybrid scheme performed better than CPLEX and standard branch and price alone.