Nirmala Achuthan

An Improved Branch-and-Cut Algorithm for the Capacitated Vehicle Routing Problem

The capacitated vehicle routing problem (CVRP) deals with the distribution of a single commodity from a centralized depot to a number of specified customer locations with known demands. The CVRP considered in this paper assumes common vehicle capacity, fixed or variable number of vehicles, and an objective to minimize the total distance traveled by all the vehicles. This paper develops several new cutting planes for this problem, and uses them in an exact branch-and-cut algorithm. Two of the new cutting planes are based on a specified structure of an optimal solution and its existence. Computational results are reported for 1,650 simulated Euclidean problems as well as 24 standard literature test problems; solved problems range in size from 15--100 customers. A comparative analysis demonstrates the significant computational benefit of the proposed method.

A new subtour elimination constraint for the vehicle routing problem

Abstract Vehicle Routing Problems (VRP) are concerned with the delivery of a single commodity from a centralized depot to a number of specified customer locations with known demands. In this paper we consider the VRP characterized by: fixed or variable number of vehicles, common vehicle capacity, distance restrictions, and minimization of total distance travelled by all vehicles as the objective. We develop an exact algorithm based on a new subtour elimination constraint. The algorithm is implemented using the CPLEX package for solving the relaxed subproblems. Computational results on 1590 simulated problems and 10 literature problems (without distance restrictions) are reported and a comparative analysis is carried out.

On mixed Ramsey numbers

For positive integers m and n the classical ramsey number r(m, n) is the least positive integer p such that if G is any graph of order p then either G contains a subgraph isomorphic to Km or the complement G of G contains a subgraph isomorphic to Kn. Some authors have considered the concept of mixed ramsey numbers. Given a graph theoretic parameter f, an integer m and a graph H, the mixed ramsey number v(f; m; H) is defined as the least positive integer p such that if G is any graph of order p, then either f(G) ⩾ m or G contains a subgraph isomorphic to H. In this paper we consider the problem of determining the mixed ramsey numbers for vertex linear arboricity and some other generalizations of chromatic number. We discuss the above problem for various structures H such as the complete graph, the claw, the path and the tree. Further, we study the generalized mixed ramsey number v(f;m1, m2,…, m1; Hl + 1, Hl + 2,…, Hk), where the edge set of the complete graph is partitioned into k sets.

On the Nordhaus-Gaddum Problem for the k-Defective Chromatic Number of a Graph

The Nordhaus-Gaddum [5] problem associated with the parameter k(G) is to nd sharp bounds for k(G) + k(G) and k(G): k(G) as G ranges over the class G(p) of all graphs of order p. Maddox (4) suggested the conjecture: For a graph G of order p, k(G) + k(G) d p 1 k+1 e + 2. Achuthan et al [1] disproved the above conjecture by constructing graphs G of order p such that, k(G) + k(G) = d p 1 k+1 e + 3: Achuthan et al (1) established a weak upper bound for k(G) + k(G), as G ranges over G(p). Furthermore, they proved Maddoxs conjecture for k = 1, over the class of P4 free graphs. In this paper, we obtain a sharp upper bound for the sum, 2(G) + 2(G) where G is a P4free graph of order p.

Single Item Multi-period Multi-retailer Inventory Replenishment Problem with Restricted Order Size

Abstract In this paper, we discuss the Multi-Period Multi-Retailer Inventory Replenishment Problem (MPMRIRP) denned by the following features: The planning horizon is T periods. There are n retailers denoted by J 1 ≤ J ≤ n and a central warehouse denoted by the index 0. Let pj,t, and Kj,t, denote respectively the purchasing price per unit and the fixed ordering cost of the item for warehouse/retailer j 0 ≤ j ≤ n, during period t 1 ≤ t ≤ T. Let Dj,t, be the demand of customer j, during period t. The warehouse replenishes its stock through bulk purchase and supplies to all the n retailers to satisfy their demands. The ordering cost is incurred whenever the order quantity is positive. Furthermore, a holding cost is incurred at the rate of hj,t, per unit of item held in inventory, at the end of period t, with the retailer j/warehouse, 0 ≤ j ≤ n. The order quantity is restricted to be less than or equal to Wj for retailer j/warehouse, 0 ≤ j ≤ n. We will assume that the initial and final physical inventory of the item at the stores (retail and warehouse) is zero. Stock out situations are not allowed at any of the retailers or warehouse. The problem is to determine the replenishment schedule for the single item for all the retailers and the warehouse so as to minimize the total cost incurred during the planning horizon of T periods.