In this paper, we focus on how cost allocation can be used as a means to create incentives for collaboration among companies, with the aim of reducing the total transportation cost. The collaboration is assumed to be preceded by a simultaneous invitation of the companies to collaborate. We make use of concepts from cooperative game theory, including the Shapley value, the Nucleolus and the EPM, and develop specific cost allocation mechanisms aiming to achieve large collaborations among many companies. The cost allocation mechanisms are tested on a case study that involves transportation planning activities. Although the case study is from a specific transportation sector, the findings in this paper can be adapted to collaborations in other types of transportation planning activities. Two of the cost allocation mechanisms ensure that any sequence of companies joining the collaboration represents a complete monotonic path, that is, any sequence of collaborating companies is such that the sequences of allocated costs are non-increasing for all companies.

When a set of players cooperate, they need to decide how the collective cost should be allocated amongst them. Cooperative game theory provides several methods or solution concepts, that can be used as a tool for cost allocation. In this note, we consider a specific solution concept called the Equal Profit Method (EPM). In some cases, a solution to the EPM is any one of infinitely many solutions. That is, it is not always unique. This leads to a lack of clarity in the characterization of the solutions obtained by the EPM. We present a modified version of the EPM, which unlike its precursor ensures a unique solution. In order to illustrate the differences, we present some numerical examples and comparisons between the two concepts.

We present a sequential linear programming, SLP, algorithm in which the traditional line-search step is replaced by a multi-dimensional search. The algorithm is based on inner approximations of both the primal and dual spaces, which yields a method which in the primal space combines column and constraint generation. The algorithm does not use a merit function, and the linear programming subproblem of the algorithm differs from the one obtained in traditional methods of this type, in the respect that linearized constraints are taken into account only implicitly in a Lagrangiandual fashion. Convergence to a point that satisfies the Karush-Kuhn-Tucker conditions is established. We apply the new method to a selection of the Hoch-Schittkowski’s nonlinear test problems and report a preliminary computational study in a Matlab environment. Since the proposed algorithmcombines column and constraint generation, it should be advantageous with large numbers of variables and constraints.

We study a class of graph problems which includes as special cases the Held and Karp 1-tree problem, and the minimum spanning tree problem with one degree constraint studied by Glover and Klingman. For another special case, we note a mistake in a paper by Ramesh, Yoon and Karwan.

Given an undirected graph with edge costs and both revenues and weights on the vertices, the traveling salesman subtour problem is to find a subtour that includes a depot vertex, satisfies a knapsack constraint on the vertex weights, and that minimizes edge costs minus vertex revenues along the subtour.

We propose a decomposition scheme for this problem. It is inspired by the classic side-constrained 1-tree formulation of the traveling salesman problem, and uses stabilized column generation for the solution of the linear programming relaxation. Further, this decomposition procedure is combined with the addition of variable upper bound (VUB) constraints, which improves the linear programming bound. Furthermore, we present a heuristic procedure for finding feasible subtours from solutions to the column generation problems. An extensive experimental analysis of the behavior of the computational scheme is presented.

We present an optimizing column generation procedure for solving the vehicle routing problem with a fixed heterogeneous fleet of vehicles; if the method is used in a truncated fashion it turns into a heuristic. The method is based on a new mathematical formulation, which includes a new type of valid inequalities, strengthened by the use of Chvátal-Gomory rounding, and a Lagrangian dualization of this formulation. The dual problem is attacked by subgradient optimization and a near-optimal dual solution obtained is used for enumerating routes that are promising candidates for being used in an optimal solution. These routes are collected in a set partitioning problem, which is finally solved, and an upper bound to the optimal objective value is obtained. The method is evaluated on a set of small-scale test instances. The valid inequalities improves the lower bound significantly: the improvement depends on the ratio between total customer demand and total vehicle capacity. The qualities of the upper bounds varies quite much among the instances.

In this paper we suggest an optimization model and a solution method for a shipment planning problem. This problem concerns the simultaneous planning of how to route a fleet of ships and the planning of which products to transport in these ships. The ships are used for moving products from oil refineries to storage depots. There are inventory levels to consider both at the refineries and at the depots. The inventory levels are affected by the process scheduling at the refineries and demand at the depots. The problem is formulated using an optimization model including an aggregated representation of the process scheduling at the refineries. Hence, we integrate the shipment planning and the process scheduling at the refineries. We suggest a solution method based on column generation, valid inequalities, and constraint branching. The solution method is tested on data provided by the Nynas oil refinery company and solutions are obtained within 4 hours, for problem instances of up to 3 refineries, 15 depots, and 4 products when considering a time horizon of 42 days. © 2004 Elsevier B.V. All rights reserved.

Many classes of combinatorial and mixed integer optimization problems are attacked with decomposition methods. One technique is to perform subgradient optimization on a Lagrangean dual problem: another is to perform column generation on a Dantzig-Wolfe problem, or equivalently, cut generation on its linear programming dual. These techniques both have advantages and disadvantages. In this paper these techniques are combined into a two-phase method, with the purpose of overcoming drawbacks of the techniques.

The two-phase method consists of a prediction phase and a solution phase. In the prediction phase, subgradient optimization is performed: subgradients found are stored and used as starting columns for the solution phase. (Optionally, non-binding restricitions can be predicted based on information from the prediction phase.) The columns found are used to construct a Dantzig/Wolfe master problem. In the solution phase, column generation is performed if needed. A solid theoretical bases for the two-phase method is presented which includes strong asymptotical results. ln practise, truncated usage must be performed: practical guidelines are given in the paper.

The two-phase method is tested on two applications: a multicommodity network flow problem and a convexified version of the traveling salesman subtour problem. Two categories of numerical experiments are presented. ln the first category, various aspects of truncated usage of the theory are illustrated. In the second category, the two-phase method is tested on a relatively large number of test problems.

An overall conclusion of our work is that the two-phase method can perform significantly better, in terms of CPU-times, compared to a (stabilized) Dantzig-Wolfe algorithm. ln general, whenever the subproblems are computationaly inexpensive, compared to the Dantzig-Wolfe master programs, the two-phase method might be an interesting alternative to use instead of pure Dantzig-Wolfe.

In this paper we present a tabu search heuristic which can be used for scheduling the production at an oil refinery. The scheduling problem is to decide which production modes to use at the different processing units at each point in time. The problem is a type of lot-sizing problem where costs of changeovers, inventories and production are considered. In the suggested tabu search heuristic we explore the use of variable neighbourhood, dynamic penalty and different tabu lists. Computational results are presented for different versions of the heuristic and the results are compared to the best-known lower bound for a set of scheduling scenarios.