Kurt Jörnsten

Cost Allocation in Collaborative Forest Transportation

Transportation planning is an important part of the wood flow chain in forestry. There are often several forest companies operating in the same region and co-ordination between two or more companies is rare. However, there is an increasing interest in collaborative planning as the potential savings are large, often in the range 5-15%. A key question is how savings should be distributed among the participants. In this paper we investigate a number of possibilities based on economic models including Shapley value, the nucleolus, separable and non-separable costs, shadow prices and volume weights. We also propose a new allocation method based on finding as equal relative profits as possible among the participants. A case study including eight forest companies is described and analyzed.

Innovative Applications of O.R.Cost allocation in collaborative forest transportation

Transportation planning is an important part of the supply chain or wood flow chain in forestry. There are often several forest companies operating in the same region and collaboration between two or more companies is rare. However, there is an increasing interest in collaborative planning as the potential savings are large, often in the range 5–15%. There are several issues to agree on before such collaborative planning can be used in practice. A key question is how the total cost or savings should be distributed among the participants. In this paper, we study a large application in southern Sweden with eight forest companies involved in a collaboration. We investigate a number of sharing mechanisms based on economic models including Shapley value, the nucleolus, separable and non-separable costs, shadow prices and volume weights. We also propose a new allocation method, with the aim that the participants relative profits are as equal as possible. We use two planning models, the first is based on direct flows between supply and demand points and the second includes backhauling. We also study how several time periods and geographical distribution of the supply and demand nodes affect the solutions. Better planning within each company can save about 5% and collaboration can increase this about another 9% to a total of 14%. The proposed allocation method is shown to be a practical approach to share the overall cost/savings.

Enhancing an algorithm for set covering problems

Abstract In this note we present enhancements to a previously published algorithm for the optimal solution of set covering problems. These enhancements relate to the use of a Langrangean heuristic, feasible solution exclusion constraints, Gomory f-cuts and an improved branching strategy. Computational results, for problems involving up to 400 rows and 4000 columns, indicate that the enhanced algorithm gives better computational results than the original algorithm.

Weighted k-cardinality trees: Complexity and polyhedral structure

We consider the k-CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with k edges, having minimum weight. Applications of this problem arise in oil-field leasing and facility layout. Although the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k-CARD TREE and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex hull of the integer solutions is studied.

A bilinear approach to the pooling problem

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples

A new Lagrangian relaxation approach to the generalized assignment problem

Abstract In this paper we present a new Lagrangian relaxation approach to the generalized assignment problem (GAP). The new approach is based on a reformulation of GAP into an equivalent problem, which is then relaxed by traditional Lagrangian relaxation techniques. The reformulation is created by introducing a set of auxiliary variables and a number of coupling constraints. By relaxing the coupling constraints we get subproblems where both types of constraint structures present in the GAP are active. The lower bound from the Lagrangian dual of this approach is shown to be at least as strong as that from the best of the traditional Lagrangian relaxation approaches. An important feature is also the possibility of generating upper bounds quite easily. The new approach has been tested on 10 problems of size m = 4, n = 25, i.e. 100 0 1 -variables, and the results have been compared with the traditional approaches.

Tabu Search for General Zero-One Integer Programs Using the Pivot and Complement Heuristic

Tabu search techniques have been applied to a wide class of optimization problems. Tabu search is a meta procedure that requires a subroutine to find local optima. In most examples of tabu search the subroutine used is problem specific. However, in order to apply tabu search successfully to general 0-1 integer programs, the subroutine must be general. The pivot and complement heuristic is a procedure that frequently finds feasible solutions for general 0-1 integer programs. In this paper we implement several elements of strategies proposed by tabu search method that result in general heuristics for 0-1 integer programs that could be included in an integer programming code. These heuristics employ the ZOOM/XMP implementation of the pivot and complement heuristic as a subroutine. Computational results indicate that these heuristics provide good bounds for many problems. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Computational results from a new Lagrangean relaxation algorithm for the capacitated plant location problem

Abstract In this paper we present an algorithm for the capacitated plant location problem based on a reformulation obtained by the introduction of auxiliary variables. By relaxation of the coupling constraints two separate subproblems are obtained. This is the basis for the so called ‘variable splitting’ approach. We present a complete algorithmic procedure including solution procedures for subproblems, heuristics for the generation of feasible solutions and suggestion for branching rules. We also present numerical results for capacitated plant location problems of the size x × n where m ∈ [10,20], n ∈ [20,50].

Equilibrium prices supported by dual price functions in markets with non-convexities

The issue of finding market clearing prices in markets with non-convexities has had a renewed interest due to the deregulation of the electricity sector. In the day-ahead electricity market, equilibrium prices are calculated based on bids from generators and consumers. In most of the existing markets, several generation technologies are present, some of which have considerable non-convexities, such as capacity limitations and large start-up costs. In this paper we present equilibrium prices composed of a commodity price and an uplift charge. The prices are based on the generation of a separating valid inequality that supports the optimal resource allocation. In the case when the sub-problem generated as the integer variables are held fixed to their optimal values possess the integrality property, the generated prices are also supported by non-linear price functions that are the basis for integer programming duality.

On bicriterion minimal spanning trees: an approximation

Abstract In this paper we focus on the problem of computing the set of efficient spanning trees for a given network where each arc carries two attributes. This problem is N P -complete. We discuss two heuristics, namely, neighbourhood search (which is a well-known method) and adjacent search, a new method. They both approximate the set of efficient spanning trees. The difference lies in which kind of spanning trees are generated in each iteration. For neighbourhood search, all spanning trees which are adjacent to at least one spanning tree in the current approximation set are considered. Adjacent search is similar to neighbourhood search except that only spanning trees which are adjacent to at least two spanning trees in the current approximation set are considered. Based on computational results it is concluded that adjacent search is a reasonable alternative to neighbourhood search, especially for large problems.