Maud Göthe-Lundgren

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.

Shipment planning at oil refineries using column generation and valid inequalities

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.

An optimization model for refinery production scheduling

Abstract In this paper we describe a production planning and scheduling problem in an oil refinery company. The problem concerns the planning and the utilization of a production process consisting of one distillation unit and two hydro-treatment units. In the process crude oil is transformed to bitumen and naphthenic special oils. The aim of the scheduling is to decide which mode of operation to use in each processing unit at each point in time, in order to satisfy the demand while minimizing the production cost and taking storage capacities into account. The production cost includes costs for changing mode and for holding inventory. We formulate a mixed integer linear programming model for the scheduling problem. The model can be regarded as a generalized lot-sizing problem, where inventory capacities are considered and more than one product is obtained for some modes of operation. A number of modifications and extensions of the model are also discussed. It is shown how the optimization model can be used as a viable tool for supporting production planning and scheduling at the refinery, and that it is possible to analyze scheduling scenarios of realistic sizes. It is also shown that the model can support shipment planning and strategic decisions concerning new products and investments in storage capacity.

On the nucleolus of the basic vehicle routing game

In the vehicle routing cost allocation problem the aim is to find a good cost allocation method, i.e., a method that according to specified criteria allocates the cost of an optimal route configuration among the customers. We formulate this problem as a co-operative game in characteristic function form and give conditions for when the core of the vehicle routing game is nonempty.One specific solution concept to the cost allocation problem is the nucleolus, which minimizes maximum discontent among the players in a co-operative game. The class of games we study is such that the values of the characteristic function are obtained from the solution of a set of mathematical programming problems. We do not require an explicit description of the characteristic function for all coalitions. Instead, by applying a constraint generation approach, we evaluate information about the function only when it is needed for the computation of the nucleolus.

The Heterogeneous Vehicle-Routing Game

In this paper, we study a cost-allocation problem that arises in a distribution-planning situation at the Logistics Department at Norsk Hydro Olje AB, Stockholm, Sweden. We consider the routes from one depot during one day. The total distribution cost for these routes is to be divided among the customers that are visited. This cost-allocation problem is formulated as a vehicle-routing game (VRG), allowing the use of vehicles with different capacities. Cost-allocation methods based on different concepts from cooperative game theory, such as the core and the nucleolus, are discussed. A procedure that can be used to investigate whether the core is empty or not is presented, as well as a procedure to compute the nucleolus. Computational results for the Norsk Hydro case are presented and discussed.

The traveling salesman game: An application ofcost allocation in a gas and oil company

In this article, a cost allocation problem that arises in a distribution planning situation atthe Logistics Department at Norsk Hydro Olje AB is studied. A specific tour is considered,for which the total distribution cost is to be divided among the customers that are visited.This problem is formulated as a traveling salesman game, and cost allocation methods basedon different concepts from cooperative game theory, such as the nucleolus, the Shapleyvalue and the t-value, are discussed. Additionally, a new concept is introduced: the demandnucleolus. Computational results for the Norsk Hydro case are presented and discussed.

A strong lower bound for the Node Weighted Steiner Tree Problem

In this paper, we study the Node Weighted Steiner Tree Problem (NSP). This problem is a generalization of the Steiner tree problem in the sense that vertex weights are considered. Given an undirected graph, the problem is to find a tree that spans a subset of the vertices and is such that the total edge cost minus the total vertex weight is minimized. We present a new formulation of NSP and derive a Lagrangean bound which used together with a heuristic procedure solves the NSP. Computational results are reported on a large set of test problems and optimality is proven for all the generated instances.

A set covering reformulation of the pure fixed charge transportation problem

Abstract The pure fixed charge transportation problem is reformulated into an equivalent set covering problem with, in general, a large number of constraints. Two constraint generation procedures for its solution are suggested; in both of these, new constraints are generated by the solution of ordinary maximum flow problems. In the first procedure, which is an optimizing Benders-type scheme, each set covering problem is solved by a simple heuristic. Upper and lower bounds to the optimal value of the transportation problem are provided throughout the procedure so that it can be terminated at e-optimality. The second procedure, which is a heuristic, is based on the restricted Lagrangean concept. In this case the generated constraints are Lagrangean relaxed with fixed multiplier values. These are chosen so that the optimal solution of an initial Lagrangean subproblem, being a minimum cost network flow problem, remains optimal. At termination, the heuristic provides a lower bound and a feasible solution to the transportation problem. Moreover, the computational cost of this procedure is very low; hence it is well suited for incorporating in a branch and bound scheme. A possible branching technique is given. Computational results are presented for both the Benders-type and the branch and bound schemes.

A stabilized column generation scheme for the traveling salesman subtour problem

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.