Rafael Epstein

Harvest Scheduling Subject to Maximum Area Restrictions: Exploring Exact Approaches

We consider a spatial problem arising in forest harvesting. For regulatory reasons, blocks harvested should not exceed a certain total area, typically 49 hectares. Traditionally, this problem, called the adjacency problem, has been approached by forming a priori blocks from basic cells of 5 to 25 hectares and solving the resulting mixed-integer program. Superior solutions can be obtained by including the construction of blocks in the decision process. The resulting problem is far more complex combinatorially. We present an exact algorithmic approach that has yielded good results in computational tests. This solution approach is based on determining a strong formulation of the linear programming problem through a clique representation of a projected problem.

Use of Or Systems in the Chilean Forest Industries

The Chilean forestry sector is composed of private firms that combine large timber-land holdings of mostly pine plantations and so me eucalyptus with sawmills and pulp plants. Since1988, to compete in the world market, the main Chilean forest firms, which have sales of about

A Combinational Auction Improves School Meals in Chile

1 billion, have started implementing OR models developed jointly with academics from the University of Chile. These systems support decisions ondaily truck scheduling, short-term harvesting, location of harvesting machinery and access roads, and medium- and long-term forest planning. Approaches used in solving these complex problems include simulation, linear programming withcolumn generation, mixed-integer LP formulations, and heuristic methods. The systems have led to a change in organizational decision making and to estimated gains of at least US

A Combinatorial Heuristic Approach for Solving Real-Size Machinery Location and Road Design Problems in Forestry Planning

20 million per year.

Optimizing Long-Term Production Plans in Underground and Open-Pit Copper Mines

Chiles school system is using mathematical modeling to assign catering contracts in a singleround sealed-bid combinational auction. The Chilean state spends around US

A system for the design of short term harvesting strategy

180 million a year to feed 1,300,000 students from low income families, making this one of the largest state auctions. To improve the quality of the assignment in the auction process, we constructed an integer linear programming model to decide contract awards optimally among different concession holders. The model completely changed the nature of the process in three crucial aspects. First, it gave transparency and objectivity to the complete process, generating competition among firms. Second, it allowed the companies to build flexible territorial bids to include their scale economies, leading to efficient resource allocation. Finally, the model indeed found an optimal solution, which is not easy because the assignment problem was NP-complete with more than 10,000 binary variables. This new methodology improved the price-quality ratio of the meals with yearly savings of around US

Solving multiple scenarios in a combinatorial auction

40 million--equivalent to the cost of feeding 300,000 children during one year.

A Strategic Empty Container Logistics Optimization in a Major Shipping Company

The location and operation of harvest machinery, along with the design and construction of access roads, are important problems faced by forestry planners, making up about 55% of total production costs. One of the main challenges consists of finding a design that will minimize the cost of installation and operation of harvest machinery, road construction, and timber transport, while complying with the technical restrictions that apply to the operation of harvesting equipment and road construction. We can model the network design problem as a mixed-integer linear programming problem. This model is fed with cartographic information, provided by a geographic information system (GIS), along with technical and economic parameters determined by the planner. We developed a specialized heuristic for the problem to obtain solutions that enable harvesting economically profitable volumes at a low cost. This methodology was programmed into a computer system known as PLANEX and is being applied in nine forestry companies that report important benefits from its use.

The Supply Chain in the Forest Industry: Models and Linkages

We present a methodology for long-term mine planning based on a general capacitated multicommodity network flow formulation. It considers underground and open-pit ore deposits sharing multiple downstream processing plants over a long horizon. The purpose of the model is to optimize several mines in an integrated fashion, but real size instances are hard to solve due to the combinatorial nature of the problem. We tackle this by solving the relaxation of a tight linear formulation, and we round the resulting near-integer solution with a customized procedure. The model has been implemented at Codelco, the largest copper producer in the world. Since 2001, the system has been used on a regular basis and has increased the net present value of the production plan for a single mine by 5%. Moreover, integrating multiple mines provided an additional increase of 3%. The system has allowed planners to evaluate more scenarios. In particular, the model was used to study the option of delaying by four years the conversion of Chiquicamata, Codelcos largest open-pit mine, to underground operations.

The impact of environmental constraints on short term harvesting: Use of planning tools and mathematical models

Short term harvesting requires decisions on which stands to harvest, what timber volume to cut, what bucking patterns (how to cut up the logs) to apply to logs in order to obtain products that satisfy demand and which harvesting machinery to use. This is an important problem in forest management and difficult to solve well in satisfying demand, while maximizing net profits. Traditionally, foresters have used manual approaches to find adequate solutions, which has shortcomings both in time spent by analysts and the quality of solutions. Since demand for timber products is defined by length, diameter and quality of each piece, this leads to a complex combinatorial problem in matching supply (standing trees) and demand. We developed one of the few reported approaches for solving the short term harvesting problem based on a computerized system, using a linear programming approach. Determining adequate bucking patterns is not trivial. We develop a column generation approach to generate such patterns. The subproblem is a specially designed branch and bound scheme. The generation of bucking patterns implemented within the LP formulation led to a significant improvement of solutions. We believe this is the first system implemented with this level of detail. This system has been advantageously implemented in several forest companies. The results obtained show improvements obtained by the firms of 5-8% in net revenues over traditional manual approaches