Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2742540, 11 pages

https://doi.org/10.1155/2017/2742540

## On Best Corrected Mixture Problems in Metallurgy: A Case Study

Département de Mathématiques, Université de Sherbrooke, 2500 Boulevard de l’Université, Sherbrooke, QC, Canada J1K 2R1

Correspondence should be addressed to F. Dubeau

Received 15 October 2016; Revised 28 January 2017; Accepted 30 January 2017; Published 27 February 2017

Academic Editor: Risto Lahdelma

Copyright © 2017 F. Dubeau and M. E. Ntigura Habingabwa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider three simple mixture problems occurring in metallurgy. The first problem considered is the classic minimum cost mixture problem. For the second problem, we consider finding a correction to a given mixture, a premix, without considering the cost of this premix. We only consider the cost and the weight of the quantity used as a correction. We show that the minimum cost correction does not correspond to the minimum weight correction, and we built the Pareto curve that gives all intermediate solutions between these two extreme solutions. Finally, the third problem is the correction problem for a nonfree premix. The correction is done to obtain a minimum cost corrected mixture.

#### 1. Introduction

Using materials, such as scrap metal for recycling, in order to form blends with desired specifications on some basic chemical elements is a well-known mixture problem met in foundries. People working in these fields bring to our attention the three different but related problems that we are going to consider in this paper. The first problem, called the* basic problem*, is to make a minimum cost mixture while checking the constraints on the specifications. It is a classic mixture problem solved by linear programming. For the second problem, called the* correction problem*, we start with a given mixture, called here a* premix*, which does not satisfy all the constraints on the specifications. Without considering the cost of this premix, we look for a* correction*, which is itself a mixture of available materials, that we will add to the premix in such a way that the resulting* corrected mixture*, which is the premix and the correction added together, satisfies all the constraints on the specifications. The correction can be done at minimum cost or at minimum weight. This problem is an example of a bicriteria linear programming problem, and we are going to identify its Pareto set which gives the link between minimum cost solution and minimum weight solution. Finally, the third problem is similar to the preceding correction problem, but now we take into account the cost of the given premix. We look for a minimum unit cost corrected mixture. This is an example of a linear-fractional programming problem. It is important to point out that the questions raised in this paper and the data used in the examples are inspired from a real situation.

Mixture problems are important applications in Operations Research [1]. One such application is the diet problem which received a lot of attention [2–7]. A variant of this problem is the pooling problem [8]. Also other approaches could be used to analyze similar mixture problems; for example, the fuzzy approach could be used to introduce uncertainties on the data [9].

#### 2. Basic Problem

##### 2.1. Problem Description

For the basic problem, there are materials that can be used to produce a mixture. Let be the weight of the th material in the final mixture (). The total weight of the mixture isThe total cost of the mixture of weight is thenwhere is the unit costs of the th material (for ).

The specifications of the mixture and of the materials are the amount (proportion % of the content) of known basic chemical elements (aluminium, carbon, chromium, copper, iron, etc.) contained in a unit weight of the material. Typically, there may be between and basic chemical elements that characterize a chemical mixture. Each th material is characterized by specifications noted by .

The specifications of the mixture are determined by a linear combination of the specifications of the elements of the mixture. If we denote by , the th specification of the mixture, we haveThe specifications of the mixture are subjected to two types of constraints.

*Type 1. *Each specification must be in a well-defined range; that is to say,for , where and are, respectively, the minimum and maximum value allowed for the th specification of the mixture.

*Type 2. *Specifications can be connected or related by linear relations of the formfor . For example, in Figure 1, we illustrate a situation where there are links between the specifications of two chemical elements denoted as and . The pair of specifications must be within the region bounded by the given straight lines.