Solution of Mixture Problem Prioritized Raw Materials Using Mixed Integer Linear Programming Mehmet Akif Şahman*, Adem Alpaslan Altun + , Abdullah Oktay Dündar # , Ali Yaşar & * Guneysinir Vocational School, University of Selcuk, 42490 Konya, Turkey, asahman@selcuk.edu.tr + Faculty of Technology, University of Selcuk, 42075 Konya, Turkey, altun@selcuk.edu.tr # Akoren Vocational School, University of Selcuk, 42460 Konya, Turkey, aodundar@selcuk.edu.tr & Guneysinir Vocational School, University of Selcuk, 42490 Konya, Turkey, aliyasar@selcuk.edu.tr Abstract— Mixture problem is the problem of obtaining one or more products from by-products having different contents under the required constraints. Mixture problems should be organized in a way to optimize a given objective function. This is generally a cost optimization, fulfilling specific constraints and conditions and this kind of problems mostly solved with linear programming models. Decision makers want to have the solutions for mixture problems under the different scenarios. One of these scenarios is to use the raw materials which have different priorities in the solution. In this study, a new mixed integer linear programming (MILP) model which finds a solution according to high priorities is developed. The priorities are determined according to raw materials availabilities between 1 and 4. The developed model is practiced on a feed mix problem as a mixture problem and the model is coded by object-oriented programming language C#. Index Terms— Mixing Problem, Linear Programming, Feed Mixing, Optimisation Application, Cost Optimisation I. INTRODUCTION Feed mixture problem is mixing raw materials which have different features of feed formulation in accordance with the nutrient requirements, animal characteristics. The mixture is made according to the type of animal, age and breeding purposes and it has two important features including quality and cost for producer/decision maker. Usually linear solution techniques are used to solve problems of feed mixture. First study about mathematical analysis of feed mixture was conducted by Frederic V. Waugh in 1951 using Linear Programming (LP) [1]. Later, depending on the type and feature of animals, a number of researchers have used the LP in order to optimize the cost of the feed mixture [2-6]. Also different methods such as Goal Programming (GP) [7, 8], Multi Goal Programming (MGP) [9, 10], Multi Objective Programming (MOP) [11-13] are used. When the previous studies were examined, it was seen that Mixed Integer Linear Programming (MILP) method had never been used in the solution of feed mix problems. MILP problems are the problems in which some of the variables are restricted to be integers. While integers may have values in a specific interval, they may also have such values as 0-1 (true/false). Calculation of integer problems is difficult and complicated due to their combinational structure. As for MILP problems, they include variables having both integer values and continuous values. Thus, the calculation of MILP problems is more complicated than that of integer problems. Problems of this type belong to the Non-deterministic Polynomial-time (NP) class[14]. When the studies on feed formulation were examined, it has not been seen that selection of raw materials were made or modeled by giving priority. While feed mixtures were being prepared, ability of decision maker to see the feed formulations with a broad perspective and produce solutions in different situations have strategic importance for the companies. According to the state of raw materials in feed mills, it is sometimes requested to be included in the mixture as a higher priority than other raw materials. The reasons can be defined as; • Abundance of available raw materials, • Logistical strain of needed raw materials • Supply rate of usable needed materials, •The importance of contribution of material for the desired formulation. In this study, decision maker identifies a priority between 1 and 4 to available raw materials. 1 shows that, the raw material has a high priority and 4 shows the raw material has low priority. These priorities can be changed at any time by decision makers. Raw materials which have high priority in feed formulation are forced into the mixture primarily. Feed formulation is done under the required constraints with priorities of decision maker. International Journal of Advanced Research in Engineering Vol 1(3) Oct-Dec 2015 26 IJARE