Mathematical modeling

Assume: Z= value of overall measure of performancex j = level of activity j (j=1, 2, ….. , n)c j = increase in Z that result from each unit increase in activity jb i = amount of resource i that is available to activity j (i=1, 2,…, m)a ij = amount of resource i consumed by each unit of activity j.The general form of allocating resources to activities in order to form the generalmathematical model of linear programming is as follows:Resource12.mResources usage per unit of activity1 2 …… nAmount of resource availablea 11 a 12 …… a 1nb 1a 21 a 22 …… a 2nb 2……. .... …… …a m1 a m2 ..….. a mn b mContribution to Z c 1 c 2 ……. c nSo, the general model is to select the values of x 1 , x 2 , ….., x n so as toMaximizeZ = c 1 x 1 + c 2 x 2 + …… + c n x nSubject to a 11 x 1 + a 12 x 2 + …… + a 1n x n ≤ b 1a 21 x 1 + a 22 x 2 + …… + a 2n x n ≤ b 2.a m1 x 1 + a m2 x 2 + …… + a mn x n ≤ b mx 1 ≥ 0, x 2 ≥ 0, …..andx n ≥ 0 non-negativity constraintGenerally, the function being maximized is called the objective function. The first mrestrictions are called the constraints. The x j ≥ 0 restrictions are non-negativity constraints.2.3.3 Solving the Model GraphicallyModels are useful as they allow for a formal definition of a problem and can be useful inthinking about a problem. For a model with only two variables, it is possible to solve theproblem by drawing the feasible region and determining how the objective is optimizedon that region. That process is illustrated here. The purpose of this exercise is to give youintuition and understanding of linear programming models and their solution. In any realapplication, you would use a computer to solve even two variable problems (Excel'sSolver is outlined in the next section to find solutions for linear programming problems).Systems Analysis 17 Dr. Emad Elbeltagi