1. 1 Optimization Techniques Constrained Optimization by Linear Programming updated 10.11.01 NTU SY-521-N SMU EMIS 5300/7300 Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow

2. 2 Nature of Linear Programming

3. 3 The Product Mix Problem Among the most common in linear programming The problem is to find out which products to include in the production plan and in what quantities these should be included (product mix) in order to maximize the profit. A solution tells more generally, in effect, how to allocate scarce resources.

4. 4 The Blending Problem Involve the determination of the best blend of available ingredients to form a certain quantity of a product under strict specifications required to meet a designated level of output or given specifications Especially important in the process industries such as petroleum, chemicals and food, and in fields where a certain level of service is desired at minimum cost In blending, the cost of the ingredients is to be minimized, while adhering to certain specifications and using certain ingredients (resources).

5. 5 The Blending Problem An attempt is made to use as few resources as possible to provide a given product or service level Note: A blending problem is also considered a problem of allocating resources in the best manner and in both cases we try to achieve the highest ratio of outputs to inputs; that is, to maximize productivity.

6. 6 Formulation of the Linear Programming Model Management science models, are composed of three components: The decision (controllable) variables The environment (uncontrollable) variables The result (dependent) variables The LP model is composed of the same components, but they assume different names

7. 7 Decision Variables The decision variables in LP depend on the type of LP problem being considered. They can be the quantities of the resources to be allocated, or the number of units to be produced. The decision maker is searching for the value of these unknown variables which will provide an optimal solution to the problem. The Decision Variables (controllable) The Constraints (uncontrollable variables and relationships) Objective Function (result variables)

8. 8 The Objective Function A linear programming model attempts to optimize a single goal, written as a linear function; for example maximize 5x 1 + 7x 2 + x 3. That is, it attempts to find either the maximum level of a desirable goal, such as total share of the market or total profit, or the minimum level of some undesirable outcome, such as total cost. Two Step Approach 1. Select a primary goal whose level is to be maximized or minimized. 2. Transform the other goals into constraints, which must only be satisfied.

9. 9 The Constraints The decision maker is searching for the values of the values of the decision variables that will maximize (or minimize) the value of the objective function. Such a process is usually subject to several uncontrollable restrictions, requirements, or regulations that are called constraints. These constraints are expressed as linear equations and/or inequalities.

10. 10 Formulation of a Linear Programming Model Every LP problem is composed of: Decision variables - The variables whose values are unknown and are searched for. Usually they are designated by x 1, x 2, … Objective function - This is a mathematical expression, given as a linear function, that shows the relationship between the decision variables and a single goal (or objective) under consideration. The objective function is a measure of goal attainment.

11. 11 Formulation of a Linear Programming Model Optimization - Linear programming attempts to either maximize or minimize the values of the objective function. Profit or cost coefficients - The coefficients of the variables in the objective function are called the profit (or cost) coefficients. They express the rate at which the value of the objective function increases by including in the solution one unit of each of the decision variables.

12. 12 Formulation of a Linear Programming Model Constraints - The maximization (or minimization) is performed subject to a set of constraints. Therefore, linear programming can be defined as a constrained optimization problem. These constraints are expressed in the form of linear inequalities (or, sometimes inequalities). They reflect the fact that resources are limited (e.g., in a product-mix problem) or the specific product requirements (e.g., in a blending problem).

13. 13 Formulation of a Linear Programming Model Input-output (technology) coefficients - The coefficients of the constraints’ variables are called the input-output coefficients. They indicate the rate at which a given resource is depleted or utilized. They appear on the left hand side of the constraints. Capacities - The capacities (or availability) of the various resources, usually expressed as some upper or lower limit, are given on the right hand side of the constraints. The right-hand side also expresses minimum requirements. Non-Negativity - It is required that only non- negative (zero or positive) values of the decision variables be considered.

14. 14 Formulation of a Linear Programming Model The general linear programming model can be presented in the following mathematical terms a ij = input-output coefficients c j = cost (profit) coefficients b i = capacities (right hand side) x j = decision variables Find a vector (x 1, …, x n ) which minimizes (or maximizes) a linear objective function F(x) where: F(x) = c 1 x 1 + c 2 x 2 + … + c j x j + … + c n x n

15. 15 Formulation of a Linear Programming Model subject to the linear constraints: a 11 x 1 + a 12 x2 + … + a 1n x n  b 1 a 21 x 1 + a 22 x2 + … + a 2n x n  b 1 … … … … a i1 x 1 + a i2 x2 + … + a in x n  b i … … … … a m1 x 1 + a m2 x2 + … + a mn x n  b m and the non-negativity constraints: x 1  0, x 2  0, …

16. 16 Advantages of Linear Programming Model Linear programming is a tool that can be used to solve allocation type problems. Such problems are very common and extremely important. Solution is difficult due to the fact that an infinite number of possible solutions may exist. Linear programming not only provides the optimal solution but does so in a very efficient manner. It provides additional information concerning the values of the resources that are allocated.

17. 17 Limitations of Linear Programming Due to Assumptions The applicability of linear programming is limited by several assumptions. As in all mathematical models, assumptions are made for reducing the complex real-world problem into a simplified form. The major ones are: Certainty - It is assumed that all data involved in the linear programming problem are known with certainty Linear objective function - It is assumed that the objective function is linear. This means that per unit cost, price or profit are assumed to be unaffected by changes in production methods or quantities produced or sold.

18. 18 Limitations of Linear Programming Due to Assumptions Linear constraints - The constraints are also assumed to be linear. This means that all the input-output coefficients are considered to be unaffected by a change of methods, quantities, or utilization level. Nonnegativity - Negative activity levels (or negative production) are not permissable. It is required, therefore, that all decision variables take nonnegative values.

19. 19 Limitations of Linear Programming Due to Assumptions Additivity - It is assumed that the total utilization of each resource is determined by adding together that portion of the resource required for the production of each of the various products or activities. The assumption of additivity also means that the effectiveness of the joint performance of activities, under any circumstances, equals the sum of the effectiveness resulting from the individual performance of these activities.

20. 20 Limitations of Linear Programming Due to Assumptions Divisibility - Variables can, in general, be classified as continuous or discrete. Continuous variables are subject to measurement (e.g., weight), temperature Independence - Complete independence of coefficients is assumed, both among activities and among resources. For example, the price of one produce has no effect on the price of another.

21. 21 Limitations of Linear Programming Due to Assumptions Proportionality - The requirement that the objective function and constraints must be linear is a proportionality requirement. This means that the amount of resources used, and the resulting value of the objective function, will be proportional to the value of the decision variables.

22. 22 Solving Linear Programs All solutions to a linear programming problem which satisfy all the constraints are called feasible The collection of feasible solutions is called the feasible solution space or area Any solution which violates one or more of the constraints is termed infeasible

23. 23 Solving Linear Programs The major task in applying linear programming is the formulation of the problem. Once a problem has been formulated, one of several available methods of solution can be applied The graphical method, whose main purpose is to illustrate the concepts involved in the solution process The general, computationally powerful simplex method Other approaches

24. 24 The Graphical Method of Solution The graphical method is used mainly to illustrate certain characteristics of LP problems and to help in explaining the simplex method. The only case where it has a practical value is in the solution of small problems with two decision variables and only a few constraints, or, problems with two constraints or only a few decision variables.

25. 25 Example The two models of color sets produced by the Sekido Corporation, will be designated as A and B. The company is in the market to make money; that is, its objective is profit maximization. The profit realized is $300 from set A and $250 from set B. Obviously, the more A sets produced and sold, the better. The trouble is that there are certain limitations which prevent Sekido Company from producing and selling thousands of sets. These Limitations are: 1.Availability of only 40 hours of labor each day in the production department (a labor constraint). 2.A daily availability of only 45 hours of machine time (a machining constraint). 3.Inability to sell more than 12 sets of model A each day (a marketing constraint). (the constraints are on the next slide)

26. 26 Example First let A = x 1 and B = x 2. So we need to maximize z = 300x 1 + 250x 2 with these constraints 2x 1 + 1x 2 40 (labor constraint) 1x 1 + 3x 2 45 ( machining constraint) 1x 1 + 0x 2 12 (marketing constraint) x 1 0nonnegativity x 2  0nonnegativity

27. 27 Example - solution Graphing the feasible area - The graphical method starts with the graphing of a feasible are within which a search for the optimal solution is to be conducted. The feasible area is established through graphing all of the inequalities and equations that describe the constraints. 1st quadrant feasible x 1  0, x 2  0 4th quadrant infeasible x 1  0, x 2  0 2nd quadrant infeasible x 1  0, x 2  0 3rd quadrant infeasible x 1  0, x 2  0

28. 28 Example - solution Graphing the first constraint - This is expressed as 2x 1 + x 2  40. The steps in drawing the constraint are: Step 1: Since an inequality of the type less than or equal to has in effect two parts, we will first consider only the equality part of the constraint. In our example, it will be 2x 1 + x 2 = 40. Since an equation can be shown graphically as a straight line, it is sufficient to find the coordinates of two points to graph the entire equation as a line. To do so, x 1 is first set to zero. This will yield a point where the equation intersects the x 2 axis.

29. 29 Example - solution if we repeat this for the rest of the constraints we will get a graph that looks like this:

30. 30 Example - solution We limit our selves to the largest region that satisfies all our constraints. This region has 4 vertices, which we will use in our main equation to try and maximize it. 15 12 0 a b c d maximum

31. 31 Example - solution So our maximum value of $6,350 is when x 1 = 12 and x 2 =11.