2. Introduction to Linear Programming and Formulation Meeting 2 Course: D0744 - Deterministic Optimization Year: 2009

3. Bina Nusantara University 3 Introduction A model consisting of linear relationships representing a firm’s objective and resource constraints LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints

4. Bina Nusantara University 4 Linear Programming Linear Programming is a mathematical technique for optimum allocation of limited or scarce resources, such as labor, material, machine, money, energy and so on, to several competing activities such as products, services, jobs and so on, on the basis of a given criteria of optimality.

5. Bina Nusantara University 5 Linear Programming (LP) Problem A mathematical programming problem is one that seeks to maximize or minimize an objective function subject to constraints. If both the objective function and the constraints are linear, the problem is referred to as a linear programming problem. Linear functions are functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0). Linear constraints are linear functions that are restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant.

6. Bina Nusantara University 6 Building Linear Programming Models 1.What are you trying to decide - Identify the decision variable to solve the problem and define appropriate variables that represent them. For instance, in a simple maximization problem, RMC, Inc. interested in producing two products: fuel additive and a solvent base. The decision variables will be X1 = tons of fuel additive to produce, and X2 = tons of solvent base to produce. 2.What is the objective to be maximized or minimized? Determine the objective and express it as a linear function. When building a linear programming model, only relevant costs should be included, sunk costs are not included. In our example, the objective function is: = 40X1 + 30X2; where 40 and 30 are the objective function coefficients

7. Bina Nusantara University 7 3. What limitations or requirements restrict the values of the decision variables? Identify and write the constraints as linear functions of the decision variables. Constraints generally fall into one of the following categories: a.Limitations - The amount of material used in the production process cannot exceed the amount available in inventory. In our example, the limitations are: Material 1 = 20 tons Material 2 = 5 tons Material 3 = 21 tons available. The material used in the production of X1 and X2 are also known.

8. Bina Nusantara University 8 To produce one ton of fuel additive uses.4 ton of material 1, and.60 ton of material 3. To produce one ton of solvent base it takes.50 ton of material 1,.20 ton of material 2, and.30 ton of material 3. Therefore, we can set the constraints as follows:.4X1 +.50 X2 <= 20.20X2 <= 5.6X1 +.3X2 <=21, where.4,.50,.20,.6, and.3 are called constraint coefficients. The limitations (20, 5, and 21) are called Right Hand Side (RHS). b.Requirements - specifying a minimum levels of performance. For instance, production must be sufficient to satisfy customers’ demand

9. Bina Nusantara University 9 Linear Progamming Application Model TypeApplication

10. Bina Nusantara University 10 LP Model Formulation Decision variables mathematical symbols representing levels of activity of an operation Objective function a linear relationship reflecting the objective of an operation most frequent objective of business firms is to maximize profit most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost Constraint a linear relationship representing a restriction on decision making

11. Bina Nusantara University 11 Max/min z = c 1 x 1 + c 2 x 2 +... + c n x n subject to: a 11 x 1 + a 12 x 2 +... + a 1n x n (≤, =, ≥) b 1 a 21 x 1 + a 22 x 2 +... + a 2n x n (≤, =, ≥) b 2 a m1 x1 + a m2 x 2 +... + a mn x n (≤, =, ≥) b m x j = decision variables b i = constraint levels c j = objective function coefficients a ij = constraint coefficients

12. Bina Nusantara University 12 Formulation Modeling : Example LaborClayRevenue PRODUCT(hr/unit)(lb/unit)($/unit) Bowl1440 Mug2350 There are 40 hours of labor and 120 pounds of clay available each day Decision variables x 1 = number of bowls to produce x 2 = number of mugs to produce RESOURCE REQUIREMENTS

13. Bina Nusantara University 13 Maximize Z = $40 x 1 + 50 x 2 Subject to x 1 +2x 2 ≤ 40 hr(labor constraint) 4x 1 +3x 2 ≤ 120 lb(clay constraint) x 1, x 2 ≥ 0 Solution is x 1 = 24 bowls x 2 = 8 mugs Revenue = $1,360