LINEAR PROGRAMMING MODULE 2 BY Olagunju O E

LINEAR PROGRAMMING MODULE 2 BY Olagunju O. E. Bowen University, Iwo, Osun state.

Linear programming (LP) Module Objectives • • • Understand the concept of linear programming Outline the assumptions of LP Identify the areas of application of LP Apply basic principles of LP Formulate LP problems

Introduction of Linear programming (LP) Linear programming is a mathematical technique concerned with the allocation of scarce resources. It involves procedures to optimize the values of some desired objectives (maximize profit or minimize cost) when factors involved (such as labour hours, machine hours, raw materials) are subject to some constraints (limitations)

Assumptions in Linear Programming Model 1. 2. 3. 4. 5. Proportionality Additivity Continuity Certainty Finite choices

Applications of Linear programming Some of the application of LP are 1. Industrial Applications • • • Product Mix problems Production scheduling problems Blending problems Trim loss problems Assembly line balancing

Application contd 2. Management Applications • • • Profit planning problems Portfolio selection problems Media selection problems Transportation problems Assignment problems

Formulating of Linear Programming Problems • Step 1: Identify the decision variables. Decision variables are quantities that are inputs or factors in a system (model). The extent of inputs that are permitted in the model. (x 1, x 2, x 3……. . xn)

Formulating contd Step 2: Decide what result is required (objective) and construct the objective function. • Objective is meant to decide whether to maximize or minimize a variable. • The mathematical statement that reflect the element in achieving the objectives is called the Objective Function. • It should be expressed as a linear function of variables. • It should indicate the number of unknown decision variables

Objective function (maximization problem) If a firm produce three product A, B and C and wishes to maximize profit. If the profit per unit for products A, B and C are N 20, N 30 and N 40 respectively. The objective function becomes: Maximize 20 x 1 + 30 x 2 + 40 x 3 where x 1 = number of units of A produced X 2 = number of units of B produced X 3 = number of units of C produced

Objective function (minimization problem) A manufacturer is to market a new fertilizer which is to be a mixture of two ingredients A and B. The properties of the two ingredients are: Ingredients Analysis Phosphate Bone meal Nitrogen Lime Cost/Kg Ingredient A 10% 20% 30% 40% #12 Ingredient B 5% 40% 10% 45% #8

The manufacturer wishes to market the fertilizers at the minimum cost possible. The objective function becomes: Minimize 12 x 1 + 8 x 2 where x 1 = kilograms of ingredient A X 2 = kilograms of ingredient B

Formulating contd Step 3: Identify the existing constraints Constraints are sometimes called restriction or limitations. These are factors that limits the range of one or more decision variables Maximization problem • Resources Used ≤ Resources Available Minimization problem • Optimized Outcome ≥ Cost

Maximization Example Suppose an industry is manufacturing four types of products P 1, P 2, P 3 and P 4 which are made from two basic ingredients, Maize and Groundnut. Only 1000 tons of maize and 1500 tons of groundnut are available in a period. The following table shows the usage of the materials to produce P 1, P 2, P 3 and P 4. Product P 1 P 2 P 3 P 4 Maize 2 5 0 4 Groundnut 3. 5 4 6 0

Constraint/Limitations are as follows: Maize constraint Groundnut constraint 2 x 1 3. 5 x 1 + + where 5 x 2 4 x 2 + + 4 x 4 6 x 3 ≤ ≤ x 1 ≥ 0, x 2≥ 0, x 3≥ 0, x 1 = number of units of P 1 x 2 = number of units of P 2 x 3 = number of units of P 3 x 4 =number of units of P 4 1000 1500 x 4 ≥ 0 A maximization problem follows a pattern of being less than or equal to (≤) type. This is used because resources used (LHS) cannot exceed the available resources (RHS).

Minimization Example As in the example above (minimization), if it has been decided that the fertilizer will be sold in bags containing 100 kgs, contain at least 15% nitrogen, 8% phosphate, 25% bone meal. The manufacturer wishes to market the fertilizers at the minimum cost possible. Weight constraint Nitrogen constraint Phosphate constraint Bone meal constraint x 1 0. 3 x 1 0. 1 x 1 0. 2 x 1 where + x 2 = + 0. 1 x 2 ≥ + 0. 05 x 2 ≥ + 0. 4 x 2 ≥ x 1 ≥ 0 and x 2 ≥ 0 x 1 = kgs of ingredient A x 2 = kgs of ingredient B 100 15 8 25 In a minimization problem, constraints are of greater than or equal to (≥) type.

Take Note • Generally, sign restrictions for x 1, x 2, x 3 are ≥ 0, since real-world situations such as quantities of goods produced, quantities of raw material are non-negative. • However, exceptional cases, such as sales restrictions may be of the ≥ or ≤ type becau. se of quota or contract requirements