Linear Programming

Learning Objectives Describe the type of problem tha would lend itself to solution using linear programming Formulate a linear programming model from a description of a problem Solve linear programming problems using the graphical method Interpret computer solutions of linear programming problems Do sensitivity analysis on the solution of a linear progrmming problem

Linear Programming Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials Budgets Labor Machine time

Linear Programming Linear programming (LP) techniques consist of a sequence of steps that will lead to an optimal solution to problems, in cases where an optimum exists

Linear Programming Model Objective Function: mathematical statement of profit or cost for a given solution Decision variables: amounts of either inputs or outputs Feasible solution space: the set of all feasible combinations of decision variables as defined by the constraints Constraints: limitations that restrict the available alternatives Parameters: numerical values

Linear Programming Assumptions Linearity: the impact of decision variables is linear in constraints and objective function Divisibility: noninteger values of decision variables are acceptable Certainty: values of parameters are known and constant Nonnegativity: negative values of decision variables are unacceptable

Linear Programming A company is manufacturing two products A and B. The manufacturing time required to make them, the profit and capacity available at each work centre are given as follows: Product Matching Fabrication Assembly Profit per unit A 1 Hour 5 Hours 3 Hours 80 B 2 Hours 4 Hours 1 Hour 100 Total Capacity 720 Hours 1800 Hours 900 Hours

Linear Programming Maximize, Z = 80x1 + 100x2 subject to the constraints: x1 + 2x2 ≤ 720 5x1 + 4x2 ≤ 1800 3x1 + x2 ≤ 900 x1 , x2 ≥ 0.

Linear Programming A company produces three products P, Q, and R from three raw materials: A, B and C. One unit of product P requires 2 units of A and 3 units of B. One unit of product Q requires 2 units of B and 5 units of C and one unit of product R requires 3 units A, 2 units of B and 4 units of C. The company has 8 units of material A, 10 units of material B and 15 units of material C available to it. Profits per unit of products P, Q and R are Tk3, Tk.5, and Tk4 respectively. Formulate the program mathematically.

Linear Programming Decision Product Types of raw material Profit per variable A B C Unit (Tk) x1 P 2 3 - 3 x2 Q - 2 5 5 x3 R 3 2 4 4 Unit of materials Max 8 10 15 available

Linear Programming Maximize, Z = 3x1 + 5x2 + 4x3 subject to the constraints: 2x1 + 3x3 ≤ 8 3x1 + 2x2 + 2x3 ≤ 10 5x2 + 4x3 ≤ 15 x1 , x2, x3 ≥ 0

Linear Programming A diet conscious housewife wishes to ensure certain minimum intake of vitamins A, B, and C for the family. The minimum daily (quantity) needs of the vitamins A, B, C for the family are respectively 30, 20, and 16 units. For the supply of these minimum vitamin requirements, the housewife relies on two fresh foods. The first one provides 7, 5, 2 units of the three vitamins per gram respectively and the second one provides 2, 4, 8 units of the same three vitamins per gram respectively. The first foodstuff costs Tk.3 per gram and the second Tk.2 per gram. How many grams of each foodstuff should the housewife buy everyday to keep her food bill as low as possible ?

Linear Programming Minimize, Z = 3x1 + 2x2 subject to the constraints: 7x1 + 2x2 ≥ 30 5x1 + 4x2 ≥ 20 2x1 + 8x2 ≥ 16 x1 , x2 ≥ 0.

Linear Programming A factory manufactures two articles A and B. To manufacture the article A, a certain machine has to be worked for 1.5 hours and in addition a craftsman has to work for 2 hours. To manufacture the article B, the machine has to be worked for 2.5 hours and in addition a craftsman has to work for 1.5 hours. In a week the factory can avail of 80 hours of machine time and 70 hours of craftsman’s time. The profit on each article A is Tk.5 and that on each article B is Tk.4. If all the articles produced can be sold away, find how many of each kind should be produced to earn the maximum profit per week.

Linear Programming Decision Article Hours on Hours on Profit per variable Machine Craftsman (Tk) x1 A 1.5 2 5 x2 B 2.5 1.5 4 Hours Available 80 70 (per week) Maximum Maximum x1 = number of units of Article A x2 = number of units of Article B

Linear Programming Maximize, Z = 5x1 + 4x2 subject to the constraints: 1.5x1 + 2.5x2 ≤ 80 2x1 + 1.5x2 ≤ 70 x1 , x2 ≥ 0.

Graphical Linear Programming Graphical method for finding optimal solutions to two-variable problems Set up objective function and constraints in mathematical format Plot the constraints Identify the feasible solution space Plot the objective function Determine the optimum solution

Linear Programming Example Objective - profit Maximize, Z = 60X1 + 50X2 Subject to: Assembly 4X1 + 10X2 ≤ 100 hours Inspection 2X1 + 1X2 ≤ 22 hours Storage 3X1 + 3X2 ≤ 39 cubic feet X1, X2 ≥ 0

Linear Programming Example

Linear Programming Example

Linear Programming Example Inspection Storage Assembly Feasible solution space

Linear Programming Example Z=900 Z=300 Z=600

Solution The intersection of inspection and storage Solve two equations in two unknowns 2X1 + 1X2 = 22 3X1 + 3X2 = 39 X1 = 9 X2 = 4 Z = $740

Constraints Redundant constraint: a constraint that does not form a unique boundary of the feasible solution space Binding constraint: a constraint that forms the optimal corner point of the feasible solution space

Solutions and Corner Points Feasible solution space is usually a polygon Solution will be at one of the corner points Enumeration approach: Substituting the coordinates of each corner point into the objective function to determine which corner point is optimal.

Slack and Surplus Surplus: when the optimal values of decision variables are substituted into a greater than or equal to constraint and the resulting value exceeds the right side value Slack: when the optimal values of decision variables are substituted into a less than or equal to constraint and the resulting value is less than the right side value

Simplex Method Simplex: a linear-programming algorithm that can solve problems having more than two decision variables

MS Excel Worksheet for Microcomputer Problem Figure 6S.15

MS Excel Worksheet Solution Figure 6S.17

Sensitivity Analysis Range of optimality: the range of values for which the solution quantities of the decision variables remains the same Range of feasibility: the range of values for the fight-hand side of a constraint over which the shadow price remains the same Shadow prices: negative values indicating how much a one-unit decrease in the original amount of a constraint would decrease the final value of the objective function