Linear Programming Management Science Linear Programming • It is mathematical modeling technique designed to help managers in planning and decision making relative to resource allocation. Properties of Linear Programming One objective Function One or more constraints Alternative courses of action Objective function and constraints are linearproportionality and additivity • Certainty • Divisibility • Nonnegative variables • • • • Formulating LP Problems • Completely understand problem being faced. the managerial • Identify the objective and the constraints. • Define the decision variables Decision variable – a controllable input in LP Model • Use the decision variables to write mathematical expression for the objective function and the constraints. Graphical Solutions to LP Problems Step 1 – Plot each constraint on a graph. Step 2 – Determine the feasible region Feasible region is the set of points that satisfy all the constraints. Step 3 – Find the optimal solution - Isoprofit/Isocost Line Solution Method - Corner Point Solution Method Example Flaire Furniture Company produces inexpensive Required tables and chairs. The production process for each is similar in that both require a certain number of • Write a complete mathematical statement of the problem, hours of carpentry work and a certain number of labor hours in the painting and varnishing including the objective function department. The following data are available for and the constraints. (Let T = the production of tables and chairs: number of tables to be produced Department Hours required to produce one unit Tables Chairs Carpentry 4 3 Painting and Varnishing 2 1 Profit per unit P70 P50 During the current production period, 240 hours of carpentry time are available and 100 hours in painting and varnishing time are available. Each table sold yields a profit of $70; each chair produced is sold for a $50 profit. • • and C = number of chairs to be produced). Find the values of T and C at the optimal solution. Find the optimal value of the objective function. πππππππ§π πΆππ π‘ 2π₯1 + 3π₯2 Constraints: 5π₯1 + 10π₯2 ≥ 90 4π₯1 + 3π₯2 ≥ 48 0.5π₯1 ≥ 0 π₯1 , π₯2 ≥ 0 Slack and Surplus οSlack -It is the amount of resource that is not used . -It is the amount by which the left side of a ≤ constraint is smaller than the right side -less than or equal to constraint Slack=(Amount of resource available)-(Amount of resourced used) Slack and Surplus οSurplus -It is the amount of resource that is not used . -It is the amount by which the left side of a ≥ constraint is larger than the right side -greater than or equal to constraint Slack=(Actual Amount)-(Minimum Amount) Special Cases in LP 1. Infeasibility 2. Unboundedness 3. Redundancy 4. Alternate Optimal Solutions Special Cases in LP Infeasibility - The situation in which no solution to the linear programming problem satisfies all the constraints. - Lacks feasible solution region - Occurs when constraints conflict with one another Special Cases in LP Infeasibility Constraints π₯1 + 2π₯2 ≤ 6 2π₯1 + π₯2 ≤ 8 π₯1 ≥ 7 Special Cases in LP Unboundedness -the situation when the value of the solution may be definitely large or infinitely small without violating any of the constraints -Feasible region is open ended -One or more constraints is missing -Implies that the problem has been improperly formulated Special Cases in LP Unboundedness Constraints π₯1 ≥ 5 π₯2 ≤ 10 π₯1 + 2π₯2 ≥ 10 π₯1 , π₯2 ≥ 0 Special Cases in LP Redundancy -The situation when one constraint may be more binding or restrictive than another and thereby negate its need to be considered -Redundant Constraint – does not affect the feasible solution region Special Cases in LP Redundancy Constraints π₯1 + π₯2 ≤ 20 2π₯1 + π₯2 ≤ 300 π₯1 ≤ 25 π₯1 , π₯2 ≥ 0 Special Cases in LP Alternate Optimal Solutions -The situation when more than one solution provides the optimal value for the objective function -Multiple optimal solutions are possible in LP problems Special Cases in LP Alternate Optimal Solution Constraints 6π₯1 + 4π₯2 ≤ 24 π₯1 ≤ 3 π₯1 , π₯2 ≥ 0
