Problem 1. Consider the Example Problem on page 93 of the text book.
Problem Statement
The Xeko Tool Company is considering bidding on a job for two airplane wing parts. Each wing part must be processed through three manufacturing stages-stamping, drilling, and finishing-for which the company has limited available hours. The linear programming model to determine how many of part 1 (x1) and part 2 (x2) the company should produce in order to maximize its profit is as follows:
Objective: maximize z (x1, x2) = $650x1 + 9$10x2
Subject to:
4.0x1 + 7.5x2 <= 105 (stamping, hrs) (1)
6.2x1 + 4.9x2 <= 90 (drilling, hrs) (2)
9.1x1 + 4.1x2 <= 110 (finishing, hrs) (3)
x1, x2 >= 0 (non-negativity) (4)
Problem 1.1. LP Graphical Solution
Follow the steps one by one as follows:
Step 1. Assume that initially we have only constraint 1 and 4. Draw the feasible region using constraint 1 and find the Optimal value. Draw the objective function at two different locations at the graph.
Step 2. Assume that we have now constraints 1, 2, and 4. Draw the feasible region using constraints 1 and 2 and find the optimal solution. Draw the objective function at two different locations at the graph.
Step 3. Assume that now we have constraints 1, 2, 3 and 4. Draw the feasible region using all constraints and find the optimal solution. Draw the objective function at two different locations at the graph.
Step 4. Conduct sensitivity analysis on the objective function coefficients.
Step 4.1 What is the range for the objective function coefficient for wing part x1 by fixing the corresponding coefficient of the wing part x2 at its current value of $910.
Draw the objective function at two different locations on the graph.
Step 4.2 What is the range for the objective function coefficient for wing part x2 by fixing the corresponding coefficient of the wing part x1 at its current value of $650. Draw the objective function at two different locations on the graph.
Step 5. Conduct sensitivity analysis for the resource (hours of manufacturing production) constraints
Step 5.1 Increase the value of the stamping optimal binding constraint by one hour. Draw the feasible region and find the optimal value.
Step 5.2. Decrease the value of the stamping optimal binding constraint using the original constraint value by one hour. Draw the feasible region and find the optimal value.
Step 5.3 Increase the value of the drilling optimal binding constraint by one hour. Draw the feasible region and find the optimal value.
Step 5.4. Decrease the value of the drilling binding constraint using the original constraint by one hour. Draw the feasible region and find the optimal value.
Step 5.5. Increase the stamping constraint by a maximum value such that it is still an optimal binding constraint. Draw the feasible region and find the optimal value.
Step 5.6. Increase the drilling constraint by a maximum value such that it is still an optimal binding constraint. Draw the feasible region and find the optimal value.
Step 5.7. Decrease the stamping constraint by a maximum value such that it is still an optimal binding constraint. Draw the feasible region and find the optimal value.
Step 5.8. Decrease the drilling constraint by a maximum value such that it is still an optimal binding constraint. Draw the feasible region and find the optimal value.