[Page 93] This example demonstrates the transformation of a linear programming model into standard form, sensitivity analysis, computer solution, and shadow prices. Problem Statement The Xecko Tool Company is considering bidding on a job for two airplane wing parts . Each wing part must be processed through three manufacturing stagesstamping, drilling, and finishingfor which the company has limited available hours. The linear programming model to determine how many of part 1 ( x _{ 1 } ) and part 2 ( x _{ 2 } ) the company should produce in order to maximize its profit is as follows : maximize Z = $650 x _{ 1 } + 910 x _{ 2 } subject to 4 x _{ 1 } + 7.5 x _{ 2 } 105 (stamping, hr.) 6.2 x _{ 1 } + 4.9 x _{ 2 } 90 (drilling, hr.) 9.1 x _{ 1 } + 4.1 x _{ 2 } 110 (finishing, hr.) x _{ 1 } , x _{ 2 } 
Solve the model graphically. 
Indicate how much slack resource is available at the optimal solution point. 
Determine the sensitivity ranges for the profit for wing part 1 and the stamping hours available. 
Solve this model by using Excel. Solution 

The slack at point B , where x _{ 1 } = 5.97 and x _{ 2 } = 10.82, is computed as follows: 4(5.97) + 7.5(10.82) + s _{ 1 } = 105 (stamping, hr.) s _{ 1 } = 0 hr. 6.2(5.97) + 4.9(10.82) + s _{ 2 } = 90 (drilling, hr.) s _{ 2 } = 0 hr. 9.1(5.97) + 4.1(10.82) + s _{ 3 } = 110 (finishing, hr.) s _{ 3 } = 11.35 hr. [Page 94] 
The sensitivity range for the profit for part 1 is determined by observing the graph of the model and computing how much the slope of the objective function must increase to make the optimal point move from B to C . This is the upper limit of the range and is determined by computing the value of c _{ 1 } that will make the slope of the objective function equal with the slope of the constraint line for drilling, 6.2 x _{ 1 } + 4.9 x _{ 2 } = 90: The lower limit is determined by computing the value of c _{ 1 } that will equate the slope of the objective function with the slope of the constraint line for stamping, 4 x _{ 1 } + 7.5 x _{ 2 } = 105: Summarizing, 485.33 c _{ 1 } 1,151.43 The upper limit of the range for stamping hours is determined by first computing the value for q _{ 1 } that would move the solution point from B to where the drilling constraint intersects with the x _{ 2 } axis, where x _{ 1 } = 0 and x _{ 2 } = 18.37: The lower limit of the sensitivity range occurs where the optimal point B moves to C , where x _{ 1 } = 8.87 and x _{ 2 } = 7.15: Summarizing, 89.10 q _{ 1 } 137.76. 
