Question
Klein Industries manufactures three types of portable air compressors: small, medium, and large, which have unit profits of $20.50, $34.00, and $42.00, respectively. The projected monthly sales are:
Small Medium Large
Minimum 14,000 6,200 2,600
Maximum 21,000 12,500 4,200
The production process consists of three primary activities: bending and forming, welding, and painting. The amount of time in minutes needed to process each product in each department is shown below:
Small Medium Large Available Time
Bending/forming 0.4 0.7 0.8 23,400
Welding 0.6 1.0 1.2 23,400
Painting 1.4 2.6 3.1 46,800
QUESTION: What is the shadow price associated with increasing the minimum projected sales for the large compressor (rounded to 2 decimal places)?
Explanation / Answer
Small Medium Large Total Profit Unit profit $20.50 $34.00 $42.00 $651,221.43 Monthly Sales Small Medium Large Minimum 14,000 6,200 2,600 Maximum 21,000 12,500 4,200 Checking points Minutes Small Medium Large Available Time Time Used Available-Used Bending/Forming 0.40 0.70 0.80 23,400 12882.85714 10,517 Welding 0.60 1.00 1.20 23,400 19014.28571 4,386 Painting 1.40 2.60 3.10 46,800 46800 0 Total production time 2.40 4.30 5.10 Production 16157.14 6200.00 2600.00 Checking points Min 2157.14 0.00 0.00 Max 4842.86 6300.00 1600.00 Total Profit ($) Total Profit ($) = 20.50 small + 34.00 medium + 42.00 large Production constraints: 14,000 <= small <= 21,000 6,200 <= medium <= 12,500 2,600 <= large <= 4,200 Time constraints: 0.40 small + 0.70 medium + 0.80 large <= 23,400 0.60 small + 1.00 medium + 1.20 large <= 23,400 1.40 small + 2.60 medium + 3.10 large <= 46,800 Small Medium Large Unit profit $20.50 $34.00 $42.00 Production time per unit 2.40 4.30 5.10 Profit per production time $8.54 $7.91 $8.24
Engine: Simplex LP Solution Time: 0.032 Seconds. Iterations: 2 Subproblems: 0 Max Time Unlimited, Iterations Unlimited, Precision 0.000001, Use Automatic Scaling Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative
Cell Name Original Value Final Value $E$5 Unit profit Total Profit $651,221.43 $651,221.43
Cell Name Original Value Final Value Integer $B$16:$D$16 $B$16 Production Small 16157.14 16157.14 Contin $C$16 Production Medium 6200.00 6200.00 Contin $D$16 Production Large 2600.00 2600.00 Contin
Cell Name Cell Value Formula Status Slack $E$12 Bending/Forming Available Time 23,400 $E$12>=$F$12 Not Binding 10,517 $E$13 Welding Available Time 23,400 $E$13>=$F$13 Not Binding 4,386 $E$14 Painting Available Time 46,800 $E$14>=$F$14 Binding 0 $B$16 Production Small 16157.14 $B$16<=$B$9 Not Binding 4842.857143 $B$16 Production Small 16157.14 $B$16>=$B$8 Not Binding 2157.14 $C$16 Production Medium 6200.00 $C$16<=$C$9 Not Binding 6300 $C$16 Production Medium 6200.00 $C$16>=$C$8 Binding 0.00 $D$16 Production Large 2600.00 $D$16<=$D$9 Not Binding 1600 $D$16 Production Large 2600.00 $D$16>=$D$8 Binding 0.00 Variable Cells
Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$16:$D$16 $B$16 Production Small 16157.14286 0 20.5 1E+30 1.532258064 $C$16 Production Medium 6200 -4.071428571 34 4.071428571 1E+30 $D$16 Production Large 2600 -3.392857143 42 3.392857143 1E+30 Constraints
Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$12 Bending/Forming Available Time 23400 0 0 10517.14286 1E+30 $E$13 Welding Available Time 23400 0 0 4385.714286 1E+30 $E$14 Painting Available Time 46800 -14.64285714 0 3020 6780
