Case 1 Computers and Printers - Maximization
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Kennesaw State University *
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3300
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Industrial Engineering
Date
Apr 3, 2024
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pptx
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Case 1.1: Computers and Printers – Maximizing Profit
•
The Mapple store sells Mapple computers and printers. The computers are shipped in 12-cubic-foot boxes and printers in 8-cubic-foot boxes. The Mapple store estimates that at least 30 computers can be sold each month and that the number of computers sold will be at least 50% more than the number of printers. The computers cost the store $1000 each and are sold for a profit of $1000. The printers cost $300 each and are sold for a profit of $350. The store has a storeroom that can hold 1000 cubic feet and can spend $80,000 each month on computers and printers. The objective is to maximize total profit.
•
Identify the variables. Formulate the mathematical model. Solve it graphically. How many extreme points do you see? 5, see below What is the optimal solution? (C=77.273, P=9.091) How much is total profit? Total profit= 1000*77.273 + 350*9.091 = 80,454.85 Which constraints are binding and which ones are not? Binding: capacity, budget Non-binding: computers, ratio. Any constraint crossing the optimal solution is binding. Calculate slack or surplus for each constraint. Do you see any redundant constraint? Solve it in Excel. Did you get the same optimal solution and profit?
Case 1.2: Computers and Printers – Changes in Objective Coefficients
The Mapple store sells Mapple computers and printers. The computers are shipped in 12-cubic-foot boxes and printers in 8-
cubic-foot boxes. The Mapple store estimates that at least 30 computers can be sold each month and that the number of computers sold will be at least 50% more than the number of printers. The computers cost the store $1000 each and are sold for a profit of $1000. The printers cost $300 each and are sold for a profit of $350. The store has a storeroom that can hold 1000 cubic feet and can spend $80,000 each month on computers and printers. The objective is to maximize total profit. We obtained the following mathematical model (without integer constraint):
Max 1000C + 350P
S.T.:
Computers
C>=30 Ratio
C-1.5P>=0 Budget
1000C+300P<=80000
Capacity
12C+8P<=1000 N-N
C,P>=0
We obtained a Total Profit of $80,454.55.
Using Desmos, estimate the range of optimality for profit per computer.
Not in Exam
Case 1.3: Computers and Printers – Changes in Objective Coefficients
The Mapple store sells Mapple computers and printers. The computers are shipped in 12-cubic-foot boxes and printers in 8-
cubic-foot boxes. The Mapple store estimates that at least 30 computers can be sold each month and that the number of computers sold will be at least 50% more than the number of printers. The computers cost the store $1000 each and are sold for a profit of $1000. The printers cost $300 each and are sold for a profit of $350. The store has a storeroom that can hold 1000 cubic feet and can spend $80,000 each month on computers and printers. The objective is to maximize total profit. We obtained the following mathematical model (without integer constraint):
Max 1000C + 350P
S.T.:
Computers
C>=30 Ratio
C-1.5P>=0 Budget
1000C+300P<=80000
Capacity
12C+8P<=1000 N-N
C,P>=0
We obtained a Total Profit of $80,454.55 and the following Sensitivity Report:
Variable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$B$10
C
77.27272727
0
1000
166.6666667
475
$C$10
P
9.090909091
0
350 316.6666667
50
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
$B$15
Capacity LHS
1000
11.36363636
1000
155.5555556
40
$B$16
Computers LHS
77.27272727
0
30
47.27272727
1E+30
$B$17
Ratio LHS
63.63636364
0
0
63.63636364
1E+30
$B$18
Budget LHS
80000
0.863636364
80000
3333.333333
10769.23077
Assume profit per computer is $1050 (instead of $1000). Is the optimal solution going to change?
Is the total profit going to change? If so, calculate the new total profit by hand
Re-run Solver to match your answers above.
Case 1.4: Computers and Printers – Changes in Objective Coefficients
The Mapple store sells Mapple computers and printers. The computers are shipped in 12-cubic-foot boxes and printers in 8-
cubic-foot boxes. The Mapple store estimates that at least 30 computers can be sold each month and that the number of computers sold will be at least 50% more than the number of printers. The computers cost the store $1000 each and are sold for a profit of $1000. The printers cost $300 each and are sold for a profit of $350. The store has a storeroom that can hold 1000 cubic feet and can spend $80,000 each month on computers and printers. The objective is to maximize total profit. We obtained the following mathematical model (without integer constraint):
Max 1000C + 350P
S.T.:
Computers
C>=30 Ratio
C-1.5P>=0 Budget
1000C+300P<=80000
Capacity
12C+8P<=1000 N-N
C,P>=0
We obtained a Total Profit of $80,454.55 and the following Sensitivity Report:
Variable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$B$10
C
77.27272727
0
1000
166.6666667
475
$C$10
P
9.090909091
0
350 316.6666667
50
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
$B$15
Capacity LHS
1000
11.363636
36
1000
155.5555556
40
$B$16
Computers LHS
77.27272727
0
30
47.27272727
1E+30
$B$17
Ratio LHS
63.63636364
0
0
63.63636364
1E+30
$B$18
Budget LHS
80000
0.8636363
64
80000
3333.333333
10769.23077
Assume profit per computer is $1200 (instead of $1000). Is the optimal solution going to change?
Is the total profit going to change? If so, calculate the new total profit by hand
Re-run Solver to match your answers above.
Case 1.5: Computers and Printers – Changes in Objective Coefficients
The Mapple store sells Mapple computers and printers. The computers are shipped in 12-cubic-foot boxes and printers in 8-
cubic-foot boxes. The Mapple store estimates that at least 30 computers can be sold each month and that the number of computers sold will be at least 50% more than the number of printers. The computers cost the store $1000 each and are sold for a profit of $1000. The printers cost $300 each and are sold for a profit of $350. The store has a storeroom that can hold 1000 cubic feet and can spend $80,000 each month on computers and printers. The objective is to maximize total profit. We obtained the following mathematical model (without integer constraint):
Max 1000C + 350P
S.T.:
Computers
C>=30 Ratio
C-1.5P>=0 Budget
1000C+300P<=80000
Capacity
12C+8P<=1000 N-N
C,P>=0
We obtained a Total Profit of $80,454.55 and the following Sensitivity Report:
Variable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$B$10
C
77.27272727
0
1000
166.6666667
475
$C$10
P
9.090909091
0
350 316.6666667
50
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
$B$15
Capacity LHS
1000
11.363636
36
1000
155.5555556
40
$B$16
Computers LHS
77.27272727
0
30
47.27272727
1E+30
$B$17
Ratio LHS
63.63636364
0
0
63.63636364
1E+30
$B$18
Budget LHS
80000
0.8636363
64
80000
3333.333333
10769.23077
Assume profit per printer is $340 (instead of $350). Is the optimal solution going to change?
Is the total profit going to change? If so, calculate the new total profit by hand
Re-run Solver to match your answers above.
Case 1.6: Computers and Printers – Changes in Objective Coefficients
The Mapple store sells Mapple computers and printers. The computers are shipped in 12-cubic-foot boxes and printers in 8-
cubic-foot boxes. The Mapple store estimates that at least 30 computers can be sold each month and that the number of computers sold will be at least 50% more than the number of printers. The computers cost the store $1000 each and are sold for a profit of $1000. The printers cost $300 each and are sold for a profit of $350. The store has a storeroom that can hold 1000 cubic feet and can spend $80,000 each month on computers and printers. The objective is to maximize total profit. We obtained the following mathematical model (without integer constraint):
Max 1000C + 350P
S.T.:
Computers
C>=30 Ratio
C-1.5P>=0 Budget
1000C+300P<=80000
Capacity
12C+8P<=1000 N-N
C,P>=0
We obtained a Total Profit of $80,454.55 and the following Sensitivity Report:
Variable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$B$10
C
77.27272727
0
1000
166.6666667
475
$C$10
P
9.090909091
0
350
316.6666667
50
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
$B$15
Capacity LHS
1000 11.36363636
1000
155.5555556
40
$B$16
Computers LHS
77.27272727
0
30
47.27272727
1E+30
$B$17
Ratio LHS
63.63636364
0
0
63.63636364
1E+30
$B$18
Budget LHS
80000 0.863636364
80000
3333.333333
10769.23077
Assume profit per printer is $250 (instead of $350). Is the optimal solution going to change?
Is the total profit going to change? If so, calculate the new total profit by hand
Re-run Solver to match your answers above.
Variables
C
P
80
0
Max
80000
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