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B1_Answer

B.1 Solve the following linear programming problem
graphically:
Maximize profit = 4X + 6Y
Subject to: X + 2Y ≤ 8
5X + 4Y ≤ 20
X, Y ≥ 0
The optimum solution is the point where profit will be maximum after solving theconstraints
To determine the optimum solution point, Overlap the feasible region of constraint A and Constraint B. The point of intersection
of constraint A and cosntraint B is the optimum solution point.
Constraint A X + 2Y ≤ 8
Constraint B 5X + 4Y ≤ 20
From the above graph, optimum solution point is X = 1.33 and Y = 3.33
Substitute these values in the objective function
Maximize Profit = 4X + 6Y
Profit = 4(1.33) + 6(3.33)
Profit 25.3
Thus, the maximum profit is $25.3

B5_Answer

Solve the following LP problem graphically:
Minimize cost = 24X + 15Y
Subject to: 7X + 11Y ≥ 77
16X + 4Y ≥ 80
X, Y ≥ 0
The objective function is to minimze cost.
Constraints in a linear programming confines the degree to which the objective function can be accomplished.
Constraints are expressed as follows
Y
Constraint A 7X + 11Y ≥ 77 20
Constraint B 16X + 4Y ≥ 80
18
To determine the feasible region, we will solve the two constraints
16
Substitute X = 0 in constraint A
7X + 11Y = 77 14
(0) + 11Y = 77
Y = 7 12
Substitute Y = 0 in constraint A 10 Constraint B
7X + 11Y = 77
7X + (0) = 77 8
x = 11
6
Substitute X = 0 in constraint B Constraint A
16X + 4Y = 80 4
(0) + 4Y = 80
Y = 20 2
Substitute y = 0 in constraint B 0 2 4 6 8 10 12 X
16X + 4Y = 80
(16X+ (0) = 80
X = 5
From the above graph, optimum solution point is X = 3.86 and Y = 4.54
Substitute these values in the objective function
Minimize Cost 24X + 15Y
Profit = 24(3.86) + 15(4.54) 3.86 4.54
X Y Sum RHS
Cost 160.86 Minimize Cost 24 15 160.86
Constraint A 7 11 77 77
Thus, the minimum cost is $160.86 Constraint B 16 4 80 80

B7_Answer

B.7 The Attaran Corporation manufactures two electrical
products: portable air conditioners and portable heaters. The
assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours
of wiring and 2 hours of drilling. Each heater must go through
2 hours of wiring and 1 hour of drilling. During the next production
period, 240 hours of wiring time are available and up to
140 hours of drilling time may be used. Each air conditioner sold
yields a profit of $25. Each heater assembled may be sold for a
$15 profit.
Formulate and solve this LP production-mix situation, and
find the best combination of air conditioners and heaters that
yields the highest profit.
Let X be air conditioner
Y be heater
Maximize Profit = 25X + 15Y
subject to
Wiring 3X + 2Y ≤ 240
Drilling 2X + Y ≤140
40 60
X Y Sum RHS
Maximize profit 25 15 1900
Wiring 3 2 240 240
Drilling 2 1 140 140
Thus, profit is maximized when X = 40 and Y = 60
Profit $1900

Sensitivity Report B7

Microsoft Excel 14.0 Sensitivity Report
Worksheet: [answer.xlsx]B7_Answer
Report Created: 22-06-2015 13:22:48
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$23 2X + Y ≤140 40 0 25 5 2.5
$C$23 60 0 15 1.6666666667 2.5
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$D$26 Wiring Sum 240 5 240 40 30
$D$27 Drilling Sum 140 5 140 20 20

Answer Report B7

Microsoft Excel 14.0 Answer Report
Worksheet: [answer.xlsx]B7_Answer
Report Created: 22-06-2015 13:22:48
Result: Solver found a solution. All Constraints and optimality conditions are satisfied.
Solver Engine
Engine: Simplex LP
Solution Time: 0.015 Seconds.
Iterations: 2 Subproblems: 0
Solver Options
Max Time Unlimited, Iterations Unlimited, Precision 0.000001, Use Automatic Scaling
Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative
Objective Cell (Max)
Cell Name Original Value Final Value
$D$25 Maximize profit Sum 0 1900
Variable Cells
Cell Name Original Value Final Value Integer
$B$23 2X + Y ≤140 0 40 Contin
$C$23 0 60 Contin
Constraints
Cell Name Cell Value Formula Status Slack
$D$26 Wiring Sum 240 $D$26<=$E$26 Binding 0
$D$27 Drilling Sum 140 $D$27<=$E$27 Binding 0

B11_Answer

B.11 The Sweet Smell Fertilizer Company markets bags
of manure labeled “not less than 60 lb dry weight.” The packaged
manure is a combination of compost and sewage wastes. To
provide good-quality fertilizer, each bag should contain at least
30 lb of compost but no more than 40 lb of sewage. Each pound
of compost costs Sweet Smell 5¢ and each pound of sewage costs
4¢. Use a graphical LP method to determine the least-cost blend
of compost and sewage in each bag.
Let X be compost and Y be sewage
Minimize Cost = 5X + 4Y
subject to
X + Y ≥ 60 - Constraint A
X ≥ 30 - Constraint B Y
Y ≤ 40 - Constraint C
X, Y≥ 0 60
Constraint A
Determine the feasible region for constraint A. Solve equation X + Y = 60 to determine the feasible region 50 Constraint B
Susbtitute X = 0 in constraint A 40
X + Y = 60
(0) + Y = 60 30
Y = 60
20
Susbtitute Y = 0 in constraint A
X + Y = 60 10
X + (0) = 60
X = 60 0
10 20 30 40 50 60 X
30.00 30.00
X Y Sum RHS
Minimize Cost 5 4 270
Constraint A 1 1 60 60
Constraint B 1 0 30 30
Constraint C 0 1 30 40
Cost is minimized when X = 30 and Y = 30
Minimum Cost 5X + 4Y
5(30) + 4(30)
Minimum cost $270

B21_Answer

B.21. Par, Inc., produces a standard golf bag and a deluxe
golf bag on a weekly basis. Each golf bag requires time for cutting
and dyeing and time for sewing and finishing, as shown in the following
table:
HOURS REQUIRED PER BAG
PRODUCT CUTTING AND DYEING SEWING AND FINISHING
Standard bag 1/2 1
Deluxe bag 1 2/3
The profits per bag and weekly hours available for cutting and
dyeing and for sewing and finishing are as follows:
PRODUCT PROFIT PER UNIT ($)
Standard bag 10
Deluxe bag 8
ACTIVITY WEEKLY HOURS AVAILABLE
Cutting and dyeing 300
Sewing and finishing 360
Par, Inc., will sell whatever quantities it produces of these two
products.
a) Find the mix of standard and deluxe golf bags to produce per
week that maximizes weekly profit from these activities.
b) What is the value of the profit? 
Let X be standard bag and Y be deluxe bag
Maximize Profit = 10X + 8Y
subject to
1/2 X + 1Y ≤ 300
= X + 2Y ≤ 600 cutting and dyeing constraint
1X + 2/3Y ≤ 360
= 3X + 2Y ≤ 1080 Sewing and finishing constraint
240 180
X Y Sum RHS
Maximize Profit 10 8 3840
cutting and dyeing constraint 1 2 600 600
Sewing and finishing constraint 3 2 1080 1080
a) The mix is when X = 240 and Y = 180 which maximizes profit
b) Maximum profit $3,840

Answer Report B21

Microsoft Excel 14.0 Answer Report
Worksheet: [answer.xlsx]B21_Answer
Report Created: 22-06-2015 13:32:44
Result: Solver found a solution. All Constraints and optimality conditions are satisfied.
Solver Engine
Engine: Simplex LP
Solution Time: 0.016 Seconds.
Iterations: 2 Subproblems: 0
Solver Options
Max Time Unlimited, Iterations Unlimited, Precision 0.000001, Use Automatic Scaling
Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative
Objective Cell (Max)
Cell Name Original Value Final Value
$D$38 Maximize Profit Sum 0.00 3840.00
Variable Cells
Cell Name Original Value Final Value Integer
$B$36 0.00 240.00 Contin
$C$36 Sewing and finishing constraint 0.00 180.00 Contin
Constraints
Cell Name Cell Value Formula Status Slack
$D$39 cutting and dyeing constraint Sum 600 $D$39<=$E$39 Binding 0
$D$40 Sewing and finishing constraint Sum 1080 $D$40<=$E$40 Binding 0

Sensitivity Report B21

Microsoft Excel 14.0 Sensitivity Report
Worksheet: [answer.xlsx]B21_Answer
Report Created: 22-06-2015 13:32:44
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$36 240 0 10 2 6
$C$36 Sewing and finishing constraint 180 0 8 12 1.3333333333
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$D$39 cutting and dyeing constraint Sum 600 1 600 480 240
$D$40 Sewing and finishing constraint Sum 1080 3 1080 720 480