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LPP, Transportation, Assignment problems

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39. Max Z = \$0.30x + \$0.90y
Subject to : 2x + 3.2y 160
4x + 2y 240
y 40
x, y
Solve for the quantities of x and y which will maximize Z. What is the value of the slack variable associated with constraint 2?

40. The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are \$2 per bottle, and profits for dark beer are \$1 per bottle. What is the constraint for malt?

41. The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are \$3.00 per case and profits for diet soft drink are \$2.00 per case. How many cases of regular and how many cases of diet soft drink should Whoppy produce to maximize daily profit?

42. The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x1), bright red nail polish (x2), basil green nail polish(x3), and basic pink nail polish(x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for fire red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand for bright red, green and pink nail polish bottles combined is at least 50 bottles.

MAX 100x1 + 120x2 + 150x3 + 125x4

Subject to 1. x1 + 2x2 + 2x3 + 2x4 108
2. 3x1 + 5x2 + x4 120
3. x1 + x3 25
4. x2 + x3 + x4 50
x1, x2 , x3, x4 0

Optimal Solution:
Objective Function Value = 7475.000
Variable Value Reduced Costs
X1 8 0
X2 0 5
X3 17 0
X4 33 0

Constraint Slack / Surplus Dual Prices
1 0 75
2 63 0
3 0 25
4 0 -25

Objective Coefficient Ranges

Variable Lower Limit Current Value Upper Limit
X1 87.5 100 none
X2 none 120 125
X3 125 150 162
X4 120 125 150

Right Hand Side Ranges

Constraint Lower Limit Current Value Upper Limit
1 100 108 123.75
2 57 120 none
3 8 25 58
4 41.5 50 54

a) To what value can the per bottle profit on fire red nail polish drop before the solution (product mix) would change?
b) By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?

43. A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS (tablespoons) of almond paste. An almond- filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today's production run. Bear claw profits are 20 cents each, and almond-filled croissant profits are 30 cents each. What is the objective function?

44. Let x1 be the number of units to make and x2 be the number of units to buy. If it costs \$2 to make a unit and \$3 to buy a unit and 4000 units are needed, what is the objective function?

45. Consider the following transportation problem:
1 2 Supply
1 5 6 100
2 4 2 200
3 3 6 150
4 9 7 50
Demand 250 250

What is the objective function?

46. Consider a capital budgeting example with 5 projects from which to select. Let x1 = 1 if project a is selected, 0 if not, for a = 1, 2, 3, 4, 5. Conditions are independent. Projects cost \$100, \$200, \$150, \$75, and \$300 respectively. The budget is \$450. Write the appropriate constraint for the following condition : If project 3 is chosen, project 4 must be chosen.

47. A logistics specialist for Wiethoff Inc. must distribute cases of parts from 3 factories to 3 assembly plants. The monthly supplies and demands, along with the per-case transportation costs are:

Destination Assembly Plant
1 2 3 Supply
Source A 5 9 16 200
Factory B 1 2 6 400
C 2 8 7 200
Demand 120 620 60

Using the optimal solution, how many cases of parts should be shipped from factory C to assembly plant 1? What is the cost of shipping these units from factory B to assembly plant 1?

48. A logistics specialist for Wiethoff Inc. must distribute cases of parts from 3 factories to 3 assembly plants. The monthly supplies and demands, along with the per-case transportation costs are:

Destination Assembly Plant
1 2 3 Supply
Source A 5 9 16 200
Factory B 1 2 6 400
C 2 8 7 200
Demand 120 620 60

What are the total monthly transportation costs for the optimal solution?

49. Consider the following transportation problem:
1 2 Supply
1 5 6 100
2 4 2 200
3 3 6 150
4 9 7 50
Demand 250 250

How many supply-side constraints are there? Write the supply-side constraints.

50. Consider the following transportation problem:
1 2 Supply
1 5 6 100
2 4 2 200
3 3 6 150
4 9 7 50
Demand 250 250

How many demand-side constraints are there? Write the demand-side constraints

Please show complete work to all problems.
1. A large book publisher has five manuscripts that must be edited as soon as possible. Five editors are available for doing the work, however their working times on the various manuscripts will differ based on their backgrounds and interests. The publisher wants to use an assignment method to determine who does what manuscript. Estimates of editing times (in hours) for each manuscript by each editor is:

Editor
Manuscript A B C D E

1 12 8 10 16 13
2 9 10 14 13 9
3 17 14 9 18 12
4 15 7 11 9 18
5 12 18 22 11 27
What is the total minimum editing time?

2. A large book publisher has five manuscripts that must be edited as soon as possible. Five editors are available for doing the work, however their working times on the various manuscripts will differ based on their backgrounds and interests. The publisher wants to use an assignment method to determine who does what manuscript. Estimates of editing times (in hours) for each manuscript by each editor is:

Editor
Manuscript A B C D E

1 12 8 10 16 13
2 9 10 14 13 9
3 17 14 9 18 12
4 15 7 11 9 18
5 12 18 22 11 27
a) How many supply-side constraints are needed?
b) How many demand-side constraints are needed?
c) How many variables are involved in this assignment method?