Problem #1:

The company recently contracted to supply gasoline distributors in southern Ohio, and it has $600,000 available to spend on the necessary expansion of its fleet of gasoline tank trucks. Three models of gasoline tank trucks are available.

The company estimates that the monthly demand for the region will be 550,000 gallons of gasoline. Because of the size and speed differences of the trucks, the number of deliveries or round trips possible per month for each truck model will vary. Trip capacities are estimated at 15 trips per month for the Super Tanker, 20 trips per month for the Regular Line, and 25 trips per month for the Econo-Tanker. Based on maintenance and driver availability, the firm does not want to add more than 15 new vehicles to its fleet. In addition, the company has decided to purchase at least three of the new Econo-Tankers for use on short-run, low-demand routes. As a final constraint, the company does not want more than half the new models to be Super Tankers.

If the company wishes to satisfy the gasoline demand with a minimum monthly operating expense, develop a mathematical model that will help the company find out how many trucks of each model should be purchased.

Problem#2:
The Porsche Club of America sponsors driver education events that provide high performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries
manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame.

Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 40,000 pounds of the high alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:

MAXIMIZE (200DRB + 280DRW)
Subject to:
20DRB + 25DRW ≤ 40000 (Steel Available)
40DRB + 100DRW ≤ 120000 (Manufacturing time (in minutes))
60DRB + 40DRW ≤ 96000 (Assembly time (in minutes))
DRB ≥ 0 and DRW ≥ 0
where:
DRB = Number of units of model DRB roll bars to produce

DRW = Number of units of DRW model roll bars to produce

The Management Scientist solution is shown below:
OPTIMAL SOLUTION
Objective Function Value = 424000

a) What is the optimal solution and what is the total profit?

b) Which constraints are binding? Justify your answer

c) Which resources were used up and which constraint shows extra capacity and how much? Justify your answer and explain in terms of the problem

d) Interpret the dual prices corresponding to constraints 1 and 3

e) Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.

f) If the current objective function coefficient of variable model DRB is decreased by $25, will the optimal solution change? How about the objective function value? Justify your answers.

g) If the available manufacturing time is increased by 500 hours, what effect would this have on the total profit? Explain

Problem#3:
Consider the following linear programming model:
Maximize (2A + 3B)
Subject to:
5A + 5B ≤ 400 (Constraint 1)
-A + B ≤ 10 (Constraint 2)
A + 3B ≤ 90 (Constraint 3)
A ≥ 0 and B ≥ 0
The figure below shows a graph of the constraint lines.

a) On the graph below, place a number (1, 2 or 3) next to each constraint line to identify which constraint it represents

b) Shade in the feasible region on the graph

c) Use the objective function to identify the optimal extreme point . (Explain and show your work)

d) What is the optimal solution? (Explain and show your work)

Attachment:- Assignment.rar