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# Problems on Linear programming

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Objective Function: Maximize 6x + 8y
Constraints:
x + 2y <= 16
3x + 2y <= 24
2x + y >= 2
x , y >= 0
a) Plot the inequalities and shade in the feasible region for all three constraints.
b) Plot an iso-objective line for the Objective Function.
c) Compute the coordinates of the Optimal Corner Point and Mark on the graph. (Give the coordinates of the Optimal Corner Point.)
d) Calculate the Maximum value possible for this feasibility region.
2.
Formulate the following problem, and then use Excel to solve it.
The Wilkerson Motor company must determine how many Automobiles and SUV's to sell next quarter to increase their profits. The firm makes \$650 profit for each SUV it sells and \$470 profit for each Automobile it sells. Next quarter sales of both types of vehicles together must total of 450 or more. It is required that they sell at least 50 more SUV's than Automobiles. Dealer preparation requires 2.5 hours for each SUV and 1.25 hour for each Automobile. Next quarter the company has up to 1000 hours of shop time available for preparation of both vehicles.
a) Select decision variable symbols and define what quantities each of them represent in the description of the problem above.
b) Write the objective function and state what quantity is computed by this equation.
c) Write the three constraint equations (the non-negativity constraint is assumed).
3.

This is the formulated model of an electric network with three power plants P1, P2, P3. This model will determine additional hours of operation per week for each power plant. The power plants are trying to maximize the amount of power output, but need to limit the amount of emissions of Sulfur and Hydrocarbons. To be effective these plants are also required to produce a minimal amount of Superheated Steam.
Objective Function:
Maximize Z = 15 P1 + 35 P2 + 18 P3 = millions of kilowatt power
Constraints:
(1) P1 + 2 P2 + P3 <= 30 = tons of Sulfur
(2) 3 P1 + 4 P2 + P3 <= 100 = tons of Hydrocarbons
(3) 10 P1 + 10 P2 + 6 P3 >= 140 = tons of Superheated Steam
a) Enter and Solve this problem with the Excel Solver software.
b) What is the optimal power output of these plants, and how many additional hours should each of the three plants operate in a week?
c) How much below the limiting amount of 30 tons of Sulfur was produced?
d) How many tons of Hydrocarbons was produced?
e) The minimum requirement of Superheated Steam is 140 tons. What is the actual Total level of Superheated Steam produced?
f) Plants P1 and P2 are determined to be under utilized. Their combined additional hours are now required to be at least 10 hours. Reformulate (add this constraint) and resolve.
What is the new optimal solution?
HINT: For this problem you will have two separate Excel printouts for part A and part F.