Problem 1:

Solve each of these problems by computer and obtain the optimal values of the decision variables and the objective function.

1. Maximize \(4 x_{1}+2 x_{2}+5 x_{3}\)
   Subject to
   \(\begin{aligned}
   1 x_{1}+2 x_{2}+1 x_{3} & \leq 25 \\
   1 x_{1}+4 x_{2}+2 x_{3} & \leq 40 \\
   3 x_{1}+3 x_{2}+1 x_{3} & \leq 30 \\
   x_{1}, x_{2}, x_{3} & \geq 0
   \end{aligned}\)
2. Maximize \(10 x_{1}+6 x_{2}+3 x_{3}\)

Subject to

\(\begin{array}{r}

1 x_{1}+1 x_{2}+2 x_{3} \leq 25 \\

2 x_{1}+1 x_{2}+4 x_{3} \leq 40 \\

1 x_{1}+2 x_{2}+3 x_{3} \leq 40 \\

x_{1}, x_{2}, x_{3} \geq 0

\end{array}\)

Problem 2: Let us introduce the problem we will use to demonstrate the simplex method. HighTech Industries imports electronic components that are used to assemble two different models of personal computers. One model is called the Deskpro, and the other model is called the Portable. HighTech's management is currently interested in developing a weekly production schedule for both products.

The Deskpro generates a profit contribution of $50 per unit, and the Portable generates a profit contribution of $40 per unit. For next week's production, a maximum of 150 hours of assembly time can be made available. Each unit of the Deskpro requires 3 hours of assembly time, and each unit of the Portable requires 5 hours of assembly time. In addition, HighTech currently has only 20 Portable display components in inventory; thus, no more than 20 units of the Portable may be assembled. Finally, only 300 square feet of warehouse space can be made available for new production. Assembly of each Deskpro requires 8 square feet of warehouse space; similarly, each Portable requires 5 square feet.

To develop a linear programming model for the HighTech problem, we will use the following decision variables:

\(\begin{aligned}

&x_{1}=\text { number of units of the Deskpro } \\

&x_{2}=\text { number of units of the Portable }

\end{aligned}\)

Problem 3: A manager wants to know how many units of each product to produce on a daily basis in order to achieve the highest contribution to profit. Production requirements for the products are shown in the following table.

Material 1 costs $5 a pound, material 2 costs $4 a pound, and labor costs $10 an hour. Product A sells for $80 a unit, product B sells for $90 a unit, and product C sells for $70 a unit. Available resources each day are 200 pounds of material 1, 300 pounds of material 2 , and 150 hours of labor.

The manager must satisfy certain output requirements: The output of product A should not be more than one-third of the total number of units produced: the ratio of units of product A to units of product \(B\) should be 3 to 2 ; and there is a standing order for 5 units of product A each day. Formulate a linear-programming model for this problem, and then solve.

Problem 4: The Navy has 9,000 pounds of material in Albany, Georgia which it wishes to ship to three installations: San Diego, Norfolk, and Pensacola. They require 4,000, 2,500, and 2,500 pounds respectively. The following gives the shipping costs per pound for truck, railroad, and airplane transit.

Government regulations require equal distribution of shipping among the three carriers. Formulate and solve a linear program to determine the shipping arrangements (mode, destination, and quantity) that will minimize the total shipping cost.

Problem 5: Given the following linear program:

\(\begin{array}{ll}

\text { MIN } z=5 x_{1}+2 x_{2} \\

\text { s.t. } & 2 x_{1}+5 x_{2} \geq 10 \\

& 4 x_{1}-x_{2} \geq 12 \\

& x_{1}+x_{2} \geq 4 \\

& x_{1}, x_{2} \geq 0

\end{array}\)

1. Solve graphically for the optimal solution.
2. How does one know that although \(x_{1}=5, x_{2}=3\) is a feasible solution for the constraints, it will never be the optimal solution no matter what objective function is imposed?

Price: $19.71

Solution: The downloadable solution consists of 11 pages, 871 words and 3 charts.
Deliverable: Word Document