Exam 3: Linear Programming: Basic Concepts and Graphical Solutions

If a graphically solvable linear program is unbounded, then it can always be converted to a regular bounded problem by removing a constraint.

Constraint A: $3 X_{1}+X_{2} \leq 90$ Constraint B: $3 X_{1}+2 X_{2} \leq 120$ Constraint C: $X_{1}-X_{2} \leq 0$ All variables are required to be non-negative. Let the objective function be Max: $X_{1}+X_{2}$ Optimal solution (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $X_{2}$ ) will be

Constraint A: $-3 X_{1}+X_{2} \leq 90$ Constraint B: $4 X_{1}-2 X_{2} \geq 120$ Constraint C: $X_{1}+2 X_{2} \leq 150$ All variables are required to be non-negative. Let the objective function be Max: $2 X_{1}-3 X_{2}$ Optimal solution and corresponding objective function value (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $X_{2}$ ) will be

XYZ Inc. produces two types of printers - regular and high-speed. Regular uses 2 units of recycled plastic per unit, and high-speed uses 1 unit of recycled plastic per unit of production. XYZ is committed to using at least 5,000 units of recycled plastic per month. A critical machine is needed to manufacture the printers. Each unit of regular requires 5 units of time in this machine, and each unit of high-speed requires 3 units of time. The total time available in this machine per month is 15,000 units. Let $X_{1}$ be the number of units of regular produced per month and $X_{2}$ the number of units of high-speed produced per month. Imposing both of these constraints and non-negativity constraints, one of the feasible corner points is (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $X_{2}$ )

Constraint A: $-3 X_{1}+X_{2} \leq 90$ Constraint B: $4 X_{1}-2 X_{2} \geq 120$ Constraint C: $X_{1}+2 X_{2} \leq 150$ All variables are required to be non-negative. Let the objective function be Min: $2 X_{1}-3 X_{2}$ . Corner points of the feasible region include (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $\left.X_{2}\right)$

Constraint A: $-3 X_{1}+X_{2} \leq 90$ Constraint B: $4 X_{1}-2 X_{2} \geq 120$ Constraint C: $4 X_{1}+10 X_{2} \leq 40$ All variables are required to be non-negative. Let the objective function be Min: $2 X_{1}-3 X_{2}$ . Optimal solution and the corresponding objective function to this problem (assuming the first number in parenthesis is $X_{1}$ and the second number in the parenthesis is $X_{2}$ ) will be

Vikram is planning his summer vacation so he can maximize his earnings per week. He has an opportunity to mow two types of lawns-household and commercial. Each household takes 1.5 hours to mow and requires $\$ 3.00$ of raw materials (gas, mower rental, etc.). Each commercial job requires 4.0 hours and $\$ 6.00$ worth of raw materials. The maximum number of household jobs available is 10 per week; the maximum number of commercial jobs available is 8 . Having a maximum of 50 hours per week, he wants to take at least 4 household and 3 commercial jobs. He charges $\$ 20.00$ per household and $\$ 40.00$ per commercial lot; variable costs are his only expense. Build a linear programming model to maximize his net contribution per week. Solve the problem using graphical method.

Constraint A: $X_{1}+X_{2} \leq 7$ Constraint B: $2 X_{1}+5 X_{2} \leq 20$ All variables are required to be non-negative. The corner points of the feasible region for this constraint set (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $X_{2}$ ) are

Constraint A: $2 X_{1}+3 X_{2} \leq 90$ Constraint B: $2 X_{1}-3 X_{2} \leq 120$ The feasible region with these two constraints and non-negativity constraints

Constraint A: $3 X_{1}+X_{2} \geq 90$ Constraint B: $3 X_{1}+2 X_{2} \leq 120$ Constraint C: $X_{1}-X_{2} \geq 0$ All variables are required to be non-negative. Let the objective function be Max: $X_{1}+X_{2}$ Non-binding constraint corresponding to the optimal solution in this problem will be

In any graphically solvable linear program, it is enough if we examine all the corner points of the feasible region to find an optimal solution.

XYZ Inc. produces two types of printers - regular and high-speed. Regular uses 2 units of recycled plastic per unit, and high-speed uses 1 unit of recycled plastic per unit of production. XYZ is committed to using at least 5,000 units of recycled plastic per month. A critical machine is needed to manufacture the printers. Each unit of regular requires 5 units of time in this machine, and each unit of high-speed requires 3 units of time. The total time available in this machine per month is 15,000 units. Let $X_{1}$ be the number of units of regular produced per month and $X_{2}$ the number of units of high-speed produced per month. The appropriate constraint/s will be

Constraint A: $2 X_{1}+2 X_{2} \leq 14$ Constraint B: $4 X_{1}+10 X_{2} \leq 40$ All variables are required to be non-negative. Let the objective function be Max: ${ }^{2 X_{1}+3 X_{2}}$ . The optimal solution and the corresponding objective function value to this problem (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $X_{2}$ ) will be

Consider the following constraints and choose the correct answer: Constraint A: $2 X_{1}^{2}+3 X_{2} \leq 100$ Constraint B: $2 X_{1}+3 X_{2} \leq 1000$

Constraint A: $-2 X_{1}+2 X_{2} \leq 14$ Constraint B: $4 X_{1}-10 X_{2} \leq 40$ All variables are required to be non-negative. Let the objective function be Max: $2 X_{1}+3 X_{2}$ . The optimal solution and the corresponding objective function value to this problem (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $X_{2}$ ) will be

Per unit contribution for a new cooker goes down by $5 \%$ as a salesman tries harder and harder to sell the cooker. For this situation

If a graphically solvable linear program has multiple optimal solutions, it implies that two or more corner points are optimal.

XYZ Inc. produces two types of printers - regular and high-speed. Net contribution is $\$ 50.00$ per unit from regular and $\$ 70.00$ per unit from high-speed. Regular uses 2 units of recycled plastic per unit, and high-speed uses 1 unit of recycled plastic per unit of production. XYZ is committed to using at least 5,000 units of recycled plastic per month. A critical machine is needed to manufacture the printers. Each unit of regular requires 10 units of time in this machine and each unit of high-speed requires 3 units of time in this machine. The total time available in this machine per month is 15,000 units. Let $X_{1}$ be the number of units of regular produced per month and $X_{2}$ the number of units of high-speed produced per month. Imposing both of these constraints and non-negativity constraints, the optimal solution to this problem (assuming the first number in parenthesis is $X_{1}$ and the second number in parenthesis is $X_{2}$ ) will be

XYZ Inc. produces two types of paper towels-regular and super-soaker. Manufacturing has imposed a constraint that the total monthly production of regular should be at least as many as the monthly production of super-soakers. Let $X_{1}$ be the number of units of regular produced per month and $X_{2}$ the number of units of super-soaker produced per month, the appropriate constraint/s will be

Constraint A: $3 X_{1}+X_{2} \geq 90$ Constraint B: $3 X_{1}+2 X_{2} \leq 120$ Constraint C: $X_{1}-X_{2} \geq 0$ All variables are required to be non-negative. Let the objective function be Max: $X_{1}+X_{2}$ The objective function value corresponding to the optimal solution will be