Exam 3: Linear Programming: Basic Concepts and Graphical Solutions

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. 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

Divisibility assumption in linear programming implies

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. 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

XYZ Inc. produces two types of paper towels-regular and super-soaker. Regular uses 2 units of recycled paper per unit of production, and super-soaker uses 3 units of recycled paper per unit of production. The total amount of recycled paper available per month is 10,000 . 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

If a linear program has a feasible solution with unbounded objective function value, then it must be an unbounded problem.

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

In any graphically solvable linear program, if two points are feasible, then any weighted average of the two points (where weights are non-negative and add up to 1.0) will also be feasible.

If a linear programming problem has redundant constraints, then the optimal solution without the redundant constraints will be the same as the optimal solution with the redundant constraints.

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: $X_{1}+X_{2}$ . 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

In a graphically solvable linear program with an optimal solution, if one of the constraints is changed from "= type" to " $\geq$ type", all other things remaining the same, the problem may become infeasible.

XYZ Inc. produces two types of paper towels-regular and super-soaker. Marketing has imposed a constraint that the total monthly production of regular should be no more than twice 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

If a graphically solvable linear program is unbounded, then the feasible region must be open at least in one direction.

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: $X_{1}+X_{2}$ The objective function value corresponding to the optimal solution will be

In a graphically solvable linear program with an optimal solution, if one of the constraints is changed from "= type" to " $\leq$ type", all other things remaining the same, the problem may become infeasible.

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 objective function value corresponding to 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

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

XYZ Inc. produces two types of exercise machines-Multi-purpose, which can be used for cardiac fitness, fat loss, weight reduction etc., and Cardiac, just for cardiac fitness. Each unit of Multi-purpose generates a contribution of $\$ 300.00$ , while each unit of Cardiac produces a contribution of $\$ 200.00$ . Each Multi-purpose requires 7 man-hours of assembly time and 2 man-hours of packing and shipping time. It also uses 5 pounds of specialized steel in limited supply. Each unit of Cardiac requires 5 man-hours of assembly time and 1 man-hour of packing and shipping time. It also uses 7 pounds of specialized steel in limited supply. Assembly man-hours available per month are 500; packing and shipping manhours available per month are 150 . The number of units of specialized steel available per month is 600 pounds. The sales department has imposed a restriction limiting the number of Multi-purpose machines produced to not exceed double the number of Cardiac machines produced per month. Formulate a linear programming model that would maximize profit subject to the constraints. Solve the problem using the graphical method.

All linear programming problems with only two variables and a maximum of 10 constraints may be solved using graphical method.

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. 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 $X_{2}$ ) is