Exam 3: Linear Programming: Basic Concepts and Graphical Solutions

John has two midterm exams-one in English and the other in Introduction to Business-coming up exactly 4 days from now. He figures that he has a maximum of 30 hours that could be spent preparing for these exams. For every hour spent on English he expects to get 5 points. For every hour spent on Introduction to Business he expects to get 6 points. He needs to score a minimum of 50 points in English. He does not have to get more than 84 points in Introduction to Business to hit his target grade. He values the score he gets in English at $80 \%$ of the value he assigns to Introduction to Business. Formulate a linear program that maximizes the sum of properly weighted scores in both midterms, while satisfying the constraints. Solve the problem using the graphical method.

Constraint A: $-2 X_{1}+2 X_{2} \geq 14$ 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

Additivity assumption in linear programming implies

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 optimal solution (assuming the first number in parenthesis is $X_{1}$ and the second number in the parenthesis is $X_{2}$ ) will be

In a graphically solvable linear program, if one of the constraints is changed from " $\leq$ type" to "= type", all other things remaining the same, then the optimal solution will never change.

Certainty assumption in linear programming implies

XYZ Inc. produces two types of paper towels-regular and super-soaker. Manufacturing has imposed a constraint that the total monthly production of regular and super-soaker should be in the ratio of 2:3. 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

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 10 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: $2 X_{1}+3 X_{2} \leq 90$ Constraint B: $2 X_{1}-3 X_{2} \leq 120$ The feasible region with these two constraints and non-negativity constraints

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. The total amount of recycled plastic available per month is 5,000. 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 10,000 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

Assumptions of linear programming include

XYZ Inc. produces two types of printers, called 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 10,000 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 $X_{2}$ )

Kathy is on a diet, which allows her to eat Food A (a tonic) and Food B (an elixir). Food A contains 80 grams of protein, 180 grams of vitamins, and 100 calories per ounce. Food B contains 20 grams of protein, 80 grams of vitamins, and 200 calories per ounce. The minimum nutritional requirements are 120 grams of protein and 360 grams of vitamins. The daily intake should not exceed 1600 calories. Each pound of Food A costs 4.80; each pound of Food B costs $\$ 2.40$ . Formulate a linear program specifying the decision variables, constraints, and the objective function that would meet Kathy's nutritional requirements at minimal cost. Solve the problem using the graphical method.

If a graphically solvable linear program is infeasible, that implies there is no point in the graph satisfying all constraints.

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

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

All linear programming problems with only two variables may be solved using graphical method.

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 . They also have a binding contract to use at least 8,000 units of recycled paper per month with a local pollution control organization. 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

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

In any graphically solvable linear program, no point other than one of the corner points of the feasible region can ever be optimal.