Deck 3: Linear Programming: Basic Concepts and Graphical Solutions

Question 1

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

Answer 1

True

Question 2

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

Answer 2

False

Question 3

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.

Answer 3

True

Question 4

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

Question 5

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

Question 6

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

Question 7

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

Question 8

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

Question 9

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.

Question 10

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.

Question 11

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

Question 12

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.

Question 13

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.

Question 14

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.

Question 15

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.

Question 16

Assumptions of linear programming include

A) linearity
B) additivity
C) divisibility
D) certainty
E) all of the above

Question 17

Divisibility assumption in linear programming implies

A) resources can be divided among products
B) products can be divided among customers
C) decision variables may take on integer values
D) decision variables may take on fractional values

Question 18

Additivity assumption in linear programming implies

A) resources can be added together to generate products
B) products can be added at any time
C) resource usage for each level of decision variable can be added up to get total usage
D) total profit for each decision variable can be added up to get total profit for the model
E) C and D

Question 19

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

A) a linear programming model will be appropriate
B) an integer linear programming will be appropriate
C) an integer linear program may be appropriate
D) a linear program will not be appropriate
E) none of the above

Question 20

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

A) constraint A can be used in a linear program
B) constraint $\mathrm{B}$ can be used in a linear program
C) neither can be used in a linear program
D) both may be used in a linear program
E) both may be used in an integer linear program, but not in linear program

Question 21

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

A) constraint A can be used in a linear program
B) constraint B can be used in a linear program
C) neither can be used in a linear program
D) both may be used in a linear program
E) both may be used in an integer linear program, but not in a linear program

Question 22

Certainty assumption in linear programming implies

A) available resources, profit and other coefficients are known with certainty
B) all constraints on the system have been included in the model.
C) A and B
D) neither A nor B
E) the right problem has been formulated with certainty

Question 23

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

A) $X_{1} \leq 2 X_{2}$
B) $2 X_{1} \leq X_{2}$
C) $X_{1} \leq 0.5 X_{2}$
D) $X_{1}-X_{2} \leq 0$
E) $X_{1}-0.5 X_{2} \geq 0$

Question 24

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

A) $X_{1} \geq 2 X_{2}$
B) $2 X_{1} \geq X_{2}$
C) $X_{1} \geq 0.5 X_{2}$
D) $X_{1}-X_{2} \leq 0$
E) $X_{1}-X_{2} \geq 0$

Question 25

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

A) $2 X_{1} \geq 3 X_{2}$
B) $2 X_{1} \leq 3 X_{2}$
C) $2 X_{1}=3 X_{2}$
D) $2 X_{1}-3 X_{2}=0$
E) C or D, since they both mean the same thing

Question 26

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

A) $2 X_{1}+3 X_{2}=10000$
B) $180 X_{1}+80 X_{2} \geq 360$
C) $2 X_{1}+3 X_{2} \geq 10000$
D) $2 X_{1}=3 X_{2}$
E) A, B, and C

Question 27

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

A) $2 X_{1}+3 X_{2} \geq 8000$
B) $2 X_{1}+3 X_{2} \leq 10000$
C) $2 X_{1}+3 X_{2}=10000$
D) A or B but not both
E) A and B

Question 28

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

A) $2 X_{1}+X_{2}=5000$
B) $2 X_{1}+X_{2} \leq 5000$
C) $5 X_{1}+3 X_{2} \leq 10000$
D) A and C
E) B and C

Question 29

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

A) $2 X_{1}+X_{2} \geq 5000$
B) $2 X_{1}+X_{2} \leq 5000$
C) $5 X_{1}+3 X_{2} \leq 15000$
D) A and C
E) B and C

Question 30

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 $X_{2}$ )

A) $(0,0)$
B) $(2000,0)$
C) none exists
D) $(0,5000)$
E) $(2500,0)$

Question 31

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 $X_{2}$ )

A) $(0,0)$
B) $(2000,0)$
C) none exists
D) $(0,5000)$
E) $(1500,0)$

Question 32

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 $X_{2}$ )

A) $(0,0)$
B) $(2500,0)$
C) none exists
D) $(0,5000)$
E) $(1500,0)$

Question 33

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

A) $(0,0)$
B) $(2500,0)$
C) none exists
D) $(0,5000)$
E) $(1500,0)$

Question 34

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

A) 350000
B) 125000
C) none exists
D) 250000
E) 35000

Question 35

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

A) is closed
B) open and hence does not have an optimal solution
C) open but has an optimal solution
D) non-existent (any objective with this as the constraint region will be infeasible)
E) is closed and will have an optimal solution for any linear objective function

Question 36

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

A) is closed
B) open and hence does not have an optimal solution
C) open but has an optimal solution
D) non-existent (any objective with this as the constraint region will be infeasible)
E) is closed and will have an optimal solution for any linear objective function

Question 37

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

A) $(0,0),(0,7),(7,0)$
B) $(0,0),(0,4),(7,0)$
C) $(0,0),(0,4),(10,0)$
D) $(0,0),(0,4),(5,2),(10,0)$
E) $(0,0),(0,4),(5,2),(7,0)$

Question 38

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

A) $(0,0),(0,7),(7,0)$
B) $(0,0),(0,4),(7,0)$
C) $(0,0),(0,4),(10,0)$
D) $(0,0),(0,4),(5,2),(10,0)$
E) $(0,0),(0,4),(5,2),(7,0)$

Question 39

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

A) $(0,0),(0,7),(7,0)$
B) $(0,0),(0,4),(7,0)$
C) $(0,0),(0,4),(10,0)$
D) $(0,0),(0,4),(5,2),(10,0)$
E) $(0,0),(0,4),(5,2),(7,0)$

Question 40

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

A) $(0,0),(0,7),(7,0)$
B) $(0,0),(0,4),(7,0)$
C) $(0,0),(0,4),(5,2),(10,0)$
D) $(0,0),(0,4),(5,2),(7,0)$
E) none of the above

Question 41

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

A) $(0,0),(0,7),(7,0)$
B) $(0,0),(0,4),(7,0)$
C) $(0,0),(0,4),(5,2),(10,0)$
D) $(0,0),(0,4),(5,2),(7,0)$
E) none of the above

Question 42

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

A) $(0,0)$ and 0
B) $(0,4)$ and 18
C) $(5,2)$ and 16
D) $(0,7)$ and 21
E) $(10,0)$ and 20

Question 43

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

A) $(0,0)$ and 0
B) $(0,4)$ and 18
C) $(5,2)$ and 16
D) $(0,7)$ and 21
E) $(10,0)$ and 20

Question 44

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

A) $(0,0)$ and 0
B) the problem is unbounded
C) $(5,2)$ and 16
D) $(0,7)$ and 21
E) $(10,0)$ and 20

Question 45

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

A) $(0,0)$ and 0
B) $(0,4)$ and 18
C) problem is infeasible
D) solution is unbounded
E) $(10,0)$ and 20

Question 46

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

A) $(0,0)$ and 0
B) $(0,4)$ and 4
C) $(5,2)$ and 7
D) $(0,7)$ and 7
E) C and D

Question 47

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 $\left.X_{2}\right)$

A) $(0,0)$ and $(0,75)$
B) $(0,0)$ and $(-30,0)$
C) $(54,48)$ and $(150,0)$
D) $(0,-60)$ and $(0,90)$
E) A and B

Question 48

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

A) $(0,0)$ and 0
B) $(30,0)$ and 60
C) $(54,48)$ and -36
D) $(150,0)$ and 300
E) $(0,90)$ and -270

Question 49

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

A) $(0,0)$ and 0
B) $(30,0)$ and 60
C) $(54,48)$ and -36
D) $(150,0)$ and 300
E) $(0,90)$ and -270

Question 50

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

A) $(24,24)$
B) $(0,60)$
C) $(20,30)$
D) $(22.5,22.5)$
E) $(0,90)$

Question 51

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

A) 48
B) 60
C) 50
D) 45
E) 90

Question 52

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

A) $(24,24)$
B) $(0,60)$
C) $(20,30)$
D) $(22.5,22.5)$
E) $(0,90)$

Question 53

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

A) 48
B) 60
C) 50
D) 45
E) 90

Question 54

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

A) constraint A
B) constraint B
C) constraint $\mathrm{C}$
D) both $\mathrm{A}$ and $\mathrm{B}$
E) no redundant constraint

Question 55

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

A) constraint A
B) constraint B
C) constraint $\mathrm{C}$
D) both $\mathrm{A}$ and $\mathrm{B}$
E) no non-binding constraint

Question 56

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.

Question 57

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.

Question 58

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.

Question 59

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.