Deck 7: Integer Programming

Question 1

A two-variable pure integer programming problem cannot be solved by the graphical method.

Answer 1

False

Question 2

A two-variable pure integer programming problem can only be solved by the simplex method.

Answer 2

False

Question 3

A feasible solution to a two-variable pure integer programming problem can always be found by first solving the corresponding linear programming problem (i.e. problem obtained by ignoring the integrality constraints) and by rounding down the fractional values in the optimal solution to the linear programming problem.

Answer 3

False

Question 4

A feasible solution to a two-variable pure integer programming problem can always be found by first solving the corresponding linear programming problem (i.e. problem obtained by ignoring the integrality constraints) and by rounding to the nearest integer all fractional values in the optimal solution to the linear programming problem.

Question 5

Since the number of feasible solutions to a pure integer programming problem is a lot less than the number of feasible solutions to the corresponding linear programming problem (i.e. problem obtained by ignoring the integrality constraints), a pure integer programming problem must be easier to solve.

Question 6

Problem $\mathrm{A}$ is a two-variable linear programming problem with a maximization objective function. Problem B is a two-variable pure integer programming problem obtained from Problem A by requiring the variables to be integers and leaving all other things unchanged. If Problem A has an optimal solution, then Problem B must have an optimal solution.

Question 7

Problem $\mathrm{A}$ is a two-variable linear programming problem with a maximization objective function. Problem B is a two variable pure integer programming problem obtained from Problem A by requiring the variables to be integers and leaving all other things unchanged. If Problem A has an optimal solution with integer values for the variables, then Problem B must have an optimal solution.

Question 8

Problem A is a two-variable linear programming problem with a maximization objective function. Problem B is a two variable pure integer programming problem obtained from Problem A by requiring the variables to be integers and leaving all other things unchanged. If Problem A has an optimal solution with integer objective function value, then Problem B must have an optimal solution.

Question 9

Problem $\mathrm{A}$ is a two-variable linear programming problem with a maximization objective function. Problem B is a two-variable pure integer programming problem obtained from Problem A by requiring the variables to be integers and leaving all other things unchanged. If Problem A has an optimal solution with integer values for the variables, then the same solution must be optimal for Problem B as well.

Question 10

Sensitivity analysis on the RHS of integer programming problems is relatively easy.

Question 11

Sensitivity analysis on the objective function coefficients is very hard, if not impossible.

Question 12

Fixed charge problems can be formulated as linear programs, though a 0-1 integer program will yield the answer faster.

Question 13

A fixed charge problem typically uses $M$ , which stands for a very large positive number in its formulation.

Question 14

A vehicle routing problem is a special case of the well-known traveling salesman problem

Question 15

A vehicle routing problem is a special case of the well-known fixed charge problem.

Question 16

A common feature of set covering, knapsack, and location problems is that all of them use 0-1 variables.

Question 17

A common feature of set covering, knapsack, and location problems is that all of them use mixed integer programming formulation requiring no $0-1$ variables.

Question 18

If a problem contains data on profit as well as cost, it has to be broken down into two problems-one involving cost minimization and the other involving profit maximization.

Question 19

If a problem contains data on profit as well as cost, it cannot be formulated as an integer programming problem. We have to pick one or the other and then do the formulation.

Question 20

Set covering problem talks about a salesman and how he/she could cover a set of sales territories at minimal cost.

Question 21

A fixed charge problem models shops that incur a fixed cost for producing any positive quantity of a product and incur 0 cost if nothing is produced.

Question 22

In general, we consider more feasible solutions in solving integer linear programming problems. as compared to solving linear programming problems by simplex method.
1.

Question 23

For a typical integer programming problem, the number of feasible solutions

A) increases linearly
B) increases exponentially
C) decreases exponentially
D) decreases linearly

Question 24

For a pure 0-1 integer programming problem with 3 variables, the maximum number of potential solutions is

A) 9
B) 27
C) 16
D) 8

Question 25

Branching in the branch and bound method refers to

A) adding a constraint
B) removing a constraint
C) either adding or removing a constraint
D) removing a constraint and extending the feasible set

Question 26

In modeling a shopping mall construction problem, there are four potential locations giving rise to four 0-1 decision variables, denoted as $X_{1}, X_{2}, X_{3}, X_{4}$ , which take a value of 1 if a mall is constructed and 0 otherwise. Identify the correct set of constraint/s to satisfy the following condition/s: at most only one mall may be constructed among locations 1 and 3

A) $\mathrm{X}_{1}+\mathrm{X}_{3} \leq$ 1
B) $\mathrm{X}_{1}+\mathrm{X}_{3} \geq$ 1
C) $X_{1}+X_{3}<$ 1
D) $X_{1}+X_{3}=$ 1

Question 27

In modeling a shopping mall construction problem, there are four potential locations giving rise to four 0-1 decision variables, denoted as $\mathrm{X}_{1}, \mathrm{X}_{2}, \mathrm{X}_{3}, \mathrm{X}_{4}$ , which take a value of 1 if a mall is constructed and 0 otherwise. Identify the correct set of constraint/s to satisfy the following condition/s: at least one mall may be constructed among locations 1 and 3

A) $\mathrm{X}_{1}+\mathrm{X}_{3} \leq$ 1
B) $\mathrm{X}_{1}+\mathrm{X}_{3} \geq$ 1
C) $X_{1}+X_{3}>$ 1
D) $X_{1}+X_{3}=$

Question 28

In modeling a shopping mall construction problem, there are four potential locations giving rise to four 0-1 decision variables, denoted as $X_{1}, X_{2}, X_{3}, X_{4}$ ,which take a value of 1 if a mall is constructed and 0 otherwise. Identify the correct set of constraint/s to satisfy the following condition/s: exactly one mall should be constructed among locations 1 and 3

A) $\mathrm{X}_{1}+\mathrm{X}_{3} \leq$ 1
B) $\mathrm{X}_{1}+\mathrm{X}_{3} \geq$ 1
C) $X_{1}+X_{3}>$ 1
D) $X_{1}+X_{3}=$ 1

Question 29

In modeling a shopping mall construction problem, there are four potential locations giving rise to four 0-1 decision variables denoted as $X_{1}, X_{2}, X_{3}, X_{4}$ , which take a value of 1 if a mall is constructed and 0 otherwise. Identify the correct set of constraint/s to satisfy the following condition/s: if a mall is constructed in location 2, then a mall should be constructed in location 4

A) $\mathrm{X}_{2}-\mathrm{X}_{4} \leq$ 0.
B) $-X_{2}+X_{4} \geq$ 0.
C) $\mathrm{X}_{2}-\mathrm{X}_{4} \leq$ 0
D) $-\mathrm{X}_{2}+\mathrm{X}_{4} \leq$ 0

Question 30

In modeling a traveling salesman problem, let $X_{\mathrm{ijk}}$ be a 0-1 decision variable, which takes a value of 0 if in the kth leg the tour corresponding to the solution does not go from node $\mathrm{i}$ to node $\mathrm{j}$ . $\mathrm{X}_{\mathrm{ijk}}$ is set to be equal to 1 if the tour goes from node $\mathrm{i}$ to node $\mathrm{j}$ in the kth leg. In a problem with just 4 nodes and without loss of generality, assume that the tour starts and ends in node 1. Mark the correct constraint to make sure that the tour starts in node 1

A) $\mathrm{X}_{121}+\mathrm{X}_{131}+\mathrm{X}_{141} \leq 1$
B) $X_{121}+X_{131}+X_{141} \geq 1$
C) $X_{121}+X_{131}+X_{141}=1$
D) $X_{214}+X_{314}+X_{414}=1$

Question 31

In modeling a traveling salesman problem, let $X_{\mathrm{ijk}}$ be a 0-1 decision variable, which takes a value of 0 if in the kth leg the tour corresponding to the solution does not go from node $\mathrm{i}$ to node $\mathrm{j}$ . $\mathrm{X}_{\mathrm{ijk}}$ is set to be equal to 1 if the tour goes from node $i$ to node $\mathrm{j}$ in the kth leg. In a problem with just 4 nodes and without loss of generality, assume that the tour starts and ends in node 1. Mark the correct constraint to make sure that the tour ends in node 1

A) $\mathrm{X}_{214}+\mathrm{X}_{314}+\mathrm{X}_{414} \leq 1$
B) $X_{214}+X_{314}+X_{414} \geq 1$
C) $X_{214}+X_{314}+X_{414}=1$
D) $\mathrm{X}_{121}+\mathrm{X}_{131}+\mathrm{X}_{141}=1$

Question 32

In modeling a traveling salesman problem, let $X_{i j k}$ be a 0-1 decision variable, which takes a value of 0 if in the kth leg the tour corresponding to the solution does not go from node $\mathrm{i}$ to node $\mathrm{j}$ . $\mathrm{X}_{\mathrm{ijk}}$ is set to be equal to 1 if the tour goes from node $\mathrm{i}$ to node $\mathrm{j}$ in the kth leg. In a problem with just 4 nodes and without loss of generality, assume that the tour starts and ends in node 1 . If, in a particular solution, $X_{213}=1$ , it means that the tour corresponding to the solution

A) goes from node 3 to 1 in the 2nd leg
B) goes from node 2 to 1 in the 3rd leg
C) goes from node 1 to 2 in the 3rd leg
D) goes from node 1 to 3 in the 2 nd leg

Question 33

An airport limousine service which parks all its limos at the airport can minimize its cost by using a proper order to pick up passengers from their houses and return to the airport using a

A) set covering problem
B) traveling salesman problem
C) knapsack problem
D) fixed charge problem

Question 34

An Avon lady carrying her tote containing makeup materials can maximize her profit from one trip to the rural Mississippi hinterland if she models the process of loading her bag (with the "right" materials having maximum profitability per unit volume) by using a

A) set covering problem
B) traveling salesman problem
C) knapsack problem
D) fixed charge problem

Question 35

A problem that has fixed cost as well as variable cost per unit, depending on the level of that activity, is usually formulated as a

A) set covering problem
B) knapsack problem
C) fixed charge problem
D) traveling salesman problem

Question 36

Problems with variables that are yes-no types are called

A) pure integer programming problems
B) linear programming problems
C) yes-no type problems
D) $0-1$ problems
E) binary linear problems

Question 37

Problems in which some variables are required to be integers are called

A) pure integer programming problems
B) mixed integer programming problems
C) linear and integer programming problems
D) $0-1$ problems
E) yes-no type problems

Question 38

Problem $\mathrm{A}$ is a two-variable linear programming problem with a maximization objective function. Problem B is a two-variable pure integer programming problem obtained from Problem A by requiring the variables to be integers and leaving all other things unchanged. Assuming that both problems have optimal solutions, the objective function value of problem A will always be the objective function value of Problem B.

A) greater than
B) less than
C) equal to
D) greater than or equal to
E) less than or equal to

Question 39

Problem A is a two-variable linear programming problem with a minimization objective function. Problem B is a two-variable pure integer programming problem obtained from Problem A by requiring the variables to be integers and leaving all other things unchanged. Assuming that both problems have optimal solutions, the objective function value of problem A will always be the objective function value of Problem B.

A) greater than
B) less than
C) equal to
D) greater than or equal to
E) less than or equal to

Question 40

Problem $\mathrm{A}$ is a two-variable linear programming problem with a minimization objective function. Problem B is a two-variable mixed integer programming problem obtained from Problem A by requiring one of the variables to be integer and leaving all other things unchanged. Assuming that both problems have optimal solutions and that Problem A has a unique non-integer optimal solution, the objective function value of problem A will always be the objective function value of Problem B.

A) greater than
B) less than
C) equal to
D) greater than or equal to
E) less than or equal to

Question 41

A retail outlet wants to replace a broken ice cream vending machine with one of two models. The purchase of model 1 is captured by variable X1. Taking a value of 1 and 0 would represent that machine 1 is not purchased. X2 is similarly defined. One of the constraints for this problem would be

A) $\mathrm{X}_{1}+\mathrm{X}_{2}<$ 1
B) $\mathrm{X}_{1}+\mathrm{X}_{2} \leq$ 1
C) $\mathrm{X}_{1}+\mathrm{X}_{2} \geq$ 1
D) $X_{1}+X_{2}=$ 1
E) $X_{1}+X_{2}>$ 1

Question 42

A retail outlet wants to replace a broken ice cream vending machine with one of two models. Purchase of model 1 machine is captured by variable $X_{1}$ taking a value of 1 and 0 if not purchased. $X_{2}$ is similarly defined. They would also want to allow for the possibility of getting rid of the old machine with no replacement. One of the constraints for this problem would be

A) $\mathrm{X}_{1}+\mathrm{X}_{2}<$ 1
B) $\mathrm{X}_{1}+\mathrm{X}_{2} \leq$ 1
C) $\mathrm{X}_{1}+\mathrm{X}_{2} \geq$ 1
D) $X_{1}+X_{2}=$ 1
E) $X_{1}+X_{2}>$ 1

Question 43

A retail outlet wants to replace a broken ice cream vending machine with one of two models. The purchase of model 1 is captured by variable X1, taking a value of 1 and 0 if not purchased. X2 is similarly defined. They would not be opposed to buying both machines, though buying one of them is a must. One of the constraints for this problem would be

A) $\mathrm{X}_{1}+\mathrm{X}_{2}<$ 1
B) $\mathrm{X}_{1}+\mathrm{X}_{2} \leq$ 1
C) $X_{1}+X_{2} \geq$ 1
D) $X_{1}+X_{2}=$ 1
E) $\mathrm{X}_{1}+\mathrm{X}_{2}>$ 1

Question 44

A shopping complex is considering building a self-service vending machine island. The mall can buy a vending machine for dispensing various types of ready-to-pop corn. Purchase of this machine is captured by variable $\mathrm{X} 1$ , taking a value of 1 and 0 if not purchased. There is also a machine available for popping any type of corn vended by the vending machine. Purchase of this machine is captured by variable X2, taking a value of 1 and 0 if not purchased. Assume that buying just one of these machines does not do much good. That is, either the mall should buy both or buy neither. The constraint that implements this will be

A) $\mathrm{X}_{1}-\mathrm{X}_{2}<$ 0
B) $\mathrm{X}_{1}-\mathrm{X}_{2} \leq$ 0
C) $\mathrm{X}_{1}-\mathrm{X}_{2} \geq$ 0
D) $X_{1}-X_{2}=$ 0
E) $X_{1}-X_{2}>$

Question 45

A shopping complex is considering building a self-service vending machine island. The mall can buy a vending machine for dispensing various types of ready-to-pop corn. Purchase of this machine is captured by variable $\mathrm{X} 1$ , taking a value of 1 and 0 if not purchased. There is also a machine available for popping any type of corn vended by the vending machine. Purchase of this machine is captured by variable X2, taking a value of 1 and 0 if not purchased. If they decide to buy the popcorn popper, then they must have the ready-to-pop corn vending machine. However, if they only buy the ready-to-pop corn vending machine, the customers can take the corn home or pop it in a microwave already available, if the popcorn popper is not purchased. The constraint that implements this will be

A) $X_{1}-X_{2}<$ 0
B) $\mathrm{X}_{1}-\mathrm{X}_{2} \leq$ 0
C) $\mathrm{X}_{1}-\mathrm{X}_{2} \geq$
D) $\mathrm{X}_{1}-\mathrm{X}_{2}=$ 0
E) $X_{1}-X_{2}>$

Question 46

A Mississippi farmer is considering adding a special breed of bull to his farm. Let $X_{1}=1$ imply that the bull will be purchased and 0 otherwise. If it is purchased, then the pounds of organic soy purchased per day (denoted by $\mathrm{X}_{2}$ ) and the pounds of organic corn purchased per day (denoted by $\mathrm{X}_{3}$ ) should be at least 50. If the bull is not purchased, this constraint will be immaterial. The constraint that implements this would be ( $\mathrm{M}$ stands for a very big positive number and assume non-negative variables)

A) $X_{2}+X_{3} \geq 50+M * X_{1}$
B) $X_{2}+X_{3} \geq 50+M *\left(1-X_{1}\right)$
C) $X_{2}+X_{3} \geq 50 * X 1$
D) $X_{2}+X_{3} \geq 50 *\left(1-X_{1}\right)$

Question 47

A Mississippi farmer is considering adding a special breed of bull to his farm. Let $X_{1}=1$ imply that the bull will be purchased and 0 otherwise. If it is purchased, then the pounds of organic soy purchased per day (denoted by $\mathrm{X}_{2}$ ) and the pounds of organic corn purchased per day (denoted by $\mathrm{X}_{3}$ ) should be at most 50. If the bull is not purchased, this constraint will be immaterial. The constraint that implements this would be ( $\mathrm{M}$ stands for a very big positive number and assume non-negative variables)

A) $X_{2}+X_{3} \leq 50+M^{*} X_{1}$
B) $X_{2}+X_{3} \leq 50+M *\left(1-X_{1}\right)$
C) $X_{2}+X_{3} \leq 50 * X_{1}$
D) $X_{2}+X_{3} \leq 50 *\left(1-X_{1}\right)$

Question 48

A Mississippi farmer is considering adding a special breed of bull to his farm. Let $X_{1}=1$ imply that the bull will be purchased and 0 otherwise. If it is purchased, then the pounds of organic soy purchased per day (denoted by $\mathrm{X}_{2}$ ) and the pounds of organic corn purchased per day (denoted by $\mathrm{X}_{3}$ ) should be at least 50 . If the bull is not purchased, then $2^{*}$ the pounds of organic soy purchased $-3^{*}$ the pounds of organic corn purchased should be at least 30 . The constraint/s that implements this would be (M stands for a very big positive number and assume non-negative variable)

A) $X_{2}+X_{3} \geq 50+M^{*}\left(1-X_{1}\right)$ and $2 X_{2}-3 X_{3} \geq 30+M^{*} X_{1}$
B) $X_{2}+X_{3} \geq 50 X_{1}$ and $2 X_{2}-3 X_{3} \geq 30\left(1-X_{1}\right)$
C) $X_{2}+X_{3} \leq 50 * X_{1}$ and $2 X_{2}-3 X_{3} \leq 30\left(1-X_{1}\right)$
D) $X_{2}+X_{3} \leq 50+M^{*} X_{1}$ and $2 X_{2}-3 X_{3} \leq 30+M^{*}\left(1-X_{1}\right)$

Question 49

A Mississippi farmer is considering adding a special breed of bull to his farm. Let $X_{1}=1$ imply that the bull will be purchased and 0 otherwise. If it is purchased, then the pounds of organic soy purchased per day (denoted by $\mathrm{X}_{2}$ ) and the pounds of organic corn purchased per day (denoted by $\mathrm{X}_{3}$ ) should be at most 50. If the bull is not purchased, then $2 *$ the pounds of organic soy purchased $-3^{*}$ the pounds of organic corn purchased should be at most 30. The constraint/s that implements this would be (M stands for a very big positive number and assume non-negative variables)

A) $X_{2}+X_{3} \geq 50+M^{*} X_{1}$ and $2 X_{2}-3 X_{3} \geq 30+M^{*}\left(1-X_{1}\right)$
B) $X_{2}+X_{3} \geq 50 X_{1}$ and $2 X_{2}-3 X_{3} \geq 30\left(1-X_{1}\right)$
C) $X_{2}+X_{3} \leq 50 * X_{1}$ and $2 X_{2}-3 X_{3} \leq 30\left(1-X_{1}\right)$
D) $X_{2}+X_{3} \leq 50+M^{*}\left(1-X_{1}\right)$ and $2 X_{2}-3 X_{3} \leq 30+M^{*} X_{1}$

Question 50

A Mississippi farmer is considering producing a special kind of $0.25 \%$ fat milk. The potential buyer demands a minimum lot size of 1000 gallons. Let $X_{1}=$ represent the gallons of $0.25 \%$ milk produced. Let $\mathrm{Y}_{1}$ be a 0 or 1 variable; if it is 0 , it signifies that the $0.25 \%$ milk was not produced at all. If it is 1 , it signifies that 1000 or more gallons of $0.25 \%$ milk was produced. The constraint/s that implements this would be ( $\mathrm{M}$ stands for a very big positive number and assume non-negative variables.)

A) $X_{1} \geq 1000$
B) $\mathrm{X}_{1} \geq 1000 \mathrm{Y}_{1}$
C) $X_{1} \geq 1000+M^{*} Y_{1}$
D) $X_{1} \geq 1000+M^{*}\left(1-Y_{1}\right)$

Question 51

In a plant location problem with 5 potential locations, each with an associated 0-1 decision variable, $X_{1}$ , $X_{2}, X_{3}, X_{4}$ ,and $X_{5}$ , which take a value of 1 if a plant is located in that location and 0 otherwise, the constraint to make sure that at least 3 plants are put up will be

A) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}>3.0$
B) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5} \geq 3.0$
C) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}=3.0$
D) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5} \leq 3.0$
E) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}<3.0$

Question 52

In a plant location problem with 5 potential locations, each with an associated $0-1$ decision variable, $X_{1}$ , $X_{2}, X_{3}, X_{4}$ ,and $X_{5}$ , which take a value of 1 if a plant is located in that location and 0 otherwise, the constraint to make sure that at most 3 plants are put up will be

A) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}>3.0$
B) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5} \geq 3.0$
C) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}=3.0$
D) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5} \leq 3.0$
E) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}<3.0$

Question 53

In a plant location problem with 5 potential locations, each with an associated 0-1 decision variable, $X_{1}$ , $\mathrm{X}_{2}, \mathrm{X}_{3}, \mathrm{X}_{4}$ , and $\mathrm{X}_{5}$ , which take a value of 1 if a plant is located in that location and 0 otherwise, the constraint to make sure that exactly 3 plants are put up will be

A) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}>3.0$
B) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5} \geq 3.0$
C) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}=3.0$
D) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5} \leq 3.0$
E) $X_{1}+X_{2}+X_{3}+X_{4}+X_{5}<3.0$

Question 54

Wilkinson Auto Dealership is contemplating selling one or more of the following types of automobilessedans, SUV's, and trucks. There is a fixed license fee per year for doing business in each line, $\$ 100,000$ , $\$ 250,000$ , and $\$ 50,000$ , respectively. The profit contribution exclusive to the fixed cost is $\$ 250.00$ per unit for sedans, $\$ 700.00$ per unit for SUV's, and $\$ 150.00$ per unit for trucks. The company is planning the placement of orders with the manufacturer for next year. Dealer preparation takes 2 hrs/sedan, 3.0 hrs/SUV, and 1.5 hrs/truck. They have 4400 hrs of preparation time next year. Sedans take 1 unit of space, SUV's take 1.5 units of space, and trucks take 1.1 units of space. 1200 units of space are available. How many sedans, SUV's, and trucks should be ordered in order to maximize total profit contribution less fixed costs incurred? Define the decision variables, constraints, and the objective function for this problem. (Allow the order quantities to be fractional).

Question 55

A cargo plane has two compartments for storing cargo, front and back. These compartments have capacity limits on both weight and space, as summarized below.

They have three available cargos for an upcoming flight. The details are given below:

For maintaining balance, the total weight of the cargo loaded in the back must be more than the total cargo loaded in the front. Each cargo must be accepted in full and loaded entirely in front or back. The objective is to determine which must be accepted and where it should go in the plane, so as to maximize total profit. Formulate this as an integer linear program.

Question 56

Market research corporation of Toledo Inc. (MRCT) has to finalize recommendation for the 2006 advertising campaign for Miracle Motors Inc. (MM), a major hydrogen-powered automobile manufacturer. MRCT has developed 5 distinct campaigns for MM Inc. MM Inc. has 7 distinct customer segments, and it wants all of its customers to be "hit" by at least one of its campaigns. It wants to minimize the total cost of reaching its customer base through these campaigns.

Question 57

Clean Energy Inc. has a planning horizon of 3 years and has 5 potential projects. Each year, the outlay required by each project is known. The amount available for investment in any of these projects each year is also assumed to be known. Each project has an associated expected contribution to total sales whose net present value is also assumed to be known. Determine the project/s that should be undertaken so as to maximize the total net present value of sales subject to budget constraints of each year.

Question 58

A jumbo jet factory can produce two types of planes\_-UnitMach and BigMach. UnitMach takes 6 man weeks of labor and 7 units of critical machine time. BigMach takes 8 man weeks of labor and 12 units of critical machine time. The total man weeks of labor available during the planning horizon are 50; total units of machine time available during the planning horizon are 70 . Contribution of UnitMach is $\$ 10,000$ and BigMach is $\$ 12000$ . Assuming that integer quantities of products should be produced, find the total contribution maximizing solution.