Exam 7: Integer Programming

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.

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)

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

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

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)

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.

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).

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)

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

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.

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

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.

Branching in the branch and bound method refers to

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.)

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

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

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.