Exam 5: Integer Programming

The college dean is deciding among three equally qualified (in their eyes, at least) candidates for his associate dean position. If this situation could be modeled as an integer program, the decision variables would be cast as 0-1 integer variables.

Saba conducts regular tours of his favorite city in the world, Paris. Each semester he selects among the finest students in the university and escorts them to the City of Lights. In addition to a world-class education on conducting business in Europe, he arranges a number of cultural outings for them to help them immerse themselves in all that France has to offer. He collects an extra $100 from each student for this purpose and limits his tour group to ten lucky individuals. Some of the events (and their prices) he proposes to the students include: Eiffel Tower visit, $40 per student, E Paris Sewer spelunking, $20 per student, S Half day passes to the Louvre, $60 per student, L Bon Beret tour, $50 per student, B So much to do and so little time! -If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a ________ constraint.

In choosing four electives from the dazzling array offered by the Decision Sciences Department next semester, the students that had already taken the management science class were able to craft a model using a ________ constraint.

In a total integer model, all decision variables have integer solution values.

Future Plastics manufactures plastic products for industrial use worldwide. In order to meet demand, they are considering setting up a facility in each region in order to lower transportation cost and to possibly avoid duties that could be imposed if the product is imported from another region. The disadvantage of this approach is that plants are sized to meet local demand and may not fully exploit economies of scale. Therefore, Future Plastics is also interested in determining the appropriate size of the facility to build in each location and are choosing between facilities with capacities of 5 or 10 million. The fixed costs of each facility as well as the cost of shipping between regions is shown in the table below. The decision variables are defined as follows: Xij = quantity shipped from supply region i to demand region j. i = 1, 2, 3, 4 and j = 1, 2, 3, 4. Yik = 1 if facility k is selected for supply region i; 0 otherwise, where i = 1, 2, 3, 4 for each supply region; k = 1 (low capacity facility) or 2 (high capacity facility) -Which of these constraints will ensure that a low capacity facility is not built in South America?

Types of integer programming models are ________.

Due to increased sales, a company is considering building three new distribution centers (DCs) to serve four regional sales areas. The annual cost to operate DC 1 is $500 (in thousands of dollars). The cost to operate DC 2 is $600 (in thousands of dollars.). The cost to operate DC 3 is $525 (in thousands of dollars). Assume that the variable cost of operating at each location is the same, and therefore not a consideration in making the location decision. The table below shows the cost ($ per item) for shipping from each DC to each region. Region 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3 The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. -Define the decision variables for this situation.

A rounded-down integer solution can result in a less than optimal solution to an integer programming problem.

In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot be selected. Which of the alternatives listed below correctly models this situation?

Saba conducts regular tours of his favorite city in the world, Paris. Each semester he selects among the finest students in the university and escorts them to the City of Lights. In addition to a world-class education on conducting business in Europe, he arranges a number of cultural outings for them to help them immerse themselves in all that France has to offer. He collects an extra $100 from each student for this purpose and limits his tour group to ten lucky individuals. Some of the events (and their prices) he proposes to the students include: Eiffel Tower visit, $40 per student, E Paris Sewer spelunking, $20 per student, S Half day passes to the Louvre, $60 per student, L Bon Beret tour, $50 per student, B So much to do and so little time! -Which constraint is most appropriate if the students can choose only three of these activities?

Future Plastics manufactures plastic products for industrial use worldwide. In order to meet demand, they are considering setting up a facility in each region in order to lower transportation cost and to possibly avoid duties that could be imposed if the product is imported from another region. The disadvantage of this approach is that plants are sized to meet local demand and may not fully exploit economies of scale. Therefore, Future Plastics is also interested in determining the appropriate size of the facility to build in each location and are choosing between facilities with capacities of 5 or 10 million. The fixed costs of each facility as well as the cost of shipping between regions is shown in the table below. The decision variables are defined as follows: Xij = quantity shipped from supply region i to demand region j. i = 1, 2, 3, 4 and j = 1, 2, 3, 4. Yik = 1 if facility k is selected for supply region i; 0 otherwise, where i = 1, 2, 3, 4 for each supply region; k = 1 (low capacity facility) or 2 (high capacity facility) -The constraint for the North American supply region is:

The Wiethoff Company has a contract to produce 10,000 garden hoses for a customer. Wiethoff has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same. Machine Fixed Cost to Setup Production Run Variable Cost per Hose Capacity 1 750 1.25 6000 2 500 1.50 7500 3 1000 1.00 4000 4 300 2.00 5000 -Write the objective function.

The production planner for Airbus showed his boss the latest product mix suggestion from their slick new linear programming model: 12.5 model 320s and 17.4 model 340s. The boss looked over his glasses at the production planner and reminded him that they had several half airplanes from last year's production rusting in the parking lot. No one, it seems, is interested in half of an airplane. The production planner whipped out his red pen and crossed out the .5 and .4, turning the new plan into 12 model 320s and 17 model 340s. This production plan is definitely feasible.

Consider a capital budgeting example with five projects from which to select. Let x1 = 1 if project a is selected, 0 if not, for a = 1, 2, 3, 4, 5. Projects cost $100, $200, $150, $75, and $300, respectively. The budget is $450. -Write the appropriate constraint for the following condition: If project 3 is chosen, project 4 must be chosen.

Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution to an integer programming problem.

If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a mutually exclusive constraint.

Saba conducts regular tours of his favorite city in the world, Paris. Each semester he selects among the finest students in the university and escorts them to the City of Lights. In addition to a world-class education on conducting business in Europe, he arranges a number of cultural outings for them to help them immerse themselves in all that France has to offer. He collects an extra $100 from each student for this purpose and limits his tour group to ten lucky individuals. Some of the events (and their prices) he proposes to the students include: Eiffel Tower visit, $40 per student, E Paris Sewer spelunking, $20 per student, S Half day passes to the Louvre, $60 per student, L Bon Beret tour, $50 per student, B So much to do and so little time! -What is the full set of constraints if the following situations occur? The Eiffel Tower visit needed to take place at the same time has the half day at the Louvre. Students taking the Paris Sewer tour had to wear the special sanitary beret available only from the Bon Beret tour. Saba applies for university tarvel funds and supplements the students' accounts with an extra $30 each.

Saba conducts regular tours of his favorite city in the world, Paris. Each semester he selects among the finest students in the university and escorts them to the City of Lights. In addition to a world-class education on conducting business in Europe, he arranges a number of cultural outings for them to help them immerse themselves in all that France has to offer. He collects an extra $100 from each student for this purpose and limits his tour group to ten lucky individuals. Some of the events (and their prices) he proposes to the students include: Eiffel Tower visit, $40 per student, E Paris Sewer spelunking, $20 per student, S Half day passes to the Louvre, $60 per student, L Bon Beret tour, $50 per student, B So much to do and so little time! -If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a ________ constraint.

The Wiethoff Company has a contract to produce 10,000 garden hoses for a customer. Wiethoff has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same. Machine Fixed Cost to Setup Production Run Variable Cost per Hose Capacity 1 750 1.25 6000 2 500 1.50 7500 3 1000 1.00 4000 4 300 2.00 5000 -Write a constraint to ensure that if machine 4 is used, machine 1 will not be used.

Consider the following integer linear programming problem: Max Z =   3x1 + 2x2 Subject to:  3x1 + 5x2 ? 30       5x1 + 2x2 ? 28         x1 ? 8        x1, x2 ? 0 and integer The solution to the linear programming formulation is: x1 = 5.714, x2 = 2.571. What is the optimal solution to the integer linear programming problem? State the optimal values of decision variables and the value of the objective function.