Exam 10: Integer Programming, Goal Programming, and Nonlinear Programming

A model containing a linear objective function and linear constraints but requiring that one or more of the decision variables take on an integer value in the final solution is called ________

The constraint X₁ + X₂ ≤ 1 with 0 -1 integer programming allows for either X₁ or X₂ to be a part of the optimal solution, but not both.

A package express carrier is considering expanding the fleet of aircraft used to transport packages. There is a total of $220 million allocated for purchases. Two types of aircraft may be purchased - the C1A and the C1B. The C1A costs $25 million, while the C1B costs $18 million. The C1A can carry 60,000 pounds of packages, while the C1B can only carry 40,000 pounds of packages. The company needs at least eight new aircraft. In addition, the firm wishes to purchase at least twice as many C1Bs as C1As. Formulate this as an integer programming problem to maximize the number of pounds that may be carried.

A package express carrier is considering expanding the fleet of aircraft used to transport packages. Of primary importance is that there is a total of $350 million allocated for purchases. Two types of aircraft may be purchased - the C1A and the C1B. The C1A costs $25 million, while the C1B costs $18 million. The C1A can carry 60,000 pounds of packages, while the C1B can only carry 40,000 pounds of packages. Of secondary importance is that the company needs at least 10 new aircraft. It takes 150 hours per month to maintain the C1A, and 100 hours to maintain the C1B. The least level of importance is that there are a total of 1,200 hours of maintenance time available per month. (a) First, formulate this as an integer programming problem to maximize the number of pounds that may be carried. (b) Second, rework the problem differently than in part (a) to suppose the company decides that what is most important to them is that they keep the ratio of C1Bs to C1As in their fleet as close to 1.2 as possible to allow for flexibility in serving their routes. Formulate the goal programming representation of this problem, with the other three goals having priorities P₂, P₃, and P₄, respectively.

Goal programming differs from linear programming in which of the following aspects?

Which of the following is a category of mathematical programming techniques that doesn't assume linearity in the objective function and/or constraints?

Data Equipment Inc. produces two models of a retail price scanner, a sophisticated model that can be networked to a central processing unit and a stand-alone model for small retailers. The major limitations of the manufacturing of these two products are labor and material capacities. The following table summarizes the usages and capacities associated with each product. The typical LP formulation for this problem is: Maximize P = $160 X₁ + $95 X₂ Subject to: 8 X₁ + 5 X₂ ≤ 800 20 X₁ + 7 X₂ ≤ 1500 X₁, X₂ ≥ 0 However, the management of DEI has prioritized several goals that are to be attained by manufacturing: (1) Management had decided to severely limit overtime. (2) Management has established a profit goal of $15,000 per day. (3) Due to the difficulty of obtaining components from non-routine suppliers, management wants to end production with at least 50 units of each component remaining in stock. (4) Management also believes that they should produce at least 30 units of the network model. Given the above additional information, set this up as a goal programming problem.

A bakery produces muffins and doughnuts. Let x₁ be the number of doughnuts produced and x₂ be the number of muffins produced. The profit function for the bakery is expressed by the following equation: profit = 4x₁ + 2x₂ + 0.3x₁² + 0.4x₂². The bakery has the capacity to produce 800 units of muffins and doughnuts combined and it takes 30 minutes to produce 100 muffins and 20 minutes to produce 100 doughnuts. There is a total of 4 hours available for baking time. There must be at least 200 units of muffins and at least 200 units of doughnuts produced. How many doughnuts and muffins should the bakery produce in order to maximize profit?

Bastille College is going to purchase new computers for both faculty and staff. There are a total of 50 people who need new machines - 30 faculty and 20 staff. The cost of the basic personal computer with monitor and disk drive is $2,000, while the deluxe version with VGA and advanced processor is $3,500. Due to internal politics, the number of deluxe computers assigned to staff must be less than half the number of deluxe computers assigned to faculty. The College feels that it must purchase at least 5 deluxe computers for the faculty; if possible, it would like to purchase as many as 20 deluxe computers for the faculty. Staff members do feel somewhat "put upon" by having a limit placed upon the number of deluxe machines purchased for their use, so the College would like to purchase as many deluxe machines for the staff as possible (up to 10). The budget is $100,000. Develop a goal programming formulation of this problem that treats each of the requirements stated above as an equally weighted goal.

Consider the following 0 - 1 integer programming problem: If we wish to add the constraint that X must be positive, and that only Y or Z but not both can be positive, how would the additional constraint(s) be written?

In an integer programming problem, if it is desired to have variable X be exactly twice the value of variable Y, the constraint would be written:

Nonlinear programming is the case in which objectives and/or constraints are nonlinear.

A mathematical programming model that permits decision makers to set and prioritize multiple objective functions is called a

Table 10-6 -Table 10-6 represents a solution for an integer programming problem. If one uses the optimal solution presented, how much slack is there in the first equation?

Goal programming is characterized by

The following objective function is nonlinear: Max 5X + (1/8)Y - Z.

Data Equipment Inc. produces two models of a retail price scanner, a sophisticated model that can be networked to a central processing unit and a stand-alone model for small retailers. The major limitations of the manufacturing of these two products are labor and material capacities. The following table summarizes the usages and capacities associated with each product. The typical LP formulation for this problem is: Maximize $160 X₁ + $95 X₂ Subject to: 8 X₁ + 5 X₂ ≤ 800 20 X₁ + 7 X₂ ≤ 1500 X₁, X₂ ≥ 0 However, the management of DEI has prioritized several goals that are to be attained by manufacturing: (1) Since the labor situation at the plant is uneasy (i.e., there are rumors that a local union is considering an organizing campaign), management wants to assure full employment of all its employees. (2) Management has established a profit goal of $12,000 per day. (3) Due to the high prices of components from nonroutine suppliers, management wants to minimize the purchase of additional materials. Given the above additional information, set this up as a goal programming problem.

A model containing a linear objective function and requiring that one or more of the decision variables take on an integer value in the final solution is called

A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a cost of $2,400 and goal number 2 was to have no idle time for workers in the factory. The optimal solution to this problem resulted in a cost of $2,400 and no idle time. What was the value for the objective function for this goal programming problem?

Which of the following functions is nonlinear?