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

Define deviational variables.

Goal programming is characterized by

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: 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. Management has established a profit goal of $12,000 per day. 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.

Classify the following problems as to whether they are pure-integer, mixed-integer, zero-one, goal, or nonlinear programming problems. (a)Maximize Z = 5 X1 + 6 X1 X2 + 2 X2 Subject to: 3 X1 + 2 X2 ≥ 6 X1 + X2 ≤ 8 X1, X2 ≥ 0 (b)Minimize Z = 8 X1 + 6 X2 Subject to: 4 X1 + 5 X2 ≥ 10 X1 + X2 ≤ 3 X1, X2 ≥ 0 X1, X2 = 0 or 1 (c)Maximize Z = 10 X1 + 5 X2 Subject to: 8 X1 + 10 X2 = 10 4 X1 + 6 X2 ≥ 5 X1, X2 integer (d)Minimize Z = 8 X12 + 4 X1 X2 + 12 X22 Subject to: 6 X1 + X2 ≥ 50 X1 + X2 ≥ 40

Allied Manufacturing has three factories located in Dallas, Houston, and New Orleans.They each produce the same 281 products and ship to three regional warehouses - #1, #2, and #3.The cost of shipping one unit of each product to each of the three destinations is given in the table below: There is no way to meet the demand for each warehouse.Therefore, the company has decided to set the following equally weighted goals: (1)each source should ship as much of its capacity as possible, (2)the number shipped to each destination should be as close to the demand as possible, (3)the capacity of New Orleans should be divided as evenly as possible between warehouses #1 and #2, and (4)the total cost should be less than $1,400.Formulate this as a goal program, which includes a strict requirement that capacities cannot be violated.

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.

If conditions require that all decision variables must have an integer solution, then the class of problem described is an integer programming problem.

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.

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

A goal programming problem had two goals (with no priorities assigned).Goal number 1 was to achieve a cost of $3,600 and goal number 2 was to complete the task in 400 hours or fewer.The optimal solution to this problem resulted in a cost of $3,600 and a completion time of 420 hours.What was the value for the objective function for this goal programming problem?

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

Goal programming permits multiple objectives to be satisfied.

The Elastic Firm has two products coming on the market, Zigs and Zags.To make a Zig, the firm needs 10 units of product A and 15 units of product B.To make a Zag, they need 20 units of product A and 15 units of product B.There are only 2,000 units of product A and 3,000 units of product B available to the firm.The profit on a Zig is $4 and on a Zag it is $6.Management objectives in order of their priority are: (1)Produce at least 40 Zags. (2)Achieve a target profit of at least $750. (3)Use all of the product A available. (4)Use all of the product B available. (5)Avoid the requirement for more product A. Formulate this as a goal programming problem.

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.

Agile Bikes has manufacturing plants in Salt Lake City, Dallas, and Chicago.The bikes are shipped to retail stores in Los Angeles, New York, Miami, and Seattle.Information on shipping costs, supply, and demand is given in the following table: What type of mathematical programming is required to solve this problem?

State the advantage of goal programming over linear programming.

A transportation problem is an example of

If we wish to develop a stock portfolio wherein we maximize return and minimize risk, we would have to use

The three types of integer programs are: pure integer programming, impure integer programming, and 0-1 integer programming.

The global optimum point may be superior to nearby points, but the local optimum point is the true optimal solution-n to a nonlinear programming problem.