Exam 23: Linear Programming

A linear programming problem has three constraints,plus nonnegativity constraints on X and Y.The constraints are: 2X + 10Y ≤ 100;4X + 6Y ≤ 120;6X + 3Y ≥ 90. What is the largest quantity of X that can be made without violating any of these constraints?

What combination of x and y will yield the optimum for this problem? Maximize $3x + $15y,subject to (1)2x + 4y ≤ 12 and (2)5x + 2y ≤ 10 and (3)x,y ≥ 0.

A financial advisor is about to build an investment portfolio for a client who has $100,000 to invest.The four investments available are A,B,C,and D.Investment A will earn 4 percent and has a risk of two "points" per $1,000 invested.B earns 6 percent with 3 risk points;C earns 9 percent with 7 risk points;and D earns 11 percent with a risk of 8.The client has put the following conditions on the investments: A is to be no more than one-half of the total invested.A cannot be less than 20 percent of the total investment.D cannot be less than C.Total risk points must be at or below 1,000. Let A be the amount invested in investment A,and define B,C,and D similarly. Formulate the linear programming model.

A stereo mail order center has 8,000 cubic feet available for storage of its private label loudspeakers.The ZAR3 speakers cost $295 each and require 4 cubic feet of space;the ZAR2ax speakers cost $110 each and require 3 cubic feet of space;and the ZAR4 model costs $58 and requires 1 cubic foot of space.The demand for the ZAR3 is at most 20 units per month.The wholesaler has $100,000 to spend on loudspeakers this month.Each ZAR3 contributes $105,each ZAR2ax contributes $50,and each ZAR4 contributes $28.The objective is to maximize total contribution.Formulate this problem as a linear program.

Rienzi Farms grows sugar cane and soybeans on its 500 acres of land.An acre of soybeans brings a $1000 contribution to overhead and profit;an acre of sugar cane has a contribution of $2000.Because of a government program no more than 200 acres may be planted in soybeans.During the planting season 1200 hours of planting time will be available.Each acre of soybeans requires 2 hours,while each acre of sugar cane requires 5 hours.The company seeks maximum contribution (profit)from its planting decision. a.Formulate the problem as a linear program. b.Solve using the corner-point method.

What is the usefulness of a shadow price (or dual value)?

What is the region that satisfies all of the constraints in linear programming called?

In a linear programming formulation,a statement such as "maximize contribution" becomes a(n):

Suppose that a constraint is given by X + Y ≤ 10.If another constraint is given to be 3X + 2Y ≥ 15,and if X and Y are restricted to be nonnegative,determine the corners of the feasible solution.If the profit from X is 5 and the profit from Y is 10,determine the combination of X and Y that will yield maximum profit.

What combination of x and y will yield the optimum for this problem? Minimize $3x + $15y,subject to (1)2x + 4y ≤ 12 and (2)5x + 2y ≤ 10 and (3)x,y ≥ 0.

________ is an analysis that projects how much a solution might change if there were changes in the variables or input data.

Suppose that the feasible region of a maximization LP problem has corners of (0,0), (10,0), (5,5),and (0,7).If profit is given to be $X + $2Y what is the maximum profit the company can earn?

The requirements of linear programming problems include an objective function,the presence of constraints,objective and constraints expressed in linear equalities or inequalities,and ________.

Suppose that a constraint for assembly time has a shadow price of $50/hour for 15 hours in either direction and that all available assembly time is currently used (would require overtime to do more).If the salary of workers is $30 and they receive 50% extra pay for overtime what should management do?

The property manager of a city government issues chairs,desks,and other office furniture to city buildings from a centralized distribution center.Like most government agencies,it operates to minimize its costs of operations.In this distribution center,there are two types of standard office chairs,Model A and Model B.Model A is considerably heavier than Model B,and costs $20 per chair to transport to any city building;each model B costs $14 to transport.The distribution center has on hand 400 chairs-200 each of A and B. The requirements for shipments to each of the city's buildings are as follows: Building 1 needs at least 100 of A Building 2 needs at least 150 of B. Building 3 needs at least 100 chairs,but they can be of either type,mixed. Building 4 needs 40 chairs,but at least as many B as A. Formulate this problem as a linear program.(Hint: there are eight decision variables because we need to know how many of each chair (A and B)to deliver to each of the four buildings).

The feasible region in the diagram below is consistent with which one of the following constraints?

Suppose that the shadow price for assembly time is $5/hour.The allowable increase for the assembly time constraint is 40 hours,and the allowable decrease is 30 hours.If all assembly hours were used under the initial LP solution and workers normally make $4/hour but can work overtime for $6/hour,what should management do?

The diet problem is known in agricultural applications as the: