Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Q1E (page 239)

Consider the following linear program.

maximize 5x+3y

5x-2y≥0x+y≤7x≤5x≥0y≥0

Plot the feasible region and identify the optimal solution.

Short Answer

Expert verified

The maximum value is 31 at (5,2) is the optimal solution and the feasible region is plotted.

Step by step solution

01

Explain feasible region and optimal solution

The feasible region is the region where the optimal solutions can be found between the axis. Optimal solution is the most appropriate solution of the problem.

02

Plot the feasible region and find the optimal solution

Consider that the obj (x,y).=5x+3y Assume that the x is 0, and y is 0, then,

obj (0,0) =0 .

Knowing that on x-axis ,y is 0,

obj5,0=5×5+3×0obj5,0=25+0obj5,0=25Since,x+y≤7x≤5,Considerx=2,y=5obj2,5=2×5+3×5obj2,5=10+15obj2,5=25Now,Considerx=5,y=2obj5,2=5×5+3×2obj5,2=25+6obj5,2=31

The feasible region is plotted as follows

Therefore, the maximum value is 31 at (5,2) is the optimal solution and the feasible region is ploted.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For the following network, with edge capacities as shown, find the maximum flow from S to T, along with a matching cut.

A cargo plane can carry a maximum weight of 100 tons and a maximum volume of 60 cubic meters. There are three materials to be transported, and the cargo company may choose to carry any amount of each, up to the maximum available limits given below.

  • Material 1 has density 2tons/cubicmeters, maximum available amount 40 cubic meters, and revenue \(1,000 per cubic meter.
  • Material 2 has density 1ton/cubicmeters,maximum available amount 30 cubic meters, and revenue \)1,200 per cubic meter.
  • Material 3 has density 3tons/cubicmeters, maximum available amount 20 cubic meters, and revenue $12,000 per cubic meter.

Write a linear program that optimizes revenue within the constraints.

Give an example of a linear program in two variables whose feasible region is infinite, but such that there is an optimum solution of bounded cost.

There are many common variations of the maximum flow problem. Here are four of them.

(a) There are many sources and many sinks, and we wish to maximize the total flow from all sources to all sinks.

(b) Each vertex also has a capacity on the maximum flow that can enter it.

(c) Each edge has not only a capacity, but also a lower bound on the flow it must carry.

(d) The outgoing flow from each node u is not the same as the incoming flow, but is smaller by a factor of (1-∈U), whererole="math" localid="1659789093525" ∈u is a loss coefficient associated with node u.

Each of these can be solved efficiently. Show this by reducing (a) and (b) to the original max-flow problem, and reducing (c) and (d) to linear programming.

Consider the following network (the numbers are edge capacities).

(a)Find the maximum flow fand a minimum cut.

(b)Draw the residual graphGf (along with its edge capacities). In this residual network, mark the vertices reachable fromS and the vertices from whichT is reachable.

(c)An edge of a network is called a bottleneck edge if increasing its capacity results in an increase in the maximum flow. List all bottleneck edges in the above network.

(d)Give a very simple example (containing at most four nodes) of a network which has no bottleneck edges.

(e)Give an efficient algorithm to identify all bottleneck edges in a network.

See all solutions

Recommended explanations on Computer Science Textbooks

Algorithms in Computer Science

Read Explanation

Issues in Computer Science

Read Explanation

Theory of Computation

Read Explanation

Computer Programming

Read Explanation

Functional Programming

Read Explanation

Game Design in Computer Science

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free