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Chapter 7: Q1E (page 239)

Consider the following linear program.

maximize 5x+3y

5x-2y鈮�0x+y鈮�7x鈮�5x鈮�0y鈮�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脳5+3脳0obj5,0=25+0obj5,0=25Since,x+y鈮�7x鈮�5,Considerx=2,y=5obj2,5=2脳5+3脳5obj2,5=10+15obj2,5=25Now,Considerx=5,y=2obj5,2=5脳5+3脳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.

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Most popular questions from this chapter

Question: A linear program for shortest path. Suppose we want to compute the shortest path from node s to node t in a directed graph with edge lengths le>0.

a) Show that this is equivalent to finding an s - tflow fthat minimizes 鈭�elefesubject to size (f) = 1. There are no capacity constraints.

b) Write the shortest path problem as a linear program.

c) Show that the dual LP can be written as

role="math" localid="1659250472483" maxxs-xtxu-xv鈮�luvforall(u,v)鈭�E

d) An interpretation for the dual is given in the box on page 223. Why isn鈥檛 our dual LP identical to the one on that page?

Hall鈥檚 theorem. Returning to the matchmaking scenario of Section 7.3, suppose we have a bipartite graph with boys on the left and an equal number of girls on the right. Hall鈥檚 theorem says that there is a perfect matching if and only if the following condition holds: any subset sof boys is connected to at least |s|girls.

Prove this theorem. (Hint: The max-flow min-cut theorem should be helpful.)

Show that the change-making problem (Exercise) can be formulated as an integer linear program. Can we solve this program as an LP, in the certainty that the solution will turn out to be integral (as in the case of bipartite matching)? Either prove it or give a counterexample.

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

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.
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