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Chapter 7: Q2E (page 239)

Question: Duckwheat is produced in Kansas and Mexico and consumed in New York and California. Kansas produces 15 shnupells of duckwheat and Mexico 8. Meanwhile, New York consumes 10 shnupells and California 13. The transportation costs per shnupell are \(4 from Mexico to New York, \)1 from Mexico to California, \(2 from Kansas to New York, and \)3 and from Kansas to California. Write a linear program that decides the amounts of duckwheat (in shnupells and fractions of a shnupell) to be transported from each producer to each consumer, so as to minimize the overall transportation cost

Short Answer

Expert verified

Answer:

The minimum transportation cost: 4MN+MC+2KN+3KC.

MN+KN=10

MC+KC=13

MN+MC=8

KN+KC=15

MN鈮�0, MC鈮�0, KN鈮�0, KC鈮�0 will be therequired linear program

Step by step solution

01

Defining the variables

For our convenience, let us denote each country by its first alphabet:

M=Mexico

K=Kansas

N=New York

C=California

Let the number of shnupells shipped from Kansas to New York be 鈥淜N鈥� and its transportation cost is $2.So, the transportation cost will become 2KN.

Let the number of shnupells shipped from Kansas to California be 鈥淜C鈥� and its transportation cost is $3.So, the transportation cost will become 3KC.

Let the number of shnupells shipped from Mexico to New York be 鈥淢N鈥� and its transportation cost is $4.So, the transportation cost will become 4MN.

Let the number of shnupells shipped from Mexico to California be 鈥淢C鈥� and its transportation cost is $1.So, transportation cost will become 1MC.

02

Defining the constraints which will be the desired Linear Program

The constraints can be formed as follows:

The minimum transportation cost: 4MN+MC+2KN+3KC.

MN+KN=10

MC+KC=13

MN+MC=8

KN+KC=15

MN鈮�0, MC鈮�0, KN鈮�0, KC鈮�0 will be therequired linear program.

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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.

Suppose someone presents you with a solution to the max-flow problem on some network. Give a linear-time algorithm to determine whether the solution does indeed give a maximum flow.

Find necessary and sufficient conditions on the reals a and b under which the linear program

max鈥�x+yax+by鈮�1x,y鈮�0

(a) Is infeasible.

(b) Is unbounded.

(c) Has a unique optimal solution.

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.

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

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