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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

Moe is deciding how much Regular Duff beer and how much Duff Strong beer to order each week. Regular Duff costs Moe \(1 per pint and he sells it at \)2 per pint; Duff Strong costs Moe $1.50 per pint and he sells it at per pint. However, as part of a complicated marketing scam, the Duff company will only sell a pint of Duff Strong for each two pints or more of Regular Duff that Moe buys. Furthermore, due to past events that are better left untold, Duff will not sell Moe more than 3,000 pints per week. Moe knows that he can sell however much beer he has. Formulate a linear program for deciding how much Regular Duff and how much Duff Strong to buy, so as to maximize Moe鈥檚 profit. Solve the program geometrically.

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.

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-鈭�U), whererole="math" localid="1659789093525" 鈭�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.

You are given the following points in the plane:

(1,3),(2,5),(3,7),(5,11),(7,14),(8,15),(10,19)

.You want to find a lineax+by=c that approximately passes through these points (no line is a perfect fit). Write a linear program (you don鈥檛 need to solve it) to find the line that minimizes the maximum absolute error,max1鈮�i鈮�7|axi+byi鈭�c|

A vertex cover of an undirected graph G = (V,E) is a subset of the vertices which touches every edge鈥攖hat is, a subset S鈯�Vsuch that for each edge {U,V}鈭�E, one or both of u, v are in S. Show that the problem of finding the minimum vertex cover in a bipartite graph reduces to maximum flow. (Hint: Can you relate this problem to the minimum cut in an appropriate network?)
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