Solid Waste Engineering: A Global Perspective, Si Edition
Solid Waste Engineering: A Global Perspective, Si Edition
3rd Edition
ISBN: 9781305638600
Author: William A. Worrell, P. Aarne Vesilind, Christian Ludwig
Publisher: Cengage Learning
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Chapter 3, Problem 3.16P
To determine

The objective function and the constraint equations for minimizing the transportation cost for the community.

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Answer to Problem 3.16P

The objective function for minimizing the transportation cost for the community is [x1A+x2A+x3A+x1C+x2C+x3C].

The constraint equations are i=13xikBk and k=12xik=Wi . Here xik0 for all i and k .

Explanation of Solution

Given:

The community is shown in the figure below:

    Solid Waste Engineering: A Global Perspective, Si Edition, Chapter 3, Problem 3.16P

    Figure (1)

Assume that the cost of disposal is the same at both the disposal sites, and therefore, the disposal cost can be ignored.

Write the expression for the general objective equation for transportation cost.

C=i=1Nk=1K( x ik c ik ) ...... (I)

Here, the transportation cost is C , the number of disposal sites is K , the number of source of refuse is N , the cost per quantity of hauling the waste from source to disposal site is cik , the quantity of the waste hauled from source to disposal site is xik , the number of source of waste hauled is i, and the number of disposal site is k .

In Figure-(1) there are three routes, this means there are three sources and two disposal sites A and B.

Substitute 3 for N and 2 for K in the Equation-(I).

C=i=13k=12( x ik c ik )

From the given data disposal cost can be ignored.

Thus, expand the above equation to find the transportation cost.

C=[x1A+x2A+x3A+x1C+x2C+x3C]

Write the 1st constraint equation from Equation-(I).

The waste hauled is equal to or less than the capacity of the disposal sites.

i=13xikBk ...... (II)

Here, the disposal site is Bk .

Expand Equation (II) to find the 1st constraint equations.

(x 1A+x 2A+x 3A)BA(x 1C+x 2C+x 3C)BC

Write the 2nd constraint equation from Equation-(I).

The waste hauled is equal to the capacity of the disposal sites.

k=12xik=Wi ...... (III)

Here, the disposal site is Wi .

Expand Equation (III) to find the 2nd constraint equations.

(x 1A+x 1C)=WA(x 2A+x 2C)=WC(x 3A+x 3C)=WD

The quantity of the waste hauled out must be positive.

xik0 for all i and k .

Conclusion:

Thus, the objective function for minimizing the transportation cost for the community is [x1A+x2A+x3A+x1C+x2C+x3C] .

The constraint equations are i=13xikBk and k=12xik=Wi . Here xik0 for all i and k .

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