Chapter 3, Problem 3.16P

To determine

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

Answer to Problem 3.16P

The objective function for minimizing the transportation cost for the community is $\left[x_{1\text{A}}+x_{\text{2A}}+x_{\text{3A}}+x_{1\text{C}}+x_{\text{2C}}+x_{\text{3C}}\right].$

The constraint equations are ${\displaystyle \sum _{i=1}^3{x}_{ik}}\le B_k$ and ${\displaystyle \sum _{k=1}^2{x}_{ik}}=W_i$ . Here $x_{ik}\ge 0$ for all $i$ and $k$ .

Explanation of Solution

Given:

The community is shown in the figure below:

    

    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.

$C={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{k=1}^K\left(x_{ik}\cdot c_{ik}\right)}}$ ...... (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 $c_{ik}$ , the quantity of the waste hauled from source to disposal site is $x_{ik}$ , 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).

$C={\displaystyle \sum _{i=1}^3{\displaystyle \sum _{k=1}^2\left(x_{ik}\cdot c_{ik}\right)}}$

From the given data disposal cost can be ignored.

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

$C=\left[x_{1\text{A}}+x_{\text{2A}}+x_{\text{3A}}+x_{1\text{C}}+x_{\text{2C}}+x_{\text{3C}}\right]$

Write the $1^{\text{st}}$ constraint equation from Equation-(I).

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

${\displaystyle \sum _{i=1}^3{x}_{ik}}\le B_k$ ...... (II)

Here, the disposal site is $B_k$ .

Expand Equation (II) to find the $1^{\text{st}}$ constraint equations.

$\begin{array}{l}\left(x_{1\text{A}}+x_{\text{2A}}+x_{\text{3A}}\right)\le B_{\text{A}}\\ \left(x_{1\text{C}}+x_{\text{2C}}+x_{\text{3C}}\right)\le B_{\text{C}}\end{array}$

Write the $2^{\text{nd}}$ constraint equation from Equation-(I).

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

${\displaystyle \sum _{k=1}^2{x}_{ik}}=W_i$ ...... (III)

Here, the disposal site is $W_i$ .

Expand Equation (III) to find the $2^{\text{nd}}$ constraint equations.

$\begin{array}{l}\left(x_{1\text{A}}+x_{1\text{C}}\right)=W_{\text{A}}\\ \left(x_{\text{2A}}+x_{\text{2C}}\right)=W_{\text{C}}\\ \left(x_{\text{3A}}+x_{\text{3C}}\right)=W_{\text{D}}\end{array}$

The quantity of the waste hauled out must be positive.

$x_{ik}\ge 0$ for all $i$ and $k$ .

Conclusion:

Thus, the objective function for minimizing the transportation cost for the community is $\left[x_{1\text{A}}+x_{\text{2A}}+x_{\text{3A}}+x_{1\text{C}}+x_{\text{2C}}+x_{\text{3C}}\right]$ .

The constraint equations are ${\displaystyle \sum _{i=1}^3{x}_{ik}}\le B_k$ and ${\displaystyle \sum _{k=1}^2{x}_{ik}}=W_i$ . Here $x_{ik}\ge 0$ for all $i$ and $k$ .