Chapter 3, Problem 3.15P

To determine

The objective function and the constraint equations for minimizing the transportation cost from three collection routes to the two disposal sites.

Answer to Problem 3.15P

The objective function for minimizing the transpiration cost is $c={\displaystyle \sum _{\text{i}=1}^3{\displaystyle \sum _{\text{k}=1}^2{x}_{\text{ik}}{c}_{\text{ik}}}}$ .

The $1^{\text{st}}$ constraint equations are,

$\begin{array}{l}{x}_{11}+x_{21}+x_{31}\le B_1\\ x_{12}+x_{22}+x_{32}\le B_2\end{array}$

The $2^{\text{nd}}$ equations are,

$\begin{array}{l}{x}_{11}+x_{12}=W_1\\ x_{21}+x_{22}=W_2\\ x_{31}+x_{32}=W_3\end{array}$ .

The last constraint equation is $x_{\text{ij}}\ge 0$ .

Explanation of Solution

Write the expression to obtain the objective function for minimizing the transpiration cost.

$c={\displaystyle \sum _{\text{i}=1}^{\text{N}}{\displaystyle \sum _{\text{k}=1}^{\text{K}}{x}_{\text{ik}}{c}_{\text{ik}}}}$ ....... (I)

Here, the transportation cost is $c$ , the number of collecting routes in town is $\text{N}$ , the number land fill in a community is $\text{K,}$ and the cost obtained per quantity of hauling from the source $\text{i}$ to the disposal site $\text{k}$ is $c_{\text{ik}}$ .

Substitute $3$ for $\text{N}$ and $2$ for $\text{K}$ in Equation (I).

$c={\displaystyle \sum _{\text{i}=1}^3{\displaystyle \sum _{\text{k}=1}^2{x}_{\text{ik}}{c}_{\text{ik}}}}$

Write the expression for the $1^{\text{st}}$ constraint equation for minimizing the transportation cost.

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

Further solve the above equation.

$\begin{array}{l}{x}_{11}+x_{21}+x_{31}\le B_1\\ x_{12}+x_{22}+x_{32}\le B_2\end{array}$

Here, the above equation signifies that the waste hauled out from each section is less than or equal to the capacity of each of the disposal site.

Write the expression for the $2^{\text{nd}}$ constraint equation for minimizing the transportation cost.

${\displaystyle \sum _{\text{k}=1}^2{x}_{\text{ik}}\le W_{\text{i}}}$ ...... (III)

Further solve the above equation.

$\begin{array}{l}{x}_{11}+x_{12}=W_1\\ x_{21}+x_{22}=W_2\\ x_{31}+x_{32}=W_3\end{array}$

Here, the above equation signifies that the waste hauled out from each section is equal to the amount generated.

Write the expression for the last constraint equation for minimizing the transportation cost.

$x_{\text{ij}}\ge 0$

Here, above equation signifies that the waste hauled out from each section must be positive.

Conclusion:

Thus, the objective function for minimizing the transpiration cost is $c={\displaystyle \sum _{\text{i}=1}^3{\displaystyle \sum _{\text{k}=1}^2{x}_{\text{ik}}{c}_{\text{ik}}}}$ .

The $1^{\text{st}}$ constraint equations are,

$\begin{array}{l}{x}_{11}+x_{21}+x_{31}\le B_1\\ x_{12}+x_{22}+x_{32}\le B_2\end{array}$

The $2^{\text{nd}}$ equations are,

$\begin{array}{l}{x}_{11}+x_{12}=W_1\\ x_{21}+x_{22}=W_2\\ x_{31}+x_{32}=W_3\end{array}$ .

The last constraint equation is $x_{\text{ij}}\ge 0$ .