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A steel company is faced with the problem of transporting coal from two coal mines to four of its steel plants. The amount of coal available in the coal mines are $a_1, a_2$ metric tons. The amounts required at plants are $b_1, b_2, b_3, b_4$ metric tons. Exactly one truck is used for these shipments.It is possible to ship from any mine to a plant, but the truck cannot make more than $1$ trip from a mine to a plant. The problem is to determine the minimum capacity truck which can complete all these shipments. Formulate this as a LPP.

$\mathbf{\text{Decision variable:}}$

Let $x_{ij}$ be amount of coal from mine ($i=1,2$) to plant ($j=1,2,3,4$).

$\mathbf{\text{Subject to constraints:}}$

Total amount from mine $1$: $x_{11}+x_{12}+x_{13}+x_{14}=a_1$

Total amount from mine $2$: $x_{21}+x_{22}+x_{23}+x_{24}=a_2$

Amout in plants :

$x_{11}+x_{21}=b_1$

$x_{12}+x_{22}=b_2$

$x_{13}+x_{23}=b_3$

$x_{14}+x_{24}=b_4$

$x_{ij} \ge 0$

I think that is all for the constraints. Then what comes next?

1 Answers 1

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You are missing the objective function which is to minimize the truck capacity $$ \min c $$ and the 8 capacity constraints $$ x_{ij}\leq c\quad \forall\ i,j. $$ (in standard LP form you would write $x_{ij}-c\leq0$).