A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each.The cost for engaging each large van is Rs 400 and each small van is Rs 200.Not more than Rs 3000 is to be spent on the job and the number of large vans cannot exceed the number of small vans.Formulate this problem as a LPP given that the objective is to minimize cost.

Answer 1

Toolbox:

• Let $R$ be the feasible region for a linear programming problem and let $z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region.
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R

Step 1:

The data in the given problem can be summarized as follows:

For the type of large vans:

Packages =200

Cost = Rs.400

For the type of small vans:

Packages =80

Cost = Rs.200

For the minimum requirement :

Packages =1200

Cost = Rs.300

Step 2:

Let $x$ be no of large vans and $Y$ be the no of small vans,used for carrying the packages.The objective is to minimize the cost.

The mathematic formulation of the above problem is as follows :

Minimize $z=400x+200y$ subject to

$200x+80y\geq 1200$

$\Rightarrow 5x+2y\geq 30$

$400x+200y\geq 3000$

$\Rightarrow 2x+y\geq 15$

Also $x\leq y,x\geq 0,y\geq 0$