Question

There is a factory located at each of the two places $P$ and $Q$. From these location, a certain commodity is delivered to each of the three depots situated at $A,B$ and $C$. The weekly requirements of the depots are respectively 5,5 and 4 units commodity, while of the production capacity of the factories at $P$ and $Q$ are 8 and 6 units respectively. The cost of transportation per unit is given below.

To/From	Cost (in ₹)
	A	B	C
P	16	10	15
Q	10	12	10

Formulate the LPP, so that the units tranported from each factory to each depot in such an order that the transportation cost is minimum.

• A. Minimise $Z=x-7y+190$
  subject to the constraints
  $x+y\le 8$
  $x+y\ge 4$
  $x\le 5$
  $y\le 5$
  $x,y\ge 0$
• B. Minimise $Z=x+7y+190$
  subject to the constraints
  $x+y\le 8$
  $x+y\le 5$
  $x,y\ge 0$
• C. Minimise $Z=x+7y+90$
  subject to the constraints
  $x+y\le 8$
  $x+y\le 4$
  $x\le 5$
  $y\le 5$
  $x,y\ge 0$
• D. None of the above

Answer 1

Answer: (A)

The given information, can be expressed diagramatically as follows

Let the factory at $P$ transports $x$ units of commodity to depot at $A$ and $y$ units to depot at $B$. Since, the factory at $P$ has the capacity of 8 units of the commodity.
Therefore, the left out $(8-x-y)$ units will be transported to depot at $C$.
Since, the requirements are always non-negative quantities, therefore we have
$x\ge 0,y\ge 0\text{ and }8-x-y\ge 0$
$\Rightarrow x\ge 0,y\ge 0\text{ and }x+y\le 8$
Since, weekly requirements of the depot at $A$ is 5 units of the commodity and $x$ units are transported from the factory at $P$. Therefore, the remaining $5-x$ units are to be transported from the factory at $Q$.
Similarly, $5-y$ units of the commodity will be transported from the factory at $Q$ to the depot at $B$.
But the factory at $Q$ has the capacity of 6 units only, therefore the remaining $6-(5-x+5-y)=x+y-4$ units will be transported to the depot at $C$.
As, the requirements at the depots at $A,B$ and $C$ are always non-negative.
$\therefore 5-x\ge 0,5-y\ge 0\text{ and }x+y-4\ge 0$
$\Rightarrow x\le 5,y\le 5\text{ and }x+y\ge 4$
The transportation cost from the factory at $P$ to the depots at $A,B$ and $C$ are respectively, $₹16x$, ₹ $10y$ and $₹15(8-x-y)$. Similarly, the transportation cost from the factory at $Q$ to the depots at $A,B$ and $C$ are respectively $₹10(5-x)$, ₹ $12(5-y)$ and $₹10(x+y-4)$.
Therefore, the total transportation $\mathrm{cost}Z$ is given by
$Z=16x+10y+15(8-x-y)+10(5-x)+12(5-y)+10(x+y-4)$
$=x-7x+190$
Hence, the mathematical formulation of the given problem is as follows.
Minimise $Z=x-7y+190$
Subject to constraints are
$x+y\le 8$
$x+y\ge 4$
$x\le 5$
$y\le 5$
$x\ge 0,y\ge 0$