Question

A company has two factories located at $P$ and $Q$ and has three depots situated at $A, B$ and $C$. The weekly requirement of the depots at $A, B$ and $C$ is respectively $5, 5$ and $4$ units, while the production capacity of the factories at $P$ and $Q$ are respectively $8$ and $6$ units. The cost of transportation per unit is given below:

How many units should be transported from each factory to each depots $A, B, C$ respectively in order that the transportation cost is minimum?

• A. a
• B. b
• C. c
• D. d

Answer 1

The Correct Option is C

The given data can be represented as given in the following diagram :

Total weekly production of $P$ and $Q = 8 + 6 = 14$ units, and total weekly requirement at depots $A, B, C = 5 + 5 + 4 = 14$ units There is no mismatch between supply and demand. Let factory $P$ supply $x$ units per week to depot $A$ and $y$ units to depot $B$, then it supplies $8 - x - y$ units to depot $C$. Clearly, $ 0 \le x \le 5, 0\le y \le 5, 0\le 8 - x -y \le 4$. As depot $A$' s requirement is $5$ units and it receives $x$ units from factory $P$, it must receive $(5 - x)$ units from factory $Q$. Similarly, depot $B$ receives $(5 - y)$ units from factory $Q$ and depot $C$ receives $(x + y - 4)$ units from factory $Q$. Thus total transportation cost (in $?$) $ = 16x + 10y + 15(8 - x - y) + 10(5 - x) + 12(5 - y) $ $+ 10(x+ y-4)$ $= x - 7y+ 190 $ Mathematical formulation of $LPP$ is: Minimize $z = x- 7y + 190$ subject to the constraints: $ x + y\ge 4 , x + y \le 8, x\ge 0, x \le 5, y \ge 0, y \le 5$ Now, we draw the graphs of the lines $x + y = 4, x + y = 8, x = 5, y = 5, x = 0, y = 0$ The feasibler egion corresponding to these constraints is shown in the figure. It is the bounded (convex) region $ABCDEF$.

Vertices of the feasible region are $A(4, 0), B(5, 0), C(5, 3), D(3, 5), E(0, 5)$ and $F(0, 4)$ $\because Z= x- 7y+ 190$ $\therefore$ At $A(4, 0), Z = 194$ At $B(5, 0), Z = 195$ At $C(5, 3), Z = 174 $ At $D(3, 5), Z = 158 $ At $E(0, 5), Z = 155 $ At $F(0, 4), Z = 162$ Thus $Z$ is minimum at $E(0, 5)$ i.e., when $x = 0, y = 5$. Thus, for minimum transportation cost, factory $P$ should supply $0, 5, 3$ units to depots $A, B, C$ respectively and factory $Q$ should supply $5, 0, 1$ units respectively to depots $A, B, C$.