Chapter 7, Problem 23RP

Explanation of Solution

Formulating a balanced transportation problem:

A balanced transportation problem is one in which the supply is equal to the total demand.

In the given problem, the variables are as follows:

• Let the supply from plant 1 (number of units) to customer 1 be represented as p11, to customer 2 be p12, to customer 3 be p13 and to customer 4 be p14.
• Let the supply from plant 2 to customer 1 be p21, to customer 2 be p22, to customer 3 be p23, to customer 4 be p24.
• Let the supply from plant 3 to customer 1 be p31, to customer 2 be p32, to customer 3 be p33, to customer 4 be p34.

The objective is to maximize profit. Total profit can be represented by the below mentioned linear programming,

$65*p11+63*p12+62*p13+64*p14+68*p21+67*p22+65*p23+62*p24+63*p31+60*p32+59*p33+60*p34$

This is our objective function and we have to maximize it to gain maximum profit.

The constraints can be represented as follows:

$\left(i\right)p11+p21+p31=4,000$

$\left(ii\right)p12+p22+p32=3,000$

$Totalsupply=3000+5000+5000=13,000units$

In this customer 1 and 2 are using $4000+3000=7000units$

$Unitsleft=13,000-7,000=6,000units$

$\left(iii\right)p13+p23+p33>=3,000$

$\left(iv\right)p13+p23+p33+p14$