How many units of each product is to be made so as to utilise the full available raw material Solution Step 1 The situation is easily identifiable y fitne Step 2 Suppose the firm produces x units of P y units of P and z units of P Since product P requires 3 units of R P requires 7 units of R and P requires 5 units of R observe matrix B and the total number of units of R available is 330 we have 3x 7y 5z 330 for raw material R Similarly we have and 4x 9y 12z 455 for raw material R 3y 72 140 for raw material R This system of equations can be expressed in matrix form as 202 MATHEMATICS 375 4 9 12 037 330 y 455 140 X Rationalised 2023 24 Step 3 Using elementary row operations we obtain 100 X 010 00 Z This gives x 20 y 35 and z 5 Thus the firm can produce 20 units of P 35 units of P and 5 units of P to make full use of its available raw material 20 35 Remark One may observe that if the manufacturer decides to manufacture according to the available raw material and not according to the purchase orders of the two clients F and F as in Example 3 he she is unable to meet these purchase orders as F demanded 6 units of P where as the manufacturer can make only 5 units of P3 Example 5 A manufacturer of medicines is preparing a production plan of medicines M and M There are sufficient raw materials available to make 20000 bottles of M and 40000 bottles of M but there are only 45000 bottles into which either of the medicines can be put Further it takes 3 hours to prepare enough material to fill 1000 bottles of M it takes 1 hour to prepare enough material to fill 1000 bottles of M and there are 66 hours available for this operation The profit is Rs 8 per bottle for M and Rs 7 per bottle for M How should the manufacturer schedule his her production in order to maximise profit Solution Step 1 To find the number of bottles profit under the given hypotheses bed the and M in order to maximise the Step 2 Let x be the number of bottles of type M medicine and y be the number of bottles of type M medicine Since profit is Rs 8 per bottle for M and Rs 7 per bottle for M therefore the objective function which is to be maximised is given by Z Z x y 8x 7y The objective function is to be maximised subject to the constraints Refer Chapter 12 on I inear Programming