010 y 001 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 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 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 Solution Step 1 To find the number of bottles profit under the given hypotheses Z Z x y 8x 7y The objective function is to be maximised subject to the constraints Refer Chapter 12 on Linear Programming not 20 35 5 x 20000 y 40000 x y 45000 0 40 000 T 3x y 66000 x20 y20 Step 3 The shaded region OPQRST is the feasible region for the constraints 1 Fig A 2 3 The co ordinates of vertices O P Q R S and T are 0 0 20000 0 20000 6000 10500 34500 5000 40000 and 0 40000 respectively Rationalised 2023 24 x 20 000 MATHEMATICAL MODELLING S 5 000 40 000 y 40 000 1 R 10 500 34 500 203