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This clearly shows that to meet the purchase order of F and F the raw material required is 335 units of R 467 units of R and 147 units of R which is much more than the available raw material Since the amount of raw material required to manufacture each unit of the three products is fixed we can either ask for an increase in the available raw material or we may ask the clients to reduce their orders Remark If we replace n place A in Example 3 by A given by i e if the clients agree to reduce their purchase orders then A B 10 A 9 12 6 10 20 0 Similarly we have 9 12 6 10 20 0 3 4 0 793 5 12 7 Rationalised 2023 24 141 216 78 170 220 60 MATHEMATICAL MODELLING This requires 311 units of R 436 units of R and 138 units of R which are well below the available raw materials i e 330 units of R 455 units of R and 140 units of R Thus if the revised purchase orders of the clients are given by A then the firm can easily supply the purchase orders of the two clients Note One may further modify A so as to make full use of the available raw material R R R P 3 4 0 B P 7 Query Can we make a mathematical model with a given B and with fixed quantities of the available raw material that can help the firm owner to ask the clients to modify their orders in such a way that the firm makes the full use of its available raw material The answer to this query is given in the following example 201 Example 4 Suppose P P P3 and R R R are as in Example 2 Let the firm has 330 units of R 455 units of R and 140 units of R available with it and let the amount of raw materials R R and R required to manufacture each unit of the three products is given by a d 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 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 4x 9y 12z 455 for raw material R 3y 7z 140 for raw material R and This system of equations can be expressed in matrix form as

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