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0,00 Torciary unus. Optimal solution to the dual is found to be (left as an exercise for the reader) y = 0, 11 Y2 = 65 f58 3 V = 65 W max = 8,800 Now, as the R.H.S. of constraints (6.12) and (6.13) denotes monetary units, the L.H.S. must also be expressed in monetary units. In the first term 1,500 y of the first constraint, 1,500 is the number of bottles of whisky produced per day. Hence, y, must denote the cost of producing one bottle of whisky. Similarly y, denotes the cost of producing a bottle of beer and yz denotes the cost of producing a bottle of fruit juices. Y, yz and y; are called the shadow prices of whisky, beer and fruit juices respectively. They represent not the actual market prices but the true accounting values or the imputed values of the three drinks. The objective function of the dual is to maximize the total accounting values of the drinks produced per month. The two constraints of the dual ensure that the total accounting value of the daily output of each plant must remain less than the daily cost of operating the plant. The shadow prices represent the values that the company should set on its resources in order to reflect their value to society, while the constraints ensure that the internal price cannot be set to get more value from a drink than what the company puts into it. It means that in a situation of equilibrium, the laws of economics for society do not require any profit The values of y = 0, y = 11/65 and y; = 3/65 represent the shadow prices of whiskey, beer and fruit-juices respectively, y = 0 means that the accounting value of whisky is zero and that it is produced in surplus as a by-product EXAMPLE 6.1-5.1 A company makes three products X, Y and Z out of three raw materials A, B and C. The number of units of raw materials required to produce one unit of the product is as given in the table below. X Y Z A 2 1 B 1 4 2 5 1 The unit profit contribution of the products X, Y and Z is 240, 25 and 50 respectively. The number of units of raw materials available are 36, 60 and 45 respectively. (1) Determine the product mix that will maximize the total profit. (ii) From the final simplex table, write the solution to the dual and give the economic interpretation