The problem is to be formulated as two integer programming problems, one for the first year and the other for the second year.
I Year Problem
Total fund available = $10,000
For convenience rename the brand Petite Sirah as Brand I and brand Sauvignon Blanc as Brand II For Brand I the cost for grape is $0.80 per bottle and for Brand II the cost for grape is $0.70 per bottle.
It is given that one dollar spent for promoting Brand I produce a demand for 5 bottles and one dollar spent for promoting Brand II produce a demand for 8 bottles . This means the advertisement cost per bottle for Brand I is $0.20 and the advertisement cost per bottle for Brand II is $0.125.
The cost-profit structure of the two brands during the first year is as follows.
|Brand |Grape cost |Advt. cost |Total cost |Selling Price |Profit | |Brand I |$0.80 |$0.20 |$1.00 |$8.00 |$7.00 | |Brand II |$0.70 |$0.125 |$0.825 |$7.00 |$6.175 |
Suppose George decide to produce X bottles of Brand I and Y bottles of Brand II Then the total profit function to be maximised is [pic]
Total amount required is [pic].
Hence the constraint on the funds becomes C1: [pic]
Further, it is given that the proportion of Brand I should be between 40% and 60%. The corresponding constraint becomes [pic] This can be expressed as two constraints as follows
Thus the first year problem can be expressed as the following integer programming problem. Maximize [pic]
Subject to [pic]
X and Y non-negative integers
Solution of the problem using Solver of MS Excel is as follows
| |X |Y |Function |limits | | |4470.00 |6703.00 | | | |Objective fn |7.000 |6.175 |72681.025 | | |Constraint1 |1.000 |0.825 |9999.975 |10000.000 | |Constraint2 |3.000 |-7.000 |-33511.000 |0.000 | |Constraint3 |6.000 |-4.000 |8.000 |0.000 |
|Brand |Qty to be |Grape cost |Advt. cost |Total cost |Revenue |Profit | | |produced | | | | | | |Brand I |4470.000 |3576.000 |894.000 |4470.000 |35760.000 |31290.000 | |Brand II |6703.000 |4692.100 |837.875 |5529.975 |46921.000 |41391.025 | |Total |11173.000 |8268.100 |1731.875 |9999.975 |82681.000 |72681.025 |
Thus first year George has to produce 4470 bottles of Petite Sirah and 6703 bottles of Sauvignon Blanc by spending all the available funds.
He has to spent $ 8268.1 to buy grapes and $1731.875 for advertisement.
He has to produce a total of 11173 bottles of wine.
He can earn a profit of $72681.025
Total revenue available at the end of the first year is $82681.00
II Year Problem
Total fund available = $82681.00
For Brand I the cost for grape is $0.75 per bottle and for Brand II the cost for grape is $0.85 per bottle.
It is given that one dollar spent for promoting Brand I produce a demand for 6 bottles and one dollar spent for promoting Brand II produce a demand for 10 bottles . This means the advertisement cost per bottle for Brand I is $0.167 and the advertisement cost per bottle for Brand II is $0.10.