Old Tennessee Distillers, Inc., is a large distiller in rural Tennessee with a strong brand recognition for three major brands of blended whiskey. Old Tennessee blends five bourbons into the three final branded blends of whiskey. Brand A is a blend of 20% component bourbon 1, 15% component 2, 30% component 3, 10% component 4, and 25% component 5. Similarly, Brand B is produced from 10%, 40%, 20%, 10%, and 20% of the same ordered component bourbons. For Brand C, the bourbon component percentages are 30%, 10%, 10%, 20%, and 30% respectively. There are 50,000 gallons of bourbon component 1 available each week at a cost of $2.50 per gallon. Similarly, availabilities of the remaining bourbon components are 40,000, 30,000, 10,000, and 20,000 gallons respectively. Costs for these components are $2.75, $2.25, $3.00, and $3.50 per gallon respectively.
Weekly processing capacity for the final products is 125,000 gallons and the distiller wishes to operate at capacity. Minimum weekly production levels of 20,000 of Brand A, 20,000 gallons of Brand B, and 10,000 gallons of Brand C have been established for the final blends. Old Tennessee Distillers sells these products at the wholesale price of $6.50 per gallon for Brand A, $7.50 for Brand B, and $7.00 for Brand C.
a. Formulate a linear programming model that can be used to determine the optimal product mix that yields the maximize total weekly contribution to profit while meeting the individual product and weekly production blending requirements.
b. Determine the optimal production plan for Old Tennessee Distillers, Inc., using the Management Scientist software, including the quantity of each product type produced each week, the quantity of each resource used each week, and the total contribution to profit. Provide a narrative that explains the Management Scientist solution used.