McKinney’s Sweet Shop specializes in homemade candies and ice cream. McKinney’s produces its ice cream in house, in batches of 50 pounds. The first stage in ice cream making is the blending of ingredients to obtain a mix which meets pre-specified requirements on the percentages of certain constituents of the mix.

The desired composition is as follows:

The mix can be composed of ingredients from the following list:

The number of pounds of a constituent found in a pound of an ingredient is shown in the following table (the rows correspond to constituents 1 through 7 and the columns correspond to ingredients 1 through 14). Note that a pound of stabilizer contributes only to the stabilizer requirement (one pound), one pound of emulsifier contributes only to the emulsifier requirement (one pound), and that water contributes only to the water requirement (one pound).

Jack McKinney Jr. has recently acquired the shop from their father (Papa Jack).
Jack’s father has in the past used the following mixture: 9.73 pounds of Plastic Cream, 3.03 pounds of Skim Milk Powder, 11.37 pounds of Liquid Sugar, 0.44 pounds of Sugared Frozen Fresh Egg Yolk, 0.12 pounds of Stabilizer, 0.07 pounds of Emulsifier and 25.24 pounds of water. (The scale at McKinney’s is only accurate to 100ths of a pound.)

a. Jack feels that perhaps it is possible to produce the ice cream in a more costeffective manner. Jack would like to find the cheapest mix for producing a batch of ice cream, which meets the requirements specified above. How does the optimal cost compare to the cost of Papa Jack’s recipe?

b. Jack is also curious about the cost effect of being a little more flexible in the requirements. Jack wants to know the cheapest mix if the composition meets the following tolerances:

How much cost benefit is there to being more flexible?

Transcribed Image Text:

Constituent 1. Fat 2. Serum Solids 3. Sugar Solids 4. Egg Solids 5. Stabilizer 6. Emulsifier 7. Water Required Percentage (%) 16.00 8.00 16.00 0.35 0.25 0.15 59.25