Case Problem $2.$ Schneider$$s Sweet Shop

Schneider$$s Sweet Shop specializes in homemade candies and ice cream. Schneider 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:

$1.$

Fat

$16\mathrm{.}00\%$

$2.$

Serum solids

$8\mathrm{.}00\%$

$3.$

Sugar solids

$16\mathrm{.}00\%$

$4.$

Egg solids

$\mathrm{.}35\%$

$5.$

Stabilizer

$\mathrm{.}25\%$

$6.$

Emulsifier

$\mathrm{.}15\%$

$7.$

Water

$59\mathrm{.}25\%$

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

Ingredient Cost $($$$/$lb$.)$

$1.40\%$ Cream

$$1\mathrm{.}19$

$2.23\%$ Cream

$\mathrm{.}70$

$3.$ Butter

$2\mathrm{.}32$

$4.$ Plastic cream

$2\mathrm{.}30$

$5.$ Butter oil

$2\mathrm{.}87$

$6.4\%$ Milk

$\mathrm{.}25$

$7.$ Skim condensed milk

$\mathrm{.}35$

$8.$ Skim milk powder

$\mathrm{.}65$

$9.$ Liquid sugar

$\mathrm{.}25$

$10.$ Sugared frozen fresh egg yolk

$1\mathrm{.}75$

$11.$ Powdered egg yolk

$4\mathrm{.}45$

$12.$ Stabilizer

$2\mathrm{.}45$

$13.$ Emulsifier

$1\mathrm{.}68$

$14.$ Water

$\mathrm{.}00$

The number of pounds of a constituent found in a pound of an ingredient is shown in the next list. 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$).$

Constituent

Ingredient

PRIVATE

$1$

$2$

$3$

$4$

$5$

$6$

$7$

$8$

$9$

$10$

$11$

$12$

$13$

$14$

$1$

$\mathrm{.}4$

$\mathrm{.}2$

$\mathrm{.}8$

$\mathrm{.}8$

$\mathrm{.}9$

$\mathrm{.}1$

$\mathrm{.}5$

$\mathrm{.}6$

$2$

$\mathrm{.}1$

$\mathrm{.}1$

$\mathrm{.}1$

$\mathrm{.}3$

$1$

$3$

$\mathrm{.}7$

$\mathrm{.}1$

$4$

$\mathrm{.}4$

$\mathrm{.}4$

$5$

$1$

$6$

$1$

$7$

$\mathrm{.}5$

$\mathrm{.}8$

$\mathrm{.}2$

$\mathrm{.}1$

$\mathrm{.}1$

$\mathrm{.}8$

$\mathrm{.}7$

$\mathrm{.}3$

$1$

Young Jack Schneider has recently acquired the shop from his father. Jack$$s father has in the past used the following mixture: $9\mathrm{.}73$ pounds of plastic cream, $3\mathrm{.}03$ pounds of skim milk powder, $11\mathrm{.}37$ pounds of liquid sugar, $0\mathrm{.}44$ pounds of sugared frozen fresh egg yolk, $0\mathrm{.}12$ pounds of stabilizer, $0\mathrm{.}07$ pounds of emulsifier, and $25\mathrm{.}24$ pounds of water. $($The scale at Schneider$$s is only accurate to $100$th of a pound.$)$ Jack feels that perhaps it is possible to produce the ice cream in a more cost$-$effective manner. He would like to find the cheapest mix for producing a batch of ice cream, which meets the requirements specified above.

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

$1.$ Fat

$15\mathrm{.}0017\mathrm{.}00\%$

$2.$ Serum solids

$7\mathrm{.}009\mathrm{.}00\%$

$3.$ Sugar solids

$15\mathrm{.}5016\mathrm{.}50\%$

$4.$ Egg solids

$\mathrm{.}30\mathrm{.}40\%$

$5.$ Stabilizer

$\mathrm{.}20\mathrm{.}30\%$

$6.$ Emulsifier

$\mathrm{.}10\mathrm{.}20\%$

$7.$ Water

$58\mathrm{.}0059\mathrm{.}50\%$

Managerial Report

Write a managerial report which compares the cost of Papa Jack$$s approach to

$($a$)$

the cost$-$minimized approach using the desired composition and

$($b$)$

the cost$-$minimized approach with the more flexible requirements.

Include the following in your report:

The cost of $50$ pounds of ice cream under each of the three approaches

The amount of each ingredient used in the mix for each of the three approaches

A recommendation as to which approach should be used