A Company Makes Products a and B from 2 Resources

question 44

Essay

A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50- 0.09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30- 0.14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.
$\begin{array}{lllc}\text { Product } & \text { A } & \text { B } & \text { Available resources } \\\hline \text { Labor (hr/unit) } & 2 & 4 & 150 \\\text { Material (ounces/unit) } & 2 & 8 & 220 \\\text { Manufacturing cost(\$/unit) } & 30 & 20 & \\\text { Demand (units) } & 50-0.09 * P_{1} & 30-0.14 * P_{2} &\end{array}$

Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs.
The NLP for the problem is:
$\text {MAX}$ : $52.70 P _ { 1 } - 0.09 P _ { 1 } ^ { 2 } + 32.80 P _ { 2 } - 0.14 P _ { 2 } ^ { 2 } - 2100$
$\text {Subject to:}$
$\begin{array}{l}\mathrm{X}_{1}-50+0.09 \mathrm{P}_{1}=0 \\\mathrm{X}_{2}-30+0.14 \mathrm{P}_{2}=0 \\2 \mathrm{X}_{1}+4 \mathrm{X}_{2} \leq 150 \\2 \mathrm{X}_{1}+8 \mathrm{X}_{2} \leq 220 \\90 \leq \mathrm{P}_{1}, \mathrm{P}_{2} \leq 140 \\\mathrm{X}_{1}, \mathrm{X}_{2} \geq 0 \\\text { and the solution }\left(\mathrm{P}_{1}, \mathrm{P}_{2}\right)=(140.0,117.14)\end{array}$ What values should go in cells B3:E18 of the spreadsheet for this problem?
$\begin{array}{|c|l|c|c|c|c|}\hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & & & & \\\hline 2 & & \text { A } & \text { B } & & \\\hline 3 & \text { Price } & & & & \\\hline 4 & \text { Min Price } & & & & \\\hline 5 & \text { Max Price } & & & & \\\hline 6 & \text { Marginal Cost } & & & & \\\hline 7 & \text { Profit Margin } & & & & \\\hline 8 & & & & & \\\hline 9 & \text { Demand } & & & & \\\hline 10 & & & & & \\\hline 11 & \text { Total Profit } & & & & \\\hline 12 & & & & & \\\hline 13 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 14 & \text { Labor } & & & & \\\hline 15 & \text { Material } & & & & \\\hline \end{array}$ .

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Definitions:

Homo Habilis

An extinct species of early humans, known for its use of basic stone tools, living approximately 2.1 to 1.5 million years ago.

Australopithecines

A genus of hominids that lived in Africa between about 4 to 2 million years ago, known for walking upright and being a direct ancestor of modern humans.

Laetoli Site

An archaeological site in Tanzania known for its hominin footprints preserved in volcanic ash, providing evidence of early human bipedalism.

Tanzania

A country located in East Africa known for its vast wilderness areas, including the plains of Serengeti National Park and Kilimanjaro National Park.