Answer to Question #86858 in Operations Research for Yara Sayed Ahmad

Question #86858
The Make-It-Good manufacturing company produces three products, Widgets, Mingets, and Tringles.

During a given year they plan to produce a total of 13,000 units of these products.

The per unit production costs for Widgets, Mingets, and Tringles are $4, $5, and $7 respectively.

The per unit profit for the Widgets, Mingets, and Tringles is $1, $2, and $3 respectively.

If the production costs are to be $70,000 and the desired profit is $27,000, how many of each product should the company produce?

Set up the solution: define variables and determine the equations. Then, solve the system of equations using any valid method, and answer the question asked in the problem.
1
Expert's answer
2019-04-05T08:46:29-0400

Define variables and make the system of equations:


Let w - count of Widgets.

Let m - count of Mingets.

Let t - count of Tringes.


Then

4w - costs of Widgets,

     5m - costs of Mingets,

     7t - costs of Tringes.


And w - profit for Widgets

     2m - profit for Mingets

     3t - profit for Tringes.



"\\begin{cases} \nm + t + w = 13000 \\\\\n5m + 7t + 4w = 70000 \\\\\n2m + 3t + w = 2700 \n\\end{cases}"

Solve it:


"\\begin{pmatrix}\n 1 & 1 & 1 \\\\\n 5 & 7 & 4 \\\\\n 2 & 3 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n 13000 \\\\\n 70000 \\\\\n 27000\n\\end{pmatrix}"

"\\begin{pmatrix}\n 1 & 1 & 1 \\\\\n 0 & 2 & -1 \\\\\n 0 & 1 & -1\n\\end{pmatrix}\n\\begin{pmatrix}\n 13000 \\\\\n 5000 \\\\\n 1000\n\\end{pmatrix}"


"\\begin{pmatrix}\n 1 & 3 & 0 \\\\\n 0 & 2 & -1 \\\\\n 0 & -1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n 18000 \\\\\n 5000 \\\\\n -4000\n\\end{pmatrix}"





"\\begin{pmatrix}\n 1 & 3 & 0 \\\\\n 0 & 2 & -1 \\\\\n 0 & 1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n 18000 \\\\\n 5000 \\\\\n 4000\n\\end{pmatrix}"


"\\begin{pmatrix}\n 1 & 0 & 0 \\\\\n 0 & 0 & -1 \\\\\n 0 & 1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n 6000 \\\\\n -3000 \\\\\n 4000\n\\end{pmatrix}"



so m = 6000, w = 3000, t = 4000


The Answer:


Tringles - 3000 items

Widgets - 6000 items

Mingets - 4000 items


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