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Question

# Plantco produces three products. Three workers work for Plantco, and the company must determine which product(s) each worker should produce. The number of units each worker would produce if he or she spent the whole day producing each type of product are given in Table 21. TABLE 21:$\begin{matrix} \text{ } & \text{Product}\\ \text{Worker} & \text{1} & \text{2} & \text{3}\\ \text{1} & \text{20} & \text{12} & \text{10}\\ \text{2} & \text{12} & \text{15} & \text{9}\\ \text{3} & \text{6} & \text{5} & \text{10}\\ \end{matrix}$The company is also interested in maximizing the happiness of its workers. The amount of happiness “earned” by a worker who spends the entire day producing a given product is given in Table 22. TABLE 22:$\begin{matrix} \text{Worker} & \text{Product}\\ \text{ } & \text{1} & \text{2} & \text{3}\\ \text{1} & \text{6} & \text{8} & \text{10}\\ \text{2} & \text{6} & \text{5} & \text{9}\\ \text{3} & \text{9} & \text{10} & \text{8}\\ \end{matrix}$Construct a trade-off curve between the objectives of maximizing total units produced daily and total worker happiness.

Solution

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We have two objectives, maximize production and maximize worker happiness. Letting $x_{i,j}$ denote the fraction of the day worker $i$ produces product $j$ these objectives are given by the formulas

\begin{align*} \max z_1&=20x_{1,2} + 12x_{1,2}+10x_{1,3}+12x_{2,1}+15x_{2,2}+9x_{2,3}+6x_{3,1}+5x_{3,2}+10x_{3,3}\\ \max z_2&=6x_{1,2} + 8x_{1,2}+10x_{1,3}+6x_{2,1}+5x_{2,2}+9x_{2,3}+9x_{3,1}+10x_{3,2}+8x_{3,3} \end{align*}

We have the following resource restrains:

\begin{align*} x_{i,1}+x_{i,2}+x_{i,3}+&= 1\ \ \ (i=1,2,3)\\ x_{1,j}+x_{2,j}+x_{3,j}+&= 1\ \ \ (j=1,2,3)\\ x_{i,j}&\geq 0 \end{align*}

If we want to maximize production we obtain the solution

$(z_1,z_2)=(45,19)$

If we want to maximize happiness we obtain the solution

$(z_1,z_2)=\left(27,26\right)$

Adding a restraint $z_2\geq 19+i$ for $i\in\{0,1,2,\dots,7\}$ we obtain the following values for the trade-off curve:

$\begin{array}{|c|c|}\hline z_1&z_2\\\hline 45&19\\\hline 43.17&20\\\hline 41.33&21\\\hline 39.5&22\\\hline 37.67&23\\\hline 35.83&24\\\hline 34&25\\\hline 27&26\\\hline \end{array}$

Bellow we show a graph of this trade-off curve.

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