Question

# Productco produces three products. Each product requires labor, lumber, and paint. The resource requirements, unit price, and variable cost (exclusive of raw materials) for each product are given in Table 70. Currently, 900 labor hours, 1,550 gallons of paint, and 1,600 board feet of lumber are available. Additional labor can be purchased at $6 per hour, additional paint at$2 per gallon, and additional lumber at $3 per board foot. For the following two sets of priorities, use preemptive goal programming to determine an optimal production schedule. For set 1: Priority 1: Obtain profit of at least$10,500. Priority 2: Purchase no additional labor. Priority 3: Purchase no additional paint. Priority 4: Purchase no additional lumber. For set 2: Priority 1: Purchase no additional labor. Priority 2: Obtain profit of at least \$10,500. Priority 3: Purchase no additional paint. Priority 4: Purchase no additional lumber. TABLE 70:$\begin{matrix} \text{Product} & \text{Labor} & \text{Lumber} & \text{Paint} & \text{Price (\)} & \text{Variable Cost (\)}\\ \text{1} & \text{1.5} & \text{2} & \text{3} & \text{26} & \text{10}\\ \text{2} & \text{3} & \text{3} & \text{2} & \text{28} & \text{6}\\ \text{3} & \text{2} & \text{4} & \text{2} & \text{31} & \text{7}\\ \end{matrix}$

Solution

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Three products are produced at productco, each of which requires labor, lumber and paint. $900$ hours of labor, $1550$ gallons of paint and $1600$ board feet of lumber are avialable. Additional labor is priced at $\6$ per hour, additional paint at $\2$ per gallon and additional lumber at $\3$ per board foot. Ofcourse, we are to use preemptive goal programming to determine an optimal production schedule. Let us $x_1,x_2$ and $x_3$ be the decision variables that denote the number of products $1,2$ and $3$ (respectively) sold. Also let us $y_1,y_2$ and $y_3$ be the number of additional labor, paint and lumber purchased. The highest priority is to obtain a profit of at least $\10\,500.$ Then we have (with adding deviational variables):

$16x_1+22x_2+24x_3-6y_1-2y_2-3y_3+s_1^--s_1^+=10500.$

The second highest priority is to purchase no additional labor, followed by no additional paint and lumber :

$y_1+s_2^--s_2^+=0,y_2+s_2^--s_2^+=0,y_3-s_3^--s_3^+=0.$

Ofcourse, all variables are nonnegative. Therefore, our aim is to observe the following expression:

$(P_1s_1^-+P_2s_2^++P_3s_3^++P_4s_4^+)\rightarrow\min.$

For the set $2$ we established that the highest prioirty goal is to purchase no additional labor which implies:

$y_1+s_1^--s_1^+=0.$

It is followed by priority $2$ which is obtaining at least $\10\,500$ of profit:

$16x_1+22x_2+24x_3-6y_1-2y_2-3y_3+s_2^--s_2^+=10500.$

Similarly we have:

$y_2+s_3^--s-3^+=0,y_3+s_4^--s_4^+=0.$

Therefore we minimize $P_1s_1^++P_2s_2^-+P_3s_3^++P_4s_4^+.$ Thus, we are done.

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