Comfortable Hands is a company which features a product line of winter gloves for the entire family - men,women,and children.They are trying to decide what mix of these three types of gloves to produce.Comfortable Hands' manufacturing labor force is unionized.Each full-time employee works a 40-hour week.In addition,by union contract,the number of full-time employees can never drop below 20.Nonunion,part-time workers can also be hired with the following union-imposed restrictions: (1)each part-time worker works 20 hours per week,and (2)there must be at least 2 full-time employees for each part-time employee.All three types of gloves are made out of the same 100% genuine cowhide leather.Comfortable Hands has a long term contract with a supplier of the leather,and receives a 5,000 square feet shipment of the material each week.The material requirements and labor requirements,along with the gross profit per glove sold (not considering labor costs)is given in the following table.
Each full-time employee earns $13 per hour,while each part-time employee earns $10 per hour.Management wishes to know what mix of each of the three types of gloves to produce per week,as well as how many full-time and how many part-time workers to employ.They would like to maximize their net profit - their gross profit from sales minus their labor costs.Formulate a linear programming model for this problem.

Question

Comfortable Hands is a company which features a product line of winter gloves for the entire family - men,women,and children.They are trying to decide what mix of these three types of gloves to produce.Comfortable Hands' manufacturing labor force is unionized.Each full-time employee works a 40-hour week.In addition,by union contract,the number of full-time employees can never drop below 20.Nonunion,part-time workers can also be hired with the following union-imposed restrictions: (1)each part-time worker works 20 hours per week,and (2)there must be at least 2 full-time employees for each part-time employee.All three types of gloves are made out of the same 100% genuine cowhide leather.Comfortable Hands has a long term contract with a supplier of the leather,and receives a 5,000 square feet shipment of the material each week.The material requirements and labor requirements,along with the gross profit per glove sold (not considering labor costs)is given in the following table.  
Each full-time employee earns $13 per hour,while each part-time employee earns $10 per hour.Management wishes to know what mix of each of the three types of gloves to produce per week,as well as how many full-time and how many part-time workers to employ.They would like to maximize their net profit - their gross profit from sales minus their labor costs.Formulate a linear programming model for this problem.

Quizplus · Accepted Answer

The Answer is The decision variables can be denoted and defined as follows: M = number of men's gloves to produce per week,W = number of women's gloves to produce per week,C = number of children's gloves to produce per week,F = number of full-time workers to employ,PT = number of part-time workers to employ.(Alternative notation for the decision variables is x_M,x_W,x_C,x_F,and x_PT,respectively.)Also letting P (or Z)denote the total net profit per week,the linear programming model for this problem is Maximize P = 8 M + 10 W + 6 C - 13(40)F - 10(20)PT,subject to 2 M + 1.5 W + C

\le

5000 30 M + 45 W + 40 C

\le

40(60)F + 20(60)PT F

\ge

20 F

\ge

2 PT and M

\ge

0,W

\ge

0,C

\ge

0,F

\ge

0,PT

\ge

0.

Comfortable Hands Is a Company Which Features a Product Line