Answer to Question #173499 in Operations Research for ANJU JAYACHANDRAN

Question

5. A businessman needs five cabinets, 12 desks and 18 shelves cleaned out. He has two part 

time employees, Rashid and Ruby. Ruby can clean one cabinet, three desks and three 

shelves in a day while Rashid can clean one cabinet, two desks and 6 shelves in one 

day. Rashid is paid Rs. 22 per day and Ruby is paid Rs. 25 per day. In the order to 

minimize the cleaning costs, for how many days should Rashid and Ruby be 

employed? Formulate the problem as a linear programming problem and find its 

solution by the graphical method.

Answer 1

Solution.

Let be $x_1$ and $x_2$ - numbers of days, when Ruby and Rashid will be work. Then solve

$F(x_1,x_2)=25x_1+22x_2\mapsto min,$

when

$x_1+x_2 \geq 5,\newline 3x_1+2x_2\geq 12,\newline 3x_1+6x_2\geq 18,\newline x_1\geq 0,x_2\geq 0.$

Let us construct the region of feasible solutions, i.e. we will solve graphically the system of inequalities. For this, we construct each line and define the half-planes given by the inequalities.

Let us denote the boundaries of the region of the solution polygon.

Consider the objective function of the problem $F(x_1,x_2)=25x_1+22x_2.$

Let us construct a straight line corresponding to the value of the function $F(x_1,x_2)=25x_1+22x_2\mapsto min$ The vector-gradient, composed of the coefficients of the objective function, indicates the direction of maximization of F. The beginning of the vector is the point (0,0), the end is the point (25,22). We will move this straight line in a parallel manner. Since we are interested in the minimal solution, so we move the straight line to the first touch of the designated area.

The point (2,3) is finding point. Where can we find the minimum value of the objective function:

$F(x_1,x_2)=25\cdot 2+22\cdot 3=116.$

Therefore, Ruby will be work 2 days and Rashid will be work 3 days for

minimize the cleaning costs.