Question

The customer services department of a large retailer receives at least approximately 1200 emails from customers per day, all of which have to be replied to that day. The department employs two grades of staff: junior and senior. Both grades work for 8 hours per day, but junior staff can only process emails at a rate of 8 per hour compared to the senior rate of 10 per hour. Each member of staff requires a computer terminal and the department has a maximum of 25 terminals available. There is also a limited amount of desk space available (35m²). Each senior member of staff is expected to supervise junior staff, and hence senior staff are allocated more space (2m²) than junior staff (1m²). In order to ensure that proper supervision is always available, there must be at least 2 members of senior staff to every five junior members of staff. Taking into account all direct and indirect costs, the cost of employing a senior member of staff is $15 per hour, compared to $10 per hour for a junior member of staff. Formulate a linear programming model for this problem and solve it using the graphical method.   

OKAY so I am struggling to get the right graph here. if you answer this please follow these directions1) LP model where decision variables are clearly defined, the objective function and the constraints. 2) A graph showing all constraints and the feasible region. 3) All extreme corners of the feasible region computed with the optimal solution clearly marked.

I have used excel solver - Min. 10X+15Y

s.t. 64X+80Y>=1200

1X+1Y<=25

1X+2Y<=35

-2X+5Y>=0

I got (12.5, 5) as my values and 200 as my objective function but can't get together a graph to match it, help me clearly see what I am supposed to be doing- if my doing something wrong in excel also please let me know

Answer 1

Use the following table and scatter plot option to generate the constraint lines in Excel:

Once you develop this, select this (i.e. select A1 to E7) and then go to scatter plot (line, not points) option, the following lines should automatically develop.

Then I format the above graph in various ways to get the following chart. This may not be necessary but looks good.

The green area is the feasibility area and the five points shown are the corner points.

For knowing the coordinates of these five points, one will have to solve the simultaneous equations of the intersecting lines.

So, for example, point 'A' is the intersection of the lines X + Y = 25 and X + 2Y = 35

Multiply the first equation by 2 and subtract the second equation from it.

2X + 2Y - X - 2Y = 2*25 - 35
or, X = 15
Y = 25 - 15 = 10

So, A is (15, 10)

Similarly, solve for the other four points. Check the objective value for them as shown in the table:

Point	X	Y	Z = 10X + 15Y
A	15	10	300
B	125/7	50/7	285.7
C	12.5	5	200 (min)
D	0	15	225
E	0	17.5	262.5

Note that the objective is minimized at point C. o, the optimal solution is X = 12.5, Y = 5 and Min Z = 200