Question

A charity wants to contact people to collect donations. A person can be contacted morning or evening, by phone or door-to-door. The average donation resulting from each type of contact is given below:

                                                                 

                        Phone          Door-to-Door

____________________________________

Morning           $30                    $55

Evening            $35                   $65

The Charity has 200 volunteer hours in the morning and 150 volunteer hours in the evening. Each phone contact takes 8 minutes and each door-to-door contact takes 18 minutes to conduct. The Charity wants to have at least 600 phone and at least 410 door-to-door contacts.

Formulate a linear programming model that meets these restrictions and maximizes the total average donations.

(a) Define the decision variables.

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

Note: Do NOT solve the problem after formulating

Answer 1

The decision variables are the number of each type of contact in the morning and evenings.

Let's assume that the phone contact in the morning is X1 and that is evening is X2

and the door to door contact in morning is X3 and that in the evening is X4

Total revenue = 30X1+35X2+55X3+65X4

Objective function. is to maximise the revenue i.e.

Max 30X1+35X2+55X3+65X4

It represents the most optimum value of profits from judicious use of each resource.

Constraints-

8X1+18X3 <= 200x60 -----Const 1

or

8X1+18X3 <=12000

and

8X2+18X4 <=150x60

or

8X2+18X4 <=9000 ---------- Const 2

X1+X2=>600 --- Const 3

and

X3+X4 =>410 ----- Const 4

and X1, X2,X3 and X4 =>0 ---- Const 5

The first two constraint limits the activity due to limitation on number of volunteer hours in the morning and evening. The third and fourth constraint indicate the minimum number of contacts to be made through both the means. The fifth constraint is the non negativity constraint which indicates that neither the time value nor the number of volunteer hours value can be negative.