Question

3. A post office requires different numbers of full-time employees on different days of the week. Each full-time employee must work five consecutive days and then receive two days off. The following table gives the number of employees required on each day of the week is specified. The post office would like to minimize the number of full-time employees needed to satisfy these constraints. Day No. of full-time employees required 15 Monday Tuesday Wednesday Thursday Friday Saturday Sunday 16 11 (a) Formulate an LP to help the post office to find the minimum number of workers (6) Write the LP in compact form. (c) Find a feasible solution to the model and compute the objective function value

Answer 1

Solution

a)

First step

We need to select the decision variable

Let x1 is the number of workers who start working on Monday, and work till Friday

Let x2 is the number of workers who start working on Tuesday

Let x3 is the number of workers who start working on Wednesday

Let x4, x5, x6, x7 defined similarly

The main goal is to minimize the number of hired employees

b)

Minimize

z=x1 + x2 + x3 + x4 + x5 + x6 + x7

subject to

x1 +x4+x5+x6+x7 ≥ 17 (Monday constraint)

x1+x2 +x5+x6+x7 ≥ 13 (Tuesday constraint)

x1+x2+x3 +x6+x7 ≥ 15 (Wednesday constraint)

x1+x2+x3+x4 +x7 ≥ 19 (Thursday constraint)

x1+x2+x3+x4+x5 ≥ 14 (Friday constraint)

x2+x3+x4+x5+x6 ≥ 16 (Saturday constraint)

x3+x4+x5+x6+x7 ≥ 11 (Sunday constraint)

xi ≥ 0 for i=1,2,…,7 (Non negativity constraints)

c)

Next step is that we need to determine the objective function and constraints

Maximize

                z=4x+5y               (Objective function)

Subject to

                2x+3y<=0             (Constraint)

               x>=0; y>=0 (non negativity constraints)

Feasible solution

It should satisfies all of the constraints

here

x=10, y=10 is feasible

x=10, y=15 is infeasible

Optimal solution

It is the best feasible solution

The optimal solution is

x=30,y=0

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all the best