OR MODELS Multiperiod Work Scheduling

The Temporary Help Company must provide secretaries to its clients over the next year on the following estimated schedule: spring, 6000 secretary-days; summer, 7500 secretary-days; fall, 5500 secretary-days; and winter, 9000 secretary-days. A secretary must be trained for 5 days before becoming eligible for assignments to clients.

There are 65 working days in each quarter, and at the beginning of the spring season there are 120 qualified secretaries on the payroll. The secretaries are paid by the company and the not the client; they can earn a salary of $800 a month. During each quarter, the company loses 15% of it's personnel (including secretaries trained in the previous quarter).

Formulate the problem as a linear programming problem. (Hint Use as the number of secretaries hired at the beginning of season t, as the total number of secretaries at the beginning of season t.)

Solution

• Let be the number of secretaries hired in the spring, summer, fall and winter respectively.
• Let be the total number of secretaries on the payroll at the beginning of each season.
• To simplify the formulation, assume that new secretaries are hired and begin training on the first day of each season.
• There are 3 months in each quarter, so the company will owe its secretaries a total of over the coming year.
• In season t, the company hires new secretaries who can work 60 days and has qualified secretaries who can each work for 65 days. Thus, the company can provide at total of secretary-days in season t.
• For , due to the %15 attrition rate.

Linear Programming Formulation

Optimal Solution

Primal - Optimal:  Objective =    1.1484257500e+06
Solution time =    0.00 sec.  Iterations = 0 (0)

CPLEX> dis sol var -
Variable Name           Solution Value
S1                          120.000000
S2                          116.500000
S3                           99.025000
S4                          142.985729
x2                           14.500000
x4                           58.814479
All other variables in the range 1-8 are zero.

Integer Linear Programming (ILP) Formulation

minimize 2400 S1 + 2400 S2 + 2400 S3 + 2400 S4
subject to
65 S1 - 5 x1 >= 6000
65 S2 - 5 x2 >= 7500
65 S3 - 5 x3 >= 5500
65 S4 - 5 x4 >= 9000
S1 - x1 = 120
S2 - 0.85 S1 - x2 = 0
S3 - 0.85 S2 - x3 = 0
S4 - 0.85 S3 - x4 = 0

bounds
x1 <= 2000
x2 <= 2000
x3 <= 2000
x4 <= 2000
S1 <= 2000
S2 <= 2000
S3 <= 2000
S4 <= 2000
integer
x1 x2 x3 x4 S1 S2 S3 S4
end

Optimal Solution

Integer optimal solution:  Objective =    1.2048000000e+06
Solution time =    1.05 sec.  Iterations = 544  Nodes = 5691

CPLEX> dis sol var -
Variable Name           Solution Value
S1                          120.000000
S2                          120.000000
S3                          120.000000
S4                          142.000000
x2                           18.000000
x3                           18.000000
x4                           40.000000

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