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Employee training through varying course sequences

You are going to train new employees. Make sure that each trainee must successfully complete a sub-set of 6 available courses (A, B, C, D, E, F). There are four different sequences of courses that can be taken in order to achieve the required skill level. These sequences are A-E, B, C-F, and A-D-F. See the the table of information on the six courses in the file and the questions

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Employee Training - ILP
Problem definition: New employees of an organization need to be trained in order to meet demand for investment consultants. There are six courses available for the training, each charging for a trainee and having min number of trainees and max trainees that are allowed.
Each trainee needs to take a pre-defined sequence of these courses to complete the training. Objective is to allocate trainees to different sequences while minimizing total training cost. All trainees should be trained and a trainee should take at least one sequence of courses.
The problem is formulated as integer linear program in which there is a mix of integer and binary variables to find out leas cost flow through the network depicted below.

Following are the steps for formulating and solving the problem
Decision variables definition
Objective function formulation
Constraints formulation

There are two approaches to solve the problem, depending on how decision variables are defined.
Decision variables are defined as flow through each arc and using mass balance equation least cost flow through the network is identified
Decision variables are defined as flow through the sequence of activities and least cost flow is identified for each sequence
Approach 2 is used in this formulation. Note that both approaches should result in the same solution.

Step 1: Decision variables definition
There are two types of variables for the stated problem.
Allocation decision variables stating number of trainees taking up a particular sequence of course(s). This variables is constructed as an integer variable because a human being is an element here and fraction values are impractical
Course selection or choice variable which is a binary or Boolean variable
Let C={A,B,C,D,E,F} be the set of courses
Let S={1,2,3,4} be the set of course sequences
Sequence number Course sequence
1 A-E
2 B
3 C-F
4 A-D-F

The variables are defined as follows
Allocation variables: Number of employees taking a particular course sequence
Let X_1 be the number of trainees taking course sequence 1
Let X_2 be the number of trainees taking course sequence 2
Let X_3 ...

Solution Summary

The employee training through varying course sequences are examined.

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