**Linear Programming has many applications in Project Management. I present you one such application. Once you get the idea, you can apply it in so many different scenarios.**

One of the major requirement of any Project Manager is to be able to complete the project early. There can be many reason why one may need to complete the Project early. Some of the reasons could be

- Early realisation of revenues
- Reduction in Direct and Indirect Costs
- It may be due to Customer Requirement
- It may be due to Contract Commitments (There are usually bonus associated with early completion of projects).
- Time to Market Pressures
- Pressure to move the resources to other Projects
- etc.

However, trying to complete the Project Early comes with a cost. Typical examples of cost associated with trying to complete the Project early are

- Cost due to added resources to the Project
- Cost due to Overtime spent on the Project
- etc.

*So one has to balance between the benefits of completing the project early with the increased direct cost of completing the activities.*

- adding additional resources for some activities
- scheduling overtime
- outsourcing some project work
- forming a competent, dedicated, and focused core project team
- adopting a Critical-Chain approach to project management
- etc.

Consider the following Project Schedule.

*The relationship between the incremental data cost and the reduction in task time is assumed to be linear*. For each task whose duration can be reduced, we need to estimate the incremental data cost for each unit reduction and duration.

**crash time**and

**crash cost**. You can, if necessary, consider certain type of non-linear relationships between the reduction in the activity duration and the increase in direct cost but we will not do so in this example.

**For Linear Programming, any tool available could be used. However, I have used Excel as I expect most reader to have Excel available with them.**

**Now suppose there’s a bonus of 40,000 for each duration reduction in project completion time by one week.**

## First, let us try to solve this problem manually.

### We want to determine the least incremental cost for reducing the project completion time by one week at a time.

**The project completion time can be reduced only if you reduce the duration of the tasks on the critical path, which are tasks A –> B –> E –> F –> G –> H.**

**having an net gain of 40000 – 25000 = 15000**.

**net gain of 40000 – 30000 = 10000**. Now

**the project completion time is 9 weeks**.

**This, however, implies that the increase in direct cost would be 50,000 for reducing project duration by 1 week. Since the bonus is only 40,000 per week reduction, we will not reduce the duration of tasks B and C.**

**as you keep reducing the task durations, more paths become critical, and more tasks are on one or more critical paths**. The critical tasks must be managed well so that none of them can take a longer time than the stipulated duration if the project completion time is not to be delayed. So,

**reducing the project completion time typically implies that the flexibility is lost and project management becomes more difficult**.

**The above approach to reduce the product duration is conceptually sound. But for large problems, it becomes very cumbersome and difficult to implement. A Linear Programming approach to solving this problem would be most appropriate.**

**Excel**, we can use the feature of the Add-In

**Solver**to solve Liner Programming Problems.

**Objective Function**which has to be maximised or minimised,

**Variables**to be altered to meet the requirement of the Objective Function and

**Constraints**which have to be considered for optimisation.

- We need the durations to remain non-negative as a solution. So, we add one constraint for each duration referring to the Cell Address > 0. For example, if we have store the duration for Task A in Cell C4, then one constraint becomes C4 > 0.
- We need the durations to remain an Integer. So, we add one constraint for each duration variable. For example, if we have stored the duration for Task A in Cell C4, then one constraint becomes C4 is Integer.
- We need the Precedences to remain honoured in the solution. If the start date of Task A is in Cell D4 and the start date of Task B is in Cell D5, then one constraint would be D5 – D4 >= 2. However, we cannot enter a formula in the Constraints area of Solver. So, we need to add a new column where we can store the differences like D5 – D4. Say this Cell is E5. Then we can add the constraint E5 >= 2.
- We also need to add constraints to indicate that certain task durations cannot be altered. This can be done by fixing the duration of these tasks as per the current duration. For example, the duration of Task A cannot be crashed. So, we add a constraint as C4 = 2, where we assume that Cell C4 stores the duration for Task A and the duration for Task A is 2.

Categories: Technical

You must log in to post a comment.