Management Science Assignment 3 Due Date April 8, 2016
1 Ba 3623 Management Science Assignment 3due Date Apr08 2016
Write the LP mathematical formulation to find the minimum crashing cost for this problem.
Paper For Above instruction
Management science often employs linear programming models to optimize various operational problems, including project management scenarios such as crashing a project timeline to minimize cost. The process involves formulating a mathematical model that captures the relationships among project activities, their durations, costs, and the constraints imposed by project deadlines. This paper presents the linear programming formulation aimed at minimizing the crashing cost in a project scheduling problem, adhering to the general principles of management science.
In project management, "crashing" refers to shortening the duration of certain activities to accelerate the project's completion, typically at an increased cost. The objective function, constraints, and decision variables need to be precisely defined to formulate an LP model capable of determining the optimal activities to crash within given limitations.
Decision Variables
Let \( x_i \) denote the amount of crashing (or reduction in duration) applied to activity \(i\). Each activity's crashing can be represented as a continuous variable, where the value of \( x_i \) indicates how much the activity's duration is reduced from its normal duration.
Parameters
- \( c_i \): the cost per unit of crashing activity \(i\)
- \( d_i^0 \): the normal duration of activity \(i\)
- \( d_i^{min} \): the minimum duration to which activity \(i\) can be crashed
- \( t_{max} \): the desired project completion time
Objective Function
The goal is to minimize the total crashing cost, which can be expressed as:
\[ \text{Minimize} \quad Z = \sum_{i} c_i x_i \]
where \( c_i \) is the crashing cost per unit for activity \(i\), and \( x_i \) is the amount of crashing applied to that activity.
Constraints
- Duration constraints: the actual duration of each activity after crashing should not be less than its minimum possible duration:
\[ d_i^0 - x_i \geq d_i^{min} \quad \text{for all activities}\end{p>
- Project time constraint: the sum of the durations along the critical path should be less than or equal to the target project duration:
\[ \sum_{i} (d_i^0 - x_i) \leq t_{max} \]
- Non-negativity constraints:
\[ x_i \geq 0 \quad \text{for all activities} \]
Overall, this LP model seeks the optimal crashing amounts \( x_i \) for each activity to minimize total crashing costs, subject to project duration constraints and activity constraints. Solving this LP provides the project manager with the most cost-effective crash plan, ensuring the project completes within the desired timeframe while minimizing additional costs incurred through crashing activities.
Conclusion
The linear programming formulation provides a systematic method for minimizing project crashing costs while satisfying project completion time constraints. Such models are integral to decision-making in project management, allowing managers to balance the costs and benefits of crashing activities efficiently. By accurately defining decision variables, constraints, and the objective function, managers can use LP solvers to derive the optimal crashing strategy, ultimately leading to cost-effective project acceleration.
References
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