To Formulate The Linear Model We First Need To Find Out ✓ Solved

To Formulate The Linear Model We First Need To Find Out The Crash Cos

To formulate a linear model for project crashing, it is essential first to determine the crash costs associated with each activity. Crash cost refers to the additional cost incurred to reduce the duration of an activity, and it is calculated using the formula (Crash cost - Normal cost) / (Normal time - Crash time). This calculation provides the cost per unit time of crashing each activity. Once these costs are identified, we can develop a mathematical model aiming to minimize the total crash cost, subject to various constraints related to activity durations and project deadlines.

Let us define the variables pertinent to this model. Assume that xi (i = 1 to 8) represent the earliest start times of each activity, while yi (i = 1 to 8) indicate the amount of crash time for each activity. Additionally, xs and xf denote the project start and finish times, respectively. The primary objective is to minimize the total crashing cost, which can be expressed as:

Minimize : 6y₁ + 12y₂ + 4y₃ + 6.67y₄ + 10y₅ + 7.3y₆ + 5.75y₇ + 8y₈

This objective function accounts for the individual crash costs for activities 1 through 8. Each coefficient reflects the crash cost per unit time for the respective activity.

The model is subject to several constraints. Firstly, the crash duration for each activity cannot exceed its maximum permissible crash time, provided as:

• y₁ < 2 (Maximum crash time for activity 1 = 2)
• y₂ < 1
• y₃ < 2
• y₄ < 3
• y₅ < 1
• y₆ < 3
• y₇ < 4
• y₈ < 2

Furthermore, the total project finish time must not exceed the deadline, specified as:

• x_f < 15 (Project must finish within 15 units of time)

Since the project starts at time zero, the start time constraint is:

• x_s = 0

The crashing constraints are formulated based on the activity sequences, showing how activity durations relate to the start times and crash times. These include:

• x₁ - x_s + y₁ ≥ 5
• x₂ - x_s + y₂ ≥ 3
• x₃ - x₁ + y₃ ≥ 4
• x₄ - x₁ + y₄ ≥ 6
• x₅ - x₂ + y₅ ≥ 5
• x₆ - x₂ + y₆ ≥ 7
• x₇ - x₃ + y₇ ≥ 9
• x₇ - x₅ + y₇ ≥ 9
• x₈ - x₄ + y₈ ≥ 9
• x₈ - x₆ + y₈ ≥ 8

This linear programming model comprehensively captures the relationships between activity times, crash durations, and project completion constraints, enabling an optimal crashing plan that minimizes total costs while adhering to project deadlines.

Sample Paper For Above instruction

Introduction

Project management often involves optimizing resource allocation and scheduling to ensure timely completion while minimizing costs. Crashing, a technique that reduces project duration at an additional cost, plays a crucial role in this process. Accurate formulation of linear models for project crashing allows managers to identify the most cost-effective way to accelerate project timelines within resource constraints. This paper discusses the formulation of such a linear programming model, emphasizing how to calculate crash costs, define decision variables, and establish constraints to minimize total crashing expenses effectively.

Understanding Crash Cost and Its Calculation

The first step in developing a linear model for project crashing is understanding and quantifying crash costs. Crash cost is incurred when an activity's duration is shortened, and it is typically higher than the normal cost. Mathematical calculation involves the difference between crash and normal costs divided by the difference in times. For example, if activity 1 has a normal cost of $10,000 and a crash cost of $13,000, with normal and crash durations of 3 and 2 days respectively, then the crash cost per day is ($13,000 - $10,000) / (3 - 2) = $3,000 per day. Such calculations are performed for every activity to determine their respective crash costs, which are critical inputs for the optimization model.

Formulating the Objective Function

The objective of the model is to minimize the total crash cost, which is a sum of the individual crash costs for activities that are shortened. Using the crash costs per activity derived earlier, the objective function can be written as:

Minimize Z = c₁ y₁ + c₂ y₂ + ... + c_n y_n

where c_i is the crash cost per unit time for activity i, and y_i is the amount of crash time allocated to activity i. In this specific case, the coefficients are predetermined as 6, 12, 4, etc., based on the calculations of crash costs, reflecting the additional cost incurred per unit time of crashing each activity.

Establishing Constraints

The model incorporates several constraints to ensure feasibility: activity duration limits, project deadline, and sequence relations. Constraints on crash durations ensure activities are not shortened beyond their maximum crash potential. For example, if activity 1 can be crashed by at most 2 days, y₁ < 2. Similarly, the overall project finish time must not exceed the defined deadline, leading to constraints such as x_f < 15. Additionally, start and finish time relationships between activities are formulated to respect precedence relations, leading to inequalities like x₁ - xs + y₁ ≥ 5, ensuring activity 1 begins only after the project start and accounts for potential crash time.

Conclusion

Linear programming models for project crashing provide a systematic approach to minimize costs while respecting project constraints. By accurately calculating crash costs, defining decision variables for crash durations, and establishing logical constraints based on activity sequences, project managers can develop optimal crashing strategies. These models facilitate informed decision-making that balances project speed and cost, ultimately leading to efficient resource utilization and timely project completion.

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