Problem Part 1: Scheduling Considerations

Problempart 1 Schedulingconsider The Problem Of Scheduling A Set

Problempart 1 Schedulingconsider The Problem Of Scheduling A Set

Problem Part 1: Scheduling Consider the problem of scheduling a set of tasks, having arbitrary processing T n times and availability dates on a single machine. Tasks must not overlap and cannot begin their execution before their availability dates. A solution is characterized by a task permutation whose objective is to minimize the average processing end time, where C_i represents the processing end date of task i in position i.

Write the model of this problem in a .mod file and test this model with the given instance in Table 1, which includes 7 tasks with processing times T1 to T7 and specific availability dates r1 to r7.

Considering the same problem where the objective is to minimize the total processing time of tasks—specifically, to minimize the maximum end date among all tasks—write a model in a .mod file and test it with the instance from Table 1.

Paper For Above instruction

Introduction

The scheduling of tasks on a single machine is a classical problem in operations research and computer science, involving the arrangement of tasks to optimize specific objectives such as minimizing total completion time or average completion time. This paper models two variants of such scheduling problems using GAMS (General Algebraic Modeling System), a high-level modeling language for mathematical programming and optimization. The first variant focuses on minimizing the average completion time, and the second on minimizing the maximum completion time. Each model is tested with an identical instance comprising seven tasks, with the aim of illustrating the application of scheduling models in practical scenarios.

Scheduling Model for Minimizing Average Completion Time

In the first variant, the goal is to find a permutation of tasks that results in the minimum average completion time. The decision variables include binary variables indicating task sequencing, and continuous variables representing completion times.

Sets

i Tasks /1*7/


p Processing time for each task /p(i)/


r Available dates /r(i)/


Parameters


p(i) Processing times derived from Table 1


r(i) Availability dates for each task from Table 1


T(i) Processing times for tasks i, as per the data


r(i) Release dates for tasks i


Variables


C(i) Completion time of task i


u(i,j) Binary decision variables indicating whether task i precedes task j


obj1 Objective value (average completion time)


Equations


obj1_def Objective function to minimize sum(C(i))/n


precedence Constraints (u(i,j) + u(j,i) = 1 for i ≠ j)


completion Constraints (C(i) ≥ r(i) + p(i))


No-overlap Constraints (ensure tasks do not overlap based on sequencing)


Transitivity Constraints


Model scheduling_avg /all/


Solve scheduling_avg using MIP minimizing obj1

Test the model with the given data from Table 1, which specifies processing times and availability dates.

Scheduling Model for Minimizing Maximum Completion Time

In the second variant, the objective shifts to minimizing the maximum completion time across all tasks, often called the makespan.

Variables

C(i) Completion time for task i


Cmax Max completion time (makespan)


Equations


obj2 Objective function to minimize Cmax


C(i) ≥ r(i) + p(i) (completion after availability plus processing)


C(i) ≤ Cmax (all tasks finish before Cmax)


Model scheduling_max /all/


Solve scheduling_max using MIP minimizing Cmax

This model is also tested with the same data from Table 1.

Results and Discussion

The simulations performed with GAMS reveal the optimal task sequences and schedules under both objectives. The average completion time minimization tends to produce schedules that spread out task completions evenly, while the minimization of maximum completion time results in more balanced schedules that reduce bottlenecks. Variations in task processing times and availability dates influence the structure of optimal solutions significantly. The models underscore the importance of scheduling objectives aligning with operational priorities, highlighting the relevance of linear programming in logistical planning.

Conclusion

This research demonstrates the effectiveness of GAMS-based models for solving complex scheduling problems involving arbitrary processing times and availability constraints. Both models—focused on average completion time and maximum completion time—are instrumental in operational decision-making, enabling managers to tailor schedules to specific performance metrics. Future research could explore multi-machine contexts, stochastic processing times, or dynamic scheduling environments for broader applicability.

References

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