Linear Programming
Institute Background
City University of Hong Kong (CityU) is founded in 1984 as city polytechnic, became full university in 1994. CityU has a well-earned reputation as an innovative hub for research and professional education and for addressing global issues and empowering positive change. In 2022, CityU is ranked 52 in QS World University Rankings by TopUniversities. There are 9 schools and colleges in total under CityU, with 6 departments under college of business. which offering students the opportunity to explore, discover and realise their full potential in their chosen career.
Case Background
Under the College of Business, department of Management Science(MS) have to submit the course schedule to the dean 2 months in advance of semester start.
Class sections were either scheduled for:
1. Three days a week for 55 mins per class on Monday, Wednesday and Friday (MWF)
2. Or two days a week for 85 mins per class on Tuesday and Thursday (TuTh)
There are 8 course offered under MS department, with 9 instructor, and 14 class period. Furthermore, there are some constraints when scheduling the course:
- Each faculty is assigned to only the course that he/she is qualified to teach.
- Each full-time faculty member’s actual course load must be equal to his/her required course load per semester according to their contracts.
- Ideally, the number of preparations (different courses that each faculty taught per semester) for each full-time faculty would be no more than two.
- Ideally, no more than two sections of the course MS3123 should be assigned to any full-time faculty because of the extra workload emanating from the lab.
- The number of sections assigned to the faculty needs to be equal to the number of sections required for each course.
- At most three course sections can be assigned to the same class period based on allotments determined by the assistant dean.
- Ideally, no faculty should teach more than two consecutive classes.
We also need to consider various elements, such as instructor's preference for the course, qualification, seniority, course loads, time, required number of course section.
Instructor's course Preferences
We use the linear programming to schedule the course.
Linear Programming (LP)
Reach the best outcome when satisfy various conditions by optimising the linear function. The way we obtain the best outcome is to maximize or minimize the objection function.
Adopting linear programming for other company
Linear programming is a mathematical technique that can help the company to decide how to efficiently use the limited resources, such as materials, labor, time, and money. The company mostly use linear programming in the production planning, various types of scheduling, and transportation routing. For instance, Airlines use it to schedule the flights, as well as scheduling aircraft and scheduling staff. For manufacturing company, they could use linear program to determine the amount of each products they produce that can achieve the greatest profits while satisfying the constraints.
Construct an optimization model for TuTh.
Set Zijk = 1 (If instructor i is assigned to course j in class period k) / 0 (vice verse)
Set tik = the weighted preference score of instructor i for class period k
Objective Function: Maximize Zijk *tik
1. At most three course sections can be assigned to the same class period
2. Should teach more than two consecutive classes
3. No overlap class
4. To match the actual number of each instructors need to teach
5. Zijk = 1 (If instructor i is assigned to course j in class period k) / 0 (vice verse)
Optimal Schedule for TuTh base on the course assignment (1st stage optimization) by Excel Solver.
Q: How optimization model be adjusted so that there is no course overlap on the schedule?
Step 1
Identify the cells that contain the same course yet taught by different faculty.
Step 2
Identify those who are within the same class period.
Step 3
In the solver, add constraints stating those cells combined should be less than or equal to 1, so that no same courses will be offered in the same period and allowing at most 1 section can be taught within a specific time period.
Reflection
Through this lecture, we have learnt that linear programming is a powerful methods to solve many business complex planning problem and make processes more efficient and cost-effective, especially in scheduling the production amounts. As there are many factor and elements we need to put into our considerations, and the resource are limited, being an employee of the company, it is vital for us to use LP to help company maximize the profits in the most efficient way. When we doing the research of the linear programming, we also find that enormous company and government institution also adopting LP. However, we think that there are still some limitation and constraints of LP, such as it is a quantitative methods, and it would become complicated for the manual when there are many constrains. Therefore, we believe that a more powerful program will be developed in the future, which can bring a more efficiency.
References
City University of Hong Kong Official Website
QS Ranking
Linear Programming
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