Project Allocation Linear Programming Optimisation
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This document summarizes the formulation, implementation, analysis and evaluation of different integer programming models for optimizing student-project allocation. Model 1 assigns each student to exactly one topic based on their preferences while satisfying supervisor and topic capacity constraints. Model 2 adds penalties to encourage distributing students across more topics without sacrificing preferences. The results are analyzed for student satisfaction and constraint satisfaction. Alternative formulations are reviewed that consider collective satisfaction, fairness and computational cost trade-offs.
![Problem Formulation: Model #13
Objective Function
Decision Variables
[Binary for assigning topic j to student i]
𝐶𝑖𝑗
∈ {3, 2, 1} is Coefficient of Satisfaction
𝑠j
∈ ℤ+
is the extra capacity for each topic
𝑀𝑗
∈ ℤ+
is the penalty for each 𝑠𝑗
𝑥𝑖𝑗
is 0 if student i is not assigned to topic j
𝑥𝑖𝑗
is 1 if student i is assigned to topic j
where 𝑖 ∈ [1,𝐼] and j ∈ [1,𝐽]
𝑥𝑖𝑗
∈ { 0, 1 }](https://image.slidesharecdn.com/linearprogrammingoptimisation-200609230052/85/Project-Allocation-Linear-Programming-Optimisation-3-320.jpg)
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Project Allocation Linear Programming Optimisation
- 1. Optimisation of Student-Project Allocation BMAN60101 Mathematical Programming and Optimisation 2019-20 Group 4 Math Myth
- 2. TASK 1 2 3 4 5 Problem Formulation as Integer Optimisation Model Model Implementation using Excel Solver Analysis & Interpretation of Results: Difficulty & Properties of Constraints Extending Model with Optional Property Evaluation of Alternative Formulations
- 3. Problem Formulation: Model #13 Objective Function Decision Variables [Binary for assigning topic j to student i] 𝐶𝑖𝑗 ∈ {3, 2, 1} is Coefficient of Satisfaction 𝑠j ∈ ℤ+ is the extra capacity for each topic 𝑀𝑗 ∈ ℤ+ is the penalty for each 𝑠𝑗 𝑥𝑖𝑗 is 0 if student i is not assigned to topic j 𝑥𝑖𝑗 is 1 if student i is assigned to topic j where 𝑖 ∈ [1,𝐼] and j ∈ [1,𝐽] 𝑥𝑖𝑗 ∈ { 0, 1 }
- 4. Problem Formulation: Model #14 Max. Internal Supervisors’ Supervision Capacity Max. External Supervisors’ Supervision Capacity Min. External Supervisors’ Supervision Capacity 𝜎k is the maximum supervision capacity for internal supervisor, k 𝜎q is the maximum supervision capacity for external supervisor, q tkj is a binary for student being assigned to the topic j of internal supervisor, k tqj is a binary for student being assigned to the topic j of external supervisor, q s.t.:
- 5. s.t.: Problem Formulation: Model #15 Total no. of topics assigned to student i Max. Topic Capacity 𝑠j ∈ ℤ+ is the extra capacity for each topic 𝛼𝑗 is the maximum capacity for topic j
- 6. Implementation: Model #1 Survey #1 Matrix allow all combination of students and topics. E.g. Each student can be assigned to any topic xij = 1, if student is assigned to topic j xij = 0, if student is not assigned to topic j Coefficient of Satisfaction, Cij Allocation of topics to each student (Output)Coefficient of Preference Score of each student (Input) 1st choice = 3 2nd choice = 2 3rd choice = 1 6
- 7. . . Implementation: Model #1 Survey #1 . Student Supervisor Topic • Each student is assigned to exactly one topic • Student is assigned to a topic that he/she finds acceptable ID One Topic per Student Student Satisfaction 1 1 = 1 1 ≥ 1 2 1 1 3 1 3 1 1 3 1 --- --- --- --- --- 10 1 1 2 1 Topic # of Student ≤ Topic Capacity 1 1 1 2 2 2 3 0 2 --- --- --- 10 1 2 *Solver output *Solver output • Each topic has limited capacity Supervisor # of student Max Min E Margaret 1 ≤ 1 ≥ 1 Sylvia 2 4 1 Pierre 1 2 1 I Alan 2 = 2 Gabriel 2 2 Li Bai 2 2 *Solver output • Every supervisor is assigned at least to one student • No more students than maximum Supervisor capacity • Internal supervisor is prioritized 7
- 8. Analysis: Interpretation of Results (Model #1) Survey #1 Student Satisfaction Z = 23 ● Number of topics selected: 9 ● No penalty needed ● Difficulty of maximizing student satisfaction due to: - Uneven student preference distribution - Limited topic capacity - Limited supervisor capacity Topic capacity: 1 Unused penalty 8
- 9. Analysis: Interpretation of Results (Model #1) ● Supervisor Constraint : as an internal supervisor, Gabriel should be assigned to two students. ● Topic Constraint: topic 6 has limited capacity of 1 Infeasible solution because of unsatisfied constraints. Survey #2 2 Solver could not find a feasible solution 9 1
- 10. Analysis: Interpretation of Results (Model #1) The problem can be solved Survey #2 ● More capacity allocated to topic 6 ● Adding 1 penalty 10
- 11. Problem Formulation: Model #2 Optional Property: - Avoid too many students assigned to the same topic without sacrificing students’ preferences, if possible Approach: - Add a penalty to a topic that has no student’s assignment - Allow flexibility in the future by adjusting the size of the penalties 11 𝑀2𝑗 is a penalty applied to topic j when student is not assigned to the topic. 𝑦𝑗 is a binary variable 𝑀1𝑗 ≪ 𝑀2𝑗 𝑦𝑗 = 0 when topic j is assigned with at least 1 student 𝑦𝑗 = 1 when student is not assigned to topic j
- 12. . Analysis: Properties of Constraints (Model #2) Topic Topic # of Student ≤ Max. Capacity Min. Capacity 1 1 1 1 2 2 2 1 3 0 2 1 --- --- --- --- 10 1 2 1 *Solver output • Each topic has max. and min. capacity 12
- 13. Analysis: Interpretation of Results (Model #2) Student Satisfaction = 23 Survey #1 No penalty for increasing topic capacity 13 ● Reach max. topic distribution ● Same student allocation as Model #1 ● Not possible to improve the student satisfaction and topic distribution with the given data (Survey #1) and constraints 𝑀1𝑗 ≪ 𝑀2𝑗 No penalty for topics that isn’t assigned to any students
- 14. Model #1 Result: 8 topics allocated Model #2 Result: 8 topics allocated In Survey #2: Cannot avoid assigning student to same topic because of students’ topic selection but maintain student preferences ● 1 penalty for increasing topic capacity of Topic 6 (no student choose Topic 7 which is the topic of Internal Supervisor Gabriel) ● No penalty for topics that isn’t assigned to any students - Reach max. topic distribution - Same student allocation as Model #1 Survey #2 Analysis: Interpretation of Results (Model #2) 14
- 15. Evaluation: Review of Alternatives15 References: Chiarandini et al., Handling Preferences in Student-Project Allocation, 2019 A. A. Anwar and A. S. Baha, Student Project Allocation Using Integer Programming, 2003 COLLECTIVE SATISFACTION ● ORDERED WEIGHTED AVERAGING ● LEXICOGRAPHIC OPTIMISATION FAIRNESS / INDIVIDUAL WELFARE Promote egalitarianism in the outcome MINIMAX / MAXIMIN CRITERION ● Student that receives worst is assigned project with preference as high as possible ● Minimize the max preferenced value attained by any student (weighted sum of preference) PARETO EFFICIENCY Resource allocation optimisation COMPUTATIONAL COST Within a minute in the MILP Solver TRADE-OFF Conciliate equity & global utility
- 16. ALTERNATIVES Evaluation: Review of Alternatives16 References: Chiarandini et al., Handling Preferences in Student-Project Allocation, 2019 A. A. Anwar and A. S. Baha, Student Project Allocation Using Integer Programming, 2003 A. Higher collective satisfactions value -> Greedy Maximum Matchings: Assign higher score to 1st choice B. Different ways of collecting data of students preference C. Minimisation Objective: number of projects supervised by each supervisor A. More likely give result of more students get their 1st choice (e.g. 7 students get 1st choice and 3 get 3rd, rather than 4 students get 1st choice, 4 get 2nd, and 2 get 3rd) B. Increase number of choices and collect data of skills / experience, multi-attribute model with OWA C. Distribute projects more evenly among supervisors
- 17. Thank you Q&A
- 18. Appendix: Model #2 with Survey #1 18 Effect of the Penalty Size : - Both penalties are too small (e.g. penalties = 1) - Easy for the model to break the constraints Result: - Better student satisfaction of 25 - Break one constraint on topic 6 - Lower topic distribution 23 25 No allocation for Topic 7 𝑀1𝑗 = 𝑀2𝑗