Applied Integer Linear Programming for project allocation optimisation problem using Excel Solver.

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