In [1], the following problem is stated:
- There is a collection of \(n=1,000\) test questions.
- Each question covers a number of skills.
- Given is a requirement for a number of questions for each required skill (e.g., 4 questions about skill 1, 3 questions about skill 2, etc.).
- Create a test with the minimum number of questions that fulfills the requirements.
---- 11 SET qsskills covered by questions
skill_1 skill_2 skill_3 skill_4 skill_5 skill_6 skill_7 skill_8 skill_9
question3 YES
question4 YES
question5 YES
question8 YES YES
question9 YES
question11 YES
question14 YES YES
question20 YES YES
. . .
---- 11 PARAMETER arassessment requirements
skill_2 3.000, skill_6 10.000, skill_14 6.000, skill_17 4.000, skill_21 7.000, skill_22 2.000
skill_24 3.000
| Mathematical Model |
|---|
| \[ \begin{align} \text{Sets} \\ & q \text{: questions} \\ & s \text{: skills} \\ & \color{darkblue}qs_{q,s}\text{: skill is covered by question} \\ & \hline \\ \text{Data} \\ & \color{darkblue}{\mathit{ar}}_s \text{: assessment requirements}\\ & \hline \\ \text{Variables} \\ & \color{darkred}{\mathit numq} \text{: number of questions needed} \\ & \color{darkred}{\mathit question}_{s} = \begin{cases}1&\text{if question $s$ is used}\\ 0&\text{otherwise}\end{cases} \\ & \hline \\ \text{Model} \\ \min\>& \color{darkred}{\mathit numq} = \sum_{q} \color{darkred}{\mathit question}_{q} \\ & \sum_{q|\color{darkblue}qs_{q,s}}\color{darkred}{\mathit question}_{q} \ge \color{darkblue}{\mathit{ar}}_s && \forall s|\color{darkblue}{\mathit{ar}}_s\gt 0 \\ & \color{darkred}{\mathit question}_s \in \{0,1\} \end{align}\] |
COIN-OR CBC 42.2.0 ef14ea53 Feb 16, 2023 WEI x86 64bit/MS Window
COIN-OR Branch and Cut (CBC Library 2.10.8)
written by J. Forrest
Calling CBC main solution routine...
Integer solution of 19 found by feasibility pump after 0 iterations and 0 nodes (0.00 seconds)
Search completed - best objective 19, took 0 iterations and 0 nodes (0.00 seconds)
Maximum depth 0, 0 variables fixed on reduced cost
Solved to optimality (within gap tolerances optca and optcr).
MIP solution: 1.900000e+01 (0 nodes, 0.006 seconds)
Best possible: 1.900000e+01
Absolute gap: 0.000000e+00 (absolute tolerance optca: 0)
Relative gap: 0.000000e+00 (relative tolerance optcr: 0.0001)
Resolve with fixed discrete variables.
Optimal - objective value 19
---- 27 VARIABLE numq.L = 19.000number of questions needed
---- 27 VARIABLE question.L 1 if question is included in test
question9 1.000, question44 1.000, question76 1.000, question90 1.000, question149 1.000, question239 1.000
question347 1.000, question416 1.000, question446 1.000, question484 1.000, question506 1.000, question540 1.000
question579 1.000, question600 1.000, question622 1.000, question633 1.000, question771 1.000, question873 1.000
question923 1.000
A more complete solution report can look like:
---- 61 PARAMETER solution
skill_2 skill_3 skill_5 skill_6 skill_8 skill_9 skill_11 skill_12 skill_13
question9 1.000
question44 1.0001.000
question90 1.000
question149 1.000
question239 1.0001.000
question347 1.000
question416 1.0001.0001.000
question446 1.000
question540 1.000
question579 1.0001.0001.000
question600 1.000
question622 1.0001.000
question633 1.0001.000
question873 1.000
question923 1.000
total 3.0001.0002.00010.0001.0001.0003.0001.0001.000
required 3.00010.000
+ skill_14 skill_15 skill_17 skill_19 skill_20 skill_21 skill_22 skill_24
question9 1.0001.000
question44 1.000
question76 1.0001.000
question90 1.000
question149 1.0001.000
question239 1.000
question446 1.0001.000
question484 1.0001.0001.000
question506 1.0001.000
question540 1.000
question622 1.000
question771 1.0001.000
question873 1.0001.0001.000
question923 1.0001.000
total 6.0001.0004.0001.0001.0007.0002.0003.000
required 6.0004.0007.0002.0003.000
References
- Dynamic programming solution for assigning least number of questions that satisfy skills, https://stackoverflow.com/questions/75488358/dynamic-programming-solution-for-assigning-least-number-of-questions-that-satisf
Appendix: GAMS model
$ontext
Design a test with minimum number of questions such that skill coverage is obeyed. Reference: https://stackoverflow.com/questions/75488358/dynamic-programming-solution-for-assigning-least-number-of-questions-that-satisf This is an easy set covering type of problem. They often solve very fast.
$offtext
*------------------------------------------------------------------ * random data set *------------------------------------------------------------------
set q 'questions' /question1*question1000/ s 'skills' /skill_1*skill_25/ qs(q,s) 'skills covered by questions' ; qs(q,s) = uniform(0,1) < 0.04;
Parameter ar(s) 'assessment requirements'; ar(s)$(uniform(0,1)<0.2) = uniformInt(1,10); display q,s,qs,ar;
*------------------------------------------------------------------ * model *------------------------------------------------------------------
binary variable question(q) '1 if question is included in test'; variable numq 'number of questions needed';
equations cover(s) 'required number of questions with skill s' count 'number of questions in test' ;
* skip if ar(s)=0 cover(s)$ar(s).. sum(qs(q,s), question(q)) =g= ar(s); count.. numq =e= sum(q,question(q));
model m /all/; option mip=cbc; solve m minimizing numq using mip;
display numq.l, question.l;
*------------------------------------------------------------------ * solution report *------------------------------------------------------------------
parameter solution(*,*); solution(qs(q,s)) = question.l(q)>0.5; solution('total',s) = sum(qs(q,s)$(question.l(q)>0.5),1); solution('required',s) = ar(s); display solution;
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