Question A: The application of quantum computers in the optimization of credit score card combinations
In bank credit cards or related loans, before granting credit to customers, it is necessary to evaluate the credit rating of customers through various review rules, and the customers who pass the evaluation can Get credit or qualify for a loan. The rule review process is actually to score customers after one or more combined rules. These rules are called credit score cards. Each credit score card has multiple threshold settings (but only one threshold takes effect), which makes Different credit score cards correspond to different pass rates and bad debt rates under different thresholds. Generally, the higher the pass rate, the higher the bad debt rate. Conversely, the lower the pass rate, the lower the bad debt rate.
For banks, the higher the pass rate, the more customers who pass the loan qualification review, and the more interest income the bank will receive. However, a high pass rate generally corresponds to a high bad debt rate, and bad debts mean funds
. Risk of loss, so the final income of the bank can be defined as :
final income = loan interest income - bad debt loss Different bad debt rate and pass rate:
| credit score card 1 | credit score card 2 | credit score card 3 | ||||||
| threshold | Passing rate | Bad debt rate | threshold | Passing rate | Bad debt rate | threshold | Passing rate | Bad debt rate |
| 1 | 5% | 0.50% | 1 | 5% | 0.50% | 1 | 5% | 0.50% |
| 2 | 10% | 1.00% | 2 | 10% | 1.00% | 2 | 10% | 1.00% |
| 3 | 25% | 1.50% | 3 | 25% | 1.50% | 3 | 20% | 1.70% |
| 4 | 30% | 2.00% | 4 | 30% | 2.00% | 4 | 33% | 2.00% |
| 5 | 40% | 2.50% | 5 | 45% | 2.50% | 5 | 40% | 2.70% |
| 6 | 50% | 3.00% | 6 | 50% | 2.70% | 6 | 52% | 3.00% |
| 7 | 60% | 3.50% | 7 | 65% | 3.50% | 7 | 62% | 3.70% |
| 8 | 70% | 4.00% | 8 | 70% | 4.00% | 8 | 73% | 4.00% |
| 9 | 80% | 4.50% | 9 | 82% | 4.70% | 9 | 82% | 4.70% |
| 10 | 93% | 5.00% | 10 | 90% | 5.00% | 10 | 95% | 5.00% |
Problem description 1 : Process simplification and examples
Due to the complexity of the banking scene, it is often necessary to select a combination of multiple different credit score cards to achieve the best risk control strategy. The actual combination of credit score cards is a very complicated process. For the convenience of modeling, we simplify the problem as follows (this simplification is only applicable to the questions of this competition and cannot fully represent the actual scene).
Assume that the loan fund is 1,000,000 yuan, the bank loan interest income rate is 8%, and the three credit score cards listed above are used as the selected credit score card combination to measure the bank's final income.
Since each credit score card has and can only select one threshold, assuming that the threshold of credit score card 1 is set to 8, it can be seen from the table that the corresponding pass rate is 70%, the bad debt rate is 4.00%, and the threshold of credit score card 2 If it is set to 6, the pass rate is 50%, and the bad debt rate is 2.70%. If the threshold of credit score card 3 is set to 7, the pass rate is 62%, and the bad debt rate is 3.70%.
For example, if we choose the triple credit card combination strategy, then the total pass rate of the three credit score card combinations
is multiplied by the pass rate of all credit score cards, that is:
0.7×0.5×0.62 = 0.217
The total bad debt rate of the three credit score cards The average value of the corresponding bad debt rate, namely:
1/3×(0.04+0.027+0.037) = 0.0367
Based on the above conditions, the interest income of this loan is:
loan funds×interest income rate×total pass rate×(1－ total bad debt rate), that is:
1000000×0.08×(0.7×0.5×0.62)×(1－1/3×(0.04+0.027+0.037)) = 16758.18
(yuan)
The bad debt loss caused by bad debts is:
loan funds × total pass rate × total bad debt rate, namely:
1000000×(0.7×0.5×0.62)×(1/3×(0.04+0.027+0.037))=7522.666 (yuan) Then the bank’s final income is: loan interest income-bad debt loss, that is, 16758.18-7522.666 = 9235.514
(
yuan )
It can be seen that choosing different credit score cards and different threshold combinations will bring different income and losses to the bank, thus determining the final income of the bank. Therefore, the goal of the bank is to choose the most reasonable combination of credit score cards and its threshold value, so that the bank can finally earn the most money.
Question Description 2 : Introduction to the QUBO Model
The QUBO model refers to the Quadratic Unconstrained Binary Optimization model, which is a mathematical model used to solve combinatorial optimization problems. In the QUBO model, the problem needs to be converted into a decision variable as a binary variable, and the objective function is an optimization model in the form of a quadratic function.
The QUBO model can run on quantum computer hardware, and the quantum computer can be used to accelerate the solution in milliseconds. This model and acceleration method will be widely used in various industries in the future. Therefore, it is very valuable to study the quantum special algorithm based on the QUBO model at this stage. For example, typical graph coloring, traveling salesman problem, vehicle routing optimization problem, etc., can be transformed into QUBO model and solved with the help of quantum computer.
For related QUBO transformation methods and examples, please refer to the references in Appendix 2.
Question Description 3 : Question Data
Attachment 1 contains a total of 100 credit score cards, and each card can be set with one of 10 thresholds, and
There are 200 columns corresponding to their pass rate and bad debt rate, among which t_ 1 represents the pass rate of credit score card 1, a total of 10 items, h_ 1 represents the bad debt rate of credit score card 1, a total of 10 items, and so on, t_ 100 represents credit score card 100 h_100 represents the bad debt rate of credit score card 100.
According to the description of the above competition questions and the data in Appendix 1, please your team to complete the following questions 1 to 3 by building a mathematical model.
Problem 1 : Find one of the 100 credit score cards and its corresponding threshold to maximize the final income. Please model this problem, convert the model to QUBO form and solve it.
Question 2 : Assuming that the game description 3 has selected the three rules of credit score card 1, credit score card 2 and credit score card 3 given in the data set, how to set their corresponding thresholds to maximize the final income, please To model this problem, convert the model to QUBO form and solve it.
Question 3 : Select 3 credit score cards from the 100 credit score cards in the given appendix, and set a reasonable threshold to maximize the final income. Please model this problem, and convert the model to the form of QUBO and solve.
