Human Resource Scheduling Model and Algorithm with Time Windows and Multi-Skill Constraints
Abstract
:1. Introduction
2. Problem Description and Model Construction
2.1. Problem Description
2.2. Notations Description
2.3. Mathematical Model
3. Algorithm Design
3.1. Path Coding
3.2. Construction of the Initial Path
- Step 1
- Use 2-relocate, 2-opt*, and 2-exchange operators to exchange between paths to generate a new solution S′, and if , accept S′; otherwise, keep the original solution unchanged;
- Step 2
- Use the 1-relocate, or-opt, 2-opt, and 1-exchange operators to exchange in the same path to generate a new solution S′, and if , accept S′; otherwise, keep the last result;
- Step 3
- Use the 0-exchange operator to exchange the unserved points with the points in the path to obtain a new solution S′, and if , accept S′; otherwise, keep the last result;
- Step 4
- If the quality of the solution is not improved after the above three steps, the operation will jump out; otherwise, jump to Step 1.
- Step 1
- Generate n empty paths {0, 0}, and set the service personnel type to K = 2k − 1 (k is the total skill type, K is the service personnel type);
- Step 2
- Traverse all the points and u insert the points into the appropriate positions using the Solomon insert algorithm;
- Step 3
- Optimize the path using the IPP for the current solution;
- Step 4
- Determine whether all points are inserted. If all points are served, end the iteration and output the current solution; otherwise, return to Step 2.
3.3. Path Optimization
3.3.1. Remove-Random (RR) Disturbance
3.3.2. IPP-Based Tabu Search (IPP-TS)
- Step 1
- Obtain the feasible solution S generated in the process of Section 3.2 as the initial solution and set the current iteration number of iter and no_impr (the un-updated algebra) to 0.
- Step 2
- After S is disturbed by RR, the points in the set U are traversed by the Solomon insertion method, and these points are reinserted to form a new solution, and the new solution is stored as a local_best;
- Step 3
- Use IPP-TS for local search to form a new solution S′, and update the tabu table;
- Step 4
- Compare the size of f(S) and f (S’), if , update the current optimal solution ‘‘Best’’ and the local optimal solution ‘‘local_best’’, and set “no_impr” to zero, go to Step 3, otherwise, jump to Step 5;
- Step 5
- If make a judgment. Here we use the cooling strategy commonly used in SA: (α is the cooling coefficient), the probability of accepting the inferior solution: (S and S′ represent the objective function value of the current solution), and then randomly produces a random number test (0, 1). If test ≤ Psa is satisfied, the inferior solution is used to replace the current solution and it is stored as the local optimal solution local_best and let no_impr = no_impr+1, iter = iter+1;
- Step 6
- When no_impr is greater than the set value limit, the current best overrides local_best to enter the next loop, and no_impr is set to zero;
- Step 7
- Determine whether the termination condition has been reached. If iter > max terminates the current step and saves the current optimal solution best as the initial solution to the next step, otherwise go to Step 2.
3.4. Conversion of Personnel Types
3.4.1. Transformation of Personnel Types (2-Random Exchange, 2RE)
- Step 1
- Traverse each path in the solution to generate a path Rtmp that is different from the current employee type, but whether the modification is accepted is also judged by Step 2 and Step 3;
- Step 2
- Generate a random number test = rand(0,1) within an interval (0,1) judged by the following two conditions:
- If the newly generated path type is higher than the current path, and Ph < test, the new path type is accepted;
- Accept the new path type if the new path type and current type belong to the same level, or lower than the current path type, and Pl < test, the new path type is accepted.
- Step 3
- If the operations in Step 2 do not meet the probabilistic requirements, the original level remains the same, but can be combined with different skills (that is, when the original path is to master A and B skills of the 2 level staff service, it can be changed to become a level 2 employee who masters B and C skills);After Step 2 and the current step, if the path is changed, the path will not be changed again within the m iterations; (where the size of m here is consistent with the tabu step setting value in the tabu search operation);
- Step 4
- If there is a change in the Step 2 and Step 3 path, the filter is performed to traverse all the points in the current path and add points that do not meet the skill requirements to the U set; otherwise, jump Step 1 operation and continue traversing the path.
3.4.2. Skill Check (Check)
- Step 1
- Obtain the solution S produced in the process of Section 3.3 as the initial solution of this iteration, and randomly increase the R empty paths to increase the diversity of the solution;
- Step 2
- Perform RR and skill check to generate a new solution S′;
- Step 3
- Take advantage of the optimization process in the process of Section 3.3, and change the objective function to f1(S);
- Step 4
- If Step 3 satisfies the termination condition, the current optimal solution outputs and this step terminates, otherwise Step 2 is returned.
3.5. Algorithm Flow
- Step 1
- Initialization, using the Solomon insertion method and IPP to obtain the initial solution S;
- Step 2
- Path optimization, using IPP-TS combined with SA for local search for path optimization, resulting in a new solution S′ instead of the initial solution S;
- Step 3
- Personnel type conversion, the use of 2RE and Check process for path conversion;
- Step 4
- Determine whether the current iteration number of iterations Iter_num reach the maximum number of iterations N. If Iter_num < N, then Iter_num = Iter_num + 1, return to Step 2, otherwise, go to Step 5;
- Step 5
- Meet termination conditions, end the optimization, delete redundant empty path, and output the result.
4. Calculation and Analysis
4.1. Analysis of Parameters and Strategies
4.2. Example Calculation
4.3. Solving Performance
4.3.1. Impact of Algorithm Strategy
4.3.2. Effective Synthesis
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Personnel Type | Mastered Skills |
---|---|
1 | A |
2 | B |
3 | C |
4 | A,B |
5 | B,C |
6 | A,C |
7 | A,B,C |
Operational Operator | Tabu Objects | Tabu Step |
---|---|---|
2-opt, 2-opt* | (i, j), (i + 1, j + 1) | tabu1 |
or-opt | (i, i + 1, j) | tabu2 |
0-exchange, 1-exchange, 2-exchange | (i, j) | tabu3 |
2-relocate | (r 1, i) | tabu4 |
Best Solution | Average Solution | Standard Deviation | Coefficient of Variation | |
---|---|---|---|---|
0 | 803.5432 | 805.4517 | 1.338107 | 0.016 |
10 | 803.5432 | 804.8631 | 0.914442 | 0.001136 |
20 | 803.5432 | 805.2226 | 1.164162 | 0.001446 |
30 | 803.5432 | 805.3211 | 1.172981 | 0.001457 |
40 | 804.2811 | 805.0887 | 0.956384 | 0.001188 |
Best Solution | Average Solution | Standard Deviation | Coefficient of Variation | |
---|---|---|---|---|
0 | 808.925 | 843.3944 | 30.50354 | 0.036168 |
0.1 | 803.5432 | 804.8631 | 0.914442 | 0.001136 |
0.2 | 805.7826 | 807.6978 | 1.374694 | 0.001702 |
0.3 | 807.114 | 809.8734 | 1.893115 | 0.002338 |
0.4 | 809.5577 | 812.3651 | 1.767091 | 0.002175 |
Best Solution | Average Solution | Standard Deviation | Coefficient of Variation | |
---|---|---|---|---|
5 | 803.5432 | 805.8537 | 1.238797 | 0.001537 |
10 | 803.5432 | 805.307 | 1.053678 | 0.001308 |
15 | 803.5432 | 805.2574 | 1.075042 | 0.001335 |
20 | 803.5432 | 804.8631 | 0.914442 | 0.001136 |
25 | 803.5432 | 805.337 | 0.975921 | 0.001212 |
T0 | 100 | 10 | 1 | 0.1 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
O.V. 1 | Mean | C.V. 2 | O.V. 1 | Mean | C.V. 2 | O.V. 1 | Mean | C.V. 2 | O.V. 1 | Mean | C.V. 2 | |
0.9 | 803.54 | 805.18 | 0.09% | 803.54 | 805.21 | 0.10% | 804.28 | 805.29 | 0.13% | 803.54 | 805.09 | 0.12% |
0.93 | 803.54 | 805.01 | 0.10% | 803.54 | 805.17 | 0.17% | 803.54 | 804.87 | 0.12% | 803.54 | 805.11 | 0.13% |
0.95 | 803.54 | 805.52 | 0.19% | 803.54 | 805.39 | 0.13% | 803.54 | 805.19 | 0.13% | 804.28 | 805.41 | 0.16% |
0.97 | 803.54 | 805.25 | 0.14% | 803.54 | 805.14 | 0.11% | 803.54 | 805.15 | 0.14% | 804.28 | 805.07 | 0.07% |
0.99 | 804.28 | 805.45 | 0.13% | 804.28 | 805.03 | 0.09% | 804.28 | 805.24 | 0.11% | 803.54 | 805.15 | 0.16% |
0.995 | 803.54 | 805.31 | 0.18% | 803.54 | 805.05 | 0.14% | 804.28 | 805.41 | 0.16% | 803.54 | 804.95 | 0.13% |
0.999 | 803.54 | 805.24 | 0.08% | 803.54 | 804.90 | 0.09% | 803.54 | 805.58 | 0.19% | 803.54 | 804.87 | 0.10% |
0.1 | 0.3 | 0.5 | 0.7 | 0.9 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
O.V. 1 | Mean | C.V. 2 | O.V. 1 | Mean | C.V. 2 | O.V. 1 | Mean | C.V. 2 | O.V. 1 | Mean | C.V. 2 | O.V.1 | Mean | C.V. 2 | |
0.1 | 2567.60 | 2591.25 | 1.61% | 2547.08 | 2599.59 | 1.42% | 2553.17 | 2598.63 | 1.70% | 2550.42 | 2602.51 | 2.59% | 2550.07 | 2599.45 | 2.22% |
0.3 | 2640.96 | 2754.60 | 2.09% | 2565.83 | 2658.83 | 2.08% | 2564.00 | 2626.52 | 2.01% | 2552.97 | 2608.82 | 1.64% | 2567.38 | 2616.51 | 1.47% |
0.5 | 2691.27 | 2835.83 | 2.77% | 2577.94 | 2711.37 | 4.41% | 2556.58 | 2664.73 | 2.29% | 2570.94 | 2634.82 | 1.61% | 2570.91 | 2650.63 | 2.19% |
0.7 | 2711.30 | 2846.71 | 3.40% | 2570.56 | 2705.84 | 2.56% | 2566.86 | 2654.67 | 2.59% | 2588.38 | 2674.05 | 2.43% | 2558.52 | 2682.87 | 3.56% |
0.9 | 2704.76 | 2876.77 | 3.44% | 2592.08 | 2765.29 | 4.14% | 2583.40 | 2729.64 | 3.33% | 2568.42 | 2742.43 | 5.66% | 2604.02 | 2770.43 | 4.91% |
No. | Personnel Type | Personnel Salary | Path |
---|---|---|---|
1 | 7 | 300 | 0-81-78-76-71-70-73-77-79-80-0 |
2 | 7 | 300 | 0-67-65-63-62-74-72-61-64-68-66-69-0 |
3 | 7 | 300 | 0-98-96-95-94-92-93-97-100-99-2-1-75-0 |
4 | 7 | 300 | 0-57-55-54-53-56-58-60-59-0 |
5 | 7 | 300 | 0-13-17-18-19-15-16-14-12-0 |
6 | 7 | 300 | 0-20-24-25-27-29-30-28-26-23-0 |
7 | 7 | 300 | 0-5-3-7-8-10-11-9-6-4-0 |
8 | 7 | 300 | 0-32-33-31-35-37-38-39-36-34-22-21-0 |
9 | 7 | 300 | 0-43-42-41-40-44-46-45-48-51-50-52-49-47-0 |
10 | 7 | 300 | 0-90-87-86-83-82-84-85-88-89-91-0 |
path length: 822.84 | salary: 3000 | Total cost: 3822.84 |
No. | Personnel Type | Personnel Salary | Path |
---|---|---|---|
1 | 1 | 100 | 0-20-24-27-29-30-28-22-21-0 |
2 | 4 | 200 | 0-65-96-94-92-97-100-99-2-1-75-0 |
3 | 2 | 100 | 0-55-54-53-72-64-68-0 |
4 | 4 | 200 | 0-43-42-41-62-44-46-45-48-51-50-47-0 |
5 | 3 | 100 | 0-3-5-25-40-60-61-66-69-0 |
6 | 1 | 100 | 0-57-63-74-56-58-59-0 |
7 | 3 | 100 | 0-67-98-95-93-9-26-0 |
8 | 7 | 300 | 0-13-17-18-19-15-16-14-12-4-0 |
9 | 4 | 200 | 0-7-8-10-11-6-23-0 |
10 | 7 | 300 | 0-90-87-86-83-82-84-85-88-89-91-0 |
11 | 7 | 300 | 0-32-33-31-35-37-38-39-36-34-52-49-0 |
12 | 7 | 300 | 0-81-78-76-71-70-73-77-79-80-0 |
path length: 1241.87 | salary: 2300 | Total cost: 3541.87 |
Example | Salary | Path Length | LS | LS + SA | HSATS | Significance Test 3 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
O.V. 1 | Mean | C.V. 2 | O.V. 1 | Mean | C.V. 2 | O.V.1 | Mean | C.V. 2 | t Value | p Value 4 | |||
C101 | 3000 | 828.94 | 828.94 | 835.50 | 1.68% | 828.94 | 829.48 | 0.28% | 828.94 | 828.94 | 0.00% | −1.887 | 0.024 |
C102 | 3000 | 828.94 | 821.93 | 897.10 | 11.77% | 821.93 | 821.93 | 0.00% | 821.93 | 821.93 | 0.00% | N/A | N/A |
C103 | 3000 | 828.06 | 816.31 | 826.98 | 1.35% | 815.44 | 818.27 | 0.34% | 815.44 | 816.24 | 0.22% | −2.909 | 0.006 |
R101 | 5700 | 1645.79 | 1649.34 | 1546.49 | 12.34% | 1217.19 | 1539.76 | 12.11% | 1215.90 | 1537.70 | 12.10% | −1.890 | 0.023 |
R102 | 5100 | 1648.12 | 1472.81 | 1528.25 | 5.34% | 1466.60 | 1524.19 | 5.28% | 1472.81 | 1522.57 | 5.10% | −2.172 | 0.020 |
R103 | 3900 | 1292.68 | 1213.62 | 1368.05 | 5.60% | 1270.59 | 1366.39 | 5.63% | 1211.00 | 1303.12 | 9.28% | −2.389 | 0.012 |
RC101 | 4200 | 1696.94 | 1632.90 | 1653.98 | 1.11% | 1626.16 | 1646.68 | 0.65% | 1642.88 | 1643.18 | 0.03% | −2.872 | 0.007 |
RC102 | 3600 | 1554.75 | 1471.16 | 1476.83 | 0.36% | 1466.60 | 1474.05 | 0.30% | 1472.81 | 1475.10 | 0.30% | −1.669 | 0.035 |
RC103 | 3300 | 1261.67 | 1274.82 | 1318.59 | 1.26% | 1270.59 | 1313.07 | 1.45% | 1215.00 | 1221.30 | 0.26% | −28.671 | 0.000 |
C201 | 900 | 591.56 | 591.56 | 591.56 | 0.00% | 591.56 | 591.56 | 0.00% | 591.56 | 591.56 | 0.00% | N/A | N/A |
C202 | 900 | 591.56 | 591.56 | 594.02 | 1.25% | 591.56 | 592.27 | 0.66% | 591.56 | 591.56 | 0.00% | −2.883 | 0.007 |
C203 | 900 | 591.17 | 591.56 | 616.47 | 1.96% | 591.17 | 612.49 | 1.85% | 591.17 | 592.58 | 0.92% | −13.415 | 0.000 |
R201 | 1200 | 1252.37 | 1189.41 | 1193.30 | 0.59% | 1189.41 | 1194.67 | 0.76% | 1184.36 | 1193.15 | 0.63% | −3.2186 | 0.002 |
R202 | 900 | 1191.7 | 1041.10 | 1045.96 | 0.62% | 1041.10 | 1046.82 | 0.51% | 1041.10 | 1050.32 | 1.52% | −1.796 | 0.026 |
R203 | 900 | 939.54 | 879.80 | 880.51 | 0.05% | 879.80 | 881.92 | 0.42% | 879.80 | 881.72 | 0.43% | −1.652 | 0.042 |
RC201 | 1200 | 1406.91 | 1310.44 | 1350.79 | 2.47% | 1310.44 | 1318.75 | 0.84% | 1310.44 | 1319.10 | 0.91% | −1.913 | 0.022 |
RC202 | 1200 | 1367.09 | 1118.66 | 1126.11 | 0.70% | 1118.66 | 1122.16 | 0.43% | 1118.66 | 1120.64 | 0.34% | −2.525 | 0.009 |
RC203 | 1200 | 1049.62 | 937.45 | 942.65 | 0.43% | 937.45 | 941.18 | 0.44% | 928.43 | 939.97 | 0.39% | −2.682 | 0.008 |
Example | Non Multi-Skill 1 | LS 2 | LS + SA 2 | HSATS 2 | Number of Vehicles Required | Solving Time (s) | ||||||
O.V. 3 | Mean | C.V. 4 | O.V. 3 | Mean | C.V. 4 | O.V. 3 | Mean | C.V. 4 | ||||
C101 | 3828.94 | 3668.39 | 3742.94 | 1.17% | 3404.65 | 3557.09 | 3.02% | 3398.13 | 3565.57 | 3.30% | 12.03 | 13.79 |
C102 | 3828.94 | 3605.70 | 3746.78 | 2.71% | 3372.87 | 3505.16 | 2.01% | 3322.70 | 3413.64 | 2.56% | 12.40 | 18.11 |
C103 | 3828.06 | 2918.17 | 3019.06 | 1.87% | 3145.51 | 3565.80 | 5.98% | 2902.96 | 2963.92 | 0.03% | 11.73 | 20.64 |
R101 | 7345.79 | 5499.05 | 6482.29 | 6.21% | 4728.88 | 4964.37 | 2.92% | 4737.71 | 4905.07 | 3.36% | 23.30 | 13.95 |
R102 | 6748.12 | 4586.25 | 5579.13 | 8.02% | 4542.94 | 4605.69 | 0.88% | 4513.38 | 4565.15 | 0.84% | 21.17 | 18.22 |
R103 | 5192.68 | 3499.03 | 3551.96 | 0.81% | 3476.91 | 3576.80 | 3.07% | 3460.70 | 3503.75 | 0.80% | 16.10 | 20.41 |
RC101 | 5896.94 | 4408.44 | 5015.56 | 6.60% | 4432.44 | 4668.37 | 3.08% | 4346.76 | 4447.54 | 1.64% | 19.47 | 15.26 |
RC102 | 5154.75 | 3930.85 | 4250.21 | 3.25% | 3921.06 | 4137.01 | 3.08% | 3926.82 | 4033.47 | 1.44% | 17.17 | 19.37 |
RC103 | 4561.67 | 3476.22 | 3678.32 | 3.91% | 3482.32 | 3607.96 | 2.88% | 3429.77 | 3503.05 | 1.16% | 14.77 | 20.30 |
C201 | 1491.56 | 1491.56 | 1491.56 | 0.00% | 1491.56 | 1491.56 | 0.00% | 1491.56 | 1491.56 | 0.00% | 3.10 | 14.40 |
C202 | 1491.56 | 1491.56 | 1491.56 | 0.00% | 1491.56 | 1491.56 | 0.00% | 1491.56 | 1491.56 | 0.00% | 3.03 | 19.59 |
C203 | 1491.17 | 1491.17 | 1498.65 | 0.68% | 1491.17 | 1492.06 | 0.33% | 1491.17 | 1491.17 | 0.00% | 3.07 | 21.73 |
R201 | 2452.37 | 2427.48 | 2538.71 | 2.81% | 2372.59 | 2487.25 | 1.28% | 2350.23 | 2462.41 | 1.57% | 4.73 | 14.07 |
R202 | 2091.7 | 2138.74 | 2292.70 | 3.18% | 2146.09 | 2253.94 | 2.30% | 2137.84 | 2204.57 | 1.62% | 4.97 | 16.96 |
R203 | 1839.54 | 1743.87 | 1902.00 | 4.40% | 1782.73 | 1880.76 | 2.54% | 1757.78 | 1776.16 | 0.76% | 3.97 | 17.86 |
RC201 | 2606.91 | 2572.36 | 2750.00 | 2.77% | 2543.08 | 2621.48 | 2.21% | 2525.18 | 2632.17 | 2.64% | 4.73 | 17.40 |
RC202 | 2467.09 | 2373.03 | 2522.44 | 2.74% | 2362.33 | 2522.44 | 2.85% | 2362.33 | 2391.38 | 1.01% | 4.37 | 19.94 |
RC203 | 2049.62 | 1969.99 | 2111.62 | 2.42% | 1987.01 | 2071.75 | 2.83% | 1983.15 | 2060.58 | 1.86% | 3.87 | 21.65 |
Example | k = 3 | k = 4 | k = 5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
O.V. 1 | Mean | C.V. 2 | A.T. 3 | O.V. 1 | Mean | C.V. 2 | A.T. 3 | O.V. 1 | Mean | C.V. 2 | A.T. 3 | |
C101 | 3398.13 | 3565.57 | 3.30% | 9.88 | 3511.17 | 3721.28 | 3.19% | 15.94 | 3658.00 | 3792.51 | 2.57% | 17.19 |
C102 | 3322.70 | 3413.64 | 2.56% | 14.81 | 3440.76 | 3542.14 | 2.56% | 17.10 | 3621.13 | 3735.28 | 2.41% | 17.53 |
C103 | 2902.96 | 2963.92 | 1.11% | 17.14 | 3005.30 | 3087.38 | 1.24% | 19.45 | 3055.29 | 3156.36 | 1.61% | 19.88 |
R101 | 4737.71 | 4905.07 | 3.36% | 13.98 | 4849.42 | 5034.01 | 3.25% | 16.25 | 4903.55 | 5103.30 | 3.31% | 16.72 |
R102 | 4513.38 | 4565.15 | 0.84% | 18.09 | 4634.77 | 4693.55 | 0.94% | 20.34 | 4683.33 | 4770.90 | 0.92% | 20.85 |
R103 | 3460.70 | 3503.75 | 0.80% | 19.75 | 3572.01 | 3629.25 | 0.86% | 22.00 | 3650.36 | 3707.68 | 0.84% | 22.52 |
RC101 | 4346.76 | 4447.54 | 1.64% | 4.45 | 4467.53 | 4571.64 | 1.69% | 18.19 | 4547.43 | 4646.25 | 1.73% | 18.61 |
RC102 | 3926.82 | 4033.47 | 1.44% | 19.86 | 4028.88 | 4160.53 | 1.52% | 22.13 | 4116.73 | 4237.59 | 1.59% | 22.61 |
RC103 | 3429.77 | 3503.05 | 1.16% | 21.39 | 3556.40 | 3625.34 | 1.32% | 23.70 | 3610.20 | 3700.38 | 1.56% | 24.17 |
C201 | 1491.56 | 1491.56 | 0.00% | 12.43 | 1491.56 | 1491.56 | 0.00% | 14.68 | 1491.56 | 1491.56 | 0.00% | 17.49 |
C202 | 1491.56 | 1491.56 | 0.00% | 16.80 | 1491.56 | 1491.56 | 0.00% | 19.02 | 1491.56 | 1491.56 | 0.00% | 19.53 |
C203 | 1491.17 | 1491.17 | 0.00% | 19.79 | 1491.17 | 1491.17 | 0.00% | 22.02 | 1491.17 | 1491.17 | 0.00% | 22.54 |
R201 | 2350.23 | 2462.41 | 1.57% | 16.70 | 2478.35 | 2591.36 | 1.64% | 18.92 | 2578.15 | 2665.51 | 1.78% | 19.44 |
R202 | 2137.84 | 2204.57 | 1.62% | 19.35 | 2259.38 | 2326.10 | 1.52% | 21.62 | 2291.80 | 2404.34 | 1.88% | 22.13 |
R203 | 1757.78 | 1776.16 | 0.76% | 20.85 | 1859.27 | 1901.14 | 1.06% | 23.08 | 1909.08 | 1980.59 | 1.67% | 23.59 |
RC201 | 2525.18 | 2632.17 | 2.64% | 17.63 | 2644.61 | 2755.00 | 2.46% | 19.87 | 2692.69 | 2824.93 | 2.72% | 20.41 |
RC202 | 2309.00 | 2436.63 | 2.95% | 2.44 | 2415.95 | 2560.97 | 2.88% | 21.95 | 2503.37 | 2632.30 | 2.67% | 22.40 |
RC203 | 1983.15 | 2060.58 | 1.86% | 22.05 | 2091.75 | 2187.46 | 1.85% | 24.27 | 2183.92 | 2260.86 | 1.86% | 24.80 |
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Zuo, Z.; Li, Y.; Fu, J.; Wu, J. Human Resource Scheduling Model and Algorithm with Time Windows and Multi-Skill Constraints. Mathematics 2019, 7, 598. https://doi.org/10.3390/math7070598
Zuo Z, Li Y, Fu J, Wu J. Human Resource Scheduling Model and Algorithm with Time Windows and Multi-Skill Constraints. Mathematics. 2019; 7(7):598. https://doi.org/10.3390/math7070598
Chicago/Turabian StyleZuo, Zhiping, Yanhui Li, Jing Fu, and Jianlin Wu. 2019. "Human Resource Scheduling Model and Algorithm with Time Windows and Multi-Skill Constraints" Mathematics 7, no. 7: 598. https://doi.org/10.3390/math7070598