# A New Greedy Insertion Heuristic Algorithm with a Multi-Stage Filtering Mechanism for Energy-Efficient Single Machine Scheduling Problems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MILP Formulation for the Problem

_{i}and power consumption per hour p

_{i}. A machine can process, at most, one job at a time, and when it is processing a job, no preemption is allowed. Each job and the machine are available for processing at time instant 0. Machine breakdown and preventive maintenance are not considered in this paper.

_{k}and a starting time b

_{k}. The interval of period k is represented by [b

_{k}, b

_{k}

_{+1}], $k\in M$, and b

_{1}= 0 is always established. It is assumed that the C

_{max}is the given makespan and b

_{m}

_{+1}≥ C

_{max}. This means that a feasible solution always exists.

_{m}

_{+1}] to minimize total electricity cost, and the main task is to determine to which period(s) a job is assigned and how long a job is processed in each period. Hence, two decision variables are given as follows. Note that the starting time of each job can be determined by the decision variables (i.e., x

_{i}

_{,k}and y

_{i}

_{,k}).

_{i}

_{,k}: assigned processing time of job i in period k, $i\in N,$ $k\in M;$

_{k}and X

_{k}, $k\in M,$ represent the duration of period k and the total already assigned processing times in period k, respectively. The difference between d

_{k}and X

_{k}is defined as the remaining idle time of period k which is represented by I

_{k}, $k\in M.$ If I

_{k}= 0, the period k is called full.

## 3. A Greedy Insertion Heuristic Algorithm with Multi-Stage Filtering Mechanism

#### 3.1. The Characteristics of TOU Electricity Tariffs

#### 3.2. Multi-Stage Filtering Mechanism Design

_{3}< I

_{4}). It is noteworthy that the smaller the number of the job, the greater the power consumption rate.

_{k}

_{+2}> 0, it follows that period k + 2 always exists. To minimize the electricity cost for processing job i, the off-peak period with maximal remaining idle time (period k) should be given priority, compared with other off-peak, mid-peak, and on-peak periods. Thus, job i is preferred to be processed across periods k, k + 1, and k + 2. Meanwhile, the corresponding electricity cost is named as cost1. However, if job i is processed within a mid-peak period (i.e., the corresponding electricity cost is named as cost2), the electricity cost may be lower. Hence, two positions are considered and an illustration is given in Figure 5. Let c

_{A}, c

_{B}, and c

_{Γ}represent the electricity prices of off-peak, mid-peak, and on-peak periods, respectively. To select the optimal position, the key property 1 is given as follows.

**Property**

**1.**

**Proof.**

_{i}= ${x}_{i,k}+{x}_{i,k+1}+{x}_{i,k+2},$ it follows that:

_{k}

_{+2}= 0 or not. This implies that period k is the last off-peak period and period k + 2 does not exist. Hence, period k must be in the last cycle (i.e., on the last day) and there are two scenarios that should be considered as follows (the zero points of the two scenarios are different).

- To occupy off-peak periods as much as possible, a set of already inserted jobs in period k should be moved to the rightmost side, and then job i can be processed across periods k and k − 1.
- All inserted jobs in period k should be moved left so that job i can be processed within period k.

^{sm}denote the off-peak period with submaximal remaining idle time. Positions 3 and 4 are given as follows.

- 3.
- Suppose that job i is processed across periods k
^{sm}, k^{sm}+ 1, and k^{sm}+ 2. If I_{k}^{sm}is slightly smaller than I_{k}, cost3 may be less than cost1 as period k^{sm}+ 1 is a mid-peak period. Hence, Position 3 needs to be considered. - 4.
- Similar to Condition 3, job i can be processed within a mid-peak period.

**Property**

**2.**

_{k}

_{+1}= 0, job i can be directly inserted into Position 1 without moving any already inserted jobs in period k − 1.

**Proof.**

_{k}> 0, there must be no jobs processed within period k − 1. It is only possible that a job, say job j, j < i, is processed across periods k − 2 and k − 1. Therefore, x

_{j,k}

_{−2}may be the value of the maximal remaining idle time of all mid-peak periods before inserting job j. Since $\exists {k}^{\prime}\in B,$ t

_{i}≤ I

_{k}

_{’}and j < i, it is understandable that ${t}_{i}\le {I}_{{k}^{\prime}}\le {x}_{j,k-2}.$ Now, job i is to be inserted, it follows that ${t}_{i}+{x}_{j,k-1}\le {x}_{j,k-2}+{x}_{j,k-1}={t}_{j}.$ As mentioned earlier, the processing times of all the jobs do not exceed the duration of the shortest on-peak period, that is, t

_{j}≤ d

_{k}

_{−1}. Hence, t

_{i}+ x

_{j,k}

_{−1}≤ d

_{k}

_{−1}. If job i is processed across periods k and k − 1, then ${x}_{i,k-1}+{x}_{j,k-1}\le {x}_{i,k-1}+{x}_{i,k}+{x}_{j,k-1}={t}_{i}+{x}_{j,k-1}\le {d}_{k+1}.$ That is, ${x}_{i,k-1}+{x}_{j,k-1}\le {d}_{k-1}.$ Thus, ${d}_{k-1}-{x}_{j}{}_{,k-1}-{x}_{i}{}_{,k-1}={I}_{k}{}_{-1}\ge 0.$ This suggests when job i is inserted into Position 1, period k − 1 cannot be full. Hence, job i can be directly inserted into Position 1 without moving any already inserted jobs in period k − 1. Note that this property applies to Scenario 2 as well. ☐

_{k}

_{+1}> 0 or not (i.e., period k + 1 exists). Since period k + 1 is a mid-peak period, two additional positions need to be considered in comparison with Scenario 1.

- A set of already inserted jobs in period k are moved to the rightmost side of the period k + 1, and then job i is processed across periods k and k − 1.
- A set of already inserted jobs in period k should be moved to the left until job i can be processed across periods k and k + 1.

Algorithm 1: Greedy insertion heuristic algorithm with multi-stage filtering mechanism |

1. Sort all jobs in non-increasing order of their power consumption rates 2. Initialization: I _{k} = b_{k}_{+1} − b_{k}, for all 1 ≤ k ≤ m3. For i = 1 to n do3.1. If layer 1C1. If Condition 1 is satisfiedInitial the period index $kk=\mathrm{arg}{\mathrm{min}}_{k\in A}\left({I}_{k}\ge {t}_{i}\right)$ //Job i is processed within period kk. C2. Else if Condition 2 is satisfied//Job i is processed across periods k and k + 1. 3.2. Else if layer 2C3. If Condition 3 is satisfiedC3.1. If inequality (8) is not satisfied//Job i is processed across periods k, k + 1, and k + 2. C3.2. ElseInitial the period index $k{k}^{\prime}=\mathrm{arg}{\mathrm{min}}_{{k}^{\prime}\in B}\left({I}_{{k}^{\prime}}\ge {t}_{i}\right)$ //Job i is processed within period kk’. C4. Else if Condition 4 is satisfiedC4.1. If d_{k}_{+1} = 0//Calculate cost1, cost3, and cost4 and insert job i into the position with minimal insertion cost. C4.2. Else if d_{k}_{+2} = 0 and d_{k}_{+1} > 0//Calculate cost1, cost2, cost3, cost4, and cost5 and insert job i into the position with minimal insertion cost. C5. Else if Condition 5 is satisfiedInitial the period index $k{k}^{\prime}=\mathrm{arg}{\mathrm{min}}_{{k}^{\prime}\in B}\left({I}_{{k}^{\prime}}\ge {t}_{i}\right)$ //Job i is processed within period kk’. 3.3. Else if layer 3C6. If Condition 6 is satisfied//Similarly to Condition 4, it needs to calculate the insertion cost of several possible positions and insert job i into the position with minimal insertion cost. C7. Else if Condition 7 is satisfiedC7.1. If ${\mathrm{max}}_{{k}_{\u2033}\in \mathrm{\Gamma}}\left\{{I}_{{k}_{\u2033}}\right\}>{t}_{i}$Initial the period index $k{k}_{\u2033}=\mathrm{arg}{\mathrm{min}}_{{k}_{\u2033}\in \mathrm{\Gamma}}\left({I}_{{k}_{\u2033}}\ge {t}_{i}\right)$ //Job i is processed within period kk”. C7.2. Else//Job i traverses all non-full on-peak periods and insert job i into the position with minimal insertion cost. |

**Theorem**

**1.**

^{2}m|Γ|) in the worst case.

**Proof.**

^{2}m|Γ|) in the worst case. ☐

^{2}m). However, the classic greedy insertion algorithm proposed by Che et al. [9] requires O(n

^{2}m

^{2}) operations in the worst case when dealing with the same problem, because their algorithm requires all the jobs to traverse all non-full periods to find an optimum position.

## 4. Computational Results

_{max}and C

_{B}denote the given makespan and the total processing times of all the jobs, respectively. The parameter e = C

_{max}/C

_{B}(e ≥ 1) is used to measure the degree of time tightness. In these instances, C

_{max}can be obtained by the formula C

_{max}= e × C

_{B}as long as the parameter e is set. Obviously, as e increases, C

_{max}increases. Note that b

_{m}

_{+1}is calculated by ${b}_{m+1}=\lceil {C}_{max}/24)\rceil \times 24.$ Let TEC

_{F}and TEC

_{H}be the total electricity cost calculated by our algorithm (GIH-F)and Che et al.’s algorithm (GIH), respectively. The runtimes of the two algorithms are represented by CT

_{F}and CT

_{H}, respectively. The gaps between TEC

_{F}and TEC

_{H}are represented by G, G = (TEC

_{F}− TEC

_{H})/TEC

_{H}× 100%. The ratio of CT

_{H}/CT

_{F}is represented by R.

#### 4.1. A Real Case Study

_{1}= 3.5, I

_{2}= 7, I

_{3}= 4.5, I

_{4}= 1.8, I

_{5}= 1, I

_{6}= 3.5, I

_{7}= 7, I

_{8}= 4.5, I

_{9}= 1.8, I

_{10}= 1. An illustration is given in Figure 10a.

_{1}= 3.5, I

_{2}= 7, I

_{3}= 4.5, I

_{4}= 0, I

_{5}= 0, I

_{6}= 3.2, I

_{7}= 7, I

_{8}= 4.5, I

_{9}= 1.8, I

_{10}= 1. An illustration is given in Figure 10b.

_{k}

^{sm}= 0 (i.e., I

_{4}= 0), hence job 12 cannot be processed across periods k

^{sm}, k

^{sm}+ 1, and k

^{sm}+ 2, and the corresponding insertion cost is infinity. At this stage, the remaining idle time of each period is as follows: I

_{1}= 3.5, I

_{2}= 7, I

_{3}= 4.5, I

_{4}= 0, I

_{5}= 0, I

_{6}= 3.2, I

_{7}= 7, I

_{8}= 4.2, I

_{9}= 0, I

_{10}= 0. An illustration is given in Figure 10b.

_{1}= 3.5, I

_{2}= 1.8, I

_{3}= 4.5, I

_{4}= 0, I

_{5}= 0, I

_{6}= 3.2, I

_{7}= 2.2, I

_{8}= 4.2, I

_{9}= 0, I

_{10}= 0. An illustration is given in Figure 10c.

#### 4.2. Randomly Generated Instances Studies

_{i}is randomly generated from a uniform distribution (30, 210) min and the power consumption per hour p

_{i}is randomly generated in (30, 100) kW. To measure the effect of the proposed algorithm, parameter e is set as e = 1.2, 1.5, 2, 3.

_{i}≤ I

_{k}, then job i can be directly inserted into the off-peak period kk and the job no longer traverses all the non-full periods.

^{2}m) and GIH is O(n

^{2}m

^{2}). This implies that GIH-F is m times faster than GIH, theoretically, and the experimental data of the small-size instances in Table 5 can verify this conclusion. From Table 5, we can see that R and m are almost the same order of magnitude. In addition, all the small-size instances can be solved within 0.02 s using GIH-F. By and large, the computation time increases slightly with n, and parameter e has no significant effect on the computation time, which indicates that the algorithm is quite stable. In addition, it can be seen from Table 5 that the smaller the parameter e (i.e., the shorter the makespan), the higher the total electricity cost. Therefore, in a specific instance, the decision-makers can obtain a set of Pareto solutions by adjusting the makespan, and they can choose a solution according to actual needs. What is more, it is amazing to see that our algorithm not only greatly improves computation speed but also further improves the accuracy.

^{2}m

^{2}), it’s runtime will rise sharply. To ensure the feasibility of the contrast tests, we add two rules (i.e., Rule 1 and Rule 2) to improve the computational speed of GIH without changing the computational accuracy.

_{F}is significantly less than CT

_{H}

_{2}, which means that the designed filtering mechanism is efficient in dealing with large-scale instances. Specially, as n = 5000 and e = 2.0, our algorithm can solve a randomly generated instance within 1 min and maintain the same accuracy as GIH2, while GIH2 takes nearly 39 h, let alone GIH. Note that when e is set as 3.0, the given makespan is very abundant and there is no job processed within an on-peak period in our experimental environments. Thus, according to Rule 2, all the jobs do not have to traverse on-peak periods, and then CT

_{H}

_{2}is greatly reduced. Conversely, when e is set as 1.2, the number of periods decreases and the jobs are arranged very tightly. There will be many jobs inserted into the periods with higher electricity prices. Therefore, our algorithm should filter more positions with lower electricity prices and constantly judge whether the job needs to be moved. Obviously, all these operations may increase the computation time. Thus, when dealing with large-size instances and setting e to 1.2 or 1.5, our algorithm runs longer, but the computation time is still far less than GIH2.

## 5. Conclusions and Prospects

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Period Type | Electricity Price (CNY/kwh) | Time Periods |
---|---|---|

On-peak | 1.2473 | 8:00–11:30 |

18:30–23:00 | ||

Mid-peak | 0.8451 | 7:00–8:00 |

11:30–18:30 | ||

Off-peak | 0.4430 | 23:00–7:00 |

Product Model | Average Power Consumption Rate (kW) | Processing Time (h) | The Number of Parts |
---|---|---|---|

40 | 4.4 | 2.4 | 15 |

70 | 4.7 | 2.6 | 35 |

100 | 5.3 | 3.1 | 10 |

Part (Job) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Processing time (h) | 2.4 | 2.4 | 2.4 | 2.4 | 2.6 | 2.6 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 |

Power consumption rate (kW) | 4.4 | 4.4 | 4.4 | 4.4 | 4.7 | 4.7 | 5.3 | 5.3 | 5.3 | 5.3 | 5.3 | 5.3 |

Period | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Duration (h) | 3.5 | 7 | 4.5 | 8 | 1 | 3.5 | 7 | 4.5 | 8 | 1 |

Price (CNY/kwh) | 1.2473 | 0.8451 | 1.2473 | 0.443 | 0.8451 | 1.2473 | 0.8451 | 1.2473 | 0.443 | 0.8451 |

Instance | GIH | GIH-F | ||||||
---|---|---|---|---|---|---|---|---|

n | e | m | TEC_{H} | CT_{H} (s) | TEC_{F} | CT_{F} (s) | G (%) | R |

20 | 1.2 | 12.0 | 1634.1 | 0.034 | 1632.5 | 0.002 | −0.10% | 17.0 |

1.5 | 15.0 | 1370.1 | 0.037 | 1370.1 | 0.002 | 0.00% | 18.5 | |

2.0 | 19.0 | 1295.7 | 0.041 | 1295.1 | 0.002 | −0.05% | 20.5 | |

3.0 | 28.5 | 1168.0 | 0.056 | 1168.0 | 0.001 | 0.00% | 56.0 | |

30 | 1.2 | 18.0 | 2414.6 | 0.064 | 2415.7 | 0.002 | 0.05% | 32.0 |

1.5 | 20.0 | 2274.6 | 0.065 | 2274.1 | 0.002 | −0.02% | 32.5 | |

2.0 | 28.5 | 2005.0 | 0.083 | 2005.0 | 0.002 | 0.00% | 41.5 | |

3.0 | 39.5 | 1741.0 | 0.119 | 1741.0 | 0.002 | 0.00% | 59.5 | |

40 | 1.2 | 23.0 | 3342.1 | 0.096 | 3342.0 | 0.004 | 0.00% | 24.0 |

1.5 | 28.0 | 2900.1 | 0.109 | 2899.3 | 0.003 | −0.03% | 36.3 | |

2.0 | 36.0 | 2775.6 | 0.143 | 2775.0 | 0.003 | −0.02% | 47.7 | |

3.0 | 52.0 | 2380.3 | 0.194 | 2380.3 | 0.002 | 0.00% | 97.0 | |

50 | 1.2 | 27.5 | 4242.5 | 0.137 | 4242.4 | 0.005 | 0.00% | 27.4 |

1.5 | 34.0 | 3733.0 | 0.164 | 3732.6 | 0.003 | −0.01% | 54.7 | |

2.0 | 43.0 | 3243.8 | 0.212 | 3243.2 | 0.004 | −0.02% | 53.0 | |

3.0 | 64.5 | 2940.6 | 0.315 | 2940.6 | 0.003 | 0.00% | 105.0 | |

60 | 1.2 | 34.0 | 4820.8 | 0.204 | 4819.7 | 0.006 | −0.02% | 34.0 |

1.5 | 40.0 | 4536.5 | 0.224 | 4536.3 | 0.004 | 0.00% | 56.0 | |

2.0 | 52.0 | 4029.0 | 0.293 | 4028.9 | 0.004 | 0.00% | 73.3 | |

3.0 | 78.0 | 3544.1 | 0.464 | 3544.1 | 0.004 | 0.00% | 116.0 | |

70 | 1.2 | 37.5 | 6133.5 | 0.249 | 6132.2 | 0.007 | −0.02% | 35.6 |

1.5 | 46.0 | 5416.3 | 0.303 | 5416.2 | 0.007 | 0.00% | 43.3 | |

2.0 | 61.0 | 4676.0 | 0.413 | 4675.8 | 0.004 | 0.00% | 103.3 | |

3.0 | 90.0 | 4024.9 | 0.643 | 4024.9 | 0.005 | 0.00% | 128.6 | |

80 | 1.2 | 43.0 | 7073.1 | 0.321 | 7072.9 | 0.009 | 0.00% | 35.7 |

1.5 | 53.0 | 6049.6 | 0.401 | 6049.6 | 0.006 | 0.00% | 66.8 | |

2.0 | 68.5 | 5348.1 | 0.554 | 5348.1 | 0.007 | 0.00% | 79.1 | |

3.0 | 101.5 | 4514.4 | 0.868 | 4514.3 | 0.005 | 0.00% | 173.6 | |

90 | 1.2 | 48.0 | 8128.5 | 0.399 | 8128.4 | 0.009 | 0.00% | 44.3 |

1.5 | 58.0 | 6772.5 | 0.501 | 6772.4 | 0.011 | 0.00% | 45.5 | |

2.0 | 77.5 | 6172.7 | 0.697 | 6172.6 | 0.008 | 0.00% | 87.1 | |

3.0 | 104.1 | 5228.2 | 1.196 | 5228.2 | 0.009 | 0.00% | 132.9 | |

100 | 1.2 | 53.5 | 8623.5 | 0.509 | 8622.9 | 0.017 | −0.01% | 29.9 |

1.5 | 64.0 | 7607.1 | 0.614 | 7607.0 | 0.011 | 0.00% | 55.8 | |

2.0 | 86.5 | 6896.8 | 0.927 | 6896.8 | 0.014 | 0.00% | 66.2 | |

3.0 | 128.0 | 5815.2 | 1.482 | 5815.1 | 0.009 | 0.00% | 164.7 |

Instance | GIH2 | GIH-F | |||||
---|---|---|---|---|---|---|---|

n | e | m | TEC_{H} | CT_{H}_{2} (s) | TEC_{F} | CT_{F} (s) | R |

500 | 1.2 | 250.5 | 43,909.1 | 53.0 | 43,909.1 | 0.219 | 242.0 |

1.5 | 315.0 | 38,637.8 | 52.2 | 38,637.9 | 0.187 | 279.1 | |

2.0 | 417.0 | 34,417.5 | 56.3 | 34,417.5 | 0.082 | 686.6 | |

3.0 | 628.5 | 28,948.1 | 56.9 | 28,948.0 | 0.093 | 611.8 | |

1000 | 1.2 | 504.0 | 87,500.3 | 244.2 | 87,500.3 | 1.802 | 135.5 |

1.5 | 628.5 | 77,598.3 | 230.3 | 77,597.0 | 0.873 | 263.8 | |

2.0 | 840.0 | 69,199.2 | 294.1 | 69,199.2 | 0.432 | 680.8 | |

3.0 | 1256.5 | 57,923.1 | 250.7 | 57,923.1 | 0.485 | 516.9 | |

2000 | 1.2 | 1002.5 | 176,681.2 | 3910.8 | 176,680.7 | 15.701 | 249.1 |

1.5 | 1255.5 | 155,205.3 | 3114.3 | 155,206.2 | 6.503 | 478.9 | |

2.0 | 1669.0 | 137,774.1 | 4316.2 | 137,774.1 | 3.346 | 1290.0 | |

3.0 | 2501.7 | 115,661.0 | 1785.8 | 115,661.0 | 3.574 | 499.7 | |

3000 | 1.2 | 1511.7 | 263,511.1 | 19,136.9 | 263,511.6 | 46.551 | 411.1 |

1.5 | 1880.0 | 231,954.6 | 14,429.7 | 231,954.9 | 25.560 | 564.5 | |

2.0 | 2483.3 | 205,368.8 | 19,759.8 | 205,368.8 | 11.219 | 1761.3 | |

3.0 | 3780.0 | 173,630.6 | 6571.9 | 173,630.6 | 12.432 | 528.6 | |

4000 | 1.2 | 1991.7 | 352,975.8 | 59,016.2 | 352,977.0 | 107.610 | 548.4 |

1.5 | 2511.7 | 306,983.3 | 43,971.5 | 306,983.3 | 66.669 | 659.5 | |

2.0 | 3335.0 | 275,694.8 | 60,539.8 | 275,694.8 | 26.281 | 2303.6 | |

3.0 | 4986.7 | 231,148.4 | 17,014.0 | 231,148.4 | 29.728 | 572.3 | |

5000 | 1.2 | 2498.3 | 438,717.9 | 136,314.9 | 438,718.6 | 168.581 | 808.6 |

1.5 | 3131.1 | 386,546.1 | 101,071.7 | 386,548.1 | 106.764 | 946.7 | |

2.0 | 4161.1 | 341,504.3 | 139,122.7 | 341,504.3 | 50.931 | 2731.6 | |

3.0 | 6257.8 | 291,685.7 | 51,471.0 | 291,685.7 | 58.821 | 875.0 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, H.; Fang, Y.; Pan, R.; Ge, C.
A New Greedy Insertion Heuristic Algorithm with a Multi-Stage Filtering Mechanism for Energy-Efficient Single Machine Scheduling Problems. *Algorithms* **2018**, *11*, 18.
https://doi.org/10.3390/a11020018

**AMA Style**

Zhang H, Fang Y, Pan R, Ge C.
A New Greedy Insertion Heuristic Algorithm with a Multi-Stage Filtering Mechanism for Energy-Efficient Single Machine Scheduling Problems. *Algorithms*. 2018; 11(2):18.
https://doi.org/10.3390/a11020018

**Chicago/Turabian Style**

Zhang, Hongliang, Youcai Fang, Ruilin Pan, and Chuanming Ge.
2018. "A New Greedy Insertion Heuristic Algorithm with a Multi-Stage Filtering Mechanism for Energy-Efficient Single Machine Scheduling Problems" *Algorithms* 11, no. 2: 18.
https://doi.org/10.3390/a11020018