A cloud based improved method for multi-objective flexible job-shop scheduling problem

Abstract

In order to solve the multi-objective flexible job-shop scheduling problem, a novel cloud bacterial foraging optimization algorithm is proposed. In this paper, the simulation model is established to maximize the makespan and the workload of machines. The optimal bacterial individuals can preserve curing position to additional turning and the common ones swim to the direction of them to absorb location information. The cloud crossover operator and cloud mutation operator are designed to avoid the shortcoming of premature convergence. The proposed method is verified to be more effective than other existing algorithms to solve the multi-objective FJSP through the example of Kacem and testing of 6 workpieces× machines in a mould job-shop.

Keywords

Multi-objective flexible job-shop scheduling cloud computing bacterial foraging optimization crowding distance

1 Introduction

Flexible Job-shop Scheduling Problem (FJSP) is the expansion of Job-shop Scheduling Problem (JSP) [1]. There are several works on flexible job-shop scheduling problem by approaches like ant colony, wasp, bacterial foraging, etc in recent years. J. Liu [2] proposes a DAG based dynamic tasks scheduling model and a fuzzy scheduling algorithm with a lower time complexity. Xinchang Hao [3] proposes a machine selection approach to select the shorter working moment considering the load balance of machine, which can improve the quality of the initial population. K.Z. Gao [4] proposes the assignment principle for searching machine with the global minimum processing time and random arrangement work pieces. LIU Xiaobing [5] proposes the double chains quantum coding, however, this approach is lack of considering the influence on process sequence because of the machine selection. Faruk Geyik [6] presents an optimization via simulation approach to solve dynamic flexible job shop scheduling problems. This study deals with both determining the best process plan for each part and then finding the best machine for each operation in a dynamic flexible job shop scheduling environment. V. Muerza [7] adopts the hybrid DE algorithm including variable neighborhood search and coordinate azimuth search to optimize the two objective of makespan and total cost. However, the analysis of the influence factors on the efficiency is not sufficient. Mohd Helmi Ali [8] proposes a modified version of the genetic algorithm for food supply chain scheduling problems. Arit Thammano [9] presents a hybrid artificial bee colony algorithm with the criteria to minimize the maximum completion time. Faruk Geyik [10] proposes a hybrid chemical-reaction optimization (HCRO)algorithm for solving the job-shop scheduling problem with fuzzy processing time. The flexible maintenance activities under both resumable and non-resumable situations are also considered to make the problem closer to the reality. PENG Jiangang [11] proposes a cloud model evolution algorithm for FJSP based on non-dominated sorting by introducing the cloud evolutionary strategy. Yunus Demir [12] solved the FJSP with overlapping in operations. According to this approach, sublots are transferred from one machine to the next for processing without waiting at the predecessor machine. The study is carried out in two steps. In the first step, a new mathematical model is developed and compared to other models of the literature in terms of computational efficiency. P. Renna [13] creates the dynamic schedule by a pheromone-based approach, which is carried out by a multi-agent architecture. F. Zhao [14] proposes a chemotaxis-enhanced bacterial foraging optimization [15] to solve the JSP, which is based on a new chemotaxis with the differential evolution operatoradded.

Based on the above research methods, this paper proposes an improved cloud based bacterial foraging optimization algorithm (CBFOA) which can preserve curing position to additional turning and the common ones swim to the direction of them to absorb location information. In this paper, aiming at the interference management of job-shop scheduling, the interference management model is established through combining the method of cloud method and quantitative analysis in operations. Moreover, an improved bacterial foraging algorithm and disruption measurement strategies are proposed to provide theoretical basis and technical support for job-shop management.

2 Description of problem

In the description of FJSP [16], N represents the total number of workpieces to be processed, M represents the total number of machines, i represents the index of workpieces, i ∈ {1, 2,,, N}, O_i represents the ith the workpiece, n_i represents the index of process, $R_{i}^{j}$ represents the jth process of workpiece O_i, M_ij represents the machine set of R_ij, M_ij ⊆ {1, 2, M}, m represents the index of machine, m ∈ {1, 2,,, M_ij}, $S_{i}^{jm}$ descripts that R_ij can be processed on the machine m or not (1 or 0), t_ijm represents the spent time of R_ij to be processed on the machine m, b_ijm represents the start time of R_ij to be processed on the machine m and C_ijm represents the completion time for R_ij.

The primary goal for the production enterprises is to complete the production tasks timely and efficiently, therefore the primary scheduling goal for FJSP is to minimize makespan through selecting suitable machine for each process and arranging the optimal processing sequence. In this paper, several new workpieces are added after the initial scheduling, and the objective function is established as follows:

(1) Minimize the makespan f₁ $\begin{matrix} f_{1} = min (C) = min {max (C_{i} | i = \\ 1, 2, \cdot \cdot \cdot, N)} C_{ijm} = S_{ijm} b_{ijm} + S_{ijm} t_{ijm} \end{matrix}$ (1)

In Equation (1), C_i is the completion time of workpiece O_i, $C_{i}^{im}$ is the completion time for R_ij processed on machine m, $b_{i}^{jm}$ is the initial moment for R_ij to be processed on machine m. $nonumber S_{ijm} = {\begin{matrix} 1, R_{ij} is processed on the machine m; \\ 0, R_{ij} is not processed on the machine m . \end{matrix}$ (2)

b_ij: is the start time for R_ij;

(2) Minimize the total load of all the machines f₂ $f_{2} = min (\sum_{i = 1}^{N} \sum_{j = 1}^{n_{i}} \sum_{m = 1}^{M} t_{ijm} S_{ijm})$ (3)

(3) Minimize the maximum load of single machine f₃ $f_{3} = min [max \sum_{i = 1}^{N} \sum_{j = 1}^{n_{i}} t_{i^{jm}} S_{i^{jm}}]$ (4)

3 Improved algorithm for FJSP

3.1 The meaning of the symbol

The meaning of the symbol in the problem can be described as follows:

θⁱ (k, j, l): the position of the ith bacteria in the kth chemokine, the jth replication and the lthelimination;

I: the length of the search interval;

λ (i): the step of chemotaxis;

φ (i) the direction vector of unit length;

Δ (i) the random vector;

δ ∈ (0, 1) the crowding distance factor;

f_λ: the step adjustment factor;

Nc: the number of chemotaxis;

Ps: the population size;

d: the crowding distance, d = (Nc/Ps).

3.2 The improved bacterial algorithm

In order to innovate the common BFOA [17, 18] and avoid the shortcoming of the premature convergence [19, 20], the specific steps of the improved algorithm based on cloud computing are as follows:

(1) Initialization

The mapping relationship between FJSP and the algorithm based on working procedure [21] is built to encode the procedure chain and the machine chain. For example, the problem of 3 workpieces×4 machines is shown in Table 1.

Table 1
Example for 3 workpieces×4 machines

Workpiece Process Machine

M ₁ M ₂ M ₃ M ₄

O ₁ $R_{1}^{1}$ 9 — 8 7

$R_{1}^{2}$ 10 9 — 8

$R_{1}^{3}$ 3 4 3 5

O ₂ $R_{2}^{1}$ — 6 6 7

$R_{2}^{2}$ 5 5 — —

$R_{3}^{3}$ 8 9 — 9

O ₃ $R_{3}^{1}$ 6 — 6 5

$R_{3}^{2}$ 10 11 9 10

Workpiece	Process	Machine
O ₁	$R_{1}^{1}$	9	—	8	7
	$R_{1}^{2}$	10	9	—	8
	$R_{1}^{3}$	3	4	3	5
O ₂	$R_{2}^{1}$	—	6	6	7
	$R_{2}^{2}$	5	5	—	—
	$R_{3}^{3}$	8	9	—	9
O ₃	$R_{3}^{1}$	6	—	6	5
	$R_{3}^{2}$	10	11	9	10

If one valid bacterial individual is “2 3 1 3 1 1 2 2”, then the scheduling sequence of the process is $R_{2}^{1}$ , $R_{3}^{1}$ , $R_{1}^{1}$ , $R_{3}^{2}$ , $R_{1}^{2}$ , $R_{1}^{3}$ , $R_{2}^{2}$ , $R_{2}^{3}$ . When the CBFOA stops, the bacterial individual will be decoded to obtain the solution of FJSP, that is, take the scheduling process out from the bacterial encoding and select the corresponding machine which makes it complete processing the earliest.

(2) Chemotaxis

Generate the unit vector and the individual bacterial starts to turn over and swim, on the meanwhile, the position of the ith bacteria is updated as Equation 4 [17]: $θ^{i} (k + 1, j, l) = θ^{i} (k, j, l) + λ (i) φ (i)$ (5) $φ (i) = \frac{Δ (i)}{\sqrt{Δ^{T} (i) Δ (i)}}$ (6)

The notations in Equations 4 and 5 are as mentioned above.

In this paper, λ (i) is improved adaptively. Theequation of adjustable step is as follow: $λ (i) = f_{λ} [\frac{δ (δ + 1)}{δ + d} - δ] I$ (7)

In Equation 6, if d is smaller, the individual will optimize in larger step; otherwise, the individual will optimize in a smaller step. That can ensure the algorithm has strong global search ability in early and strong local search ability in the latter.

Offspring is selected through non-dominated sorting based on Pareto [22] and crowding distance [23]. The non-dominated sorting achieves the classification through calculating the parameter X_N and X_n of population P_s, the specific steps are as follows:

Step 1. Initialize the parameter X_N as θ;

Step 2. Initialize the variable X_n, where X_n is the number of individuals to dominate X;

Step 3. Calculate the dominance relationship between X and Y, X, Y ∈ P_s. If Y is dominated by X, then X_N = X_N∪ { Y }, otherwise, X_n = X_n + 1. When X_n = 0, X is considered to be non-dominated individuals, it is marked as X_r = 1, then R₁ = R₁ ∪ {X};

Step 4. If R_i≠ ∅ then Q =∅ and set i = 1. If Y ∈ X_N, set y_n = y_n - 1 until y_n = 0, then set Y_r = i + 1, Q = Q∪ { Y }, R_i = Q;

Step 5. Calculation will stop if R_i is empty, otherwise, turn to Step 4.

The crowding distance is introduced to maintain the diversity of the population: $L [X]_{dj} = \frac{f_{j}^{X + 1} - f_{j}^{X - 1}}{f_{j}^{n} - f_{j}^{1}}$ (8)

In Equation 7, L [X] _dj is the crowding distance of the jth objective in individual X, $f_{j}^{x}$ is the adaptive value of the jth objective in individual X, in which the value of x is 1, X - 1, X + 1 and n respectively.

(3) Cloud crossover operator

In the evolution of the population, it is assumed that the fitness of the two parents are represented as f_a and f_b, F_max is the maximum fitness of the parent population, F_min is the minimum fitness, and there is a real number of p_cr should meet the following condition: $p_{cr} = {\begin{matrix} t_{1} e^{\frac{- (f - \bar{f})^{2}}{2 y^{2}}}, & f ⩾ \bar{F} \\ t_{2} & f < \bar{F} \end{matrix}$ (9)

p_cr is the cloud crossover operator. t₁, t₂ and $\bar{F}$ are all constant in Equation 8, in which $\bar{f} = (f_{a} + f_{b}) / - 2$ , y is a random normal number about F_max and F_min. Because of adaptive and stochastic of cloud crossover operator, the crossover operation is designed to optimize the common genetic operator with it. The crossover operator will change with the average fitness of the parent, and it will also change with the normal random number y. The algorithm of cloud crossover operator is as follows:

Step 1. Calculate the average fitness of the parent E_x, written as E_x = (f_a + f_b)/ - 2;

Step 2. Act E_n as the expected value, then generate a random number of E_n’ with H_e as the standard deviation;

In which E_n = m₁ (F_max - F_min), H_e = n₁E_n, m₁ and n₁ represent the control coefficient.

Step 3. The cloud crossover operator can be obtained according to Equation 8: $P c r = {\begin{matrix} t_{1 e} \frac{1 {(f - E_{x})}^{2}}{2 {E^{'}}_{n}^{2}} & , f \geq \bar{F} \\ t_{2} & , f \geq \bar{F} \end{matrix} .$ (10)

$\bar{F}$ is the average fitness of parent, f = max(f_a, f_b).

(4) Cloud mutation operator

It is assumed that the fitness of a single parent is f_a in the process of population evolution, and then there is a real number of p_mt must accord with the following: $p_{mt} = {\begin{matrix} s_{1} e^{\frac{- (f - f_{a})^{2}}{2 y^{2}}} & , f ⩾ \bar{F} \\ s_{2} & , f < \bar{F} \end{matrix}$ (11)

p_mt is cloud mutation operator. In Equation 9, s₁, s₂ and $\bar{F}$ are all constants, F_max, F_min and y are defined in above Equation 8.

The stability tendency of cloud mutation operator is decided by the parameters of f_a, F_max and F_min. The algorithm of cloud mutation operator is as follows:

Step 1. Evaluate the mean E_x of a single parent’s fitness, recorded as E_x = f_a;

Step 2. Act E_n as the expected value, to generate a normal random number E_n’ with H_e as the standard deviation; in which E_n = m₂ (F_max - F_min), H_e = n₂E_n, m₂ and n2 are control coefficients;

Step 3. Calculate the cloud mutation operator according to Equation 9. $P m t = {\begin{matrix} s_{1 e} \frac{- {(f - E_{x})}^{2}}{2 {E^{'}}_{n}^{2}} & , f \geq \bar{F} \\ s_{2} & , f \geq \bar{F} \end{matrix} .$ (12)

The constant $\bar{F}$ represents the average value of the parent population fitness.

The key of the design is to improve the convergence speed of the algorithm and improve the result of the search with cloud crossover operator, cloud mutation operator and cloud generator.

4 Experimental and verification

4.1 Test on Kacem instance

In order to verify the effectiveness of the proposed algorithm, the researcher selected the classical example of Kacem [24] to execute 20 times independently under Matlab7.0. The result of the comparison between the proposed algorithm and BFOA [17] HBCA [25] for 20 workpieces× 10 machines is shown in Table 2. The optimal solution of (311, 2374, 299) is dominated by three of sixty solutions obtained with CBFOA; the eight optimal solutions obtained with the algorithm in literature [24] are all controlled by the solutions obtained with CBFOA except (310, 2286, 298). T_x is the average completion time, M_t is the total load of the machines and M_x is the maximum load of the machine.

Table 2
The comparison of different algorithm for Mk09

BFOA HBCA CBFOA

(T_x, M_t, M_x) (T_x, M_t, M_x) (T_x, M_t, M_x) (T_x, M_t, M_x) (T_x, M_t, M_x)

(311,2374,299) (310,2286,298) (308,2298,298) (321,2270,306) (338,2247,332)

(313,2286,310) (308,2298,296) (321,2284,301) (339,2242,324)

(318,2280,308) (309,2294,309) (322,2268,305) (339,2252,326)

(324,2278,308) (310,2289,298) (322,2277,299) (344,2246,332)

(410,2264,348) (310,2281,301) (324,2260,315) (344,2248,333)

(416,2256,358) (312,2289,300) (324,2261,314) (346,2243,329)

(424,2239,384) (314,2288,299) (324,2274,299) (346,2250,334)

(430,2229,400) (315,2278,308) (325,2258,325) (347,2246,330)

(315,2287,299) (325,2262,313) (347,2240,332)

(315,2278,313) (326,2263,315) (351,2240,344)

(316,2279,304) (326,2255,310) (353,2236,330)

(317,2286,297) (328,2254,322) (360,2234,350)

(317,2223,305) (329,2256,312) (402,2230,387)

(318,2270,312) (329,2250,319) (418,2221,399)

(318,2269,312) (330,2254,322) (430,2224,431)

(319,2268,311) (330,2252,311) (432,2226,424)

(320,2265,313) (331,2248,319) (433,2216424)

(320,2270,300) (331,2250,318) (434,2225,426)

(321,2260,302) (332,2246,322) (442,2219,440)

(321,2262,310) (335,2259,308) (448,2212,446)

BFOA	HBCA	CBFOA
(T_x, M_t, M_x)	(T_x, M_t, M_x)	(T_x, M_t, M_x)	(T_x, M_t, M_x)	(T_x, M_t, M_x)
(311,2374,299)	(310,2286,298)	(308,2298,298)	(321,2270,306)	(338,2247,332)
	(313,2286,310)	(308,2298,296)	(321,2284,301)	(339,2242,324)
	(318,2280,308)	(309,2294,309)	(322,2268,305)	(339,2252,326)
	(324,2278,308)	(310,2289,298)	(322,2277,299)	(344,2246,332)
	(410,2264,348)	(310,2281,301)	(324,2260,315)	(344,2248,333)
	(416,2256,358)	(312,2289,300)	(324,2261,314)	(346,2243,329)
	(424,2239,384)	(314,2288,299)	(324,2274,299)	(346,2250,334)
	(430,2229,400)	(315,2278,308)	(325,2258,325)	(347,2246,330)
		(315,2287,299)	(325,2262,313)	(347,2240,332)
		(315,2278,313)	(326,2263,315)	(351,2240,344)
		(316,2279,304)	(326,2255,310)	(353,2236,330)
		(317,2286,297)	(328,2254,322)	(360,2234,350)
		(317,2223,305)	(329,2256,312)	(402,2230,387)
		(318,2270,312)	(329,2250,319)	(418,2221,399)
		(318,2269,312)	(330,2254,322)	(430,2224,431)
		(319,2268,311)	(330,2252,311)	(432,2226,424)
		(320,2265,313)	(331,2248,319)	(433,2216424)
		(320,2270,300)	(331,2250,318)	(434,2225,426)
		(321,2260,302)	(332,2246,322)	(442,2219,440)
		(321,2262,310)	(335,2259,308)	(448,2212,446)

It is solved using the BFOA, HBCA and the CBFOA in this paper respectively and the number of iterations is 100 times and 200 times. It can be seen from Fig. 1 that the speed of BFOA is the slowest, and the convergence speed of HBCA and CBFOA are almost the same after 100 iterations. However, the speed of HBCA is significantly slower than that of CBFOA after 200 iterations, and the CBFOA is better than the other two algorithms.

Fig. 1

The comparison of the three algorithm.

4.2 Example simulation

In order to further verify the performance of CBFOA to solve the multi-objective FJSP, the processing data of the actual production of a machinery factory is calculated in this paper. The data of 6 workpieces×6 machines [26, 27] FJSP is shown in Table 3, including the processing time on different machines for working procedure. In which, the symbol “— ” represents the corresponding working procedure can not be processed on the machine, unit: hour. Three Pareto optimal solutions are shown in Table 4.

Table 3
FJSP of 6 workpieces ×6 machines

Workpiece Working procedure Machine

M ₁ M ₂ M ₃ M ₄ M ₅ M ₆

O ₁ $R_{1}^{1}$ 27 22 16 17 21 18

$R_{1}^{2}$ 16 — 18 — — 14

$R_{1}^{3}$ — 30 24 26 27 —

$R_{1}^{4}$ 5 — 2 3 3 2

$R_{1}^{5}$ 9 10 8 — — 8

O ₂ $R_{2}^{1}$ 27 24 — 24 24 —

$R_{2}^{2}$ — 35 33 29 30 31

$R_{2}^{3}$ 14 — — 13 12 11

$R_{2}^{4}$ 6 — 5 4 5 3

$R_{2}^{5}$ 26 — 25 — 26 24

O ₃ $R_{3}^{1}$ 24 27 — 24 — 25

$R_{3}^{2}$ 29 — 30 24 — 25

$R_{3}^{3}$ 8 — 6 5 — —

$R_{3}^{4}$ 17 16 — — — 16

$R_{3}^{5}$ 7 — 6 — 5 6

O ₄ $R_{4}^{1}$ 7 6 — 6 8 —

$R_{4}^{2}$ — — 5 7 5 6

$R_{4}^{3}$ — 14 11 12 — 13

$R_{4}^{4}$ 19 — 18 — 17 18

$R_{4}^{5}$ 18 16 17 — 15 —

O ₅ $R_{5}^{1}$ 25 23 — — 24 23

$R_{5}^{2}$ 6 — 7 — — 7

$R_{5}^{3}$ 10 — 9 8 10 11

$R_{5}^{4}$ 17 16 — 14 — 15

$R_{5}^{5}$ 11 12 10 — — 10

O ₆ $R_{6}^{1}$ 8 — 7 9 8 8

$R_{6}^{2}$ 21 20 19 — 21 21

$R_{6}^{3}$ 13 — — 12 12 14

$R_{6}^{4}$ — 7 — — 8 7

$R_{6}^{5}$ 13 — 11 12 — 12

Workpiece	Working procedure	Machine
O ₁	$R_{1}^{1}$	27	22	16	17	21	18
	$R_{1}^{2}$	16	—	18	—	—	14
	$R_{1}^{3}$	—	30	24	26	27	—
	$R_{1}^{4}$	5	—	2	3	3	2
	$R_{1}^{5}$	9	10	8	—	—	8
O ₂	$R_{2}^{1}$	27	24	—	24	24	—
	$R_{2}^{2}$	—	35	33	29	30	31
	$R_{2}^{3}$	14	—	—	13	12	11
	$R_{2}^{4}$	6	—	5	4	5	3
	$R_{2}^{5}$	26	—	25	—	26	24
O ₃	$R_{3}^{1}$	24	27	—	24	—	25
	$R_{3}^{2}$	29	—	30	24	—	25
	$R_{3}^{3}$	8	—	6	5	—	—
	$R_{3}^{4}$	17	16	—	—	—	16
	$R_{3}^{5}$	7	—	6	—	5	6
O ₄	$R_{4}^{1}$	7	6	—	6	8	—
	$R_{4}^{2}$	—	—	5	7	5	6
	$R_{4}^{3}$	—	14	11	12	—	13
	$R_{4}^{4}$	19	—	18	—	17	18
	$R_{4}^{5}$	18	16	17	—	15	—
O ₅	$R_{5}^{1}$	25	23	—	—	24	23
	$R_{5}^{2}$	6	—	7	—	—	7
	$R_{5}^{3}$	10	—	9	8	10	11
	$R_{5}^{4}$	17	16	—	14	—	15
	$R_{5}^{5}$	11	12	10	—	—	10
O ₆	$R_{6}^{1}$	8	—	7	9	8	8
	$R_{6}^{2}$	21	20	19	—	21	21
	$R_{6}^{3}$	13	—	—	12	12	14
	$R_{6}^{4}$	—	7	—	—	8	7
	$R_{6}^{5}$	13	—	11	12	—	12

Table 4

Three Pareto solutions obtained through CBFOA

Working procedure	Makespan f₁	TTL loads of MACH f₂	Max load of MACH f₃
1	95	465	76
2	96	464	80
3	99	463	76

It can be known that the Pareto solutions obtained through CBFOA are better than that of other classic algorithms [26 –28].

5 Conclusions

An improved cloud computing based bacterial foraging algorithm based on differential evolution proposing in this paper, the following conclusion can be obtained: The CBFOA has global and local ability and can obtain more non-dominated solutions than other algorithms. It can obtain the optimal solution more quickly than other algorithms and the efficiency of algorithm is improved. The feasibility of the algorithm and its strong ability of solving can be verified through the benchmark.

However, the proposed research has not considered the influence of the subjective preference of decision makers on the final scheduling results, which may make the following research have more practical value. Therefore, how to add the analysis of subjective factor with some advanced approach such as big data should be the next research focus.

Footnotes

Acknowledgments

This work is supported by Dr scientific research fund and Natural scientific fund of Liaoning Province (No. 201601244, 20170540125), Postdoctoral Science Foundation funded Project of China (No. 2017M611231) and Social Science planning fund of Liaoning Province (2018lslktyb-016).

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Workpiece	Process	Machine
M ₁	M ₂	M ₃	M ₄
O ₁	$R_{1}^{1}$	9	—	8	7
	$R_{1}^{2}$	10	9	—	8
	$R_{1}^{3}$	3	4	3	5
O ₂	$R_{2}^{1}$	—	6	6	7
	$R_{2}^{2}$	5	5	—	—
	$R_{3}^{3}$	8	9	—	9
O ₃	$R_{3}^{1}$	6	—	6	5
	$R_{3}^{2}$	10	11	9	10

A cloud based improved method for multi-objective flexible job-shop scheduling problem

Abstract

Keywords

1 Introduction

2 Description of problem

3.1 The meaning of the symbol

3.2 The improved bacterial algorithm

Table 1 Example for 3 workpieces×4 machines Workpiece Process Machine M 1 M 2 M 3 M 4 O 1 R 1 1 9 — 8 7 R 1 2 10 9 — 8 R 1 3 3 4 3 5 O 2 R 2 1 — 6 6 7 R 2 2 5 5 — — R 3 3 8 9 — 9 O 3 R 3 1 6 — 6 5 R 3 2 10 11 9 10

4.1 Test on Kacem instance

Footnotes

Acknowledgments

References