1. Introduction

Forging industry is a high energy consumption industry and also arises serious pollution problems. However, the forging industry is one of the basic industries for people's livelihood. There is broad consensus in the forging industry to ensure the rapid development and implementation of cleaner production, energy saving, emission reduction and noise elimination [1].

The charging combination optimization belongs to a large scale combination optimization problems. It can be described as multiple knapsack problem. It's difficult to build the model and the solving procedure is complex. So the conventional methods often fail to get the optimal solution. In literature [2], the combination multi-knapsack model based on multi-furnace types, uncertain installed furnace number was proposed for the steel coil. In view of annealing production of steel coil, the mathematical model to minimize the total time of heating is established for stacking combination optimization [3].

SFLA(Shuffled Frog Leaping algorithm) was firstly proposed in 2003 by the Eusuff and Lansey to solve combinatorial optimization problems. As a bionics intelligent optimization algorithm, SFLA is integrated with the advantages of MA (memetic algorithm) based on memes (meme) evolution and PSO (particle swarm optimization) based on group behavior. Therefore, SFLA is characterized by simple concept, less parameter adjustment, fast calculation, strong global search optimization capability, and easiness to implement. At present, SFLA was mainly used to solve the multi-objective optimization problems, such as job shop schedule, pier maintenance, water distribution and other actual engineering problems.

Many scholars studied the application of shuffled frog leaping algorithm for solving combination optimization problem or multi-knapsack problem. Wang [4] makes use of the global optimal solution as the guidance of each sub population overall forward evolution based on the shuffled frog leaping algorithm for solving combination optimization problems. Cai and Li proposed an improved shuffled frog leaping algorithm, defining the similarity and distance of frog. Accordingly, a frog shift strategy was constructed [5]. Pan designed a discrete shuffled frog leaping algorithm to solve batch production line scheduling problem [6]. A multi-agent shuffled frog leaping algorithm, combined with the evolution mechanism of shuffled frog leaping algorithm, was researched to continuously apperceive the local environment [7].

Aiming at the problem of best load with furnace weight constraints, a discrete shuffled frog leaping algorithm for forging furnace was proposed by our previous research [8]. But the charging combination optimization is not solved. In this presentation, the problem regarding to the combination optimization of forging work-pieces with different holding temperature and holding time was studied. A model for optimizing the charging combination with the goal of energy saving was established. Then a discrete shuffled frog leaping algorithm based on the same furnace heating rules is designed for solution.

2. Problem description

Forging heating and temperature holding have important effect on the forging internal micro-structure homogenization. The homogenization will not be distinct if temperature holding time is too short, and the too long holding time will cause overheating or burning. For work-pieces in the same furnace, it is stipulated technically that the lowest holding temperature of work-piece in the furnace is holding temperature of the furnace batch, and the longest time of temperature holding is holding temperature time of the whole furnace batch. The shorter holding time is, the less heating furnace energies are consumed. When a batch of work-pieces with different holding time are partially combined, different batching plan will cause different holding time of furnace batches. The distribution of forgings is not only related to the holding time, but also its holding temperature as well as the furnace batch weight. The optimization goal of forging for energy-saving furnace combination is not only the holding time, but also includes furnace batch number, average loading capacity and average holding temperature.

3. Modeling

3.1. Basic assumption

1. Each work-piece only belongs to one furnace batch.

2. A batch of work-pieces are put into the furnace and heated in the same time.

3. Maximum load weight does not exceed the load capacity of heating furnace.

4. The heating furnace can reach the temperature that meet all requirements of the work-piece holding temperature.

3.2. Modeling

Providing there are N work-pieces that their weight, holding temperature and holding time are given. Those work-pieces have to be divided into B batches and each batch corresponds to a heating job. The maximum capacity of furnace is S and the heating furnace can reach the temperature that meet all requirements of the work-piece holding temperature. The ultimate holding time must also meet all work-piece furnace holding time requirements. The optimization goal is to minimize the quantity of charging batches with maximum average charging amount (i.e. minimum average charging difference), minimum average holding temperature and minimum average holding time.

3.2.1. Constrains

Definition 1: Temperature compatibility: The temperature range of work-piece i and j are [[T.sub.imin], [T.sub.imax] and [T.sub.jmin],[T.sub.j max]], if:

[[T.sub.imin], [T.sub.imax][intersection][T.sub.jmin],[T.sub.jmax]][not equal to][phi] (1)

where the work-piece i and j have temperature compatibility.

Definition 2: Time compatibility: The time range of holding temperature of work-piece i and j are [c.sub.imin], [c.sub.imax] and [[c.sub.jmin], [c.sub.jmax]], if:

[[c.sub.imin][c.sub.imax]][intersection][[c.sub.jmin][c.sub.jmax]][not equal to][phi], (2)

where the work-piece i and j have time compatibility.

Definition 3: Same furnace heating rule: If the work-piece i and j can simultaneously satisfy the temperature compatibility and time compatibility, it is claimed the two meet the same furnace heating rules, i.e. the work-piece i and j can be placed in the same furnace batch.

3.2.2. Optimization model

Definition 4: Average furnace holding temperature: The average value of holding temperature of all furnace batch in one batching, i.e.:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where T is the average temperature in the furnace charging, k is the furnace batch number, and [T.sub.b] is the holding temperature of furnace batch (Section b) (b = 1, 2, ..., k).

Definition 5: Average furnace holding time: The average value of holding time of all furnace batch in one batching plan, i.e.:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where C is the the average holding time, k is the furnace batch number, and [C.sub.b] is the holding time of furnace batch (Sectionb) (b=1,2, ..., k).

With the above assumptions and definitions, mathematical model can be established as follows:

min (k), (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where S is the heating furnace maximum load, [z.sub.b] is the furnace batch weight (Section b). Eqs. (5) and (6) are the optimization function, aiming at minimizing the furnace batch number, the average furnace loading difference, the average holding temperature and the average holding time.

Assuming:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where n denotes the work-piece number, J denotes aggregate of work-pieces (J = {l, 2, ...,n}), [J.sub.b] denotes furnace batch number (b[member of]B), B denotes aggregate of charging batches (B = {l, 2, ..., k}),j denotes work-piece number (j [member of] J), [j.sub.b] denotes aggregate of work-pieces in batch b (b [member of] B), [Z.sub.j ]denotes weight of work-piece j, and [X.sub.jb] denotes decision variable, judging whether the work-piece j belongs to the furnace batch b.

Eq. (7) indicates that each work-piece j can only be allocated to one charging batch b.

Eq. (8) is the furnace batch weight constraint, indicating the total weight of work-pieces in a batch shall not exceed the maximum capacity of the furnace.

Eq. (9) shows that the intersection of holding temperature interval of work-pieces in the same batch cannot be empty, i.e. work-pieces in one batch shall meet the temperature compatibility.

Eq. (10) indicates that the final holding temperature of a furnace batch in heating furnace is the minimum temperature that meet the requirements of all work-pieces' holding temperature.

Eq. (11) shows that the intersection of holding time interval of work-pieces in the same batch cannot be empty, i.e. work-pieces in one batch shall meet the time compatibility;

Eq. (12) indicates that the final holding time of a furnace batch in heating furnace is the minimum time that meet the requirements of all work-pieces' holding time.

Eq. (13) defines the quantity interval of charging batches by giving a lower limit for the quantity of charging batches, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] It is assumed that work-pieces can be separated and allocated to different batches, and all work-pieces in any furnace batch can satisfy the temperature compatibility.

Eq. (14) is the decision variables.

4. DSLFA design based on same furnace heating rules

This model is similar to the bin packing problem as well as the multi-knapsack problem[9]. The work-pieces are items while furnace batches are boxes in the model. It's required that total weight of each furnace batch cannot exceed the maximum weight allowed by furnace, and each item can only be put into in a box.

This paper adopts the individual updating ideas from the discrete shuffled frog leaping algorithm in reference [10]: According to the combination optimization problem of work-pieces in different holding temperature and holding time interval sets, the discrete shuffled frog leaping algorithm based on same furnace heating rules is proposed.

Steps are as follows:

1. Coding

Use classic two-dimensional array encoding. The length of individuals is the number of work-pieces, each bit means the sequence number of work-piece, and each block represents a furnace batch.

2. Fitness function selection

Aiming to smaller furnace batch quantity, less average charging difference, smaller average temperature of holding temperature, and smaller average holding time, this paper use the linear weighted comprehensive evaluation to determine the fitness function. The expression is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where [w.sub.k], [w.sub.q], [w.sub.T], [w.sub.C] are the weight coefficient of furnace batch number, average charging difference, average holding temperature, average holding time. And they satisfy the equation [w.sub.k] + [w.sub.q] + [w.sub.T] + [w.sub.C] = 1, [k.sub.max], .[K.sub.min], .[q.sub.min],.[q.sub.max], .[T.sub.min], .[T.sub.max] [T.sub.min], [C.sub.max], [C.sub.min] are the maximum and minimum number of furnace batch, the maximum and minimum average charging difference, the maximum and minimum average holding temperature, the maximum and minimum average holding time.

3. Population initialization

BF heuristic algorithm is used to generate the initial population of individuals. In the generation of an individual, a group of work-pieces has not only to satisfy the same furnace heating rules, also cannot exceed the furnace batch weight constraint. Generation of the individual steps are as follows:

Step 1: the work-piece placed in the queue Q in random and numbered sequentially, number is 1, 2,..., n.

Step 2: pick the work-piece j out of the queue Q in order, which weight is [z.sub.j] holding temperature is [T.sub.j], holding time is [C.sub.j].

Step 3: pick batch b out of queue Q' that have not been matched with work-piece j, then find out whether the work-piece j matches batch b. The remaining weight space of batch b is [DELTA][s.sub.b] = S - s, where S is the maximum load of heating furnace and [s.sub.b] is the weight of batch b. The holding temperature is [T.sub.b], which is the holding temperature interval intersection of all work-pieces, and [T.sub.b] is not [empty set]. The holding time is [C.sub.b], which is the holding time interval intersection of all work-pieces, and [C.sub.b] is not [empty set]. If all batches in queue Q' have been operated with work-piece j, a new furnace butch [b.sub.new] is set up with work-piece j in it. Update the remaining weight space of batch [b.sub.new]: [DELTA][s.sub.beww] = S - [Z.sub.j], holding temperature: [T.sub.p] = [T.sub.j] holding time: [C.sub.bnew] = [C.sub.j]. If the queue Q is empty, go to step 4, else go to step 2. If there any butch that not operated with work-piece j, calculate [delta][s.sub.b]'=[delta][s.sub.b]+[Z.sub.i], [T.sub.b]=[T.sub.j][intersection][T.sub.b], and [C.sub.b]'=[C.sub.j][intersection][C.sub.b]. If [delta][s.sub.b]' [greater than or equal to] 0, [T.sub.b]'[not equal to] [empty set], [C.sub.b]' [not equal to][empty set], remove work-piece j into batch b, update the remaining weight space of batch b: [delta][s.sub.b] = [delta][s.sub.b]' holding temperature: [T.sub.b] = [T.sub.b]', holding time: [C.sub.b] = [C.sub.b]'. If the queue Q is empty, go to step 4, else go to step 2. If anyone of [delta][s.sub.b]' < 0, [T.sub.b]' [not equal to][empty set], [C.sub.b]' [not equal to][empty set] is true, then go to step 3.

Step 4: At this time the batches in queue Q have all work-pieces. This is a partial solution in which individual is in the encoding form talking above.

Using the steps above to generate multiple individuals, the initial population is composed in size r. Please pay attention to eliminating redundant individual.

4. Generating ethnic groups

According to the fitness function in (2), all individuals are in a descending order by fitness values, which means excellent frog is in the front. Then the population is divided into m groups, each group including n frogs. There's totally number is r = m x n.

5. Ethnic group evolution

In each group, the best frogs [X.com.b] and the worst frog [X.sub.w] are chosen, as well as the optimal frog [X.sub.g] throughout the population, then separately update the worst frog in iteration.

Update individuals based on the idea of individual updates theory in discrete shuffled frog leaping algorithm [11] and the individual update method [12].

Steps are as follows:

Step 1: Select two intersection points from [X.sub.b] in random. [X.sub.b] is divided into several fragments.

Step 2: Select intersection points from [X.sub.w] in random, and insert fragment from [X.sub.b] into [X.sub.w] before the intersection point. This means some information [X.sub.b] is inserted into [X.sub.w].

Step 3: delete these furnace batch fragment in [X.sub.w] that have some redundant work-piece, then transfer these work-pieces in that fragment into the queue Q. Please make sure that the furnace batch fragment is renewed.

Step 4: the work-pieces are ranged in random order. In accordance with the BF method in this paper those work-pieces are insert into [X.sub.w], i.e. [X.sub.w]'.

Step 5: calculate the fitness value of [X.sub.w]'. If the [X.sub.w]' is better than [X.sub.w], replace [X.sub.w], else [X.sub.g], which is the best individual in the population should replace [X.sub.b] in step 1 and start the operation from step 1 to step 4 again. Now if the [X.sub.w]' is better than [X.sub.w], replace [X.sub.w], else Generates a random feasible solution to replace [X.sub.w].

Step 6: repeat the operation above until the maximum iterations to complete one ethnic group evolution.

Examples are as follows: information of [X.sub.b] and [X.sub.w] are shown in Fig. 1. Cross location is shown by arrow. The cross fragment from [X.sub.b] is plugged into [X.sub.w] at the cross point. The new individual is shown in Fig. 2. It's shown in Fig. 2 that wherein the work-piece3, 7 and 2 is redundant, so we need to delete the corresponding furnace batch that shown in Fig. 3.

At this time the work-pieces 1, 4, 5, 6 are not in batches, so using the BF method to repartition to get individual shown in Fig. 4.

6. The ethnic mixing

After evolution of all ethnic groups, all ethnic groups will be mixed to generate a new population. Then repeat the operation steps of distribution and combination, until the condition is satisfied.

The procedure of the DSFLA based on same furnace heating rules is shown in Fig. 5.

5. Case study

5.1. Types of work-pieces to be charged and their parameters

Take multi-tasks charging batch combination form certain forging company as research object. All work-pieces are classified by different weight, holding temperature and holding time. It should be taken into consideration whether these work-pieces could be heated together referring to time compatibility, temperature compatibility and load capacity. The maximum loading amount of the related parameters of the work-piece are shown in Table 1.

5.2. Traditional batching plan

Due to furnace batch weight constraint, temperature compatibility and time compatibility, it's very difficult in practice to use the traditional manual batching.

Table 2 gives a plan by the traditional manual management. Furnace batch quantity is 12. The average furnace loading amount is 5610 kg. The average holding temperature is 1182[degrees] C. The average holding time is 240 min.

5.3. Batching plan based on DSFLA

Assuming that the furnace batch quantity weight [w.sub.k], the average charging difference weight [w.sub.q], the average temperature [w.sub.T] and the average weight of holding time [w.sub.C] are 0.50, 0.25, 0.15 and 0.10, repeat the computation 50 times. The batch number in furnace obtained is 11. The average loading amount is 5847 kg. The average temperature is mainly 1163[degrees] C. The average holding time is mainly 240 min. Table 3 is a plan of 1170[degrees] C average heat preservation temperature and 240 min average holding time. Table 4 is a comparison of traditional manual batching plan and discrete shuffled frog leaping algorithm with furnace based on same furnace heating rules.

6. Conclusions

It can be known from Table 4 that using DSFLA based on the same furnace heating rules is superior to the traditional manual batching in furnace batch number, average loading amount and the average holding temperature. The average holding time of DSFLA is not lower than that of traditional manual batching plan. The energy consumption is related to furnace batch number, average loading amount, the average holding temperature and the average holding time. The influence of the first three indicators of energy-saving effect is more obvious. The traditional manual batching plan has many disadvantages such as difficulty in operation, low efficiency, less arbitrariness and inefficient energy consumption control.

The method proposed in this paper is better than the traditional manual batching in batch efficiency and the energy consumption control. Therefore, the established model with DSFLA solution was effective and better than traditional manual batching method regarding energy saving.

Acknowledgements

National Natural Science Foundation of China under Grant (No.51575280) and the Innovative Talent Training Plan Soft Science for Jiangning District (No.2014 EC09). The supports are gratefully acknowledged.

References

[1]. Han, M.L.; Wu, S.D. 2010. Developing trend and technical methods for energy saving and emission reduction in forging industry, China Metalforming Equipment & Manufacturing Technology 5: 15-20.

[2]. Zhang, X.P.; Wang, W.; Liu, Q.L.; Zhao, J. 2009. Modeling and algorithm for charging combination of steel coils, Journal of System Simulation 21(13): 3894-3901.

[3]. Wang, Z.G.; Liu, Q.L.; Wang, W. 2010. Improved modeling and optimizing algorithm for combination stacking, Control Engineering of China 17(2): 197-204.

[4]. Wang, Y.Y. 2010. Research on the combinational optimization in shuffled frog leaping algorithm Shijia-zhuang University of Economics.

[5]. Cai, L.W.; Li, X. 2010. Optimization of job shop scheduling based on shuffled frog leaping algorithm, Journal of Shenzhen University Science and Engineering 27(4): 391-395.

[6]. Pan, Y.X.; Pan, Q.K.; Sang, H.Y. 2010. Hybrid discrete shuffled frog leaping algorithm for lot streaming flowshop scheduling problem, Computer Integrated Manufacturing Systems 16(6): 1265-1271.

[7]. Wang, L.G.; Dai, Y.Q. 2013. A multi-agent shuffled frog leaping algorithm, Computer Engineering 39(7): 265-269.

[8]. Zhu, B.Q.; Lu, H.X.; Xia, Y.; Li, D.B. 2013. Research on combinatorial optimization model for forging furnace charging based on discrete hybrid leapfrog algorithm, Journal of Chinese Agricultural Mechanization 34(6): 197-201.

[9]. Dahmani, N.; Krichen, S.; Ghazouani, D. 2015. A variable neighborhood descent approach for the two-dimensional bin packing problem, Electronic Notes in Discrete Mathematics 47: 117-124. http://dx.doi.org/10.1016/j.endm.2014.11.016.

[10]. Ma, Z.G.; Shu, S.H. 2011. Shuffled frog leaping algorithm for solving multiple knapsack problem, Computer & Digital Engineering (9): 13-15.

[11]. Ahandani, M.A.; Alavi-Rad, H. 2015. Opposition-based learning in shuffled frog leaping: An application for parameter identification, Information Sciences 291(8): 19-42. http://dx.doi.org/10.1016/j.ins.2014.08.031.

[12]. Falkenauer, E. 1996. A hybrid grouping genetic algorithm for bin packing, Journal of Heuristics 2(1): 7-10. http://dx.doi.org/10.1007/BF00226291.

Zhu Baiqing, Lu Haixing, Bai Shaobu, Tong Yifei, He Fei

STUDY ON THE CHARGING COMBINATION OPTIMIZATION FOR FORGING PRODUCTION BASED ON DISCRETE SHUFFLED FROG LEAPING ALGORITHM

Summary

As a traditional high energy-consuming industry, the forging industry consumes a lot of energy. In order to solve the typical charging optimization problem regarding how to separate work-pieces with different holding temperature intervals and holding time intervals and combine them for charging in forging, a charging combination model for forging is proposed. The discrete shuffled frog leaping algorithm (DSFLA) based on the same furnace heating rules is adopted to optimize and solve the model in order to reduce energy consumption in forging. An instance is illustrated to prove the effectiveness of the proposed model and the algorithm.

Keywords: forgings, combination optimization, shuffled frog leaping algorithm,energy saving, discrete.

Received September 28, 2015

Accepted September 28, 2016

Zhu Baiqing (*), Lu Haixing (**), Bai Shaobu (***), Tong Yifei (****), He Fei (*****)

(*) Nanjing Institute of Technology, Nanjing 211167, China, E-mail:zhubq@163.com

(**) Nanjing University of Science & Technology, Nanjing 210094, China, E-mail: Ricci_April@163.com

(***) Nanjing Institute of Technology, Nanjing 211167, China, E-mail:baisb@njit.edu.cn

(****) Nanjing University of Science & Technology, Nanjing 210094, China, E-mail: tyf51129@aliyun.com

(*****) Nanjing University of Science & Technology, Nanjing 210094, China, E-mail: hefei_njust@163.com

[cross.sub.ref] http://dx.doi.org/10.5755/j01.mech.22.5.13191

Table 1
The work-piece information table

Work-piece type  Quantity of work-piece  Unit weight of work-piece
                 n, piece                [z.sub.j], kg

J1                 7                        436
J2                 2                       1180
J3                 5                        704
J4                14                        247
J5                 4                        689
J6                 5                        334
J7                 3                        797
J8                 1                       1246
J9                12                        514

Work-piece type  Holding temperature       Holding time
                 [T.sub.j], [degrees] C    [C.sub.j], min

J1                900-950                   240-300
J2               1300-1350                  300-420
J3               1320-1400                  280-400
J4                880-940                   150-250
J5                950-1000                  260-400
J6               1130-1150                  120-240
J7               1250-1320                  270-390
J8               1390-1450                  320-480
J9               1230-1300                  200-280

Work-piece type  Work-piece  Quantity of          Unit weight of
                 type        work-piece n, piece  work-piece
                                                  [Z.sub.j], kg

J1               J10         19                    198
J2               J11          6                    358
J3               J12          2                    627
J4               J13          4                    445
J5               J14         10                    239
J6               J15          8                    482
J7               J16         19                    858
J8               J17          9                    571
J9               J18          4                    266

Work-piece type  Holding temperature      Holding time
                 [T.sub.j], [degrees] C   [C.sub.j], min

J1                  800-850                 180-300
J2                 1250-1280                120-200
J3                 1320-1380                180-280
J4                 1200-1260                180-300
J5                 1100-1180                150-270
J6                 1200-1280                180-300
J7                 1300-1380                300-420
J8                 1250-1300                180-300
J9                 1150-1230                120-240

Table 2
A batch program using the traditional manual batching method

Batch number b                              1     2      3      4     5
Work-piece type J1
Work-piece type J2                                                    2
Work-piece type J3                                                    5
Work-piece type J4
Work-piece type J5
Work-piece type J6
Work-piece type J7                                              3
Work-piece type J8
Work-piece type J9                           12
Work-piece type J10
Work-piece type J11
Work-piece type J12
Work-piece type J13                           4
Work-piece type J14
Work-piece type J15
Work-piece type J16                                 9     9           1
Work-piece type J17                                             9
Work-piece type J18
Weight of batch [s.sub.b], kg               794  7722  7722  7530  6738
                                              8
Holding temperature [T.sub.j], [degrees] C  125  1300  1300  1250  1320
                                              0
Holding time [C.sub.j], min                 200   300   300   270     3

Batch number b                                6       7      8      9

Work-piece type J1                             7
Work-piece type J2
Work-piece type J3
Work-piece type J4                            14
Work-piece type J5
Work-piece type J6                                            5
Work-piece type J7
Work-piece type J8
Work-piece type J9
Work-piece type J10                                                 19
Work-piece type J11                                    6
Work-piece type J12
Work-piece type J13
Work-piece type J14                                                 10
Work-piece type J15                                    8
Work-piece type J16
Work-piece type J17
Work-piece type J18                                           4
Weight of batch [s.sub.b], kg               6510    6004   5124   3762

Holding temperature [T.sub.j], [degrees] C   900    1250   1150    800

Holding time [C.sub.j], min                  240     180    150    180

Batch number b                                10     11     12
Work-piece type J1
Work-piece type J2
Work-piece type J3
Work-piece type J4
Work-piece type J5                             4
Work-piece type J6
Work-piece type J7
Work-piece type J8                                           1
Work-piece type J9
Work-piece type J10
Work-piece type J11
Work-piece type J12                                   2
Work-piece type J13
Work-piece type J14
Work-piece type J15
Work-piece type J16
Work-piece type J17
Work-piece type J18
Weight of batch [s.sub.b], kg               2756   1254   1246

Holding temperature [T.sub.j], [degrees] C   950   1320   1390

Holding time [C.sub.j], min                  260    180    320

Table 3
Batching plan out by DSFLA based on same furnace heating rules

Batch number b                                 1     2     3     4

Work-piece type J1
Work-piece type J2                                   1     1
Work-piece type J3
Work-piece type J4
Work-piece type J5
Work-piece type J6
Work-piece type J7
Work-piece type J8
Work-piece type J9                             9                 3
Work-piece type J10
Work-piece type J11                                              6
Work-piece type J12
Work-piece type J13                            3                 1
Work-piece type J14
Work-piece type J15                            3                 5
Work-piece type J16                                  7     5
Work-piece type J17                                  1     4     2
Work-piece type J18                            2
Weight of batch [s.sub.b], kg               7939  7757  7754  7687
Holding temperature [T.sub.j], [degrees] C  1230  1300  1300  1250
Holding time [C.sub.j], min                  200   300   300   200

Batch number b                              5       6     7      8

Work-piece type J1                                         7
Work-piece type J2
Work-piece type J3                             5
Work-piece type J4
Work-piece type J5                                         4
Work-piece type J6                                                5
Work-piece type J7                             3
Work-piece type J8
Work-piece type J9
Work-piece type J10
Work-piece type J11
Work-piece type J12                            2
Work-piece type J13
Work-piece type J14                                              10
Work-piece type J15
Work-piece type J16                                  7
Work-piece type J17                                  2
Work-piece type J18                                               2
Weight of batch [s.sub.b], kg               7165  7148  5808   4592
Holding temperature [T.sub.j], [degrees] C  1320  1300   950   1150
Holding time [C.sub.j], min                  280   300   260    150

Batch number b                                9       10     11

Work-piece type J1
Work-piece type J2
Work-piece type J3
Work-piece type J4                                    14
Work-piece type J5
Work-piece type J6
Work-piece type J7
Work-piece type J8                                            1
Work-piece type J9
Work-piece type J10                            19
Work-piece type J11
Work-piece type J12
Work-piece type J13
Work-piece type J14
Work-piece type J15
Work-piece type J16
Work-piece type J17
Work-piece type J18
Weight of batch [s.sub.b], kg                3762   3458   1246
Holding temperature [T.sub.j], [degrees] C    800    880   1390
Holding time [C.sub.j], min                   180    150    320

Table 4
Comparison of DSFLA batching and manual batching

Batching plan      Quantity of   Average holding  Average
                   charging      temperature      holding time
                   batches b     T, [degrees] C   C, min
DSFLA based on
same furnace
heating rules        11           1170             240
Traditional
manual batching
method               12           1182             240
Compare DSFLA to
manual method        -1            -12               0