Study on the charging combination optimization for forging production based on discrete shuffled frog leaping algorithm

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


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.

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.

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.

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.
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.

Optimization model
Definition 4: Average furnace holding temperature: The average value of holding temperature of all furnace batch in one batching, i.e.: where T is the average temperature in the furnace charging, k is the furnace batch number, and Tb is the holding temperature of furnace batch (Section b) ( 1, 2, , bk  ).Definition 5: Average furnace holding time: The average value of holding time of all furnace batch in one batching plan, i.e.: where C is the the average holding time, k is the furnace batch number, and Cb is the holding time of furnace batch With the above assumptions and definitions, mathematical model can be established as follows: where S is the heating furnace maximum load, zb 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: 1, , where n denotes the work-piece number, J denotes aggregate of work-pieces (   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.Eq. ( 14) is the decision variables.

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.

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: ; where wk, wq, wT, wC are the weight coefficient of furnace batch number, average charging difference, average holding temperature, average holding time.And they satisfy the equation 1 w w w w     .kmax, kmin, qmax, qmin, Tmax, Tmin, Cmax, Cmin 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 zj holding temperature is Tj, holding time is Cj.
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 b s s  , where S is the maximum load of heating furnace and sb is the weight of batch b.The holding temperature is Tb, which is the holding temperature interval intersection of all work-pieces, and Tb is not  .The holding time is Cb, which is the holding time interval intersection of all work-pieces, and Cb is not  .If all batches in queue Q have been operated with work-piece j, a new furnace butch bnew is set up with work-piece j in it.Update the remaining weight space of batch bnew: 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 × n.

Ethnic group evolution
In each group, the best frogs Xb and the worst frog Xw are chosen, as well as the optimal frog Xg 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 Xb in random.Xb is divided into several fragments.
Step 2: Select intersection points from Xw in random, and insert fragment from Xb into Xw before the intersection point.This means some information Xb is inserted into Xw.
Step 3: delete these furnace batch fragment in Xw 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 Xw, i.e. w X  .
Step 5: calculate the fitness value of w X  .If the w X  is better than Xw, replace Xw, else Xg, which is the best individual in the population should replace Xb in step 1 and start the operation from step 1 to step 4 again.Now if the w X  is better than Xw, replace Xw, else Generates a random feasible solution to replace Xw.
Step 6: repeat the operation above until the maximum iterations to complete one ethnic group evolution.
Examples are as follows: information of Xb and Xw are shown in Fig. 1.Cross location is shown by arrow.The cross fragment from Xb is plugged into Xw 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. The new frog is better than ?
The new frog is better than ?

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°C.The average holding time is 240 min.

Batching plan based on DSFLA
Assuming that the furnace batch quantity weight wk, the average charging difference weight wq, the average temperature wT and the average weight of holding time wC 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°C.The average holding time is mainly 240 min.Table 3 is a plan of 1170°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.

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.

STUDY ON THE CHARGING COMBINATION OPTI-MIZATION FOR FORGING PRODUCTION BASED ON DISCRETE SHUFFLED FROG LEAPING ALGORITHM S u m m a r y
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.
is assumed that work-pieces can be separated and allocated to different batches, and all workpieces in any furnace batch can satisfy the temperature compatibility.
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 , remove work-piece j into batch b, update the remaining weight space of batch b: queue Q is empty, go to step 4, else go to step 2. If anyone of 0 is true, then go to step 3.

Fig. 1 Fig. 2 Fig. 3 Fig. 4 X
Fig. 1 The individual information and the position of the cross point

Fig. 5
Fig. 5 Flowchart of DSFLA based on same furnace heating rules 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 Table1.

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