Lei et al. [26] proposed a two-string representation. For energy-efficient UPMSP with n jobs, m machines, Rmax units of additional resource and PM, its solution consists of a machine assignment string [Mh1,Mh2,⋯,Mhn] and a scheduling string [θ1,θ2,⋯,θn], where Mhi is the assigned machine for job Ji and θi is real number.

The decoding procedure is described as follows:

(1) Obtain job permutation [π1,π2,⋯,πn] by sorting all jobs in ascending order of θi.

(2) Start with π1, for each job πi, assign it to its machine Mhπi according to the first string, decide if job πi can be inserted into idle period and deal with PM as done in paper [16].

For the example in Section 2, a solution consists of [M2,M2,M1,M1,M2,M1,M2,M1] and [0.1,0.3,0.7,0.57,0.62,0.23,0.85,0.41, the obtained job permutation is [1,3,7,5,6,2,8,4], when J6 is allocated on M1, if no additional resource constraint is considered, J6 can be processed between [6,10], however, the sum of the additional resource is 11, the additional resource constraint is violated, so J6 is processed on [10,14]. For J4, if it is processed directly after J8, C4=25>u1, so PM is first executed and then J4 is processed. The obtained schedule is shown in Fig. 1.

β initial solutions are produced by heuristics. Heuristic 1 is used to produce solution x1 and described as follows. For each Ji, min{pki,1≤k≤m} is decided, a machine Mhπi with phii=min{pki} and the smallest phii×ehi are chosen; then a scheduling string is randomly generated. Heuristic 2 is used for solution x2 and shown below. For each job Ji, compute min{pki×ek,1≤k≤m} and then select Mhπi with the smallest phii and phii×ehi=min{pki×ek}. The scheduling string of x2 is also stochastically generated.

Heuristic 3 is used for each of β−2 solutions: randomly produce a scheduling string, for each job Ji, if rand<0.5, then decide a machine Mhπi as done in heuristic 1 for each Ji; else determine a machine Mhπi as done in heuristic 2 for each Ji. Where rand is random number following uniform distribution on [0, 1].

N−β initial solutions are stochastically gotten. Employed bee swarm EB consists of randomly chosen N/2 solutions from P and onlooker bee swarm OB is composed of the remained N/2 solutions.

Six neighborhood structures are used. 𝒩1 is used to move a randomly chosen job Ji on a machine with the biggest completion time to a machine Mk with the smallest pki×ek. 𝒩2 is similar with 𝒩1, Ji is moved to Mk with the smallest pki in 𝒩2. 𝒩3 is shown below. Decide max{phii×ehi,1≤i≤n} and a job Jj with phjj×ehj=max{phii×ehi} and move Jj to a machine Mk with pkj×ek=min{plj×el,1≤l≤m}. 𝒩4 is adopted to exchange a randomly selected job on a machine Mk with the biggest completion time and a randomly chosen job on a stochastically decided Ml,l≠k. 𝒩5 is shown below. Randomly choose Mk and swap two randomly selected jobs Ji,Jj on Mk, that is, θi,θj are exchanged. 𝒩6 is described as follows. Randomly decide a machine Mk and two randomly selected jobs Ji,Jj on Mk, then insert θi into position j on scheduling string.

Algorithm 1 describes the detailed steps of dynamical employed bee phase, where cni and rankxi indicate the number of searches on a generation and rank of xi decided by non-dominated sorting [46], 𝒩g(x) is the set of neighborhood solutions of x produced by 𝒩g. The set Ω is used to store historical optimization data. When Ω is updated with x, x is added into Ω and all solutions of Ω are compared and all dominated ones are removed.

Global search between xi,y is shown below. Execute two-point crossover on machine assignment strings of xi and a randomly chosen y∈EB, and obtain a new one z, if z≻xi or z,xi are non-dominated each other, then let traili=0, xi is used to renew Ω and z substitutes for xi; otherwise, traili=traili+1, perform two-point crossover on scheduling strings of xi and a randomly selected y∈EB, obtain a new one z and update xi, traili and Ω according to the above condition.

Neighborhood search NS1 on xi is described as follows. Randomly decide 𝒩g, produce z∈𝒩g(xi), update xi, traili and Ω in terms of conditions of global search.

Solution migration is described below. Define Δ={xi∈EB|rankxi=1,It≤traili}, then perform non-dominated sorting on OB, sort all solutions of OB in the ascending order of rank, for some solutions with same rank, sort them in the ascending order of traili, select the first |Δ| solutions, for each chosen solution xi, a multiple neighborhood search is applied and let traili=0.

For solution xi, multiple neighborhood search is executed below. Let g=1, repeat the following steps until g=7: produce a solution z∈𝒩g(xi), if z≻xi or z,xi are non-dominated each other, then z substitutes for xi and let g=7; otherwise g=g+1.

Neighborhood search NS2 on xi is shown as follows. (1) Select the machine Mk with the biggest completion time and randomly choose a job Ji assigned to Mk, Then, repeat the following steps: insert Ji into each possible position on Mk and obtain a solution z until z≻xi. (2) Determine a machine Mk with the biggest energy consumption and randomly select a job Ji on Mk, repeat the following steps: move Ji to Ml,l≠k and obtain a solution z until z≻xi. In NS2, if z≻xi is not met, then traili=traili+1; else traili=0.

In dynamical employed bee phase, some dominated solutions with rankxi>1 have cni=0, and their computing resources are reallocated to non-dominated solutions with rankxi=1. As a result, cni for some solutions exceed 1 and cni of other solutions are 0. This indicates that the search times for solutions are dynamically adjusted based on solution quality. Additionally, solution migration is triggered when all non-dominated solutions satisfy It≤traili. In this case, some best solutions of OB are moved to EB and solutions of EB are dynamically adjusted when the given condition is met, Therefore, dynamic adjustment is applied in both scenarios.

Four search operators SO1−SO4 are given. SO1 is described below. For a solution 0, select a 𝒩g according to an adaptive process and produce a solution z∈𝒩g(xi), if z≻xi, then xi is used to update Ω and z substitutes for xi; if z,xi are non-dominated each other, then z is applied to renew Ω; if xi≻z, then randomly select y∈EB, multiple neighborhood search acts on y and xi is replaced with y.

Adaptive process is depicted below. Choose a neighborhood structure by roulette selection based on Pseg; if rand>Q, then randomly choose a neighborhood structure; suppose 𝒩a is chosen, produce a new solution z∈𝒩a(xi), if z≻xi, then counta=counta+2; if z,xi are non-dominated each other, then counta=counta+1, where Q is threshold.

SO2 is shown as follows. For a solution xi∈OB, let α=0, execute variable neighborhood descent (VND) shown in Algorithm 2, if α=0, then perform multiple neighborhood search on xi.

SO3 is done in the following way. For a solution xi∈OB, randomly choose y∈EB, perform global search between xi,y as done in Lines 3–7 of Algorithm 1; then execute multiple neighborhood search on xi. SO4 has the same steps as SO3; however, y∈Ω in SO4.

In SO1−SO4, when multiple neighborhood search acts on xi, for each z, if xi cannot be replaced with z, then traili=traili+1; otherwise, traili=0.

In SO1, an adaptive process is adopted to select neighborhood structure adaptively, SO2 is an adaptive combination of VND and multiple neighborhood search, SO3, SO3 are combination of global search and multiple neighborhood search.

In onlooker bee phase, for each 𝒩g, set initial countg=1 and define selection probability Pseg. (14)Pseg=countg/∑l=16countl

Algorithm 3 describes dynamical onlooker bee phase on generation gen, where if SO3 is chosen in Line 4, then y∈EB is randomly decided; if SO4 is selected, then y∈Ω is chosen randomly, in Lines 7, 9, when the chosen operator is executed, the decided y in Line 4 is directly used, EvoOBgen denotes the evolution quality. (15)EvoOBgen=∑xi∈OBnewxigenwhere newxigen is defined below. For xi when an operator SOl acts on xi on generation, gen if new solution z≻xi, then newxigen=newxigen+2; if z,xi are non-dominated each other, newxigen=newxigen+1.

Feedback is dynamical process used in control. In this study, feedback is applied to decide one of SO1−SO4 dynamically, for each xi∈OB, on generation gen, if EvoOBgen−1<EvoOBgen−2, then random select one operator of SO1,SO2 and perform the chosen operator on xi∈OB; otherwise, execute the chosen operator on generation gen−1 on xi∈OB.

In dynamical onlooker bee phase, for each xl, if rankxl>1 and xl≻y, then computing resource of xl is shifted to non-dominated solution xj∈OB, feedback is used to dynamical decide search operator by selecting a new one if EvoOBgen−1<EvoOBgen−2 or using search operator of generation gen−1, that is, search operator on generation gen is decided or affected by evolution on the previous two generations, obviously, computing resource and search operator are dynamically adjusted.

In Algorithm 1, the search operator is combination of global search and neighborhood search NS1, in Algorithm 3, SO3,SO4 are composed of global search and multiple neighborhood search, SO1,SO2 are neighborhood search-based operator; moreover, these operators are dynamically selected by feedback, these operators can be useful to make good balance between exploration and exploitation.

The detailed steps of DABC are shown in Algorithm 4.

Scout phase is described as follows. For each solution xi∈P, if traili≥Limit, then xi is used to update Ω and then replaced with a randomly produced solution and traili=0.

Unlike the previous ABC [36–40], DABC has the following new features. (1) The initial population is produced by three heuristics. (2) Dynamical employed bee phase is implemented by using computing resource shifting and solution migration. (3) Four search operators are used and dynamical onlooker bee phase is performed by applying computing resource shifting and feedback. The above dynamical optimization mechanisms such as solution migration and feedback can decide the number of searches and adjust solutions of swarms and search operator dynamically, as a result, search efficiency can be improved. On the other hand, many new things are required to be implemented when DABC is used, this may be a disadvantage of DABC.

Fanjul-Peyro et al. [8] provided 300 instances, which can be divided into 30 types and the size of each type is depicted as n×m, n∈{8,12,16,20,25,30,50,150,250,350} and m∈{2,4,6}. For each type n×m, five ways are used for generating pki and two ways are applied for rki, 10 instances n×m×1,⋯,n×m×10 are generated. Fanjul-Peyro et al. [8] described seven ways for pki,rki and the related data can be obtained directly from http://soa.iti.es/problem-instances (accessed on 24 May 2024). Rmax=5m. We generate PM data as follows, wk is integer selected from the same interval as pki, uk=round(wk+3.5×maxi=1,2,⋯,n{pki}). Where round(x) is an integer being closet to x.

Metric 𝒞 [47] is used to compare the approximate Pareto optimal set respectively obtained by algorithms. (16)𝒞(L,B)=|{b∈B:∃h∈L,h≻b}||B|

Metric ρ is the ratio of |{x∈Ωl|x∈Ω∗}| to |Ω∗| [48], where Ωl is non-dominated set of Algorithm l, the reference set Ω∗ consists of the non-dominated solutions in the union of non-dominated sets of all algorithms.

Metric DIR [49] is used to measure the convergence performance by computing the distance of the non-dominated set Ωl relative to a reference set Ω∗.

(17)DIR(Ωl)=1|Ω∗|∑y∈Ω∗min{σxy|x∈Ωl}0where σxy is the distance between a solution x and a reference solution y in the normalized objective space.

Lei et al. [23] proposed NICA for multi-objective UPMSP with PM. Shahidi-Zadeh et al. [3] presented a multi-objective harmony search (MOHS) for UPMSP. Zhang et al. [34] developed CEA for energy-efficient UPMSP with makespan and total energy consumption. These algorithms can be used to solve energy-efficient UPMSP with additional resource and PM after related steps on additional resource and PM are added into decoding procedure; moreover, they have promising advantages in solving UPMSP, so they are chosen as comparative algorithms.

ABC is used to show the effect of new strategies of DABC. ABC is constructed as follows: in employed bee phase, Lines 1–10 with cni=1 for each xi∈P of Algorithm 1 are executed; in onlooker bee phase, a solution xi∈P is selected by binary tournament and the above Lines 1–10 are executed, scout phase of DABC is adopted in ABC.

DABC has following parameters: N,It,β,R,Q,Limit and stopping condition. Stopping condition is first decided independently as done in [18], we found by experiments that DABC converges well when 0.3n s CPU time reaches. We also obtained that when 0.3n s CPU time is applied, all comparative algorithms also converge fully, so stopping condition is set as 0.3n s CPU time for all algorithms.

An empirical method was used to determine the settings for other parameters by using the instance 50×20×5. Table 1 shows the levels of each parameter. The orthogonal array L27(36) is tested. DABC with each combination runs 10 times on the chosen instance.

Fig. 2 shows the results of ρ and S/N ratio, which is defined as −10×log10⁡(ρ2). It can be found from Fig. 2 that DABC with following combination N=100,It=5,β=10,R=10,Q=0.3,Limit=10 produces better results than DABC with other combinations, moreover, we tested the above combination on all instances, the results reveal that the above combination is still effective, so the above parameter settings are adopted.

ABC has N=100,Limit=10 and the above stopping condition.

Parameter settings of three comparative algorithms are directly selected from References [3,23,34] except that the stopping condition. To compare fairly, all algorithms should be stopped under the same condition, so MOHS, CEA and NICA are given the same stopping condition as DABC. We conducted experiments on other parameters of comparative algorithms, the experimental results show that each comparative algorithm with parameter settings from [3,23,34] can produce better results than the same algorithm with other parameter settings, so the original parameter settings are still used.

DABC, its three comparative algorithms and ABC are compared. Each algorithm randomly runs 10 times for each instance. Tables 2–9 describe the corresponding results of five algorithms. D, A, N, M, C denote DABC, ABC, NICA, MOHS and CEA. Fig. 3 shows the distribution of non-dominated solutions obtained by all algorithms.

An effective way is applied to show results of five algorithms on 300 instances. In Tables 2–5, for each type n×m, four groups of data are given, 𝒞(C,D) and 𝒞(D,C) are computed for each instances, 10 𝒞(C,D) are sorted in the ascending order, the first group is the smallest 𝒞(C,D) and its corresponding 𝒞(D,C), the second is the fifth 𝒞(C,D) and its 𝒞(D,C), the third is the tenth 𝒞(C,D) and its corresponding 𝒞(D,C), let α1=α2=0, for 𝒞(C,D), 𝒞(D,C) of each instance of n×m, if 𝒞(C,D)<𝒞(D,C), then α1=α1+1; if 𝒞(C,D)>𝒞(D,C), then α2=α2+1; if 𝒞(C,D)=𝒞(D,C), then α1=α1+1, α2=α2+1, the fourth group consists of α1,α2.

For type 16×6, 10 pairs of 𝒞(C,D), 𝒞(D,C) are listed below. (0.4, 0.6), (0.143, 0.824), (0.188, 0.684), (0.4, 0.444), (0, 0.125), (0, 0.857), (0.2, 0.333), (0, 1), (0.286, 0.7), (0.25, 0.273), obviously, α1=10, α2=0, which means that 𝒞(C,D) is less than 𝒞(D,C) on 10 instances.