Process Integr Optim Sustain (2017) 1:251–267 https://doi.org/10.1007/s41660-017-0020-3
ORIGINAL RESEARCH PAPER
A Mixed-integer Multi-objective Maintenance and Production Workforce Planning Model Desmond Eseoghene Ighravwe 1 & Sunday Ayoola Oke 2,3
Received: 2 February 2017 / Revised: 17 October 2017 / Accepted: 20 October 2017 / Published online: 12 November 2017 # Springer Nature Singapore Pte Ltd. 2017
Abstract This study tackles the dilemma of producing a cooperative workforce planning involving maintenance and production departments. Its focus is on instituting workforce equilibrium between maintenance and production departments. This is necessary in order to ensure the optimal utilisation of manufacturing resources (time, equipment and funds). This maintenance-production problem was modelled using a mixed-integer multi-objective approach. The model minimises the workforce, rework and scrap, inventory holding cost and machine usage costs while maximising the average achieved machine availability. The model accounts for expected products’ demand, the amount of expected defective products, workforce size, rest period and finished good inventory. Due to the nonlinear relationships among the maintenanceproduction variables, a big bang-big crunch (BB-BC) algorithm and genetic algorithm (GA) were selected as solution methods for the model. The proposed model performance was validated in a household utensils manufacturing plant under the conditions of workers’ rest and without rest periods considerations. Based on the results obtained, the BB-BC algorithm performed better than the GA in terms of fitness function and computational time. The model performance with workers’ rest period consideration generated higher achieved * Sunday Ayoola Oke
[email protected] 1
Department of Mechanical Engineering, Faculty of Engineering and Technology, Ladoke Akintola University of Technology, Ogbomoso, Oyo State, Nigeria
2
Department of Mechanical Engineering, Faculty of Engineering, University of Lagos, Lagos, Nigeria
3
Industrial and Production Engineering Unit, Department of Mechanical Engineering, College of Engineering, Covenant University, Ota, Nigeria
machine availability value than when workers’ rest period was not considered. The optimised maintenance-production variables results confirmed the model’s capability to generate acceptable solutions for the case study. Keywords Maintenance-production problem . Human resource management . Inflation and interest rates . Big bang-big crunch algorithm . Genetic algorithm
Introduction Previous research has affirmed that contemporary times are characterised by organisations seeking to optimise available resources amidst a broad range of internal and external challenging factors (Barnett and Blundell 1981; Ntuen and Park 1999; Moore et al. 2005; Lopez and Carteno 2006). Thus, models that account for this resource control factors would be of help to the business community (Mansour 2011; Othman 2012). The development of such models will improve maintenance and production activities when compared with decision makers’ intuition. On a broader perspective, a model that integrates maintenance and production activities provide more useful insights into workforce planning than models which consider these activities separately (Cheng et al. 2016; Fakher et al. 2017). Currently, several models exist on the joint optimisation of maintenance and production variables (Hajej et al., 2015; Xia et al., 2015; Liao, 2016, Nasr et al., 2016; Wu et al., 2016; Hu et al., 2017). However, very few studies have considered worker’ rest period and interest rates on allocated workforce fund. Also, there is no reference in literature that indicates the incorporation of inflation rate in an integrated maintenanceproduction model (Weil et al. 1995; Lucic and Teodorovic 1999; Greenwood and Gupta 2000; Santos et al. 2009;
252
Pleumpirom and Amornsawadwatana 2012). The inclusion of these knowledge gaps into maintenance-production models will increase their complexity. This makes conventional optimisation models solution (simplex and big-M) methods to be less attractive as solution methods for such models. Since these methods lack the capacity to handle nonlinear optimisation models. Nonlinear optimisation models have been successfully solved using meta-heuristics (evolutionary and swarm algorithms). Currently, there is a gap in testing the potentials of big bang-big crunch (BB-BC) algorithm as a solution method for nonlinear maintenance-production models. Based on these identified knowledge gaps, the contributions of this are as follow: the applicability of a BB-BC algorithm as a solution method for a nonlinear maintenance-production model and the introduction of workers’ differences (status, turnover rates and rest period) and economic factors (inflation and interest rates). These contributions were achieved by developing a maintenance-production workforce mixed-integer multi-objective model. The model minimises workforce cost, reworked products and scraps, minimises inventory holding and machine usage costs while maximising average achieved machine availability.
Literature Review Although maintenance (Tse 2002; Sanchez et al. 2009) and production functions are different in forms, their objectives are interwoven. Maintenance aims at maintaining equipment in at acceptable conditions at reasonable costs while the production function produces physical goods with the maintained equipment at acceptable quality levels. In order to improve these functions, different models have been developed by researchers. Some of these models are simulation (Ntuen and Park 1999; Duffuaa et al. 2001), integer programming (Alfares 1999), stochastic programming (Duffuaa and AlSultan 1997) and mixed-integer programming (Judice et al. 2005) models. For instance, Barnett and Blundel (1981) investigated trade demarcation of a workforce at optimal level using simulation model. Moore et al. (2007) applied Delphi method and benefit-cost analysis in investigating the impact of workforce quality on aircraft maintenance. Ramirez-Hernandez et al. (2007) reported the translation of time from calendar-based to non-calendar-based in programmes involving the scheduling of preventive maintenance tasks. Ighravwe and Oke (2014) optimised maintenance workforce cost and productivity under full- and part-time working conditions. Sanchez et al. (2009) used a multiobjective genetic algorithm (MOGA) to optimised unavailability and cost criteria in maintenance and testing of machines under uncertainty conditions. Chen (2007) applied a combined shortest processing time and maximum tardiness
Process Integr Optim Sustain (2017) 1:251–267
scheduling model in solving the problems of maximum tardiness and minimum total flow time under periodic maintenance condition. Berrichi et al. (2001) compared a multi-objective ant colony optimisation model and MOGA as solution methods for maintenance-production scheduling of parallel machines. Ntuen and Park (1999) evaluated the performance of workers when maintaining equipment, while Lopez and Carteno (2006) simulated workforce problem in a maintained reliability system. Duffuaa et al. (2001) applied a simulation approach for manpower and inventory control by partitioning maintenance problems into modules. Belmokaddem et al. (2008) studied production workforce and inventory problem using fuzzy goal programming approach. Ip et al. (2000) integrated maintenance system into material requirement planning II (MRP II). Solution approaches for solving maintenance-production workforce models have evolved in the literature. Techniques such as simulated annealing (Lucic and Teodorovic 1999), ant colony optimisation (Berrichi et al. 2001), evolution strategy (Greenwood and Gupta 2000), goal programming (Pleumpirom and Amornsawadwatana 2012) and genetic algorithm (Sortrakul and Cassady 2007; Ighravwe et al. 2015) have been tested and satisfactory results were obtained. The models that these solution methods were apply deal with the machine components of maintenance and production problems. This has created knowledge gap on the effect of human factors on manufacturing time (Sortrakul and Cassady 2007; Sanchez et al. 2009; Pleumpirom and Amornsawadwatana 2012). The issue of concurrent maintenance-production workforce optimisation using product demand, defective product, and inventory holding manufacturing time as constraints is another knowledge gap in maintenance-production workforce literature. In addition, workers’ difference has not been considered during maintenance-production workforce planning. Human factor consideration is required during maintenanceproduction workforce planning in order account for nonmachine activities; this will entail the analysis of workers’ skills, service rates and statuses (i.e., workers’ differences) for optimal results to be obtained during production and maintenance planning. With this in mind, this study proposes a framework which amalgamates maintenance and production functions. The framework places emphasis on the workforce size and the effect of workers’ differences in order to accomplish maintenance and production goals.
Research Methodology The proposed framework for the maintenance-production workforce problem is based on multi-objective mixed-integer nonlinear programming approach (Fig. 1). The model presented considered four objective functions using maintenance and
Process Integr Optim Sustain (2017) 1:251–267 Fig. 1 Proposed framework for maintenance-production variables optimisation
253 Research motivation Recognition of the need for joint optimisation of maintenance-production workforce using bigbang big-crunch (BB-BC) and genetic algorithm in order to enhance industrial performance Research objective Minimisation of workforce cost, rework expenses and overtime cost as well as fatigue and recovery rate Model parameters and variables Maintenance and production time, cost for maintenance and production as well as expected rest periods
Solution methods Non-linear programming, weighted goal programming, BB-BC and GA Observations Quality of solution and computation time of the proposed solution methods Results Optimal size of maintenance and production workforce, rework rate and recovery period Conclusion Development of a model that addressed the problem of rework, fatigue and recovery level rate among maintenance-production workforce in manufacturing systems
production variables as constraints (Fig. 1). The model objective functions are converted into a single function using a weighted-goal programming approach and a payoff table (Fig. 1). This is necessary in order to generate Pareto front for the model’s objectives (Wu 2008). The model’s objective functions were then converted into soft constraints and solved using a BB-BC algorithm and GA (Fig. 1). Model’s Nomenclature The parameters, variables and indices used to formulate the proposed maintenance-production workforce model are given as follows: Parameters D R λi bave INVave bave υcg α λi1 λi2
amount of defective (qty) amount of rework products (qty) arrival rate of corrective maintenance work (items/h) average break period (h) average inventory (qty) average rest period for maintenance-production workers (h) center-of-mass a decision variable at iteration step or generation g constant parameter cost of defective products that cannot based on machine i outputs (N) dost of defective products that are reworked based on machine i outputs (N)
D 0 f 1g 0
f 2g f pðgÞ Τ G MAMT M MTBM M βi 0 x1g 0
x2g d −q xj P υpg F ℏi dþ q U P wq
demand (qty) fitness of the first randomly elected parent at generation g fitness of the second randomly elected parent at generation g fitness value of particle p at iteration step or generation g inflation rate iteration step or generation mean active maintenance time (h) mean time between corrective maintenance (h) mean time between maintenance (h) mean time between preventive maintenance (h) corrective maintenance time for machine i (h) mutant vector for the first randomly elected parent at generation g mutant vector for the second randomly elected parent at period generation g negative deviation variable for the qth objective function number of workers in department j particle particle p value for a decision variable at iteration or generation g prevalent interest rates preventive maintenance time for machine i (h) positive deviation variable for the qth objective function production rate (qty) processing time (h) relative importance of the qth objective function
254
Ui qi Sit p t1 t2 cijkt
vijkt
vijkt
cijkt
cj vit at C1 ϕ rjk
Process Integr Optim Sustain (2017) 1:251–267
quantity of products produced from machine i per hour (qty) quantity of products produced from machine i before routine maintenance is carried out (qty) total manufacturing time for machine i at period t (h) total number of particles total service time for planned maintenance activities (h) total service time for unplanned maintenance activities (h) unit cost of a regular worker allocated to machine i belonging to department j from section k at period t (N) unit cost for the rest period embarked upon by a regular worker allocated to machine i belonging to department j from section k at period t (N) unit cost for the rest period embarked upon by a trainee allocated to machine i belonging to department j from section k at period t (N) unit cost of a trainee allocated to machine i belonging to department j from section k at period t (N) unit cost of a worker in department j (N) unit cost for using machine i per hour at period t (N) unit inventory holding cost at period t (N) workforce cost of a simple work-system (N) workforce priority factor workers’ turnover rate of workers in department j belonging to section k
Decision variables bijkt
bijkt
Sijkt
S ijkt
MTBMit di xijkt
amount of time used as rest period by a regular worker allocated to machine i belonging to department j from section k at period t (h) amount of time used as rest periods by a trainee allocated to machine i belonging to department j from section k at period t (h) daily time expected from a regular worker allocated to machine i belonging to department j from section k at period t (h) daily time expected from a trainee allocated to machine i belonging to department j from section k at period t (h) mean time between maintenance on machine i at period t (h) number of defective products produced from machine i (qty) number of regular workers allocated to machine i belonging to department j from section k at period t
xijkt
number of trainees allocated to machine i belonging to department j from section k at period t processing time of machine i at period t (h) quality of finished goods inventory at period t (qty) quantity of goods that undergo rework on machine i (qty)
Pit It Ri
Indices j i t k N M T K
department in a manufacturing system machine planning period section in a particular department total number of departments total number machines total number of planning periods total number of sections in a department
Model’s Assumptions The following assumptions are made during the development of the proposed model: i. Defective goods can be reworked or reused in generating quality goods. ii. Maintenance and production works are not subcontracted. iii. A company incurs losses as a result of time allocated for recovery periods. iv. The amounts of opening and closing finished goods inventories are the same. v. Trainee and regular workers’ service rates are not equal.
Optimisation Model Objective Functions This study presents four objective functions in tackling the problem of equilibrium between maintenance and production departments in term of workforce size, cost and time. Also, the issue of finished inventory cost is considered. Workforce Cost The budgeted funds for workforce are expected to cover fixed costs, normal salaries and bonuses. Workforce budget can be determined by forecasting the expected workforce cost for maintenance and production workers. Mathematically, a simple work-system workforce cost (C1) is given as Eq. (1), while a complex manufacturing work-system workforce costs is given as Eq. (2). By
Process Integr Optim Sustain (2017) 1:251–267
255
Quality goods
Closing inventory
Demand Opening inventory
Goods available for sales
Goods produced
Defective goods produced
Inventory Holding and Machines Usage Cost Given that machines are available for use, machines usage is subjected to the amount of production activities and inventory in a manufacturing system. These costs (i.e., unit inventory holding and machines usage costs) are other types of manufacturing costs. These costs are expressed as Eq. (7). T
M
t¼1
Fig. 2 Interaction among finished goods inventory, goods produced defective goods and product demand
considering the cost of trainees in a manufacturing system, Eq. (2) becomes Eq. (3). When workers embarked on official and unofficial break (i.e., rest), an organisation incurs a loss. To compute the amount of losses due to equipment not been used for production activities, the cost of idle time for rest period is considered (Eq. 4). The aggregation of Eqs. (3) and (4) gives Eq. (5). This equation represents the standard costs of workers under an instance that funds are not obtained as a loan from a bank. N
C1 ¼ ∑ c jx j
T
C 3 ¼ ∑ at I t þ ∑ ∑ vit Pit
ð7Þ
i¼1 t¼1
where at represents the unit inventory holding cost at period t. By considering inflation (τ) and prevalent interest ( f ) rates (Ardalan 2000), Eqs. (5), (6) and (7) are extended to incorporate τ and f (Eqs. 8, 9 10). Min f 1 : C 1
ðτ−f Þ ð1 þ f Þ
ð8Þ
Min f 2 ¼ C 2
ðτ−f Þ þ C2 ð1 þ f Þ
ð9Þ
Min f 3 ¼ C 3
ðτ−f Þ þ C3 ð1 þ f Þ
ð10Þ
ð1Þ
j¼1 M
N
K
T
C 1 ¼ ∑ ∑ ∑ ∑ cijkt xijkt
ð2Þ
i¼1 j¼1 k¼1 t¼1
M N K T ~ ¼ ∑ ∑ ∑ ∑ cijkt xijkt þ cijkt xijkt C
ð3Þ
i¼1 j¼1 k¼1 t¼1 $ M N K T C ¼ ∑ ∑ ∑ ∑ vijkt bijkt xijkt þ vijkt bijkt xijkt
ð4Þ
i¼1 j¼1 k¼1 t¼1 $
~ C C 1 ¼ Cþ
ð5Þ
Average Achieved Machine Availability The fourth objective function is the maximisation of average achieved machine availability (Irason et al. 1996). Achieved machine availability is a function of mean time between maintenance (MTBM) and the mean active maintenance time (MAMT). The value of a machine MTBM (Eq. 11) comprises of its mean time between corrective maintenance M and mean time between preventive maintenance M . By considering the allocated maintenance time and rest periods of maintenance workforce, MAMT for each period is given as Eq. (12).
where cj represents the unit cost of a worker in department j, and xj represents number of workers in department j.
1 MTBM ¼ −1 −1 M þ M
Rework and Scrap Costs During rework activities, an organisation incurs extra cost for equipment that is reused for rework production activities, while sub-standard products or parts which cannot be reworked are considered as scraps. This institutes a loss to an organisation and increases the unit cost of production. The expression for rework and scrap costs is given as Eq. (6).
∑ ∑ ∑
M
T
C 2 ¼ ∑ ∑ ðð1−Ri Þλi1 þ λi2 Ri Þd i U i Pit
ð6Þ
i¼1 t¼1
where λi1 represents the cost of defective products that cannot be reworked based on machine i outputs, λi2 represents the cost of defective products that are reworked based on machine i outputs, and Ui represents the quantity of products produced on machine i per hour.
M
MAMT ¼
K
T
ð11Þ
ðS i2kt −bi2kt Þxi2kt þ xi2kt −bi2kt xi2kt
i¼1 k¼1 t¼1 M
K
T
∑ ∑ ∑
xi2kt þ xi2kt
ð12Þ
i¼1 kþ1 t¼1
Based on Eqs. (11) and (12), the achieved machine availability under rest periods for maintenance workers is given as Table 1
Comparison of BB-BC algorithm and GA
Parameter
BB-BC
GA
Worst solution Best solution Average solution Computational time (minutes)
7.33 × 1015 6.07 × 1015 6.34 × 1015 43.92
7.52 × 1015 6.82 × 1015 7.07 × 1015 76.05
256 Table 2 Case 1
2
Process Integr Optim Sustain (2017) 1:251–267 Compromise solutions for the various periods Period
f1
f2
f3
1
3,092,600.00
489,067.41
106,445,415.90
0.582
450
3,178,047.00
511,666.97
109,051,120.17
0.585
461
3 4
3178,047.00 3,102,039.00
511,666.97 493,203.08
109,051,120.17 106,928,146.72
0.585 0.580
461 451
1 2
3,240,654.00 3186,222.50
501,728.40 513,609.40
107,914,835.04 109,356,028.63
0.585 0.586
455 464
3
3102,039.75
490,562.02
106,619,016.37
0.583
451
4
3186,222.50
516,841.94
109,636,975.69
0.585
464
M
T
MTBM it i¼1 t¼1 MAMT
ð13Þ
Max f 4 ¼ 1− ∑ ∑
Model Constraints Demand and Finished Goods Inventory Constraints With the knowledge of the number of items produced per unit time, the total production time required to produce a set of goods can be estimated. This is given as the number of units produced per unit time and the amount of time used for production activities (Eq. 14). Under ideal conditions, the total product produced should be equal to product’s demand. An ideal condition is not feasible in manufacturing environments due to the defective items in lots. The presence of defective items in a production lot results in scraps and rework activities. The number of defective products in manufacturing systems varies from time to time. However, on the long run, it will follow a Gaussian distribution when the sample is large. Using this knowledge, we assume that the number of reworks is within a certain range of the total volume of defective products produced. UP ¼ D
ð14Þ
where U, P and D represent production rate, processing time and demand, respectively.
Month
1 2 3 4 5 6
Workforce size
2
Eq. (13).
Table 3
f4
A pictorial illustration of the interrelationships among finished goods inventory, goods produced, defective goods and product demand is shown in Fig. 2. The combination of quality goods and opening inventory constitutes the amount of goods available for sales. The difference between product’s demand and the goods available for sales gives the closing inventory of a system. The difference between the quantity of goods produced and quality goods gives the number of defective products of a system. The mathematical expression of the relationships in Fig. 2 is given as Eqs. (15) and (16). The relationship between starting and closing inventories is given as Eq. (17). The average inventory constraint is given as Eq. (18). m
I t−1 þ ∑ ðU i Pit −d þ RÞ−I t −Dt ¼ 0 ∀t∈T
ð15Þ
i¼1
Given that d = diUiPit and R = Rid, Eq. (15) becomes Eq. (16) m
I t−1 þ ∑ ð1−d i þ d i Ri ÞU i Pit −I t −Dt ¼ 0
∀t ∈T
i¼1
I1 ¼ IT
ð16Þ ð17Þ
T
∑ I t =T ¼ INV ave
ð18Þ
t¼1
where d and R represent the amount of defective and rework products, respectively.
Optimal inventory levels at the end of each sub-period Period 1
Period 2
Period 3
Period 4
Case 1
Case 2
Case 1
Case 2
Case 1
Case 2
Case 1
Case 2
15,599 12,574 11,817 22,211 11,015 10,813
15,811 12,786 12,030 22,423 11,227 11,925
16,020 12,994 12,238 22,631 11,435 11,233
16,059 13,034 12,278 22,671 11,475 11,273
15,975 12,950 12,194 22,587 11,391 11,189
15,624 12,599 11,843 22,236 11,040 10,838
11,669 12,643 11,887 22,281 11,085 10,883
16,060 13,034 12,278 22,672 11,475 11,274
Process Integr Optim Sustain (2017) 1:251–267 Average monthly production time on the machines
Table 4
Period Case Machine 1 1
1
2
3
4
257
2
3
4
5
6
1
17.13 16.13 17.13 16.13 18.13 16.13
2
17.13 16.13 15.13 16.13 18.13 17.13
2
1 2
17.17 16.17 17.17 16.17 18.17 16.17 16.17 16.17 15.17 16.17 18.17 17.17
1
1 2
17.21 16.21 17.21 16.21 18.21 16.21 16.21 16.21 15.21 16.21 18.21 17.21
2
1
17.22 16.22 17.22 16.22 18.22 16.22
1
2 1
16.22 16.22 15.22 16.22 18.22 17.22 17.20 16.20 17.20 18.20 18.20 16.20
2
2 1
16.20 16.20 15.20 16.20 18.20 17.20 17.13 16.13 17.13 16.13 18.13 16.13
2
16.13 16.13 15.13 16.13 18.13 17.13
1
1 2
17.14 16.14 17.14 16.14 18.14 16.14 16.14 16.14 15.14 16.14 18.14 17.14
2
1 2
17.20 16.22 17.22 16.22 18.22 16.22 16.22 16.22 15.22 16.22 18.22 17.22
where xmin and xmax represent the minimum and maximum workforce size, respectively, ϕ represents workforce priority factor, and rjk represents workers’ turnover rate of workers in department j belonging to section k. Workforce Service and Break Time Constraints With the knowledge of the average service time of maintenance and production workers, the total time used by maintenance and production workforce is given as Eqs. (22) and (23), respectively. Total routine maintenance time is a function of quantity of product produced before routine maintenance is carried out (Eq. 24). The total time required for corrective maintenance activities is a function of corrective maintenance time (βi), arrival rate of corrective maintenance work (λi) and the total period in which equipment is used for production activities (Eq. 25). M
si2kt xi2kt þ si2kt xi2kt ≤ t 1 þ t 2
K
∑ ∑ M
i¼1 j¼1 k¼1
N
K
∑ ∑ j¼1 k¼1
N K xðiþ1Þjkt þ xðiþ1Þjkt −ϕ ∑ ∑ xijkt þ xijkt ≤ 0
∀i∈M ; ∀t∈T
j¼1 k¼1
ð20Þ
xijkt ¼ 1−rjk xijkt‐1 þ xijkt‐1
∀i∈M ; ∀ j∈N ; ∀k∈K; ∀t∈T ð21Þ
Table 5 Case
ð22Þ
M si1kt xi1kt þ si1kt xi1kt ≤ ∑ ð1−d i þ d i Ri ÞU i Pit
K
∑ ∑ Workforce Size Constraints Proper sharing of manufacturing time between maintenance and production departments helps in improving the capacity of a system to meet the demand for their products. The allocations of manufacturing time depend partly on the workforce size in these departments (Eq. 19). This equation holds for a semi-automated manufacturing system. To further understand the time sharing problem, Eq. (20) is considered. The change in workforce size as a result of trainees becoming regular workers and worker’s turnover is given as Eq. (21). M N K xmin ≤ ∑ ∑ ∑ xijkt þ xijkt ≤ xmax ∀t∈T ð19Þ
∀t∈T
i¼2 k¼1
i¼1 k¼1
∀t∈T
i¼1
ð23Þ M
∑ ℏi ðð1−d i þ d i Ri ÞU i Pit Þ i¼1
t1 ¼
ð24Þ
qi M
t 2 ¼ ∑ λi βi Pit
ð25Þ
i¼1
where qi represents the quantity of products produced from machine i before routine maintenance is carried out, ℏi routine maintenance time for machine i, t1 and t2 represent the total time for planned (routine) and unplanned (corrective) maintenance activities, respectively. The summation of the total amount of production and maintenance workforce time used at each planning period is expected to be within the assigned minimum and maximum manufacturing time (Eq. 26). Manufacturing time management is further improved by specifying each machine production time (Eq. 27).
Optimal percentage of defective and reworked products on the machines Period 1 Machine 1
Period 2 Machine 2
Defective products from the machines (%) 1 0.43 0.71 2 0.44 0.72 Rework products on the machines (%) 1 72.51 57.51 2 73.04 58.04
Period 3
Period 4
Machine 1
Machine 2
Machine 1
Machine 2
Machine 1
Machine 2
0.44 0.44
0.72 0.72
0.44 0.43
0.72 0.71
0.43 0.44
0.71 0.72
73.56 73.66
58.56 58.66
73.45 72.57
58.45 57.57
72.69 73.66
57.69 58.66
258
Process Integr Optim Sustain (2017) 1:251–267
Table 6 Contribution of workers service time with respect to achieved machine availability
Appendix 1. bmin ≤ bijkt ≤ bmax
Category of maintenance Case 1 Case2 workers SS Contribution SS
Contribution
Mechanical (regular)
44.06
0.24
39.04
0.18
Electrical (regular)
90.16
0.49
100.25 0.46
Mechanical (trainees) Electrical (trainees)
17.14 31.41
0.09 0.17
16.76 64.10
Total
182.77 1.00
U i;min ≤ U i ≤ U i;max xijk;min ≤ xijkt ≤ xijk;max S ijk;max ≤ S ijkt ≤ S ijk;max Pi;min ≤ Pit ≤Pi;max
0.08 0.29
i∈M ; j∈N ; k∈K; t∈T
ð31Þ ð32Þ
i∈M i∈M ; j∈N ; k∈K; t∈T
ð33Þ
i∈M ; j∈N ; k∈K; t∈T
ð34Þ
i∈M ; t∈T
MTBM i;min ≤ MTBM it ≤ MTBM i;max
ð35Þ i∈M ; t∈T
ð36Þ
220.15 1.00
where SS represents sum of square deviation from mean value.
Solution Methods M
N
K
mmin ≤ ∑ ∑ ∑
sijkt xijkt þ sijkt xijkt ≤ mmax
ð26Þ
i¼1 j¼1 k¼1 T
T
t¼1
t¼1
∑ Pði−1Þt − ∑ Pit ≤ 0
ð27Þ
i∈M
The duration in which a particular machine is used during production has a direct relationship with the amounts of wear and tear the machine experience. This has a direct relationship with equipment MTBM. To address this problem, the average time for equipment MTBM is expected to be within an acceptable limit (Eq. 28). M
∑ MTBM it i¼1
≤ MTBM ave
M
ð28Þ
∀t∈T
BB-BC Algorithm
where MTBMave represents the average MTBM value. In order to control workers activities, decision makers often specify the minimum amount of time that workers are expected to put-in for an activity (Eq. 29). The average break period for the maintenance-production workers is given as Eq. (30). The Eq. (30) measures the average break period by considering the total number of workers’ groups (i.e., the denominator). The Eq. (30) is valid when different maintenance groups are assigned to a machine. i N K h ∑ ∑ xijkt sijkt −bijkt þ xijkt sijkt −bijkt ≤ S it
ð29Þ
The evolution of BB-BC algorithm is due to the classical pioneering work by Osman and Eksin (2006). Successful applications of the BB-BC algorithm have been documented on voltage stability, damage detection, software development and economic dispatch (Erol and Eksin 2006; Labbi and Attous 2010; Kripka and Kripka 2008; Azad et al., 2011; Prudhvi 2011; Sakthivel and Mary 2013; Tabrizian et al. 2013). This algorithm was inspired by the theories of BB-BC, which guided the earth’s evolution. The idea of BB-BC hinges on the interplay among gravitational energy, kinetic energy, internal forces, distance moved by particles and the final collapse. Kripka and Kripka (2008) as well as Prudhvi (2011) reported
ð30Þ
Table 7 Contribution of rest periods with respect to achieved machine availability
∀i∈M ; ∀t∈T
j¼1 k¼1
M
N
Since linearisation of a model leads to increase in a solution method computational time and computer space, this study did not linearised the proposed model (Wu 2008). Our focus is on the selection of a meta-heuristic for the proposed model. Meta-heuristics are selected because of the complex interrelations among the maintenance-production variables (Fakher et al. 2017). GA and BB-BC algorithm are considered as solution methods for the proposed model. The reason for GA selection is based on its capacity to generate promising solutions for maintenance-production problems (Fakher et al. 2017). BB-BC algorithm is selected due to its low computational time and competitive solution attributes (Labbi and Attous 2010).
K
M
N
K
∑ ∑ ∑ bijkt
∑ ∑ ∑ bijkt
i¼1 j¼1 k¼1
i¼1 j¼1 k¼1
MNK
þ
MNK
¼ bave
∀t∈T
Where Sit represents the total manufacturing time for machine i at period t, and bave represents average break period. Variables Limits The limits for the decision variables in the proposed model are expressed as Eqs. 31, 32, 33, 34, 35 and 36, while the proposed optimisation model is presented in
Category of maintenance workers
SS
Contribution
Mechanical (regular) Electrical (regular) Mechanical (trainees) Electrical (trainees) Total
0.51 0.26 0.94 1.70 3.41
0.15 0.08 0.27 0.50 1.00
Process Integr Optim Sustain (2017) 1:251–267
259 14600 Amount of products (Dozen)
Fig. 3 Average inventory level in the different periods
14400 14200 14000 13800
Case 1
13600
Case 2
13400 13200 13000 12800 Plan 1
Plan 2
Plan 3
Plan 4
Plan
that as soon as particle spread began, gravitational forces arose, which hinged on the masses of two bodies as well as the distance flanked by them. With this expansion, a decrease of each particle’s gravitational force is experienced. This makes the kinetic energy of growth to dissipate quickly. The kinetic energy is conquered by the gravitational energy concerning particles; this makes a particle to shrink. At this level, every particle caves into a sole particle identified as big crunch. This concept forms the theoretical background of a BB-BC algorithm that uses two common approaches to generate decision variables values (Eqs. 37 and 38). These equations are combined with the concept of center-of-mass in generating optimal values for different systems (Rao and Yesuratnam 2012; Tabrizian et al. 2013). The centre-of-mass for decision variables are either taken as a function of the current decision variable value (Eq. 40) and their fitness or as the global solution decision variables. min υpg ¼ υg−1 þ α υmax ð37Þ g −υg min υcg þ rα υmax g −υg ð38Þ υpg ¼ gþ1
Period Case Month 1
2 3 4
υcg ¼
p¼1
ð39Þ P
∑ 1= f pðg−1Þ
p¼1
where α represents a constant parameter, r is a random variable within the range of − 1 and 1, g, p p represent iteration step or generation, particle and total number of particles, respectively, υpg and υcg represent particle p value for a decision variable and the center-of-mass of a decision variable at iteration or generation g, υmax and υmax represent the maximum g g and minimum values of a decision variable, and f pðgÞ represents the fitness value of particle p at iteration step or generation g. Genetic Algorithm Over the years, the potential of GA as a solution method for nonlinear optimisation models has received significant attention from researchers (Berrichi et al., 2001; Sanchez et al. 2009; Ighravwe et al. 2015). Its implementation procedures are similar to other evolutionary algorithms (such as differential evolution). These procedures are mutation, reproduction and selection operations (Michalewicz, 1996; Engelbrecht 2007; Jebari and Madiafi 2013):
Sum of average production time
Table 8
1
P
∑ υpðg−1Þ =f pðg−1Þ
1 2 1 2 1 2 1 2
34.260 33.340 33.420 33.440 33.400 33.260 33.280 33.420
Month 2 32.260 32.340 32.420 32.440 32.400 32.260 32.280 32.440
Month 3 32.260 32.340 32.420 32.440 32.400 32.260 32.280 32.440
Month 4 32.260 32.340 32.420 32.440 34.400 32.260 32.280 32.440
Month 5 36.260 36.340 36.420 36.440 36.400 36.260 36.280 36.440
Month 6 33.260 33.340 33.420 33.440 33.400 33.260 33.280 33.440
Mutation Operation This operation is used to improve the genetic characteristics of an individual in a population based on a mutation rate. A mutation rate helps to minimise the distortion of individuals with highly fit characteristics (Engelbrecht 2007). Michalewicz (1996) presented an expres0 sion for generating a mutant vector (x1g ) as Eq. (40).
0
x1gþ1 ¼
8 < x1g þ Δ g; xmax −x1g :
x1g þ Δ g; x1g −xmin
Head ð40Þ Tail
260
Process Integr Optim Sustain (2017) 1:251–267
Reproduction Operation This operation is used to generate new off-springs for a particular generation. Based on GA literature, two parents can be used to create an off-spring. Wright (1991) reported that the value of an off-spring is given as Equation (Eq. 41). This expression is subject to Eq. (42). 0 0 0 x″g ¼ U ð0; 1Þ x2g −x1g þ x2g ð41Þ 0
0
f 1g > f 2g 0
ð42Þ 0
where x1g and x2g represents a mutant vector for the first and second randomly elected parents, respectively at generation g 0 0 and f 1g and f 2g represent the fitness of the first and second randomly elected parents, respectively at generation g. Selection Operation This operation is used to build a reproduction pool by selecting the fitness individuals (solutions). This is achieved by comparing parents and off-springs solutions (Engelbrecht 2007). Some frequently used selection methods are the roulette wheel, stochastic sampling, linear rank, exponential rank, hall of fame and tournament selections (Jebari and Madiafi, 2013).
Problem Description and Model Application The proposed model was applied in a manufacturing company which specialises in the production of household utensils in Nigeria. The utensils industry, which is an important aspect of the metal industry, labelled as non-ferrous, has a huge contribution to Nigeria’s economy yearly. Generally, the economic activities of the industry entail the sales of commodities such as stockpots, saucepans, soup ladles, pots, skimmer and stock pot, cutlery, kitchen kettles and frying pans. The manufacturing company market is so established that it produces and sells a broad range of SMEs (small and medium scale enterprises) the frying pan, saucepans and kitchen kettles at both the local and international markets, in the neighbouring West-African states. The selected company products profile excludes soup ladles, skimmer and stock pot. The company’s main source of revenue is the sales of frying pans and kitchen kettles. The company produces household utensils using approximately 16,000 t of molten metal annually. The rest periods for workers in the company lies between 15 and 20 min per shift while their busy period is approximately 94% of the allocated time in a particular period. The number of working hours per day for regular workers is between 8 and 10 h per shift while the minimum working day in a month is 26 days. This information was combined with the information in Table 9 during the implementation of the proposed model. The company’s maintenance-production maximum and minimum workforce size is 33 and 78 workers, respectively. The ratio of mechanical and electrical maintenance activities is approximately
60:40. The maintenance department is expected to carry out preventive and corrective maintenance between every 15 to 30 min, while the average MTBM among the machines is approximately 20 min. The arrival rate of number of machine breakdown follows a Poisson distribution. Breakdown maintenance occurs at least once between 48 to 60 h during active production time. To obtain a realistic result, we select 48 h as the arrival time for machine breakdown in the system and fixed the maximum repair time as 1 h. A two-machine case with a production ratio of 60:40 and a four-planning period was considered. Each planning period contains 6 months. The application of the proposed model under rest period is considered as case 1, while without rest period is considered as case 2. This study parametric setting for the BB-BC algorithm and GA is presented in Table 10. Generation of Trade-off Solutions Multi-objective handling methods such as ɛ-constraint method, weighting sum method, multi-attribute ability theory and goal programming have been used to generate compromise solutions for multi-objective models. Due to goal programming ease of understanding and implementation, it has gained wide acceptance than other multi-objective handling methods (Wu, 2008). Thus, it is selected in the current study. This method entails optimising each objective one after the other while converting other objectives as soft constraints (Wu, 2008). The BB-BC algorithm was used to design a payoff table for cases 1 and 2 (Table 11). Based on the information in Appendix 1, the objective functions in the proposed model were converted into a single objective function. For example, the single objective function and soft constraints for case 1—period 1 is given as Eqs. (43), (44), (45), (46) and (47). Equal weights were assumed for the objective functions (i.e., wq = 0.25). At this stage, the performance of the BB-BC algorithm was compared with GA (Table 1), the results obtained showed that the BB-BC algorithm performed better than the GA (Fig. 4). It should be noted that the computational time in Table 1 depends on a computer system’s configuration and a programmer skill. The compromise solutions generated by the BB-BC algorithm are presented in Tables 2, 3, 4 and 5. f ¼
w1 d þ w2 d þ w3 d þ 1 2 3 þ þ 878; 085:88 323; 640:32 64; 135; 758:21 þ
w4 d −4 0:04
ð43Þ
f 1 −d þ 1 ¼ 2; 896; 945:50
ð44Þ
f 2 −d þ 2 ¼ 214; 676:18
ð45Þ
f 3 −d þ 3 ¼ 106; 439; 736:69
ð46Þ
Process Integr Optim Sustain (2017) 1:251–267
f 4 þ d −4 ¼ 0:579 dþ q
261
ð47Þ
d −q
and represents the positive and negative deviawhere tion variables of the qthobjective function, and wq represents the relative importance of the qth objective function. Workforce Cost The total workforce cost for the four periods using case 1 approach is less than that of case 2 approach (Table 2).
decision makers. Based on the analysis of variance of the results in Table 13, regular electrical maintenance workers contribute more to achieved machine availability for cases 1 and 2 (Table 6). The regular mechanical workers and the trainee electrical maintenance workers contribution to achieved machine availability for case 1 and 2 were second, respectively (Table 6). From Table 7, the trainee maintenance workers’ rest periods (77%) contributes more to achieved machine availability than regular maintenance workers’ rest periods (23%).
Discussion of Results
Inventory Cost
The results in Tables 2, 3, 4 and 5 are within close ranges from one another across the different periods and cases. Thus, it can be deduced that the compromise solutions generated using the BB-BC algorithm are satisfactory.
Apart from period 3 which had inventory and machine usage cost for case 2 that was less than that of case 1, other periods had higher values. The expected average inventory at four planning periods (plans) for cases 1 and 2 are shown in Fig. 3. This result showed that the company is expected to pay more for inventory holding costs at period 2 based on case 1. At period 4—case 1 and 4—case 2 generated the lowest and highest amount of average inventory, respectively (Table 3). In terms of the total average inventory produced, periods 4 and 2 had the lowest and highest values, respectively (Table 3). Pairwise comparisons of the average inventory values in Fig. 3 show that using case 2 in designing the company’s inventory system will result in more inventories than case 1 in the company. To establish whether there are statistical differences in the mean values of inventory between the two cases at 95% significance level, t test was carried out using the information (Table 3). Before we proceed to evaluate the differences in the mean values, we determine whether the variances between the two cases are equal or not using F test. A calculated F value of 1.023 was obtained as against a critical F value as 2.014. This showed that the difference between the variances is statistically significant. Two-sample of unequal variances t test was used to analyse the differences between means. A t calculated of − 0.367 was obtained. This value is less than the t table value (2.014). Thus, it can be inferred statistically that there is no difference between the two cases mean values. It may suffice to say that application of the proposed model with rest periods produced practicable results that were similar a condition when rest periods is not considered by decision makers.
Workforce Size and Cost Based on cases 1 and 2 results, the average workforce size per period is 456 and 459 workers, respectively. The workforce size period-wise comparison showed that case 1 generated less number of total workers for all the period (Table 2). The combination of cases 1 and 2 workforce size results showed that the average workforce size per period is 478 workers. The total number of regular workforce size generated for periods 1, 2 and 4 using case 1 were less than that of case 2. At period 3, case 2 generated less number of total regular workers than case 1 (Table 12). The total number of trainees in period 1 using cases 1 and 2 were the same, while case 1 generates less number of total trainees for periods 2 and 4 than case 2. At period 3, case 2 generated a lesser number of total trainees than case 1 (Table 12). The average trainees’ size from cases 1 and 2 for all the periods are equal, while period 3 average regular workers size from cases 1 and 2 are also equal (Table 12). The average regular and trainee workers required by the system from cases 1 and 2 results were the same (Table 12). Cases 1 and 2 results showed that the average workforce cost per period were N 3092,600:00 and N 3178,784:00, respectively. The combined values of the average workforce cost from cases 1 and 2 was of N 3,158,233.97. Service Time, Rest Period and Achieved Machine Availability The incorporation of rest periods improved the achieved machine availability when compared with the application of the proposed model without rest periods provision. For cases 1 and 2, the average machine availability was 58.3 and 58.48%, respectively. This implies that creating allowances for rest periods during the design of maintenance-production workforce period will enhance the performance of workers against an instance where rest period provision is not considered by
Production Time and Defective Products Case 1 total value for rework and scrap costs (N 501,401.11) is less than that of case 2 (N 505,685.44). Apart from period 3, rework and scrap usage costs under case 2, other periods had higher values when compared with case 1 results. Although, the average monthly production time of the machines varies from one period to another (Table 4), the sum of the average
262
Process Integr Optim Sustain (2017) 1:251–267
production time of the machines followed the same pattern except for the first period (Table 8). Period 1—case 1 and period 3—case 2 had the same average production time (16.63 h). Case 2 average production time for periods 2 and 4 were the same (16.72 h). From Table 5, the average amount of defective products expected from machines 1 and 2 are 0.57 and 0.58%, respectively. These values suggest that more inspections should be given to machine 2.
The Implications of Workforce Planning Research in Practice This research has extended the frontier of knowledge with respect to developing a model that addresses maintenanceproduction problems in terms of workers’ cost, service time and rest periods. A case analysis has been reported to establish the validity of the suggested procedures in a utensil products manufacturing company. This served as a mark of the relevance of the model in practice. For progress and satisfaction in implementing the steps and model in industry, it is essential to attain full support of the management (plant manager, engineers and supervisors). Also, staff that is directly controlling the floor operations needs to be committed to have worthwhile results. Arising from the presented investigation is a number of issues (implications) to the practicing plant manager and engineers. The following are the necessary insights: &
&
The task of maintenance-production workforce planning requires skill and care from those concerned. Thus, scientific instruments such as the one proposed here could help to obtain timely and accurate results. This will help to reduce the possibility of a wrong decision on hiring and firing. Thus, the current research aids the industrial manager in the determination of the proper mix of maintenance-production workers for the upkeep of a plant. This propels the progress of workforce planning. Furthermore, the research documented here puts more energy into the industrial managers, by revealing the projected benefits of the plan in terms of economic advantages. This will serves as a motivator for the plant manager and other top management staff to strengthen their commitment to the scientific tool implementation in their organisation. This study results showed what class of workforce and section (production or maintenance) financial and nonfinancial resources should be diverted to in order to have the most enhanced operations in the utensils company.
The proposed model has the following limitations: The quality of solutions from the proposed model depends on the weights that are assigned to the goals in model and solution method. Also, the proposed model cannot be used to assign
task. The model does not consider the impact of training on workers’ performance.
Conclusions This study has developed a nonlinear multi-objective model for maintenance-production workforce problems in manufacturing systems. The applicability of the proposed model with and without provisions for rest periods was demonstrated using the combination of real and simulated datasets. This offered us the opportunity to test the applicability of the BB-BC algorithm and GA as solution methods for the proposed model. From the results obtained, the proposed model has the potential of optimising workforce size, rest periods, number of defective products and production time. Furthermore, there were variations in the model’s results under rest and without rest periods as well as among the planning periods. In addition, the BB-BC algorithm results were better than that of the GA results in terms computational time and solution quality. Thus, we deduced that BB-BC algorithm is a suitable solution method for the maintenance-production model. From the proposed model application, the following salient points emerge: The model without rest period consideration generates high values for average achieved machine availability when compared with the proposed model results with rest period consideration (see Table 1); the maximum average achieved machine availability across the different periods were close (see Table 4); and the model’s results period-wise were relatively stable and the results obtained were satisfactory. The maintenance-production workforce problem in manufacturing systems has several areas that require further investigation. There is the need to investigate the performance of other meta-heuristics such as bacterial forage algorithm as solution method for this problem. Study on maintenanceproduction predictive models evaluation is required.
Appendix 1 Proposed model Objective functions. Min f 1 : C 1 ½ðτ− f Þ=ð1 þ f Þ Min f 2 ¼ ½ðτ− f Þ=ð1 þ f ÞC 2 þ C 2 Min f 3 ¼ ½ðτ−f Þ=ð1 þ f ÞC 3 þ C 3 M
T
MTBM it i¼1 t¼1 MAMT
Max f 4 ¼ 1− ∑ ∑
Process Integr Optim Sustain (2017) 1:251–267
263 Table 9 (continued)
Subject to m
I t−1 þ
∑ ð1−d i þ d i Ri Þ U i Pit − I t −Dt ¼ 0 ∀t∈T
S/N
Maintenance time (h)
Production time (h)
Demand (dozen) *100
7
1426.13
2852.27
1160
8 9
1595.40 5291.73
3194.80 10,483.47
1404 1362
10 11
4095.00 3579.80
8190.00 7159.60
741 1039
12 Period 3
3010.00
6020.00
755
13 14
3793.39 2227.20
7586.78 4454.40
940 1337
15
1519.60
3039.20
1355
16 17
2165.15 1821.40
4330.29 3642.80
1244 1509
i¼1
I1 ¼ IT
Period 2
T
∑ I t =T ¼ INV avrage t¼1 N
K
∑ ∑
N
K
xiþ1jkt þ xiþ1jkt −ϕ ∑ ∑
j¼1 k¼1
xijkt þ xijkt ≤0∀i∈M ; ∀t∈T
j¼1 k¼1
M
N
K
xmin ≤ ∑ ∑ ∑
xijkt þ xijkt ≤ xmax
∀t∈T
i¼1 j¼1 k¼1
xijkt ¼ 1−rjk xijkt‐1 þ xijkt‐1 M
∀i∈M ; ∀ j∈N ; ∀k∈K; ∀t∈T
si2kt xi2kt þ si2kt xi2kt ≤ t 1 þ t 2
K
∑ ∑
∀t∈T
i¼2 k¼1 M
K
∑ ∑
M si1kt xi1kt þ si1kt xi1kt ≤ ∑ ð1−d i þ d i Ri ÞU i Pit ∀t∈T
i¼1 k¼1
i¼1
M
N
K
mmin ≤ ∑ ∑ ∑
sijkt xijkt þ sijkt xijkt ≤ mmax
i¼1 j¼1 k¼1 T
T
t¼1
t¼1
∑ Pi−1t − ∑ Pit ≤ 0 M
∑ MTBM it i¼1
≤ MTBM ave ∀t∈T M i N K h ∑ ∑ xijkt sijkt −bijkt þ xijkt sijkt −bijkt ≤S it ∀i∈M ; ∀t∈T
18
1580.74
3161.48
1110
Period 4 19 20 21 22 23
1225.00 2058.27 1472.03 3296.67 5407.6
2450.00 4116.53 2944.07 6593.33 10815.2
1202 1215 1195 1550 1173
24
3354.95
3554.95
1129
Asterisks denote multiplication
Appendix 3
j¼1 k¼1
M
N
K
M
∑ ∑ ∑ bijkt i¼1 j¼1 k¼1
þ
MNK
N
K
∑ ∑ ∑ bijkt i¼1 j¼1 k¼1
MNK
¼ bave
Table 10
∀t∈T
BB-BC algorithm
Eqs. (30) to (36).
r α Population size Maximum iteration
Appendix 2
Table 9 S/N
Period 1 1 2 3 4 5 6
BB-BC algorithm and GA parametric settings
Real life data for manufacturing variables Maintenance time (h)
Production time (h)
Demand (dozen) *100
1319.38 2418.45 4735.00 2177.33 929.19 1931.67
2638.77 4835.20 9470.40 4354.67 1858.39 3863.33
775 775 1026 1334 864 1891
GA ±1 0.01 30 100
Mutation rate Reproduction rate Population size Maximum iteration
0.5 0.4 30 100
264
Process Integr Optim Sustain (2017) 1:251–267
Appendix 4 Payoff table for different periods and cases
Table 11 Period
Case
1
1
2
2
1
2
f1 f2 f3 f4 Solution space f1 f2 f3 f4 Solution space f1 f2 f3 f4 Solution space f1 f2 f3 f4
3
1
2
4
1
2
Solution space f1 f2 f3 f4 Solution space f1 f2 f3 f4 Solution space f1 f2 f3 f4 Solution space f1 f2 f3 f4 Solution space
F1
f2
f3
f4
3,090,000.00 3,775,031.38 2,896,945.50 3,111, 731.00
431,873.00 538,316.50 214,676.18 493,671.77
106,439,736.69 126,859,573.00 170,575,494.90 106,982,285.49
0.576 0.536 0.579 0.575
2,896, 945.50–3,775,031.38 3,090,000.00 3,774, 911.12 2,887,254.25 3,082,908.75
214,676.18–538,316.50 430,577.54 538,316.50 214,594.19 486,470.06
106,439,736.69–170,575, 494.90 106,267,853.04 126,848,334.50 170,560,448.14 106,140,856.80
0.536–0.579 0.576 0.538 0.579 0.576
2,887,254.25–3,774,911.12 3,082,908.75 3,774,924.57 2,906,385.25 3,139,036.75 2,906,385.25–3,774,924.57 3,092,260.00 3,774,940.35 2,906,385.25
214,594.19–538,316.50 432,809.45 538,316.50 217,330.19 499,977.77 217,330.19–538,316.50 430,023.79 538,316.50 216,759.50
106,140,856.80–170,560,448.14 105,899,429.15 126,849,592.05 171,060,790.56 107,713,138.98 105,899,429.15–171,060,790.56 106,194,276.72 126,851,066.20 170,956,725.63
0.538–0.579 0.577 0.536 0.578 0.573 0.536–0.578 0.576 0.536 0.578
3,178,047.75 2,906,385.25–3,774,940.35 3,102,039.75 3,775,005.12 2,906,385.25 3,111,731.0 2,906,385.25–3,775,005.12 3,073,469.00 3,764,489.59 2,906,385.25 3,111,731.00 2,906,385.25–3,764,489.59 3,121,422.25 3,786,025.88 2,906,385.25 3,111,731.00 2,906,385.25–3,786,025.88 3,020,488.25
514,390.93 216,759.50–538,316.50 433,658.64 538,316.50 219,810.52 493,671.77 219,810.52–538,316.50 419,363.57 511,443.25 219,772.53 495,249.96 219,772.53–511,443.25 438,821.35 565,318.50 219,649.00 493,669.89 219,649.00–565,318.50 469,382.72
109,359,987.63 109,359,987.63–170,956,725.63 106,676,096.06 126,857,119.25 171,511,263.20 106,982,285.49 106,676,096.06–171,511,263.20 104,765,437.90 125,869,301.42 171,504, 387.50 107,166,072.26 104,765,437.90–171,504,387.50 107,355,849.60 127,881,514.79 171,426,703.87 106,982,465.42 106,982,465.42–171,426,703.87 111,274,602.35
0.570 0.536–0.578 0.575 0.536 0.577 0.575 0.536–0.577 0.579 0.537 0.577 0.574 0.537–0.579 0.574 0.534 0.577 0.575 0.534–0.577 0.566
3,889,008.02 2,906,385.25 3186,222.5 2,906,385.25–,889,008.02
828,958.25 219,962.87 516,847.86 219,962.87–828, 958.25
136,963,598.27 171,538,838.25 109,637,643.21 109,637,643.21–171,538,838.25
0.518 0.577 0.569 0.518–0.577
Process Integr Optim Sustain (2017) 1:251–267
265
Appendix 5 8.00E+15
Fig. 4 Convergence plot for the solution methods
BB-BC GA
Fitness value
7.50E+15 7.00E+15 6.50E+15 6.00E+15 5.50E+15 5.00E+15 1
7
13
19 25
31
37 43
49 55
61 67
73 79
85 91
97
Iteration steps
Appendix 6 Table 12 Period
Workforce distribution under different periods and cases Case 1
Case 2
Regular
1
2
3
4
Trainees
PRD
MEC
ELE
10
12
34 37 41 27
13 15 18 16
40 11
Regular
PRD
MEC
ELE
13
7
6
13 10 8 11
9 9 9 8
4 3 3 6
16 12
13 13
8 7
35 37 42
14 15 18
13 10 9
29 40 11 35 39 42 29 40 10
16 16 12 14 15 18 16 16 12
34 37 42 27 40
13 15 18 16 16
Trainees
PRD
MEC
ELE
PRD
MEC
ELE
4
11
12
4 5 6 5
35 38 42 27
14 15 18 16
13
7
6
4
13 10 8 11
9 9 9 8
4 3 3 6
4 5 6 5
5 6
2 4
40 11
16 13
13 13
8 7
5 6
2 4
9 9 10
4 4 3
4 5 6
35 38 42
14 15 18
13 10 9
9 10 10
4 4 3
4 5 6
11 13 13 13 10 9 11 13 13
9 8 7 9 9 10 8 8 7
6 5 6 4 4 3 6 5 6
5 3 4 4 5 6 5 2 4
29 40 10 34 37 42 27 40 11
16 16 12 13 15 18 16 16 13
11 13 13 13 10 8 11 13 13
9 8 7 9 9 9 8 8 7
6 5 6 4 3 3 6 5 6
5 3 4 4 5 6 5 2 4
13 10 8 11 13
9 9 9 8 8
4 3 3 6 5
4 5 6 5 2
34 39 42 29 40
14 15 18 16 16
13 10 9 11 13
9 10 10 9 8
4 4 3 6 5
4 5 6 5 3
266
Process Integr Optim Sustain (2017) 1:251–267
Appendix 7 Table 13 Period
Average monthly time required from workers per day (h) Case 1
Case 2
Regular
1
2
3
4
Trainees
Regular
Trainees
PRD
MEC
ELE
PRD
MEC
ELE
PRD
MEC
ELE
PRD
MEC
ELE
7.99
8.02
8.48
7.3
7.23
7.08
8.01
8.05
8.5
7.23
7.25
7.11
8.09
8.85
7.74
6.8
6.9
6.58
8.1
7.88
7.74
6.82
6.94
6.6
7.69 7.5
8.15 8.28
7.55 8.43
7.35 6.95
7.07 6.99
7.1 7.23
7.71 7.52
8.17 8.3
7.57 8.46
7.37 6.97
7.09 7.01
7.12 7.25
8.01
7.64
8.73
7.24
6.88
6.72
8.04
7.67
8.75
7.27
6.9
6.74
8.08 0.8
7.78 8.03
8.12 8.06
7.36 7.34
7.36 7.27
6.78 7.13
8.1 8.04
7.8 8.7
8.14 8.52
7.38 7.34
7.38 7.28
8.14 7.13
8.51
8.12
7.9
6.84
7
6.62
8.12
7.9
7.79
6.84
6.96
6.62
7.72 7.54
8.19 8.32
7.6 8.46
7.4 6.99
7.11 7.04
7.14 7.24
7.73 7.55
8.2 8.32
7.6 8.48
7.4 6.99
7.12 7.04
7.14 7.27
8.06 8.12
7.69 7.82
8.77 8.16
7.29 7.4
6.93 7.4
6.76 6.83
8.06 8.13
7.69 7.82
8.77 8.17
7.29 7.4
6.93 7.4
6.76 6.83
8.03 8.12 7.72 7.54
8.06 7.89 8.19 8.31
8.51 7.78 7.79 8.47
7.33 6.84 7.39 6.99
7.27 6.95 7.11 7.03
7.12 6.61 7.14 7.26
8 8.08 7.69 7.5
8.03 7.86 8.16 8.28
8.48 7.75 7.56 8.44
7.3 6.8 7.36 6.95
7.23 6.92 7.08 7
7.09 6.58 7.1 7.23
8.05 8.12 8 8.08 7.69
7.69 7.82 8.03 7.86 8.16
8.77 8.16 8.48 7.75 7.56
7.28 7.4 7.3 6.81 7.36
6.92 7.4 7.23 6.92 7.08
6.76 6.82 7.09 6.58 7.1
8.02 8.08 8.04 8.13 7.73
7.65 7.78 8.07 7.9 8.2
8.73 8.12 8.52 7.79 7.6
7.25 7.36 7.34 6.84 7.4
6.89 7.36 7.27 6.94 7.12
6.72 6.78 7.13 6.62 7.14
7.51 8.02
8.28 7.65
8.44 8.7
6.96 7.25
7 6.89
7.23 6.72
7.55 8.06
8.32 7.69
8.48 8.77
6.99 7.29
7.04 6.93
7.27 6.76
8.09
7.79
8.13
7.39
7.39
6.79
8.13
7.84
8.17
7.4
7.4
6.83
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