Robust optimization of a mathematical model to design a dynamic cell formation problem considering labor utilization
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Abstract
Cell formation (CF) problem is one of the most important decision problems in designing a cellular manufacturing system includes grouping machines into machine cells and parts into part families. Several factors should be considered in a cell formation problem. In this work, robust optimization of a mathematical model of a dynamic cell formation problem integrating CF, production planning and worker assignment is implemented with uncertain scenariobased data. The robust approach is used to reduce the effects of fluctuations of the uncertain parameters with regards to all possible future scenarios. In this research, miscellaneous cost parameters of the cell formation and demand fluctuations are subject to uncertainty and a mixedinteger nonlinear programming model is developed to formulate the related robust dynamic cell formation problem. The objective function seeks to minimize total costs including machine constant, machine procurement, machine relocation, machine operation, intercell and intracell movement, overtime, shifting labors between cells and inventory holding. Finally, a case study is carried out to display the robustness and effectiveness of the proposed model. The tradeoff between solution robustness and model robustness is also analyzed in the obtained results.
Keywords
Dynamic cell formation problem Scenariobased robust optimization Mixedinteger nonlinear model Worker assignmentIntroduction
Today, global competitive environment has persuaded manufacturing practitioners to deliver lowcost and highquality products. Some recently applied approaches have been put into practice to cope with the ever growing manufacturing costs, such as location, material handling system, and energy. One of these recent manufacturing approaches is Group Technology (GT). GT is one of the main building blocks to implementing JustInTime (JIT) philosophy. This approach is based upon grouping parts and machines together with respect to their similarities in production processes, functionalities, etc. The aspect of GT which associates with the configuration of manufacturing firms is cellular manufacturing system (CMS). The most outstanding benefit of CMS can be noted as reduction in some production factors, such as lot sizes, lead times, workinprocess inventories and setups, while higher level of investment is inevitable to implement this system. Designing of a CMS involves four main steps. The first step associates with cell formation problem which comprises assigning parts to their families and machines to their corresponding machine cells based on some features, such as similar geometric design or processing requirements. Second, intracell and intercell layouts are defined through Group Layout (GL). This step determines the location of machines and cells in the shop floor. Third, Group Scheduling (GS) is accomplished to schedule parts within part families. Finally, required resources such as labors and material handling devices are assigned to the manufacturing cells.
It has been clarified by Wu et al. (2007) that these four steps are interrelated and in other words, the solution for each step influences the other one. Thus, simultaneously solving these problems has to be applied by the researchers; that is, the matter not been paid attention enough. Nevertheless, due to the complexity and NPcomplete nature of CF, GL, and GS decisions, most researchers have addressed two or three decisions sequentially or independently. However, the benefits gained from CMS implementation are highly affected by how theses stages of the CMS design have been performed in collaboration with each other.
Shorter product life cycles are an increasingly significant issue in CM. As a result, neglecting new products emerging at future imposes subsequent unplanned changes to the CMS design and causes production disruptions and unexpected costs. Hence, those changes should be incorporated in the design process. To come up with a solution to handle those changes, the dynamic cellular manufacturing system (DCMS) was introduced in which it is assumed that the product mix or volume changes of demands can be predicted in a multiperiod planning horizon (Rheault et al. 1995).
Most DCMS models assume that the input parameters are deterministic and certain. However, in practical situations many parameters are uncertain and imprecise. DCMS design has to be implemented in many environments based on some parameters with uncertain values. However, there are few studies on designing cellular manufacturing systems under dynamic and uncertain conditions. These studies can be divided into four classes as fuzzy programming approach, stochastic programming approach, scenariobased programming approach, and robust optimization approach in terms of uncertainty expression type in the problem. Different robust optimization approaches have been introduced in the recent years to deal with the uncertainty of the data. In this study, a scenariobased robust optimization approach is used to cope with uncertainty and to find a solution that is robust with regard to data uncertainties in part demand, intercell and intracell movement cost, machine purchase cost, selling machine revenue, machine fixed/variable cost, machine relocation cost, intercell movement labor cost, process variable cost and inventory holding cost. It is the first time that this vast coverage of input parameters in a DCMS are considered uncertain to be handled by a robust optimization approach.
The aims of this study are twofold. The first one is to formulate a new mathematical model with an extensive coverage of important manufacturing features including batch intracell/intercell movement, production planning strategies (i.e., internal production, inventory holding, and lost sale as underfulfilled demand), selling/purchase machine, labor movement, labor assignment, labor capacity, machine relocation, regular/overtime machine capacity, cell size limit, flexible operation sequence, machine/labor processing time, and uncertain scenariobased parameters (i.e., part demand and miscellaneous costs). The second aim is to develop a robust model based on the deterministic proposed model using scenariobased robust optimization approach. The important concern of the employed robust methodology is to obtain an optimal CM design that is robust with regard to data uncertainties in part demand and miscellaneous costs. The objective function of the integrated model is to minimize the total costs of machine constant, machine procurement, machine relocation, machine operation, intercell and intracell movement, overtime, shifting labors between cells and inventory holding. The main constraints are operatormachinecell assignment, machine capacity, machine number equilibrium, labor capacity, cell size limit, and balancing inventory.
Recently, Kia et al. (2012) have formulated a mathematical model integrating the CF and GL decisions in a dynamic environment by considering some advantages including: (1) considering flexible configuration of cells, (2) calculating relocation cost based on the locations assigned to machines, (3) distancebased calculation of intra and intercell material handling costs and (4) considering multirows layout of equal sized facilities. One disadvantage in their work was ignoring the assignment of operators to machines located in different cells. In another study, Bagheri and Bashiri (2014) investigated the simultaneous consideration of the cell formation problem with intercell layout and operator assignment problems in a dynamic environment by formulating a mathematical model with the objectives of minimization of inter–intra cell part trips, machine relocation cost and operatorrelated issues. A main drawback in both mentioned studies was that all parameters were considered deterministic despite the fact some of them should be predicted for the future periods in a dynamic environment with high level of uncertainty.
Generally, the presented study is an extension of the previous studies Kia et al. (2012), Bagheri and Bashiri (2014) by integrating the CF, production planning (PP) and worker assignment in a mathematical model with data uncertainties in most parameters of model including part demand and miscellaneous costs which is solved by a scenariobased robust optimization approach. The robust approach is used to reduce the effects of fluctuations of the uncertain parameters with regards to all possible future scenarios.
To investigate the effect of turbulence in the values of uncertain data on the model performance and obtained solutions, a robust model is developed. Then, a case study is carried out to demonstrate the validity of the employed robust approach and verify the integrated DCMS model. The obtained results of implementing the case study also illustrate the applicability of the proposed model in real industrial cases.
The remainder of this paper is organized as follows. In “Literature review” section, the literature review is carried out. The background of the robust optimization approach employed in this study is described in “Robust optimization” section. A mathematical model is formulated integrating CF, PP and worker assignment decisions in “Mathematical model and model description” section followed using some linearization procedures. In addition, a robust model is developed in this section. “A case study” section illustrates the case study that is implemented to investigate the features of the proposed model and assess the performance of the developed robust model. Finally, conclusion is given in “Conclusion” section.
Literature review
One of the most important issues which have received less attention in the literature body of DCMS is consideration of humanrelated issues. The first mathematical model developed for humanrelated aspects of DCMS was presented by Aryanezhad et al. (2009). They developed a new mathematical model to deal with DCMS and worker assignment problems, simultaneously. The objective function of this model contains system costs including machine purchase, operating, intercell material handling, machine relocation, worker hiring, training, salary and firing costs. Balakrishnan and Cheng (2005) presented a flexible framework for modeling cellular manufacturing when product demand changes during the planning horizon.
Most CMS models assume that the input parameters are deterministic and certain. However, in practical situations, many parameters such as parts demands, processing times and machines capacities are uncertain. Robust optimization as a strong technique was used to deal with uncertainty in the systems. Robust optimization can be very efficient and useful because of generation of the good and robust solutions for any possible occurrences of uncertain parameters (Mulvey et al. 1995). The concept of robust optimization in operation research was presented by Mulvey et al. (1995). They extended a robust counterpart approach with a nonlinear function that penalizes the constraint violations and addresses uncertainties via a set of discrete scenarios. Bai et al. (1997) demonstrated that the traditional stochastic linear program fails to determine a robust solution despite the presence of a cheap robust point. They evaluated properties of riskaverse utility functions in robust optimization. They discussed that a concave utility function should be incorporated in a model whenever the decision maker is risk averse. BenTal and Nemirovski (1998) proposed a robust optimization approach to formulate continuous uncertain parameters. BenTal and Nemirovski (1998), BenTal and Nemirovski (2002) and BenTal et al. (2002) developed robust theory of linear, quadratic and conic quadratic problems. Bertsimas and Sim (2002) and Bertsimas and Thiele (2003) proposed robust optimization methods for discrete optimization in continuous spaces.
Mirzapour AlEHashem et al. (2011) studied multisite aggregate production planning problems under uncertainty by defining multiobjective robust optimization models.
Mahdavi et al. (2010) proposed a mathematical model for solving dynamic cellular manufacturing problem considering two areas of cell configuration and assigning the operators to the machines. In the proposed model, some factors have been considered including machine capacity, multiperiod planning horizon and the worker idleness time. Rafiei and Ghodsi (2013) designed a twoobjective mathematical model for solving the operator assignment and cell configuration simultaneously. Minimizing total costs of machines purchase, machine relocation and overhead, parts intracell and intercell movements and the operator intercell movements were considered in the first objective function. The second objective function increased the utilization level of the operators.
In similar studies, Kia et al. (2013), Shirazi et al. (2014) presented multiobjective mixedinteger nonlinear programming models to combine the problems of dynamic cell formation and group layout. They utilized the multirow layout for locating machines inside the cells with flexible size regarding the lot splitting feature and several other features (i.e., operation sequence, processing time, machine duplicates, and machine capacity).
Bashiri and Bagheri (2013) proposed a twophase heuristic method for cell formation and operator assigning, where in the first phase, clustering technique and in the second phase, a mathematical model is used. Kia et al. (2011) presented a mathematical model for a multiperiod CM system layout with fuzzy parameters. By taking the linear intracell machines layout, operation sequence, processing times and the machines capacity into account, the model intended to minimize the intra/intercell movements costs, the machines overhead costs and machines relocation costs.
Ghezavati et al. (2011) proposed a robust model for cell formation and group scheduling with supply chain approach. In this model, the uncertainty resulted from demand and parts processing time were expressed by stochastic scenarios with given probabilities. They formulated the problem with the objective to minimize delaying costs for parts delivery due time, the parts outsourcing costs to suppliers and the underutilization cost of machines and solved it by a hybrid metaheuristic algorithm. Paydar et al. (2013) presented a mathematical model for integration of cell formation, machine layout and production planning. They considered customer demand and machine capacity uncertain and proposed a robust model. Forghani et al. (2012) suggested a robust model to determine cell formation and group layout where the parts demand is uncertain.
Sakhaii et al. (2015) developed a robust optimization approach for a new integrated MILP model to solve a DCMS with unreliable machines and a production planning problem simultaneously. They adopted a robust optimization approach immunized against even worstcase to cope with the parts processing time uncertainty. Hassannezhad et al. (2014) performed sensitivity analysis of modified selfadaptive differential evolution (MSDE) algorithm for basic parameters of cell formation problem. First, they presented a DCMS model. Then, two basic test CF problems were introduced to assess the performance of MSDE algorithm by diverse problems sizes.
Regarding this section, it could be concluded that no study has been done on simultaneous integrating of three problems as cell configuration, production planning and operator assigning so far with uncertainty considered in the most model parameters including part demands and cost parameters.
Robust optimization
Mathematical model and model description
In this section, a new mixedinteger nonlinear programming model of a DCMS integrating CF, PP and worker assignment is presented to minimize total costs including machine constant, machine procurement, machine relocation, machine operation, intercell and intracell movement, overtime, shifting labors between cells and inventory holding respecting to the following assumptions.
Assumptions
 1.
Each part type has several operations which must be processed according to their sequence data.
 2.
Process time and manual workload time required for performing operations of a part type on various machine types are known and deterministic.
 3.
Part demands in each period are uncertain and defined in scenarios.
 4.
Timecapacity in regular time and overtime for each machine type are known and deterministic over the planning horizon.
 5.
Purchasing price and revenue from selling of each machine type are uncertain.
 6.
Constant cost of each machine type is uncertain. It covers overall service and maintenance cost. It is burdened for each machine even when a machine is idle.
 7.
Variable cost of each machine type in regular time and overtime is uncertain. It covers operating cost depending on the workload allocated to the machine.
 8.
Holding inventory is allowed and its related cost is uncertain.
 9.
In each period, the number of cells and the maximum cell size is known.
 10.
All machine types are multipurpose. Therefore, each operation of each part can be processed by more than one machine which brings flexibility for processing routes. However, each operation is allowed to be assigned to only one machine. In addition, there is no changeover cost for performing different operations by a machine.
 11.
Total number of labors is constant for all periods. Firing and hiring are not allowed.
 12.
Relocation cost of each machine between cells and shifting cost of operators between cells during successive periods are uncertain.
 13.
Batch sizes are fixed for moving parts between and within cells during planning horizon. However, intercell and intracell batches have different sizes. It is supposed that intercell and intracell transferring of batches has uncertain costs.
Indices
 c

Index for cells (c = 1,…,C).
 m

Index for machine types (m = 1,…,M).
 p

Index for part types (p = 1,…,P).
 h

Index for time periods (h = 1,…,H).
 j

Index for operations of part p (j = 1,…,Op).
 s

Index for scenarios (s = 1,…,S).
Input parameters
 L

Total number of labors.
 \(D_{phs}\)

Demand for part p in period h under scenario s.
 \(\vartheta_{phs }\)

1 if part p is planned to be produced in period h under scenario s; 0 otherwise.
 \(B_{p}^{\text{inter}}\)

Batch size for intercell movements of part p.
 \(B_{p}^{\text{intra}}\)

Batch size for intracell movements of part p.
 \(\gamma_{s}^{\text{inter}}\)

Intercell movement cost per batch under scenario s.
 \(\gamma _{s}^{\text{intra}}\)

Intracell movement cost per batch under scenario s. For justification of CMS, it is assumed that (\(\gamma_{s}^{\text{intra}}\) /\(B_{p}^{\text{intra}}\)) < (\(\gamma_{s}^{\text{inter}}\)/\(B_{p}^{\text{inter}}\)).
 \(\varphi_{ms}\)

Marginal cost to purchase machine type m under scenario s.
 \(h_{phs}\)

Inventory cost for holding part p at the end of period h under scenario s.
 \(W_{ms}\)

Marginal revenue from selling machine type m under scenario s.
 \(\alpha_{ms}\)

Constant cost of machine type m in each period under scenario s.
 \(\rho_{hs}\)

Constant cost of intercell labor movement in period h under scenario s.
 \(\beta_{ms}\)

Variable cost of machine type m for each unit time in regular time under scenario s.
 \(\delta_{ms}\)

Relocation cost of machine type m under scenario s.
 \(T_{mh}\)

Timecapacity of machine type m in period h in regular time.
 \(T_{mh}^{{\prime }}\)

Timecapacity of machine type m in period h in overtime.
 \(\theta_{mhs}\)

Variable cost of processing on machine type m per hour in overtime in period h under scenario s.
 UB

Maximal cell size.
 \(t_{jpm}\)

Processing time required to perform operation j of part type p by machine type m.
 \(t_{jpm}^{{\prime }}\)

Manual workload time required to perform operation j of part type p by machine type m.
 \(a_{jpm}\)

1 if operation j of part p can be processed by machine type m; 0 otherwise.
 \(p_{s}\)

Occurrence probability of scenario s.
 WT

Available time capacity per worker.
Decision variables
 \(N_{mch}\)

Number of machine type m allocated to cell c in period h.
 \(k_{mch}^{ + }\)

Number of machine type m added in cell c in period h.
 \(k_{mch}^{  }\)

Number of machine type m removed from cell c in period h.
 \(I_{mh}^{ + }\)

Number of machine type m purchased in period h.
 \(I_{mh}^{  }\)

Number of machine type m sold in period h.
 \(X_{jpmchs}\)

1 if operation j of part type p is processed by machine type m in cell c in period h under scenario s; 0 otherwise.
 \(L _{ch}\)

Number of labors assigned to cell c in period h.
 \(T_{mch}^{'}\)

Extra time needed for machine type m allocated to cell c in period h.
 \(\delta_{phs}\)

the underfulfillment of demand of part type p in period h under scenario s.
 \(I_{phs}\)

The inventory level of part p at the end of time period h under scenario s.
 \(Q_{phs}\)

Number of demand of part type p produced in period h under scenarios s.
Problem formulation
The objective function consists of nine components, given in Eqs. (1.1)–(1.9), seeks to minimize the sum of miscellaneous costs. Term (1.1) demonstrates sum of constant cost of all machines which have been used over the planning horizon for entire cells. Term (1.2) shows the total purchase cost minus selling income for entire machines during all periods. Term (1.3) indicates the variable cost of processing operations by different machines in whole cells and periods. Terms (1.4) and (1.5) calculate intercell and intracell movement costs, respectively. Term (1.6) represents the total costs for overtime working of machines which is required to produce the partial fraction of demand. Term (1.7) demonstrates the total costs of shifting labors between cells over the planning horizon. Various parameters such as labors training, wage rate of skilled labors and labors transference among the cells affect this expenditure. Term (1.8) indicates the cost of machines relocations. Finally, the last term of the objective function considers inventory holding costs. It is worth mentioning that all components (1.1)–(9) in the objective function are calculated under scenario s.
The first constraint introduced in Eq. (2) ensures that each operation of part p is allocated to only one machine capable of processing that part operation and one cell in period h on condition that part p is planned to be produced in that period. Equation (3) guarantees that an operation of a part is assigned to a machine provided that the machine is capable of processing that part operation. Equation (4) guarantees that machine capacity is not exceeded. Equation (5) calculates the number of each machine type bought or sold during each period. Equation (6) shows that the number of machines type m in cell c at the current period h equals to the number of that machines moved into cell c, plus the number of the same machine type present in the previous period and minus the number of machines removed from that cell. Equation (7) shows that summation of the extra time dedicated to all cells per machine type m cannot exceed the total capacity of machine type m in period h in overtime. Equation (8) ensures the number of labors allocated to all cells in each period is equal to the total number of available labors. Equation (9) determines the number of machines assigned to a cell in each period is less than the upper cell size limit. Equation (10) guarantees that available time capacity per worker is not exceeded. Equation (11) shows the balancing inventory constraint between periods for each part type at each period. It means that the inventory level of each part at the end of each period is equal to the quantity of production plus the inventory level of the part at the end of the previous period minus the part demand volume in the current period. Equation (12), complementary to Eq. (2), ensures that a portion of the part demand can be produced at the given period if its operations are assigned in the constraint given in Eq. (2). Logical binary, nonnegativity integer or continuous necessities for the decision variables are determined in Eqs. (13), (14) and (15).
Linearization of the proposed model
The proposed model is a mixedinteger nonlinear programming model because of absolute terms in Eqs. (1.4), (1.5) and (1.7) and the product of decision variables in Eqs. (1.3), (4) and (10).
The linearization process for absolute terms (1.4), (1.5) and (1.7) is accomplished by transforming the absolute terms into the linear form as follows:
Robust optimization formulation
The first and second terms in the objective function (28) are the expected value and variance of the objective function (27), respectively, and they measure solution robustness. The third term in (28) measures the model robustness with regards to infeasibility associated with control constraints (29) under scenario s. Equation (29) is a control constraint that is used to specify the level of inventory and the underfulfillment of part demand via violation level \(\delta_{phs}\) under scenario s. It is noted that if the total quantity of products produced in period h plus previous inventory at period h1 is greater than market demand \(D_{phs }\), then the inventory at period h will be equal to \(I_{phs} = I_{{p\left( {h  1} \right)s }} + Q_{phs }  D_{phs }\) and under minimization, the violation level \(\delta_{phs} = 0\); whereas if \(I_{{p\left( {h  1} \right)s }} + Q_{phs }\) is less than market demand \(D_{phs }\), then \(I_{phs} = 0\), and \(\delta_{phs} = D_{phs }  Q_{phs }  I_{{p\left( {h  1} \right)s}}\), demonstrating underfulfillment of part demand, thus an infeasible solution is obtained.
A case study
Case data description
A case study is conducted for a typical equipment manufacturer located in the Mazandaran province in the north of Iran. Badeleh Machinery Company was pioneered in 1988 with a factory for producing different kinds of tanked and trailed sprayers. Parallel with an increment in production rate, there came a variety of other types of machines, thus an increase in the factory’s area, as far as 15,000 meters for production section with another 15,000 meters of area left for future developments, in which 70 people consisting of workers and specialists work seven days a week. Regarding the customized demand in such case study, different scenarios in different season could be defined. Eight part types (farm equipment) consisting of (1) sprinkler, (2) Rot cultivator, (3) StalkShredder, (4) chipper, (5) Roller Chisel, (6) Borers with hydraulic inverter, (7) Borers with hydraulic inverter, and (8) Rear Hydraulic Crane Arm are produced in the company. To validate the proposed model and investigate the credibility of the employed robust optimization approach, the case study is solved using GAMS 22.0 software (solver CPLEX). First, the input data are described. Next, the obtained results are analyzed. This case study suggested in an uncertain environment includes 8 parts (p1,…,p8), six types of machines (m1,…,m6), three time periods (h1, h2, h3) and three types of cells (c1, c2, c3). For each part, three operations (j1, j2, j3) have to be processed sequentially considering processing times. The maximum available time for each worker in a time period is 40 h and the number of workers is 70. Besides, it has been assumed that the future economic scenarios will fit four probable scenarios that, respectively, are boom, good, fair and poor with the related probabilities 0.45, 0.25, 0.2, and 0.15.
Demand for eight part types in two periods under four scenarios
Dphs  Scenario  P1  P2  P3  P4  P5  P6  P7  P8 

h1  Boom  550  800  0  500  0  450  0  800 
Good  0  0  250  300  0  200  300  0  
Fair  350  500  0  0  200  0  250  250  
Poor  0  0  100  100  100  100  100  100  
h2  Boom  700  800  0  500  0  800  0  950 
Good  0  0  500  300  0  500  300  0  
Fair  500  400  0  0  300  0  200  350  
Poor  0  0  200  100  100  100  100  100  
h3  Boom  400  650  0  500  0  700  0  750 
Good  0  0  300  300  0  300  400  0  
Fair  200  400  0  0  250  0  300  200  
Poor  0  0  100  100  100  200  200  100 
Batch size for intercell and intracell movement of four part types
P1  P2  P3  P4  P5  P6  P7  P8  

B ^{inter}  35  25  20  40  45  30  35  40 
B ^{intra}  7  5  4  8  9  5  7  8 
Intercell and intracell movement cost per batch under four scenarios
Boom  Good  Fair  Poor  

\(\gamma_{s}^{\text{inter}}\)  50  40  30  20 
\(\gamma_{s}^{\text{intra}}\)  8  7  6  5 
Purchase cost of six machine types under four scenarios
\(\varphi_{ms}\)  Boom  Good  Fair  Poor 

M1  14,000  12,000  11,000  10,000 
M2  14,000  12,000  11,000  10,000 
M3  15,000  13,000  12,000  11,000 
M4  14,000  13,000  12,000  11,000 
M5  15,000  13,000  12,000  11,000 
M6  16,000  13,000  12,000  11,000 
Marginal revenue from selling six machine types under four scenarios
\(\omega_{ms}\)  Boom  Good  Fair  Poor 

M1  9800  8100  7700  7000 
M2  9800  8100  7700  7000 
M3  10,500  8700  8400  7700 
M4  9800  8100  7700  7000 
M5  9100  8000  7700  7000 
M6  11,200  9000  8400  7700 
Constant cost of six machine types in each period under 4 scenarios
\(\alpha_{ms}\)  Boom  Good  Fair  or 

M1  1400  1200  1100  1000 
M2  1400  1200  1100  1000 
M3  1500  1400  1200  1100 
M4  1400  1300  1200  1100 
M5  1300  1200  1100  1000 
M6  1600  1300  1200  1100 
Variable cost of six machine types for each unit time in regular time
\(\beta_{ms}\)  Boom  Good  Fair  Poor 

M1  9  8  7  5 
M2  9  8  7  6 
M3  8  7  6  5 
M4  8  7  6  5 
M5  9  8  7  6 
M6  8  6  5  4 
Relocation cost of six machine types under four scenarios
\(\delta_{ms}\)  Boom  Good  Fair  Poor 

M1  650  600  550  500 
M2  700  650  600  550 
M3  750  700  650  600 
M4  700  650  600  550 
M5  650  600  550  500 
M6  800  750  700  650 
Fixed cost of intercell moving of a labor in three periods under four scenarios
\(\rho_{hs}\)  Boom  Good  Fair  Poor 

H1  200  150  100  70 
H2  200  150  100  70 
H3  200  150  100  70 
Timecapacity of six machine types in regular and over time
T _{ mh }  \(T_{mh}^{{\prime }}\)  

M1  500  200 
M2  500  200 
M3  500  200 
M4  500  200 
M5  500  200 
M6  500  200 
Variable cost of processing on six machine types in overtime in three periods under four scenarios
\(\theta_{mhs}\)  Scenario  M1  M2  M3  M4  M5  M6 

h1  Boom  15  11  17  12  10  20 
Good  14  10  16  11  9  19  
Fair  13  9  15  10  8  18  
Poor  10  8  10  9  7  10  
h2  Boom  15  11  17  12  10  20 
Good  14  10  15  11  9  19  
Fair  13  9  13  10  8  18  
Poor  10  8  10  9  7  10  
h3  Boom  15  11  17  12  10  20 
Good  13  10  12  11  9  19  
Fair  12  9  11  10  8  18  
Poor  10  8  10  9  7  10 
Processing time required to perform the operations of eight part types on six machine types
t ^{ jpm }  P1  P2  P3  P4  P5  P6  P7  P8  

J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  
M1  0  0  0  0.76  0  0.39  0  0  0  0  0.83  0  0  0  0  0  0.57  0  0  0  0  0  0  0.54 
M2  0  0  0  0  0  0  0.99  0  0  0  0  0.74  0  0  0  0.72  0  0.47  0.44  0  0.28  0  0.17  0 
M3  0.73  0.93  0  0  0  0  0  0  0  0.45  0  0  0  0  0  0  0  0  0  0.97  0  0  0  0.15 
M4  0  0  0.46  0  0.81  0  0  0  0  0  0  0  0  0.26  0  0.55  0  0  0  0.47  0  0.84  0.86  0.78 
M5  0  0  0  0  0  0  0  0.48  0  0  0  0  0.12  0  0.75  0  0  0.12  0.76  0  0.86  0.2  0  0 
M6  0  0  0  0  0  0  0.36  0  0.00  0  0.78  0  0  0.45  0.59  0.81  0.48  0  0  0  0  0  0  0 
Manual workload time required to perform the operations of eight part types on six machine types
t ^{ jpm }  P1  P2  P3  P4  P5  P6  P7  P8  

J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  J1  J2  J3  
M1  0  0  0  0.076  0  0.039  0  0  0  0  0.083  0  0  0  0  0  0.057  0  0  0  0  0  0  0.054 
M2  0  0  0  0  0  0  0.099  0  0  0  0  0.074  0  0  0  0.072  0  0.047  0.044  0  0.028  0  0.017  0 
M3  0.073  0.093  0  0  0  0  0  0  0  0.045  0  0  0  0  0  0  0  0  0  0.097  0  0  0  0.015 
M4  0  0  0.046  0  0.081  0  0  0  0  0  0  0  0  0.026  0  0.055  0  0  0  0.047  0  0.084  0.086  0.078 
M5  0  0  0  0  0  0  0  0.048  0  0  0  0  0.012  0  0.075  0  0  0.012  0.076  0  0.086  0.2  0  0 
M6  0  0  0  0  0  0  0.036  0  0  0  0.078  0  0  0.045  0.059  0.081  0.048  0  0  0  0  0  0  0 
Cost of inventory holding for eight part types in three periods under four scenarios
H _{ phs }  Scenario  P1  P2  P3  P4  P5  P7  P8 

h1  Boom  17  17  25  20  19  33  25 
Good  15  15  21  18  14  25  22  
Fair  13  13  17  15  13  22  19  
Poor  10  10  10  10  10  10  10  
h2  Boom  18  22  22  22  17  32  22 
Good  15  15  18  19  12  27  19  
Fair  14  13  15  14  11  22  13  
Poor  11  10  12  11  10  12  11  
h3  Boom  17  18  20  17  20  17  29 
Good  13  13  15  15  15  15  22  
Fair  12  12  11  12  13  13  17  
Poor  10  10  10  10  11  10  10 
Results analysis
Sensitivity analysis for model Z
ω  Z 

0  2850 
100  264,416.1 
200  281,446.5 
300  292,060.5 
400  298,182.5 
500  301,576.6 
600  301,941 
700  308,054.6 
800  308,636.1 
The underfulfilled demand of eight part types in three periods under four scenarios
δ _{ phs }  Scenario  P1  P2  P3  P4  P5  P6  P7  P8 

h1  Boom  0  17  0  0  0  0  0  0 
Good  0  0  0  0  0  0  0  0  
Fair  0  0  0  0  0  0  0  0  
Poor  0  0  0  0  0  0  0  0  
h2  Boom  75  117  0  0  0  0  0  0 
Good  0  0  0  0  0  0  0  0  
Fair  0  0  0  0  0  0  0  0  
Poor  0  0  0  0  0  0  0  0  
h3  Boom  0  0  0  0  0  0  0  0 
Good  0  0  0  0  0  0  0  0  
Fair  0  0  0  0  0  0  0  0  
Poor  0  0  0  0  0  0  0  0 
Cost components of total costs [Eq. (27)] in four scenarios
Total costs  Machine constant  Machine variable  Purchasing machine  Intercell movement  Intracell movement  Inventory holding  Overtime  

Boom  275,040.3  30,000  87,218  100,000  12,571.7  1862.1  0  28,813.4 
Good  189,211.5  27,000  28,888.1  88,000  4422.7  1689.9  388.5  25,826. 1 
Fair  179,160.1  24,300  25,414.8  81,000  3313  1226.5  634.9  22,807.2 
Poor  134,714.6  22,000  9528  74,000  1266.6  1250  1569.6  19,819.9 
Table 17 illustrates the total costs based on Eq. (27) including costs of machine constant, machine variable, machine purchase, intra and intercell movement, part inventory holding and overtime under various scenarios.
According to Table 1, it is clear that the part demands and the scenariobased cell formation cost parameters are incremental from poor scenario to the boom one. As can be seen in Table 17, all cost components have increased from the poor scenario to the boom one, except the inventory holding cost. Since the part underfulfilled demand has obtained positive value for parts 1 and 2 according to Table 16 in the boom scenario, violation occurred and according to the control constraint (29), the part inventory amount and its related cost is zero in the boom scenario. However, from the boom scenario to the poor one the part inventory has increased and similarly, the part inventory cost has increased as well. Since in the good, fair and poor scenarios, \(\delta_{phs}\) equals zero, the part inventory level gets a positive level and the inventory holding cost gets a positive level as well.
In period 1, operations 1 and 2 of part 1 are processed inside cell 1 by machines 5 and 4, respectively, and operation 3 inside cell 2 by machine 6. Then, there is need for an intracell movement for operations 1 and 2 and an intercell movement for operations 2 and 3. In period 1, eight intercell movements and two intracell movement are performed for the parts processing. In period 2, seven intercell movements and three intracell movement are performed for the parts processing.
Tradeoff between solution robustness and model robustness
Comparing the effectiveness of robust model and meanvalue based model
To illustrate the robust dynamic cell formation that could be obtained by the proposed MIP model, expected values of uncertain parameters are used in the primary mixedinteger linear programming model presented in “Linearization of the proposed model” section as certain value parameters, hereafter called meanvalue based model. The results of these two models (i.e., robust model and meanvalue based model) are compared with each other at the following. Robust optimization is used to attain a robust solution against the fluctuation of uncertain parameters in the future. Note that at the inception of planning horizon, some parameters are uncertain, and only in the execution time of the plan, the real values of uncertain parameters will be realized. For this purpose, we simulate some real and conceivable scenarios that may occur after executing the cell formation in the future. We consider 10 random occurrences for the uncertain parameters and compute the objective function Z of each instance for the dynamic cell formation problem obtained by the robust and meanvalue based models.
Total objective function values obtained by the robust and meanvalue based models
Problem number  1  2  3  4  5  6  7  8  9  10 

Robust model  257,361.8  286,914.8  294,127.3  285,993.8  271,391.6  301,591.6  298,255.4  308,646.9  279,743.5  309,443 
Meanvalue based model  316,642.7  394,204.2  389,787.9  421,225.4  410,668.2  432,788.6  371,073.8  439,419.8  406,668.8  434,360.9 
In fact, the values of the objective function Z for different scenarios are closer to each other than these values for the meanvalue based model. The curve of values in the proposed method follows a more robust incline, but the fluctuation in the curve of values for the classical approach is very high. This achievement indicates that the proposed approach is efficient for any systems that the robustness of solution is important in addition to objective function value Z of production for their managers. Indeed, for such systems having a solution with minimum total objective is not adequate, but the fluctuation in real scenarios in future should be handled. Therefore, numerical results show the robustness and effectiveness of the proposed model.
Conclusion
In this study, a mathematical model based on a robust optimization approach has been presented in dynamic cell formation problem with uncertain data to integrate CF, PP and worker assignment. The robust optimization approach reduces the effect of the fluctuations of uncertain parameters under certain scenarios. In this study, the majority of cell formation parameters including cost parameters and part demand fluctuation were considered uncertain.
Next, sensitivity analysis has been presented for solution robustness and model robustness. Since the objective function has been influenced by \(\omega\), the relationship between the model robustness and solution robustness has been analyzed only for the objective function value.
The computational experiments obtained from a set of realworld data for an Iranian farm tanked and trailed sprayers manufacturer illustrated that the proposed robust model is more practical for handling uncertain parameters in the production environments. The tradeoff between optimality and infeasibility was used for obtaining robust solution based on the opinion of decisionmakers. The results showed the robustness and effectiveness of the model in realworld cell formation problem.
In addition, the results obtained by the robust MIP model indicated the advantages of robust optimization in generating more robust cell configurations with less cost over the considering expected value of uncertain parameters in a deterministic meanvalue based model. In fact, in such systems designed here as the meanvalue based model, having only solution with the minimum value of the objective function and lower costs is not sufficient rather the fluctuations in the related scenarios have to be lowered in future.
The future studies in the following of the present study can be pursued in multiobjective DCMS modeling, employing the other robust optimization methods, taking into account the setup time, defining the processing times and timecapacity of machines as uncertain, consideration of machine layout, allowing partial or total subcontracting, workload balancing among the cells, and using metaheuristics to tackle largesized problems.
References
 Aryanezhad MB, Deljoo V, Mirzapour Alehashem SMJ (2009) Dynamic cell formation and the worker assignment problem: a new model. Int J Adv Manuf Technol 41:329–342CrossRefGoogle Scholar
 Bagheri M, Bashiri M (2014) A new mathematical model towards the integration of cell formation with operator assignment and intercell layout problems in a dynamic environment. Appl Math Model 38(4):1237–1254CrossRefMathSciNetGoogle Scholar
 Bai D, Carpenter T, Mulvey J (1997) Making a case for robust optimization models. Manag Sci 43:895–907CrossRefzbMATHGoogle Scholar
 Balakrishnan J, Cheng CH (2005) Dynamic cellular manufacturing under multi period planning horizons. J Manuf Technol Manag 16(5):516–530CrossRefGoogle Scholar
 Bashiri M, Bagheri M (2013) A two stage heuristic solution approach for resource assignment during a cell formation problem. Int J Eng Trans C Asp 26(9):943Google Scholar
 BenTal A, Nemirovski A (1998) A Robust convex optimization. Math Oper Res 23:769–805CrossRefMathSciNetzbMATHGoogle Scholar
 BenTal A, Nemirovski A (2002) A robust optimization—methodology and applications. Math Program 92:453–480CrossRefMathSciNetzbMATHGoogle Scholar
 BenTal A, Nemirovski A, Roos C (2002) Robust solutions of uncertain quadratic and conicquadratic problems. SIAM J Optim 13:535–560CrossRefMathSciNetzbMATHGoogle Scholar
 Bertsimas D, Sim M (2002) Robust discrete optimization Technical Report, Sloan School of Management and Operations Research Center, MIT CambridgeGoogle Scholar
 Bertsimas D, Thiele A (2003) A robust optimization approach to supply chain management. Technical Report Working Paper MITGoogle Scholar
 Feng P, Rakesh N (2010) Robust supply chain design under uncertain demand in agile manufacturing. Comput Oper Res 37(4):668–683CrossRefzbMATHGoogle Scholar
 Forghani K, Sobhanallahi M, Mirzazadeh A, Mohammadi M (2012) A mathematical model in cellular manufacturing system considering subcontracting approach under constraints. Manag Sci Lett 2(7):2393–2408CrossRefGoogle Scholar
 Ghezavati V, Sadjadi S, Dehghan Nayeri M (2011) Integrating strategic and tactical decisions to robust designing of cellular manufacturing under uncertainty: fixed suppliers in supply chain. Int J Comput Intell Syst 4(5):837–854CrossRefGoogle Scholar
 Hassannezhad M, Cantamessa M, Montagna F, Mehmood F (2014) Sensitivity analysis of dynamic cell formation problem through metaheuristic. Procedia Technol 12:186–195CrossRefGoogle Scholar
 Kia R, Paydar MM, Jondabeh MA, Javadian N, Nejatbakhsh Y (2011) A fuzzy linear programming approach to layout design of dynamic cellular manufacturing systems with route selection and cell reconfiguration. Int J Manag Sci Eng Manag 6(3):219–230Google Scholar
 Kia R, Baboli A, Javadian N, TavakkoliMoghaddam R, Kazemi M, Khorrami J (2012) Solving a group layout design model of a dynamic cellular manufacturing system with alternative process routings, lot splitting and flexible reconfiguration by simulated annealing. Comput Oper Res 39:2642–2658CrossRefMathSciNetzbMATHGoogle Scholar
 Kia R, Shirazi H, Javadian N, TavakkoliMoghaddam R (2013) A multiobjective model for designing a group layout of a dynamic cellular manufacturing system. J Ind Eng Int 9:8CrossRefGoogle Scholar
 Mahdavi I, Aalaei A, Paydar MM, Solimanpur M (2010) Designing a mathematical model for dynamic cellular manufacturing systems considering production planning and worker assignment. Comput Math Appl 60(4):1014–1025CrossRefMathSciNetzbMATHGoogle Scholar
 Mirzapour AlEHashem SMJ, Malekly H, Aryanezhad MB (2011) A multiobjective robust optimization model for multiproduct multisite aggregate production planning in a supply chain under uncertainty. Int J Prod Econ 134(1):28–42CrossRefGoogle Scholar
 Mulvey JM, Vanderbei RJ, Zenios SA (1995) Robust optimization of largescale systems. Math Oper Res 43:264–281CrossRefMathSciNetzbMATHGoogle Scholar
 Paydar MM, SaidiMehrabad M, Teimoury E (2013) A robust optimisation model for generalised cell formation problem considering machine layout and supplier selection. Int J Comput Integr Manuf 27(8):772–786CrossRefGoogle Scholar
 Rafiei H, Ghodsi R (2013) A biobjective mathematical model toward dynamic cell formation considering labor utilization. Appl Math Model 37(4):2308–2316CrossRefGoogle Scholar
 Rheault M, Drolet J, Abdulnour G (1995) Physically reconfigurable virtual cells: a dynamic model for a highly dynamic environment. Comput Ind Eng 29(1–4):221–225CrossRefGoogle Scholar
 Sakhaii M, TavakkoliMoghaddam R, Bagheri M, Vatani B (2015) A robust optimization approach for an integrated dynamic cellular manufacturing system and production planning with unreliable machines. Appl Math Model. doi: 10.1016/j.apm.2015.05.005 Google Scholar
 Shirazi H, Kia R, Javadian N, TavakkoliMoghaddam R (2014) An archived multiobjective simulated annealing for a dynamic cellular manufacturing system. J Ind Eng Int 10:58CrossRefGoogle Scholar
 Wu X, Chu CH, Wang Y, Yue D (2007) Genetic algorithms for integrating cell formation with machine layout and scheduling. Comput Ind Eng 53(2):277–289CrossRefGoogle Scholar
 Yu CS, Li HL (2000) A robust optimization model for stochastic logistic problems. Int J Prod Econ 64(1–3):385–397CrossRefGoogle Scholar
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