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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.20 No.4 pp.522-528
DOI : https://doi.org/10.7232/iems.2021.20.4.522

# A Novel Mathematical Model for Tillage Production Planning and Facility Layout Design

Pavel M. Podolko*, Alexandra Pe. Podolko
Academy of Engineering, Peoples' Friendship University of Russia (RUDN University), Moscow, Russia
Academy of Engineering, Peoples' Friendship University of Russia (RUDN University), Moscow, Russia
*Corresponding Author, E-mail: podolko-pm@rudn.ru
August 10, 2021 September 10, 2021 September 18, 2021

## ABSTRACT

Cell formation and layout design are important steps in the implementation of production systems. Existing models focus mainly on the problem of cell formation and machine layout design and cell location in a dynamic environment, with the exception of a few recent studies. Also, in many existing articles, binary information has been used for cell formation and other information such as production volume, operation sequence and lower production costs have played a role in the structure of existing models. In this paper, a nonlinear programming model under possible dynamic conditions is presented. The objectives of the model include minimizing the total costs of intracellular and intercellular movements of parts, the existence of exceptional parts, intercellular layout of machines, operating costs, fixed costs of machines and the cost of the difference in demand estimated by the model from its expected value. A genetic algorithm is used to solve and the results are compared with the optimal solution.

## 1. INTRODUCTION

In today’s world, various service or production companies are forced to make structural adjustments due to the increased power of customer choice and expansion of competitive markets. Group technology is one of the most important philosophies of production, the goal of which is to determine, classify and allocates pieces to groups and families of parts and assign machines to cells for production of this family of parts. Cellular production systems are among the most crucial applications of group technology, recognized as an efficient tool for the improvement of production flexibility and productivity. The most important part of cellular production is the allocation of pieces and machines to cells based on their similarities in geometric properties, design and production process, known as cell formation. Machines and parts must be allocated with the minimum costs. The location of machines must be determined following the mentioned process, which is known as cellular layout (Delgoshaei et al., 2016;Wang et al., 2021). On the other hand, given the short product life cycle and rapid changes in the amount and type of demand and consequently the need to change production cells from one period to another, it seems crucial to consider a dynamic environment in designing cellular production systems. Cell reconfiguration includes costly activities, such as machinery displacement, adding new machines to the cell and removing the existing machines from the cells (Safaei et al., 2008;Roozitalab and Majidi, 2017).

Presented by Rheault et al. (1995) for the first time to increase production flexibility, dynamic cellular production systems mainly focus on changes in demand and product composition and are much more efficient, compared to traditional cell production systems. On the other hand, given the uncertain demand for products in each period, the design of dynamic cell production systems becomes much more realistic and practical by considering demand as a random variable. Balakrishnan and Cheng (2007) conducted a comprehensive review of the literature in the field of dynamic cellular production systems, especially under uncertainty. In another study, Schaller (2007) presented a comprehensive mathematical model for cell formation problems by considering a change in demand at different periods. Moreover, Bagheri and Bashiri (2014) proposed a mathematical model to solve the problem of cellular formation and operator allocation and inter-cell layout problem. In another study, Deep and Singh (2016) presented a mathematical model for concomitant cell formation and layout. In addition, Mahdavi et al. (2013) suggested an integrated model of multi-period cell formation while considering several production and operator allocation routes. The use of different methods in optimizing cell production has been observed in the latest studies by Hazrin et al. (2021), Poznyak et al. (2020), and Alimian et al. (2020). For instance, they have considered environmental pollution and the uncertain connection between cells. While different solution methods and models have been proposed to solve the cell formation problem as one of the most important issues in the design of cellular production systems, few studies have been conducted on the problem of simultaneous cell formation and layout design under the conditions of several possible periods. Therefore, the present study aims to evaluate the problem of cell formation and layout in possible dynamic conditions. Following the presentation of the mathematical model to simultaneously solve the problem of cell formation and layout in possible dynamic conditions, we will use the genetic algorithm (GA) to present a solution and will compare the results with the optimal solution using the Lingo software.

## 2. MODEL PRESENTATION

### 2.1 Model Premises

In this research, it is assumed that a certain number of operations is required for each part, and the operation time of all pieces on each machine is pre-determined. In addition, the type of parts and the probability distribution function of the demand for each piece in each period are specified. Moreover, the operational costs and capacity of each machine are determined and fixed. The parts move between cells in batches, and the cost of intracellular and intercellular movements of batches is specified and fixed. Moreover, the minimum number of cells in all periods is pre-determined and fixed, and the change of location of the machines from each cell to another is done in these periods. The cost of movement of each machine between cells depends on the sequence of cells. Each machine can perform one type of operation. Furthermore, the cost of intracellular movement of parts is not related to distance but depends on the sequence of cells. Excess inventory between periods is zero and delayed order is not allowed. In other words, the demand for each period must be met in the same period. In addition, machine efficiency and production are considered 100%. The size of each batch is specific to each product and is fixed for all periods. The type of transportation system and facility layout is Ushape and in the form of one line, which has a significant effect on the part layout cost. The operation time is specified for each part on each type of machine, and the time of availability of machines in each period is specified. The machines are rented, and the fixed cost of each machine is determined and is independent of the workload and includes maintenance and repair costs and the cost of renting facilities for each period. Moreover, this cost depends only on the type of facility and will be considered whether the car is active or idle. The desired locations for the layout of machines inside each cell are predetermined, and similar machines are assigned to one location in each programming period.

### 2.2 Indexes

• i: Machine index, i=1,2,…,m

• j: Part index, j=1,2,…,n

• k: Cell index, k=1,2,…,C

• p: Machine location index, p=1,2,…,mp

• h: Programming period index, h=1,2,…,H

### 2.3 Parameters

• Bj: Magnitude of the batch for movement of the jth part

• Cja: Cost of intracellular movement of the j-th part

• Cje: Cost of intercellular movement of the j-th part

• Cjb: Cost of one unit of return in the intracellular route of the j-th part

• Ci: Cost of one-unit intercellular movement of the i-th machine

• $C i i ' h j$ : Cost of movement of the j-th part between the i-th machine and I-th machine in the h-th period

• Mih: The location number of the i-th machine (the order of the i-th machine for layout) in the h-th period

• NC: Minimum number of cells that must be formed in each period

• NM: Maximum number of machines that must be located in one cell

• Rij: Number of operations performed on the j-th part by the i-th machine

• $f i i ' h j$ : Number of travels for moving the j-th part between the i-th machine and the i-machine in the h-th period

$f i i ' h j = { D j / B j i f R i ' j − R i j = 1 0 i f R i ' j − R i j ≠ 1$

• Ti: Capacity of the i-th machine in each period

• tij: Processing time of the j-th part by the i-th machine

• SP: Set of (i,j) pairs when aij>1

• Ai: Fixed cost of the i-th machine in each period

• Bi: Cost of adding or removing the i-th machine

• EDjh: Mean demand for the j-th part in the h-th period

• STjh: Standard deviation of demand for the j-th part in the h-th period

### 2.4 Decision Variables

• KAih: The number of i-th machine added at the beginning of the h-th period

• KRih: The number of i-th machine removed at the beginning of the h-th period

• Djh: The amount of production of the j-th part in the h-th period

• Nih: The number of i-th machine required in the hth period

Therefore, the cost of movement of the j-th part between the i-th and i'-th machine in the h-th period is:

(1)

(2)

The nonlinear modeling for this problem is as presented below based on the parameters and variables mentioned above.

$M i n i m i z e ∑ h = 1 H ∑ j = 1 n ∑ i = 1 m ∑ i ' = 1 m f i i ' h j C i i ' h j + ∑ h = 1 H ∑ k ∑ ( i , j ) ∈ s p ( U i j k h + V i j k h ) 2 + ∑ h = 2 H ∑ i = 1 m B i ( K A i h + K R i h ) + ∑ h = 1 H ∑ i = 1 m A i . N i h + ∑ h = 2 H ∑ i = 1 m ∑ j = 1 n t i j B j D j h + ∑ h = 1 H ∑ j = 1 n | D j h − E D j h | + ∑ h = 1 H − 1 ∑ i = 1 m ∑ k = 1 c ∑ k ' = 1 c | k ' − k | C i X k k ' h i N i h *$
(3)

Subject To:

$∑ k = 1 c X i k h = 1 , i = 1 , 2 , … , m ∀ h$
(4)

$∑ k = 1 c Y j k h = 1 , i = 1 , 2 , … , n ∀ h$
(5)

$∑ i = 1 m X i k h ≤ N M × I C k h , k i = 1 , 2 , … , C ∀ h$
(6)

$∑ k = 1 C I C k h ≥ N C ∀ h$
(7)

$∑ i = 1 m Z i p h = 1 , p = 1 , 2 , … , m p , ∀ h$
(8)

$∑ p = 1 m p Z i p h = 1 , i = 1 , 2 , … , m , ∀ h$
(9)

$∑ h = 1 H N i h < U B ∀ i$
(10)

$D j h ≤ E D j h + 1 / 96 S T j h ∀ j , h$
(11)

$D j h ≥ E D j h + 1 / 96 S T j h ∀ j , h$
(12)

$∑ h = 1 H D j h t j h / N i h ≤ T i$
(13)

$N i h + K A i ( h + 1 ) − K R i ( h + 1 ) = N i ( h + 1 ) ∀ j , h$
(14)

(15)

$X i k h , Y j k h , Z i p h , U i j k h , V i j k h , X k k ' h i = 0 o r 1$
(16)

### 2.5 Objectives

The present model proposed a nonlinear integer model to simultaneously determine the cell of machinery, family of parts, and facility layout under possible dynamic conditions. The model’s objective includes minimizing the total costs of intracellular and intercellular movements of parts, machinery movement, cost of adding or removing machines, the fixed costs of machines, the costs of exceptional parts, the operating costs, and the cost of the difference in demand estimated by the model from its expected value, which is explained below.

#### 2.5.1 Cost of the Difference in Demand Estimated by the Model from its Expected Value

In this model, it is assumed that the demand for each part is determined based on a statistical distribution according to previous data and experiences. In other words, the demand for each part is predicted in such a way that the objective function is also minimized in each period. Therefore, demand for parts is entered into the model as a probable variable. We need to set a confidence interval for the demand variable of each part in each period based on the dispersion parameters of the demand distribution of the part in order to achieve the objective. The lower the demand for a part, the lower the operating cost and possibly material handling but the higher its absolute value of deviation from the mean. In this article, it is assumed that the distribution of parts demand in each period follows the normal distribution. If EDjh and STjh are considered as the mean and standard deviation of normal distribution related to the j-th part in the h-th period, a (1- α)% confidence interval can be expressed for the amount of production of the j-th part in the h-th period (Djh), as shown below.

According to the mentioned confidence interval, a maximum of $E D j h + Z a 2 S T j h$ must be produced if we do not want to face shortages with a (1- α)% probability. If the confidence level is considered at (1- α) = 0.95 and is fixed in programming periods, the amount of production of the j-th part in the h-th period is obtained from the equation below:

$( 1 − α ) = 0 / 95 → α = 0.05 → a 2 = 0.025 → Z a 2 = 1 / 96$

If the demand function is continuous and of beta distribution type, three amounts must be considered for optimistic (H), pessimistic (L) and probable (M) moods for the demand. In this way, the values of EDjh and STjh are obtained from the equation below (Tables 1 and 2):

$S T j h = H − L 6 , E D i j = L + 4 M + H 6$

### 2.6 Constraints

Constraints (4) require that each type of machine is allocated to one cell. Constraints (5) are for the allocation of each part to one cell. Constraints (6) prevents the allocation of more than NM type of machine to each cell and allocation of all machines to one cell. Constraints (7) guarantee that a minimum of NC cells is formed in each period. Constraints (8) guarantee that each position in each period accepts only one machine. With constraints (9), each machine is allocated to only one location in each period. Constraints (10) guarantee that the number of similar machines in each period does not exceed the pre-specified number. Constraints (11) and (12) show that the amount of demand estimated by the model does not exceed the upper and lower limit of a 95% confidence interval. This level of reliability is determined according to the corresponding distribution function. According to constraints (13), the amount of load on each machine in each period does not exceed its capacity. Finally, constraints (13) establish balance in the number of machines existing in each period based on the added or removed machines.

## 3. SOLUTION

Since the search space for the solution increases sharply as the size of the problem increases, the hierarchical genetic algorithm is used to solve large-scale problems. Validation occurs by comparing the results with the exact solution obtained from Lingo software on a small scale.

### 3.1 Chromosome Display

Each chromosome consists of a row of real numbers, each of which represents the demand for each part and the position of that number (gene) in the chromosome represents the part number. Chromosome length is the product of the number of programming cycles multiplied by the number of parts. According to the shape of the chromosome that represents a solution for two periods, the separate representation of parts in the period is shown in Figure 2 for this solution. In the first part of each chromosome, genes indicate the machine number for layout, and the location of the gene in the chromosome demonstrates the machine sequence number. In the second part of the chromosome, each gene represents the cell number, to which a machine is allocated, and gene location shows the number of the machine. In the third part, each gene shows the cell number to which the part is allocated, and the location of the gene expresses the number of the part (Figure 1).

### 3.2 Comparison of GA Results with Optimal Solution

The results obtained from GA are compared with the results obtained from Lingo8 to present the optimal solution for a sample of 14 parts and 9 machines, as shown (Table 3). In the desired GA, the population of each generation is equal to 100 solutions (μ = 100). In addition, the probability of combination is 0.9 (pc = 0.9). The probability of mutation is equal to one hundredth (Pm = 0.01), and reaching the hundredth iteration is the condition for terminating the algorithm. There is a maximum of three machines in each cell. In addition, there is a minimum of three cells, and four operations are considered in the problem. Moreover, there is one production period. The part-machine matrix for 14 parts and 9 machines is presented below, and the results of comparing the solution obtained from GA with the optimal solution are presented in Table 4.

According to Figure 2, the difference between GA and Lingo solutions is related to the computational time and the cost of the difference in demand estimated by the model from its expected value which was presnted in Figure 5. In this problem, three optimal solutions are obtained for machine allocation, where cell 1 includes {4, 6, 7} machines and {1, 7, 9} parts, cell 2 includes {1, 2, 8} machines and {2, 5, 4, 6, 8, 10, 11, 13, 14} parts, and cell 3 includes {3, 5, 9} machines and {3, 12} parts.

## 4. CONCLUSION

In the present study, considering a dynamic environment, random demand, and costs of movement of machines from one cell to another were among the advantages of the proposed model. In addition, the GA was used to introduce the model, and the problem was solved at a small scale to compare the GA with the optimal solution of the problem. The comparison of solutions based on the quality of solution and computational time showed that the solution yielded by GA had almost the same quality as the optimal solution. However, there was a significant difference in terms of computational time in this regard. It is recommended that multifunctional machines be used and operator allocation of the dimension of travels be considered in the estimation of transportation costs in the development of models.

## Figure

Chromosome display for cell formation and machine layout.

Cost saving in optimal facility layout design.

## Table

Chromosome of parts demand for two programming periods

Separate display of the part-period

Part-machine matrix for 9×14 problem

Comparison of results for 9×14 problem

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