Int J Adv Manuf Technol (2007) 33: 345–353 DOI 10.1007/s00170-006-0461-2
ORIGINA L ARTI CLE
R. Galan . J. Racero . I. Eguia . D. Canca
A methodology for facilitating reconfiguration in manufacturing: the move towards reconfigurable manufacturing systems
Received: 12 July 2005 / Accepted: 27 September 2005 / Published online: 4 May 2006 # Springer-Verlag London Limited 2006
Abstract The launch of new products to the market is essential for companies in order to remain competitive. Products are constantly being replaced for others that fulfil the changing requirements of the customers. For achieving this, companies have to adapt their production system to the new requirements of the new products by adding or removing machines, changing the lay-out, etc. Normally, this reconfiguration implies a high investment. Reconfigurable manufacturing systems (RMSs) arise for facilitating the reconfiguration of the production system, being considered as the next step in manufacturing. This paper deals with the development of a methodology based on RMSs that allows a feasible reconfiguration of production systems. This methodology is based on the ALCA algorithm from group technology, and takes into account five requirements of products on RMSs: modularity, commonality, compatibility, reusability and demand. The selection of the product families is obtained with a mathematical model specifically formulated for this purpose. Keywords Manufacturing . Product families . Reconfiguration Abbreviations RMSs: Reconfigurable manufacturing systems . MC: Mass customisation . CMSs: Cellular manufacturing systems . ALCA: Average linkage clustering algorithm . TSP: Travelling salesman problem . BOM: Bill of materials
1 Introduction Nowadays, customers demand new products at low cost, with high quality, and highly customised. Competition is R. Galan (*) . J. Racero . I. Eguia . D. Canca School of Engineering, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain e-mail:
[email protected] Tel.: +34-954-487202 Fax: +34-954-487329
now fiercer than ever and, for surviving in this global scenario, companies are forced to compete in product variety and speed to market, as well as in price, through replacing old products constantly with new versions. Differentiation in product variety is gained with customisation, which recognises each customer as an individual together with the typical cost efficiency of mass production systems. Hence, mass customisation (MC) arises as a new manufacturing paradigm that fulfils individual customer needs with mass production efficiency. MC has been identified as a major manufacturing strategy, and its implementation requires facing with complex customer choice portfolios (product variety for customisation), economies of scale (mass efficiency), and the need for highly flexibility (quick responsiveness) in manufacturing processes [1]. Traditionally, any change in the way of working of a factory has implied an extra consumption of resources, mainly in labour and monetary terms. This consumption is incremented when the production system is changed and therefore, it is carried out when it is absolutely necessary. New configurations of the production system lead to the launch of new products to the market, thus reconfiguration has become an issue of core importance. The massive move towards reconfigurable systems will take place when new ways for an effective reconfiguration of production systems were developed. Reconfigurable manufacturing systems (RMSs) take into account the MC requirements, and they are designed to cope with situations where both productivity and the ability of the system to change its configuration for producing different products have great importance [2]. The main components of RMS are CNC machines and reconfigurable machine tools, which are a new type of modular machines, composed of a changeable structure that allows adjustment of its resources. Reference [3] considers a RMS as a manufacturing system configured to produce a family of products that share some similarities. These families consist of products that possess similarities in their functionality and may share components, production processes and architectures [4]. It
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is suggested that grouping products into families in RMSs has a positive effect on the introduction of new products [5]. The key attribute of product families is that all the components within a family may require similar production systems. Therefore, any manufacturing system that produces a component within a family can essentially produce the components of that family. RMSs have the capacity and functionality required for manufacturing a product family, allowing cost effectiveness. This research aims at the development of a methodology for selecting the best set of product families, from several sets of different possibilities, which would facilitate the move towards reconfigurable manufacturing systems. To achieve this goal, a review of the existing techniques for the formation of product families is presented. Based on them, a new method is developed that takes into account the singularities of RMSs. Then, a mathematical model is proposed for selecting the best group of families previously formed. A cost methodology is developed for the resolution of the model, which is validated with an example. Finally, the main conclusion and future works are presented.
2 Product families formation The proliferation of methods for grouping products arose with the development of cellular manufacturing systems (CMSs); the first manufacturing paradigm focused on making the cost-effective manufacture of several part types possible. Traditionally, grouping products into families and cell formation in CMSs have been closely linked. Instead of developing new methods for grouping products in RMSs a study of the methods used in CMSs is carried out with the aim of identifying one of them that can be conveniently modified in order to stand the requirements for products in RMSs. Several methods have been developed for families’ formation [6], as descriptive procedures, mathematical programming approaches, array-based clustering methods and hierarchical clustering, among others. Most of the descriptive procedures are not highly sophisticated or accurate, although they are cost-effective [7]. They may be appropriate to little and repeated problems, but they do not provide good solutions in general. The mathematical programming approaches are incompletely formulated [6] and therefore their usefulness is limited in industrial environments. All these models are computationally complex and it is unlikely that they can provide good solutions to large-scale problems. The array-based clustering methods achieve acceptable solutions with low computational cost. Although ordered matrices can be developed using array-based clustering, disjoint part families or machine groups are not identified. Besides, they have the disadvantage of depending on the initial incidence matrix configuration [7]. The hierarchical clustering agglomerative methods group together similar elements (products) in clusters
based on their attribute similarities. The coefficients that measure similarity between two parts are calculated from the incidence matrix. After that, a dendogram (inverted tree structure) shows the similarity degree for grouping parts. These methods are the most broadly implemented. They use similarity or dissimilarity coefficients among parts to obtain the groups. The most important similarity coefficient for part-family formation is the Jaccard similarity coefficient [8], which measures the similarity between a pair of products (m, n), and it is defined in terms of the machines that each product has to visit. This coefficient (Smn) may be expressed as: Smn ¼
a 0 Smn 1: aþbþc
(1)
In the above expression, a indicates the number of machines that visit both products m and n, b stands for the number of machines that visit only product m, and c the number of machines that visit only product n. Therefore, if Smn=1 then both products are processed by the same machines, and if Smn=0 then products are processed by different machines. Products with the highest similarity coefficients are grouped together. In the context of cell (and families) formation, only agglomerative clustering techniques have been used [6]. These techniques have a string effect known as “chaining”, which creates a few large clusters while leaving several parts unmerged [9]. Among those techniques, average linkage clustering algorithm (ALCA) has the least tendency to chain [10] and it is considered in this study as the most appropriate to apply. This method starts grouping products with higher coefficient of similarity. Then, a sub-matrix considering the products grouped as a family is created. The similarities between the parts are recalculated as the average values, with the following formula: PP Smn m2i n2j Sij ¼ ; (2) Ni Nj Where: i,j m,n Sij Smn Ni, Nj
families, products of family i and j, respectively, coefficient of similarity between families i and j, coefficient of similarity between products m and n, number of products in family i and j, respectively.
This procedure is repeated until all the products are grouped into a family. As a result, a dendogram illustrates the different grouping that can be formed depending on the similarity of products within a family. The ALCA algorithm takes into consideration similarities referred to whether a machine manufactures a part. For RMS that is not a sufficient criterion, because the algorithms have to consider the singularities of the new system. Those singularities are requirements of the prod-
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ucts to achieve a high level of system’s reconfigurability such as modularity, commonality, reusability, compatibility and demand. Modularity can be defined as the degree to which a product is composed of independent modules, without interactions between them [11]. Commonality is defined as a measure of how well the product uses standardized parts [12]. Compatibility can be defined as the degree to which different products can be joined to form a family of similar products. Reusability, is an economic factor that measures the use of existing design configurations while reconfiguring manufacturing elements for a new product type [5]. Finally, product demand is important because manufacturers receive orders of products for all the families, so producing a high volume of a certain family may delay the delivery time of the rest of the families [3]. Therefore, the ALCA algorithm adapted for RMS presents five similarity matrices, one per each product requirement previously identified. However, a unique matrix comprising the values of interaction among products is necessary, which may be solved using weighting techniques. With the use of those techniques, values are assigned to those five requirements to indicate their relative importance. An algorithm for this method is shown in (Fig. 1). For example, let us suppose an initial matrix for four products which represents the similarities among those products: (Table 1)
Table 1 Initial matrix
A B C
B
C
D
0.39
0.37 0.70
0.56 0.41 0.35 Start
Group together products with Max Sij
Calculate new Sij
no
All the products are grouped in the same family? yes Create the dendogram
The maximum value of Sij is 0.70 among products B and C, thus they are grouped in a family. Sij are recalculated for the new sub-matrix. This is shown in Table 2. SA;ðB;C Þ ¼ ðSAB þ SAC Þ 2 ¼ ð0:39 þ 0:37Þ 2 ¼ 0:3 SA;D ¼ 0:56 SðB;CÞ;D ¼ ðSBD þ SCD Þ 2 ¼ ð0:41 þ 0:35Þ 2 ¼ 0:38
The new maximum value of the matrix is 0.56 corresponding to products A and D. Both products are grouped and the new sub-matrix is presented in Table 3. SðA;DÞ;ðB;C Þ ¼ ðSAB þ SAC þ SDB þ SDC Þ 4 ¼ ð0:39 þ 0:37 þ 0:41 þ 0:35Þ 4 ¼ 0:38 Finally, all the products are grouped with Sij = 0.38 and the resulting dendogram is shown in Fig. 2. From the dendogram, the selection of families can be made. There are four different levels, each with different families. Upper levels are composed of several families with few products and high similarity among them. On the contrary, families in bottom levels are composed of few families with lots of products with low similarity. For example, three families can be formed with a similarity of 70% among the products in the families (level 2). The first family is composed of products B-C, the second one is composed of product A, and the last family is composed of product D. In the traditional ALCA method used in group technology, the selection of families was determined by the costs incurred when a product could not be manufactured within its cell, and the costs incurred when a machine within a cell could not manufacture a product of its associated family. In the design of a RMS, this selection depends on several factors which are described in the next section. To the best of our knowledge, the unique grouping method in the literature applied specifically for RMS is from Abdi and Labib [5], who present a methodology for grouping products in RMS based on operational similarity. It is based on the grouping approaches used in group technology, but in their research a product-operation matrix is used for describing the specifications of each product. Therefore, whether a product requires the operation produced by a machine or not is only considered. The respective matrix is composed of aij coefficients whose values are 1 if product i requires operation j and 0 otherwise. Based on this matrix, the approach investigates the clustering of product types with maximum operational Table 2 Sub-matrix
End
Fig. 1 Algorithm for the ALCA method
A B,C
B,C
D
0.38
0.56 0.38
348 Table 3 Final matrix
3.1 Mathematical model
A,D
0.38
similarities. The similarity between products is calculated through Jacard’s similarity coefficient. Products with maximum coefficient are assigned to a family. The process is repeated until the similarity coefficient is less than a threshold value previously defined.
3 Product families selection A reconfigurable manufacturing system focuses on manufacturing the product families at the same system, which is configured to produce each family. Once a family is manufactured, the system is reconfigured for producing the following family effectively. In each change of the configuration, the manufacturing company incurs in a changeover cost, which depends on the current configuration and the destination configuration [3]. The time required for the system reconfiguration is considered insignificant as the RMS should be designed for rapid reconfiguration. In the case of joining all the products in a single family, the manufacturing system is composed of all the machines required for manufacturing the products. In this situation, the company does not incur changeover costs, but idle machines and/or machines whose functionality is not fully used exist. Besides, the capacity of the machines is normally higher than the one required for manufacturing each product independently. In the case of selecting a family for each product, the company has to face the costs of system changeover, but the number of idle machines is minimised while their functionalities and capacities are both fully utilised. Finally, a smaller number of families than the number of different products may be selected. In this situation, although the changeover cost is not avoided, the changes are few and consequently the cost is low. Moreover, the idle machines are fewer than in the first case. The functionalities and capacities of the machines are not fully used but the utilisation rate is higher if compared to the first case. Concluding, the key parameters to take into consideration for selecting product families are the changeover cost of the system, cost of idle machines, cost of the underutilisation of the functionality of the machines and cost of the under-utilisation of the capacity of the machines. The selection of families can be solved calculating the cost of each level in the dendogram, and the level with the lowest cost will be selected. Thus, all the possible solutions are evaluated. A high number of products involve lots of calculations, which may require years for solving. Therefore, for the selection of the families, a model that includes the key parameters expressed above and facilitates this selection is required.
For the effective working of the RMS, a family of products is selected for being produced. When finishing, the following family is ready to be produced within a specificallydesigned production system. This process is repeated until the production of each product family. New orders for launching products to the market and the limitations of the storage capacity of the companies are two reasons that require the production of the same products again. Therefore, when the production of the last product family has been completed, the system is reconfigured for producing the first product family again, and the process is repeated. Only when the companies add or remove products from their portfolios, or their demands are very different, the methodology may select different product families and different sequences of production. This problem is very close to the travelling salesman problem (TSP), which seeks to identify an itinerary that minimises the total distance travelled by a salesman who has to visit a certain number of cities once, leaving from one of them (base city) and returning to this one. Therefore, some similarities may be highlighted. First, cities in the TSP are product families in RMS. Second, the goal in the TSP is to minimise the total distance travelled. In RMS, the goal is to minimise the total cost. Finally, in the TSP the salesman has to arrive at the base city, and in RMS when the last family has been produced the system is reconfigured for the first one. In the model proposed, level is referred to the different group of families that can be formed, as expressed above. For example, the dendogram shown in Fig. 2 presents four levels, one between 100% and 70% similitude (four families), between 70% and 56% (three families), 56% and 38% (two families), and the last between 16% and 0 (one family). This model solves a TSP in each level and selects the level which presents the minimum cost. Therefore, the problem to solve is a TSP-multilevel. As the TSP is a NPcomplete problem [13], the selection and sequence of product families is NP-complete too.
PRODUCTS B
C
A
D
L=1 70%
L=2 SIMILARITY
B,C
56%
L=3 38%
L=4 Fig. 2 Dendogram
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The notation used for model development (indices, parameters and variables) is shown in Table 4. The objective is to select the level of the dendogram that minimises both the cost for changing between families (changeover costs), and the cost for the under utilisation of the resources of each machine while producing those families. The constraints to take into account are: 1. Only one of the levels in the dendogram is selected (constraint 4). 2. All the families from the selected level will be produced, and none of the families will be produced for the rest of the levels (constraint 5). Besides, the number of variables for the changeover of the production system (T) must be equal to the number of families (constraint 6). 3. In each level l, the families are manufactured one by one following a production order, and at the end the system is configured for manufacturing the initial family. Variables T can exist before one family (constraint 7) and after one family (constraint 8) only. The same occur in the travelling salesman problem (TSP) and assignment problems. 4. Subtours within each level are not feasible. Therefore, the Miller-Tucker-Zemlin constraints are introduced [14]. This set of constraints avoids that, as for example, five families were produced in two different ways, as: 1-2-3-1-... and 4-5-4-...(constraint 9). 5. Sign constraints of binary variables (T, Q, and K) and auxiliary variables (U).
Therefore, the mathematical model is the following: Min
L1 XXX i2Fl j2Fl l¼1 j6¼i
Rij Tijl þ
L XX
Hi Qil :
(3)
i2Fl l¼1
Subject to L X
Kl ¼ 1;
(4)
Qil ¼ Nl Kl 8l ¼ 1; :::; L;
(5)
l¼1
L X i2Fl
X X i2Fl
Tijl ¼ Nl Kl 8l ¼ 1; :::; L 1;
(6)
j 2 Fl j 6¼ i
X
Tijl 1 8i 2 Fl 8l ¼ 1; :::; L 2;
(7)
Tijl 1 8j 2 Fl 8l ¼ 1; :::; L 2;
(8)
j 2 Fl j 6¼ i X i 2 Fl i 6¼ j
Table 4 Indices, parameters, and variables
Nl Tijl þ Uil Ujl Nl 1 8i 2 Fl 1
Indices i, j l Parameters L Fl Nl Rij Hi Variables Tijl Qil Kl Uil
Families Level Number of levels Set of families to produce in level l (l =1, ...,L) Number of families to produce in level l, so Nl=|Fl| Cost of reconfiguration from family i to family j (i∈Fl ; j∈Fl ; i≠j) Cost of the under-utilisation of machines’ resources while producing family i 1 if family i is produced just before family j, within level l (i∈Fl ; j∈Fl ; i≠j ; l=1, ...,L-1) 1 if family i within level l is produced (i∈Fl ; l=1, ...,L) 1 if all the families within level l are produced (l=1, ...,L) ≥0, auxiliary variables for avoiding subtours (i∈Fl-1 ; l=1, ...,L-2)
8j 2 Fl 1 j 6¼ i 8l ¼ 1; :::; L 2; Tijl ¼ f0; 1g 8i 2 Fl 8j 2 Fl j 6¼ i 8l ¼ 1; :::; L 1;
(9)
(10)
Qil ¼ f0; 1g 8i 2 Fl 8l ¼ 1; :::; L;
(11)
Kl ¼ f0; 1g 8l ¼ 1; :::; L;
(12)
Uil 0 8i 2 Fl 1 8l ¼ 1; :::; L 2:
(13)
As the number of families within level l (Nl) is L-l+1, then the number of variables and constraints may be
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calculated by the number of levels L, which fits with the number of products. In Tables 5 and 6, the number of variables and constraints of the model are presented in both compact and expanded forms. The families are manufactured in cyclical order, pointing out that it does not matter which family is manufactured first, as long as the order remains the same. Costs associated with the manufacturing order do not exist at all. The parameters required for solving the model are described in Table 4. Most of them (L, Fl, and Nl), can be taken directly from the dendogram, but the rest (costs of reconfiguration and under-utilisation) are unknown and they must be estimated. 3.2 Cost methodology For solving the model, accuracy costs are essential. The literature does not present any methodology for estimating costs required for changing modules or machines on reconfigurable manufacturing. In this section, a specific methodology for obtaining those costs that takes into accounts the singularities of RMSs is presented and developed. Instead of doing a direct and global estimation for both costs, a strategy “divide and defeat” has been followed. On one hand, reconfigurability costs have been decomposed into three different parameters, each with its own cost. Those parameters are: – – –
A single module of a machine is removed or included, A machine is removed, and A machine is included.
On the other hand, under utilisation costs have been decomposed into two parameters, which are estimated. These parameters are: – –
A module is not used when manufacturing a product from a certain family, and A machine (with all its modules) is not used when manufacturing a product from a certain family.
Costs of reconfiguration deal with incurred costs for changing the configuration of the system for producing a family to another configuration for producing the
Table 6 Number of constraints Constraints
Number of constraints Compact form
Expanded form
(4) (5) (6) (7–8)
1 L L-1 L P 2 ðlÞ
1 L L-1 L2 þ L 6
(9)
L P
l¼3 L3 3L2 þ2L 3
ðl 1Þðl 2Þ
l¼3
following family. These costs have to be calculated for each pair of families in each level of the dendogram. The last level contains one family and therefore the reconfiguration of the system is not required. This is shown in the objective function of the model, which the first term values the cost of reconfiguration until level L-1. Costs estimation starts with the identification of the components which compose the products that form both families. Those components are produced by a specific module belonging to a machine. Machines in RMSs are composed of different modules each of them develop a specific functionality. The example with four products developed in Section 2 is presented for costs estimation. Data required for the estimation are relationships among products and components, among components and the modules that produce them, and among modules and its machines. Moreover, parameters in which are decomposed the cost of reconfigurability and cost of under-utilisation of resources are estimated. Matrix in Table 7 shows the components that composed those products. This matrix is composed of coefficients aij, where: 1 if product i is composed of component j : aij ¼ 0 otherwise
Compact form
Expanded form
Modules for manufacturing those components together with their machines are presented in Table 8. The dendogram shown in Fig. 2 presents the different families that can be formed depending on the similarities among products. From level 1, four families are obtained, each of which are composed of one product. The name of each family is the name of the products that compose that family. Therefore, the families in level 1 are: A, B, C
Tijl
L P
lðl 1Þ
L3 L 3
Table 7 Products-components matrix
Qil
l¼2 L P
l
L2 þL 2
Table 5 Number of variables Variables
Kl Uil
Number of variables
l¼1
L L P
l¼3
L ðl 1Þ
L2 L2 2
A B C D
1
2
3
4
5
6
7
8
9
10
11
12
1 0 0 0
1 1 0 1
1 0 0 0
1 0 0 0
1 1 1 1
0 1 1 0
0 1 1 0
0 0 0 1
0 1 1 0
0 0 0 1
0 0 0 1
0 0 1 0
351 Table 8 Machines, modules, and components
Table 10 Cost of reconfiguration from family A to family D
Machine
Module
M1 M2
M3
M4
Modules
Components
M11 M21 M22 M23 M31 M32 M33 M41
1 2, 4, 10 3 11 5, 6 7, 9 12 8
M11 M21 M22 M23 M31 M41
and D. From level 2, three families are obtained, one composed of two products (family BC, and the rest composed of one product (families A and D). From level 3 two families appear, both composed of two products (families BC and AD). Finally, from level 4 one only family is obtained composed of the four products (family BCAD). Cost of parameters regarding reconfigurability must be estimated based on experience. The first parameter deals with if a module from a machine is used for producing one product but not other products. The second and third parameters deal with if a machine is required for producing one product but not other products. Their estimation are presented in (Table 9). An example of cost estimation for families’ reconfiguration from A to D is presented in Table 10. This table shows that machine M1 is used when producing A but not when producing D. Therefore, M1 has to be removed and its corresponding cost (β) is incurred. On the contrary, machine M4 is not required for producing A but it is required for producing D and therefore it must be included in the new system configuration together with its cost (γ). Machine M3 is required for both families, so no cost is incurred because it remains in the new system. Finally, machine M2 is composed of three different modules. Note that the machine is required for both families, so it is not removed. Module M21 is necessary for both families, and it remains in the machine (without cost). Modules M22 and M23 are required only for one family and they must be removed and included, respectively. Costs of under-utilisation of resources deal with incurred costs due to not using machines or modules. These costs have to be calculated for each family formed in any level of the dendogram. The first level of the dendogram is composed of families containing one single product. As the RMS is configured for producing a specific family, when producing a family composed of one product the system is installed with the capacity and functionality
Machine
M1 M2
M3 M4
Family A
D
1 1 1 0 1 0
0 1 0 1 1 1
Parameter
Cost
β – α α – γ
5 0 1 1 0 6 13
needed [2] allowing full utilisation of the resources installed. Consequently, the cost of under-utilisation for families composed of one product is zero. Parameters about costs of under-utilisation are estimated based on experience. The first parameter deals with if a module of a machine is not used when producing a product of the family. The second parameter deals with if a machine is not used when producing a product of the family. Their estimation can be seen in Table 11. Table 12 shows the estimation of family ADBC. For example, focusing in machine M2, when producing A the module M23 is not used, and the cost (δ) is incurred. When producing D is now the module M22 which is not used and another cost δ is incurred. However, when producing B two modules are not used (M22 and M23) and a cost 2δ must be added. Finally, for producing C the machine M2 is not necessary, so cost ɛ is required. The total cost is: δ +δ +2δ +ɛ=4δ+ɛ. Once both cost for reconfiguration and under-utilisation have been calculated, the mathematical model formulated above can be solved. A branch and bound method has been selected for solving it. The input information can be easily automated, with minor human intervention. The cost methodology requires information from the bill of materials (BOM), which is stored in the information system of the company. The BOM informs about the components of a specific product and the machine that performs the required operations. The human intervention consists of the estimation of the five parameters identified above on reconfigurability and under-utilisation costs. The values of these parameters are introduced in the information system and linked with the BOM for obtaining the costs required for the model.
Table 9 Cost estimation for reconfigurability parameters
Table 11 Cost estimation for parameters regarding under-utilisation of resources
Parameter
Symbol
Cost
Parameter
A single module of a machine is removed or included A machine is removed A machine is included
α
1
β γ
5 6
A module is not used when manufacturing δ a product from a certain family A machine (with all its modules) is not used when ɛ manufacturing a product from a certain family
Symbol Cost 1 7
352 Table 12 Cost of under-utilisation in family ADBC Module
M11 M21 M22 M23 M31 M32 M33 M41
Machine
M1 M2
M3
M4
Product A
D
B
C
1 1 1 0 1 0 0 0
0 1 0 1 1 0 0 1
0 1 0 0 1 1 0 0
0 0 0 0 1 1 1 0
Table 14 Estimation of cost of under-utilisation of the families (Hi)
Parameter
Cost
3ɛ 4δ+ɛ
21 11
– 2δ 3δ 3ɛ
0 2 3 21 58
4 Validation of the model Full cost estimation for the above example with four products is developed with the estimation of parameters presented in Tables 9 and 11. Estimation of both costs of reconfigurability among all the families and cost of under-utilisation of each individual family are developed as described above, and the final results are shown in Tables 13 and 14. These data are the values of the parameters Rij and Hi required for solving the mathematical model proposed, which presents 39 variables and 30 constraints. The model has been solved with a linear programming software, reaching the objective function a value of 38, being the activated variables the following: K2 - TBC,A,2 TA,D,2 - TD,BC,2 - QBC,2 - QA,2 and QD,2. The activation of K2 indicates that families in level 2 have been selected, which is corroborated by the activation of the rest of the variables, all of them belonging to level 2. This level is composed of three families, family BC composed of products B and C, family A composed of product A, and family D composed of product D. Besides, variables of reconfiguration (Tijl) indicate the scheduling of the families to produce. In this case, the families are produced in the following the cyclical order presented in Fig. 3. Note
Fig. 3 Order of families to produce
BC
A
B
C
D
AD
BC
ADBC
0
0
0
0
16
8
58
that it does not matter which family is produced first, the total cost remain the same. For validating the mathematical model, a study about the possible solutions of the scheduling of families in each level and its corresponding cost is required. This study is shown in Table 15, which represents all possible solutions with their corresponding costs. Note that all possible scheduling of families are included because all the families are produced in cyclical order. For example, in level 2 the scheduling BC→D→A is not presented in the table because it is included in A→BC→D. From Table 15 it can be deduced that both possible scheduling in level 2 present the minimum total cost, and any of them can be selected as the best solution. This result validates the model formulated, which has selected the scheduling A→D→BC belonging to level 2.
5 Conclusion This paper has developed a methodology that allows a feasible reconfiguration of a production system. For achieving this, a mathematical model for selecting the families of products that minimise the costs of reconfiguration and under-utilisation in RMSs has been developed. The major difficulty is the value of both costs which are unknown, and accuracy estimation is required. This has been solved dividing the costs into five parameters, which are easier to estimate. A methodology has been presented for a complete cost estimation based on those parameters. Finally, the model has been validated through the results obtained with an example. The selection of families among several possibilities is a highly complex problem (NP-complete), and therefore calculations for the resolution of the model grow exponentially together with the number of products. Therefore, the use of heuristic methods is suggested. Besides, the
A Table 15 Total costs of all possible solutions Level
Scheduling of families
Cost of reconfiguration
Cost of underutilisation
Total cost
1
A→B→C→D A→B→D→C A→C→B→D A→C→D→B A→D→B→C A→D→C→B A→BC→D A→D→BC AD→BC ADBC
40 41 40 41 40 40 30 30 30 0
0 0 0 0 0 0 8 8 24 58
40 41 40 41 40 40 38 38 54 58
D Table 13 Estimation of cost of reconfiguration among the families (Rij)
A B C D AD BC
A
B
C
D
AD
BC
0 8 14 13 – 9
7 0 7 7 – –
12 6 0 12 – –
13 8 14 0 – 9
– – – – 0 16
8 – – 8 14 0
2 3 4
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capacity of the machines have been considered unlimited. Thus, the inclusion of the capacity of the machines and the product demands in future models are required. Acknowledgements This research has been partly funded by the Spanish Ministry of Science and Technology; project SIDIFA (DPI2002-01095). The authors wish to acknowledge the Spanish Government for their support.
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