Manufacturing cell loading rules and algorithms for connected cells

Planning, Design, and Analysis of Cellular Manufacturing Systems A.K. Kamrani, H.R. Parsaei and D.H. Liles (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

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Manufacturing Cell Loading Rules and Algorithms for Connected Cells Giirsel A. Siier a, Miguel Saiza, Cihan Daglib and William Gonzalez c aIndustrial Engineering Department, University of Puerto Rico-Mayagiiez, Mayagiiez, PR 00681-5000 bEngineering Management Department, University of Missouri-Rolla, Rolla, MO 65401 CAvon Lomalinda Incorporated, San Sebastihn, PR 00755

1. INTRODUCTION Many manufacturing companies are using Cellular Manufacturing (CM) techniques and converting their conventional manufacturing facilities to manufacturing cells. In a survey of 54 US manufacturing companies, WemmerlSv and Hyer [22] found out that 60% of the companies that responded are implementing CM techniques. Half of them have only cells, whereas the remaining half have cells and dedicated machines. The average number of cells they have is 6. However, the number of cells varies between 1 and 35. This substantiates the claim that CM is gaining importance. Cellular Manufacturing can be defined as the implementation of Group Technology (GT) principles in a manufacturing environment. The central theory is that various situations that require decisions can be grouped together based on pre-selected, commonly shared criteria and that decision that applies to one situation in the group will apply to all of them in that group. The application of Group Technology to manufacturing is achieved by identifying the items with either similar design or manufacturing characteristics and grouping them into families of like items. Ham [12] said that GT is the realization that many problems are similar and by grouping similar problems, a single solution can be * This study has been supported by the National Science Foundation under Grant No. DDM-9113901 and Avon Lomalinda Incorporated.

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found to a set of problems, thus saving time and effort. Ranson [17] mentioned that GT is a logical arrangement and sequence of all facets of company operation to bring the benefits of mass production to a high variety, mixed quantity production environment. Gal]agher and Knight [8] defined it as a manufacturing philosophy that identifies and exploits the underlying sameness of items and the processes used for their manufacture. The benefits derived from Cellular Manufacturing include reduced work-in-process inventory and setup time, improved product quality, easier scheduling, better visibility of product schedule status and quicker feedback of manufacturing deficiencies.

2. CONTROL OF MANUFACTURING CELLS There are two major stages in a Cellular Manufacturing implementation: 1. Designing manufacturing cells 2. Controlling manufacturing cells Designing manufacturing cells consists of several tasks such as family formation, determination of machine requirements and cell formation, internal cell design and layout, forming product versus feasible cell matrix, etc. Having focused the entire manufacturing system into smaller units (focused factories) and designed the manufacturing cells, there is a need to devise a system to determine the items to be manufactured, how many of them should be made, or in which order they should be processed at each workstation. MRP can answer these questions, but it does not address the issue of how to produce in the most efficient way. It follows that MRP and CM are not conflicting but are complementary as specified by WemmerlSv [23]. There are two activities of special interest in Cellular Manufacturing in connection with how to control the focused units: 1. Cell Loading 2. Cell Scheduling

Cell loading involves determining to which cell(s), among the feasible cells of a focused unit, products of that family should be assigned, how many units should be produced and in what order. On the other hand, cell scheduling deals with the scheduling of operations in a cell through several workstations once a product has been assigned. This includes the determination of start times, completion times, lot sizes and transfer sizes. In some cases, finding the order of products in a cell might be included among the cell scheduling tasks. This paper focuses on cell loading aspects in CM.

99 The cell loading process in a focused unit requires three major tasks to be performed: 1. Select a product 2. Select a cell 3. Find the order of products One of the decisions to be made is to select a product from the family in question. Another decision point is to select a cell from the focused unit. If a cell is selected first and then a feasible product is chosen, this type of search is called cell priority. If the search process is reversed then it is called product priority. Another task to be performed is to find the order of products in a cell once all the products to be produced in that cell have been determined. However, this task is usually inherent in the product selection process and usually product selection order implies the sequence of the production as well. The overall objectives of a cell loading process are to minimize the work-in-process inventory, minimize tardiness, maximize the utilization of the cells and balance the load among the cells.

3. L I T E R A T U R E REVIEW Cellular Manufacturing, as an application of Group Technology, has emerged after years of development and theoretical considerations in some academic and application-oriented arenas. In recent years, the literature on GT and CM has shown an emerging interest in the areas of cell formation, layout, product standardization and coding and the application of job shop scheduling rules for some cell scheduling problems. However, little attention has been paid to the cell loading problem in the literature so far. The objective of this study to cover this gap in the literature and meet the need in the industry. Some group scheduling methods available as discussed by Ham [12] are: 1. Flowline Group Production Planning by Petrov. This technique is effective for a large-scale flowshop scheduling problem. 2. Work Loading and Scheduling in Manufacturing Cells. This method is concerned with the scheduling of work in a manufacturing cell. 3. Group Scheduling Technique (GST) by Hitomi and Ham. This technique provides the optimal solution for both group sequence and job sequence in each group for a single machine and the multi-stage manufacturing system. Greene and Sadowski [9], and Greene and Cleary [10] mentioned scheduling issues, benefits, drawbacks and system variables in a cellular manufacturing environment. Chisman [2] suggested a novel approach for a single cell for estimating sequence-dependent changeover time data and used this data in a

100 traveling salesman algorithm to find an optimal product ordering for a special case. Fry, Wilson and Breen [7] described a successful implementation of Group Technology in an industry. Taylor and Ham [21] discussed scheduling algorithms for a single part family and a set of part families. Sunduram [20] provided two heuristic algorithms to find near optimal sequences for GT shop scheduling problems to minimize makespan. Later, he compared the performance of the heuristics with integer programming solutions. Dale and Dewhurst [4] analyzed a GT cell under various conditions. They concluded that SPT minimizes the work-in-process inventory. Mosier, Elvers and Kelly [ 15] studied a GT shop using three job shop scheduling rules. Wirth, Mahmoodi and Mosier [24] mentioned that both labor scheduling and group scheduling heuristics need to be considered to control the cell effectively. In another study, Mahmoodi, Tierney and Mosier [13] compared the performance of traditional single-stage heuristics and the two-stage group scheduling heuristics. Miles and Batra [14] also simulated a manufacturing system with many similar cells. Their primary concern was to minimize the tardiness and the labor costs. Guang-Xun and Li-Hang [11] modified the job scheduling rules and used them in a GT scheduling application. Siier and Gonzalez [18] discussed the use of a fixed time bucket approach to synchronize the flow of materials in a cell when interruptions exist in the process. Espino [6] explained four cell loading algorithms to deal with the cell loading problem. Siier and Saiz [19] provided a simple classification scheme for cell loading problems.

4. P R O B L E M STATEMENT The term connected cells refers to the fact that the output from a cell becomes an input to the next cell, i.e., it takes at least two cells to complete a product. There are several possible combinations for connected cell configurations. In general, they can be grouped in two categories: 1. Pure serial systems (see Figure l a) 2. Serial-parallel mixed systems (see Figure lb) Assigning a product to a cell(s) requires careful consideration in a connected cells environment since the product will consume some of the available capacity in the succeeding cells as well. As a result, the assignment of a product cannot be made solely based on the initial cell(s) but it should consider the load on the affected cells simultaneously. In pure serial syStems, it is relatively straightforward to determine the impact in each cell because the product will go through each cell successively (in some cases, cell skipping is likely). When serial-parallel mixed systems are considered, cell loading task is further complicated since alternative routes emerge and some cells can serve as common cells to many others. This study concentrates on the common cell(s) case and

101

a) CELL 1

CELL 2

CELL n

CELL 3

b) CELL 1 CELL 2 CELL n

CELL 21 COMMON CELL

CELL 22 CELL 2n

Figure 1. Connected Cells Configuration. a) Pure Serial Systems. b) Serial-Parallel Mixed Systems. several different possible configurations are discussed in the following paragraphs. Each product family might be limited to one dedicated cell besides the common cell. In this case, cell loading is greatly simplified to the determination of the order of the products within a family. The results of the single machine scheduling literature can be used to perform the cell loading task. While this restriction may be unavoidable in some facilities, it does impose constraints on the system which will limit the machine utilization and the system flexibility. The product family versus cell matrix (Mij) describing this situation is presented in Figure 2a where (i) designates product families and (j) designates cells (Mij=l ff cell (j) is a feasible cell for product family (i); otherwise Mij=0). The number of dedicated cells and the number of product families is equal to (n). In other CM systems, a product family can be processed by any cell besides the common cell. In this situation, the cell loading task becomes more complicated than the previous case. It requires the assignment of the product families to the cells along with the order of the product families in each cell and the order of the products in each family. The results of the parallel machine scheduling literature (identical, uniform or unrelated) can be used for the cell loading purposes. This case is shown in Figure 2b where (n) represents the number of cells and (f) denotes the number of product families.

102

a) CELLS

COMMON

PRODUCT 2

FAMILIES

..

n-1

CELL

1

n-1

b) CELLS COMMON

PRODUCT 1

FAMILIES

f-1

2

..

n-1

CELL

1

1

""

1

1

1

1

1

""

1

1

1

1

1

""

1

1

1

1

1

""

1

1

1

c) CELLS

COMMON

PRODUCT FAMILIES 1

1

2

1

1

..

1

""

1

1

""

1

1

1

""

1

1

1

2

f-1

1

..

n-1

CELL

1 1

Figure 2. Possible Connected Cell Configurations. 2a) Dedicated Cells. 2b) Parallel Cells. 2c) Overlapping Cells.

103 This paper addresses the situation where the product family versus cell matrix takes any other form between the two extremes shown in Figures 2a and 2b. In other words, there is at least one product family that cannot be processed in all the cells or at least one product family that can be processed at least in two cells besides the common cell. This implies an overlapping between the cells for processing different product families, which makes the cell loading task a very complex one. The reason for the increased complexity comes from the fact that there are limitations to be considered in terms of assigning a product family to the cells. In other words, the feasible cells are the only ones a product family can be assigned. Unfortunately, the parallel machine scheduling results are no longer applicable in this situation. A typical product family versus cell matrix representing this case is given in Figure 2c. In the previous paragraphs, various manufacturing cell configurations and their impact on cell loading were discussed. It is also important to mention that capacity of the common cell(s) has to be taken into account in the cell loading task as well. The number of common cells and their processing capabilities will be important in selecting appropriate scheduling rules and/or algorithms. Furthermore, in some cases, the cell loading task might actually be centered on the common cell since it affects the production in many other cells and also most likely the capacity is tighter in the common cell. In the literature, the cell loading task in CM where there are several overlapping cells and those cells are connected through a common cell(s) has not been addressed before. There is a need to define some rules to handle this problem and then evaluate their performance. This study aims to meet this objective with an application in a real manufacturing setting.

5. THE RULES USED Eleven rules included in this study are grouped in six categories as described in the following subsections. The example problem given in Figure 3 is used throughout this section to facilitate the explanation of the rules. Figure 3a shows the current load on five cells, Figure 3b denotes the feasible cells for each product and finally Figure 3c presents the demand figures for each period for five products considered. 5.1. Search Priority: Cell Priority (CP), Product Priority (PP) Search priority determines the way the search process is carried out. If the cell priority is used, first a cell is chosen and then the search shifts to find a feasible product for that cell. Referring to the example problem, the first task would be to choose a cell among five cells by considering one of the rules described in section 5.4. Later, the search would focus on selecting a feasible product to run on the selected cell among five candidate products by considering the rules described in sections 5.2. and 5.3. The search process is reversed if the product priority is

104 used. In other words, in this case first a product is selected and later a feasible cell is searched to run the selected product.

5.2. Primary Product Rule: Earliest Due Date (EDD) The products are sorted by EDD first. When there is a tie, one of the secondary rules mentioned in section 5.3 is used to break the tie. In the example problem, products P 1, P2 and P3 have the same due date (week 1). Therefore, they have to processed before products P4 and P5, which are due in week 2. 5.3. S e c o n d a r y P r o d u c t Rules: Number of Feasible Cells (NFC), Number

of Cells Required (NCR) A secondary rule is used to select the appropriate product when there is a tie with respect to EDD rule. In the example problem considered, there is a tie among products P 1, P2 and P3 since they all have the same due date (week 1). The NFC rule checks the number of available cells among the feasible cells that each candidate product can go through at the time of a product selection. In the above example, P1, P2 and P3 have 3, 2 and 2 feasible cells, respectively. However, cell 2 and cell 5 are fully loaded in period 1. As a result, we have to consider the number of available feasible cells in selecting the product. They are 2, 1 and 1 for P1, P2 and P3, respectively. The selection will be made based on these results (minimum or maximum depending upon the rule combination used as explained in sections 5.6 and 6). If the product with maximum NFC value is preferred then P1 would be chosen. If the user decides to choose one with minimum NFC value, then the decision would be to select P2 or P3The NCR rule calculates the number of cells required to complete each candidate product by its due date and uses this information to choose the next product to be assigned. In this case, the number of cells required for P 1, P2 and P3 are 0.60, 1 and 2 cells, respectively. If the product with maximum NCR value is given higher priority, then P3 would be assigned the first. If the decision is to favor minimum NCR value, then P 1 would be chosen. These rules are applied in a dynamic manner, i.e., the values of NFC or NCR are revised after each product assignment. 5.4. P r i m a r y Cell Rules: Cell Load (CL), Number of Feasible Products

(NFP), Product Mix (PM) The CL rule chooses the cells depending on the current load. In the example given above, the current loads on cells are 20, 40, 10, 25 and 40 hours for cells 1, 2, 3, 4 and 5, respectively. The selection in this case will be made based on these results (minimum or maximum depending upon the rule combination used). If minimum CL is the basis for selection then cell 3 is selected. However, if the user wants to select the cell with maximum load, the decision would be to select either cell 2 or cell 5.

105

a) PERIOD 1 15 HR

CELL 1

.....

PERIOD 2

20 HR

!-I

_-iI .........

CELL 2 CELL 3

[ -:[

CELL 4

I ....

CELL 5 lO HR

25 HR

40 HR

80 HR

b) CELLS

PRODUCTS

P1 P2

1

2

3

1

1

1

1

1

5

1

P3 P

4

1

1

1

1

4

1

P5

1

1

c) Demand Product

Weekly Production Rate

Week 1

Week 2

P1

60

.

100

P2

40

.

40

P3

80

.

40

P4

-

60

90

P5

-

60

120

.

.

.

.

Figure 3. Example Problem. 3a) Current Load on Cells. 3b) Feasible Cells for Each Product. 3c) Demand Figures and Production Rates.

106 The NFP rule checks the number of feasible products that can be assigned to each cell at the time of a cell selection. Since there are only three products that can be assigned to cells currently, we will determine the number of feasible products for each cell considering those three products. The number of feasible products for cells 1, 2, 3, 4 and 5 are 2, 2, 2, 0 and 1, respectively. If the user decides to choose the cell with maximum NFP then cells 1, 2 or 3 are candidates. If minimum NFP is the basis for selection, then cell 5 is chosen (NFP = 0 implies that there is no product to assign to cell 4, and therefore, it is not considered at all). PM takes into account the current product mix in a cell. There are 2, 3, 1, 2 and 2 products already assigned to cells 1, 2, 3, 4 and 5, respectively. The cell selection process for this case is discussed in section 5.6. The values of these rules are updated dynamically as well.

5.5. C o m m o n Cell Capacity: M i n i m u m Load The utilization of common cell(s) is important in selecting the set of products to be considered. Whichever common cell has minimum load, the set of products that needs to be processed on that common cell is chosen and an appropriate product from that set is finally selected using the rules mentioned above, i.e., the common cell utilization directs the search toward obtaining the right set of products. Consider the additional information given in Figure 4 for the same problem. Figure 4a includes more products and common cell type information whereas Figure 4b presents the current load on common cells. Demand Figures for each product are presented in Figure 4c. Since the common cell CC1 has lower load, then the set of products that will go through CC1 is considered for product selection namely, P 1, P2, P3, P4 and P55.6. S e l e c t i o n Criterion: Maximum (MAX), M i n i m u m (MIN) In comparing the candidate products and cells, the decision might be to select the one with a maximum value or minimum value. For example, if we choose PM as the cell rule to be used, there are two possibilities. We might choose cell 3 if we would hke to level the number of products assigned to each cell since it has the minimum number of products (MIN). On the other hand, the policy might be to reduce the product mix in most of the cells by sacrificing one or two cells. In this case, the cell with maximum product mix is selected (MAX). In the same example, that would correspond to cell 2 with 3 products already assigned to it.

6. COMBINATIONS OF THE RULES The rules described in the previous section are combined in different ways and 48 possible combinations are created. Twenty-four of the rule combinations are of cell priority type and the remaining 24 are of product priority type. The Figures 5

107

a) CELLS PRODUCTS

P1

1

2

1

1

3

4

5

COMMON

CELL CC1 CCl

,,

P2 P3

1

1

.,

1

P4

1

1

1

P5

1

P6

1

1

1 ,' CCl "CC1 1 CC1 1 ~CC2 i 1 !CC2 !CC2 ,=

P7

1

I

P8

1

1

b) PERIOD 1

PERIOD 2

CC1 CC2 c) Demand Product

Week 1

Week 2

P1

60

-

,,

,,

,

Weekly Production Rate 100

P2

40

-

40

P3

80

-

40

P4

-

60

90

P5

-

60

120

P6

50

-

100

P7

40

-

80

P8

-

80

120

Figure 4. Extended Example Problem. 4a) Products versus Cells Matrix. 4b) Current Load on Common Cells. 4c) Demand Figures.

108

and 6 show the structuring of combinations of rules for the cell priority and the product priority cases, respectively. For example, the combination CP/CL/MinfEDD/NFC/Min means that this application is a cell priority case where first a cell is selected based on minimum cell load. Later, a product with EDD is searched for. If there are several products with the same due date, then the product with minimum number of feasible cells is given the highest priority. The complete description of the rule combinations is included in Appendix A.

7. T H E A S S U M P T I O N S MADE The most important assumption made in this study is the presence of connected cells through a few common cells. Each product will visit only one common cell depending upon the processing requirements. This implies that inter-cell material transfers exist. Components and raw materials arrive at one end; the parts are manufactured, the subassemblies are prepared in the cell and later sent outside the cell for further processing. Finally, they return to the cell for the final operations to be completed. Another assumption made is that the production rate of a product on any of its feasible cells is the same. In addition to these, the production can start earlier. In other words, if there is available capacity in a cell in the current period, a feasible product with a future demand can be assigned now to utilize the limited resources better. A product can be assigned to more than one feasible cell when it is necessary to complete it by its due date, i.e., lot splitting is allowed. Naturally, the number of cells a product can be assigned is limited by the maximum number of feasible cells it has. The purpose of this policy is to prevent a product from becoming tardy as long as there are feasible and available cells to assign it. Even though the objective is to minimize the number of tardy products, in the case when a product has a very large processing time, it might adversely affect the performance of the entire system by delaying the starting times of other products. A product is selected after two decision levels. If there is still a tie among the products, then the first product found with the best value is selected. In the case of a cell selection, if there is a tie among the cells with respect to the cell selection rule used, then the first cell that achieves the best result is chosen as the cell to be loaded. The order of the product selection also determines the order of processing the products in a cell. Moreover, the setup times are assumed to be sequence.independent and negligible. Therefore, they are not considered in this study.

109

ICPI I

I CLI

I

I

IMINI EDD/NFC

!

INF'! IMA:~

IMIN I

A

A

I

I I MAX~

! ! I MIN

A

A

I PMI I

EDD/NCR

EDDINFC

x

Ii,NI I~Ax

Ii

EDD/NCR

Figure 5. Structuring of Rule Combinations for Cell Priority Case

EDD/NFC

I

EDDINCR

I MAX

I

I MIN

IM;N I I ~~.~ ~!~l ~, MI I CL! II !_PI ! PMi I I_~L II NFPI I PMi I IcEI

//, _

,N I IMAx

N

/

MIN

/

M~eX

I IN =PI IPMI MIN

I IMAXI

Figure 6. Structuring of Rule Combinations for Product Priority Case

110 8. A L G O R I T H M S In this section, the algorithms for both the product and the cell priority searches are discussed. The flowchart of the algorithm for product priority search is presented in Figure 7 and its steps are given below: 1. Sort products by EDD. 2. Divide products into different sets based on common cell type required. 3. Find common cell with minimum utilization and identify corresponding set of products. 4. Select next unassigned product from the set If there is a tie, use secondary to rule (NFC or NCR) to select a product. 5. Find a cell to assign the product (CL or NFP or PM) (If none go to step 4). 6. If the product can be completed on the selected cell before its due date, then assignment process is complete. Revise cell load and common cell load information and go to step 3. Otherwise, continue. 7. Search for another cell to load the capacity requirement beyond its due date in the original cell. Repeat this step until a) load is completely assigned or b) no feasible cell is available to complete load (maintain the balance of the load as a task to be considered for later periods) 8. Revise cell load and common cell load information and go to step 3. The application of the algorithm to the example problem mentioned previously will be appropriate to complete the discussion. 1. Products by EDD: P 1, P2, P3, P6, P7, P4, P5, P82. Products on common cell 1, Sccl=(P 1, P2, P3, P4, P5) Products on common cell 2: Scc2=(P6, P7, P8) 3. Common cell with minimum utilization is CC 1. Therefore, Scc 1 is selected. 4. Products P1, P2 and P3 have the same due date. Using NCR as a secondary rule, we obtain 0.60, 1 and 2 cells for P 1, P2 and P3, respectively. Assuming that we are interested in minimum (MIN) NCR value, P 1 is chosen. 5. Feasible cells for P1 are cells 1, 2 and 3 with loads of 20, 40 and 10 hours, respectively. Obviously, cell 2 is not available for week 1. Using CL as basis for cell selection and favoring minimum (MIN) value, cell 3 is chosen as the cell to assign P 16. P 1 requires a total of 24 hours (number of cells x hours in a week). Since the completion time of P 1 on cell 3 will be 34 hours, the assignment process for P 1 is completed successfully and loads for dedicated and common cells are revised. The application continues with step 3 again.

Ill

Sort Products by EDD

$

EXvide products into sets based on common cell used

..•

Choose~cell (X) with minimum utilization

.=,

Choose next product from set (X)

Yes

Use secondary rule to break the tie

J

selecta cell by

-I

usingacellrule

No Assign the product to the cell chosen Yes

Product No Do ng

1 Select another cell for the balance of the load No Available?

Yes

Assign the

Figure 7. Flowchart for the Product Priority Approach

product

112 For the cell priority case, the order of the search process is reversed by selecting a cell to be loaded first. Once the cell has been identified, the set of products with minimum common cell utilization is determined. Finally, a feasible product from the same set is chosen using EDD and/or a secondary rule to assign to the cell identified previously.

9. E X P E R I M E N T A L CONDITIONS Experimentation has been performed in a real cellular manufacturing environment in Avon Lomalinda, Inc., a major jewelry manufacturer located in Puerto Rico. The operations are mostly labor intensive. The company has chosen low cost, light weight equipment and machinery to increase the flexibility of the manufacturing system.

9.1. Shop S t r u c t u r e All the production areas were converted to manufacturing cells by using GT concepts as shown in Figure 8. There is a total of 17 manufacturing cells grouped in four business units by using Focused Factory concepts in addition to the three common plating cells (manual, barrel and chain). The team size in a manufacturing cell varies between 20 to 30 employees depending upon the required production rate. This study focuses on the Business Unit 2 with 120 products and five cells. The products in the family are grouped in five different subfamilies. The classification of the products to the subfamilies and the allocation of the product families to the cells is given in Table 1. The last column in the Table denotes the number of products in each subfamily whereas the last row shows the number of products that each cell can process.

Table 1 Product versus Cell Matrix

Cells Subfamily Casting Earring Plastic Earring Stamping Earring Porcelain Earring Cuff Bracelet No. of Products

1 1 1 1 1 1 120

2 1 1 1 1 1 120

3 1 1 1 65

4 1 1 1 65

5 1 1 1 65

Products 33 18 37 20 12

113 9.2. Intra-cell M a t e r i a l T r a n s f e r The cells are arranged and equipped such that intra-cell material transfer follows a unidirectional flow. 9.3. Inter-cell Material T r a n s f e r Most of the operations of a product are performed in the cell. Then, the parts leave the cell to be plated in one of the plating cells depending on the process requirements. Later, the parts return to the same cell for the remaining operations to be completed as shown in Figure 8. Inter-cell material transfers indicate that the cells are connected even though some products don't visit plating cells. 9.4. T y p i c a l O p e r a t i o n s The typical operations performed in the Business Unit 2 are: casting, deburring, degating, inspection, cleaning, tumbling, putting sleeve, plating, removing sleeve, finishing and packing. 9.5. C a l c u l a t i o n of P r o d u c t i o n R a t e s An integer linear programming model has been suggested to determine the hourly production rates. Another decision variable is the number of employees to be assigned to each operation and or stage (a group of operations). The objective is to maximize the production rate (equation 1). There are three types of constraints in the model. The first constraint type (equation 2) guarantees that there are enough employees at each stage to reach the required production rate. The second constraint type (equation 3) ensures that the number of employees assigned to each stage/operation does not exceed the upper bounds. The upper bounds denote the number of machines available for each operation/stage. The machine limitations are given in Table 2. The last constraint type (equation 4) sets the upper limit for the availability of employees in the cell. The maximum number of employees that can be assigned to a cell is 30 due to space restrictions. The equations (5) and (6) provide the bounds on the decision variables. The notation used and the model formulated follows: R : hourly production rate Xi : number of employees required for stage/operation i U i : upper bound on the number of employees at stage/operation i PT i : unit processing time (hr) for stage/operation i W : upper bound on the total number of employees s : number of stages/operations

114

MANUFACTURING CELLS

CELL #1

CELL #2

........

CELL #16

I CELL I #17

I

MANUAL PLATING CELL BARREL PLATING CELL CHAIN PLATING CELL PLATING AREA

Figure 8. Pictorial Representation of the System

Table 2 Upper Bounds Used Cells Upper Bounds Number of Tumbling Machines N u m b e r of Casting Machines Number of Heat Sink Machines Maximum Team Size

3 1 12 30

3 1 12 30

3 1 30

3 1 30

3 1 30

115 Objective Function: MaxZ=R

(1)

Subject to: [(Xi)*(lfPTi)] - R > 0 i = 1,2,3 ..... s

(2)

Xi < U i i = 1 , 2 , 3 ..... s

(3)

s

Zxi_
(4)

i=l Xi, integer and positive for all i

(5)

R, integer and positive

(6)

9.6 Cell S c h e d u l i n g Considerations Although the cell loading precedes the cell scheduling task in the hierarchy of a planning process, the cell scheduling strategy needs to be considered before applying the cell loading procedure. A 4-hour time bucket-based synchronization approach is used for cell scheduling as described by Siier and Gonzalez [18]. As a result, 4-hour production rates are used in the cell loading process to be compatible with the cell scheduling approach. Four-hour production rates are calculated by simply multiplying hourly production rates determined using integer linear programming model by 4. 9.7 Data Input for Cell Loading The input data required for cell loading are demand figures, subfamily classification and 4-hour production rates. Demand forecast figures over 1990 and 1991 are grouped in four 6-month planning horizons, thus obtaining four different demand patterns, DP1, DP2, DP3 and DP4, respectively. A 6-month planning horizon consists of 13 2-week periods due to the marketing strategy of the company. An existing subfamily classification is used as is. The work schedule for manufacturing cells is 40 hours per week with only one shift and no overtime work allowed. However, the work schedule for common cells is more flexible with possibility of working overtime and/or second shift, when needed. 9.8 Software D e v e l o p e d The programs were developed in Turbo Pascal Version 6.0 along with Topaz version 3.0 database manager and objects from Object Professional version 1.1. The programs were run using an IBM compatible 486 PC.

116 10. PERFORMANCE MEASURES The four performance measures (PFM) included in this study are: 1. Number of Tardy Jobs (n T) 2. Total Tardiness (TT) 3. Maximum Tardiness (Tma x) 4. Average Cell Utilization (CUav)

11. THE BEST RULE COMBINATIONS FOR EACH PERFORMANCE MEASURE The discussion in this section is limited to the selection of the rule combinations considering its performance concerning a single performance measure only. The results obtained are based on the ranking procedure and the statistical analysis. 11.1. R a n k i n g Procedure This analysis is performed by using the simple ranking procedure. The values of the rule combinations with respect to a specified performance measure are obtained and later ranked starting from the best toward the worst. This procedure is repeated for each demand pattern independently. The complete list of ranked rule combinations for each performance measure is given in Appendix B. Having determined the ranking for four demand patterns, it is necessary to generalize the results and derive the set of rule combinations that consistently perform well with respect to a performance measure. 11.1.1. N u m b e r of Tardy Jobs The rule combinations listed below gave minimum n T values in three out of four demand patterns (DP1, DP2, DP4); 42/37/43/38, 41/25, 26, 29, 30/44, 47/48/27/28 However, the n T slightly increased when the third demand pattern was used in the order the rule combinations are presented above. The "/" between the rule combinations represents the deterioration in the value of the PFM whereas "," shows the identical performance. The cell priority search technique outperformed the product priority type since all the fourteen rule combinations recommended are of the cell priority type. The CL and PM rules both are equally represented in the set of the selected rule combinations. The NFP rule performed poorly. The NCR rule gave slightly better results than those of the NFC rule.

117 11.1.2. M a x i m u m T a r d i n e s s The following rule combinations resulted in the minimum Tma x values for the three out of four demand patterns as in the previous case;

48/30/37/25, 43/42/26, 29/38, 41/44,47 Similarly, Tma x slightly increased when the third demand pattern was used in the order the rule combinations are listed above. The performance of the CP search was superior to the PP search. Again, the NFP rule failed to produce satisfactory results. The NCR rule gave better results than those of the NFC rule. CL and PM performed equally well. 11.1.3. Total T a r d i n e s s The results obtained for TT were similar to n T and Tma x except that the order of the rule combinations slightly changed. The results are listed below;

42/37/43/38, 41/30/25/48/26, 29/44,47 The CP search outperformed the PP search and similarly, the NFP rule proved to be a poor choice for this PFM as well. The NCR rule produced better results than the NFC rule. 11.1.4. A v e r a g e Cell U t i l i z a t i o n The most difficult decision was the determination of rule combinations for CU. The four rule combinations listed below resulted in good performance only in three out of four demand patterns. This was the only PFM where the PP search produced acceptable results.

40/20,22,23 11.2. S t a t i s t i c a l A n a l y s i s To observe the variability of the set of outstanding rule combinations selected under certain variations of the first demand pattern, some replications of the experiment are conducted (A set of outstanding rule combinations consist of top 24 rule combinations with respect to a specified PFM). A normal random component is added to the demand figures. The Tables provided by Nelson [16] for the Analysis of Mean (ANOM) procedure are used to determine the necessary sample size. By the use of the Tables with (c~ =.05), power = .95, (A / o =3), and k = 24 treatments, the sample size needed is determined to be 7. As the non-normality of data is suspected, Kruskal-Wallis Test, the nonparametric counterpart of the one way analysis of variance, is conducted. Conover [3] presented the Kruskal-Wallis test as an extension of the

118

Mann-Whitney test from two independent samples to k independent samples. The hypotheses tests are: H o : All the 24 rule combinations have identical means H1 " The 24 rule combinations do not all have identical means

Table 3 Kruskal-Wallis Test Results

Performance Measure

Number of Tardy Jobs Maximum Tardiness Total Tardiness

Kruskal-Wallis

155.80 163.90 161.93

Significance Level

0 0 0

The Kruskal-Wallis Test shows that there is a significant difference between the mean responses of the rule combinations with respect to the performance measures n T, Tma x, and TT as shown in Table 3. Since the null hypothesis is rejected, the next issue is to find which rule combinations differ. A multiple comparison procedure called Fisher's least significant difference, based on the ranks rather than the data, is used to achieve this task for n T. The results show that top fourteen rule combinations have equal performance, which is very consistent with the results of the Ranking procedure.

12. THE BEST RULE COMBINATIONS C O N S I D E R I N G ALL P E R F O R M A N C E M E A SU R E S In this section, the objective is to select the rule combinations that perform well with respect to multiple-criteria. The use of the elimination techniques provides a set of decision rules to help a decision maker eliminate one or more alternatives to narrow the set of choices and perhaps even lead to a decision as mentioned by Canada and Sullivan [1]. The method is applicable when the alternatives can be measured or put in an ordinal rank.

119 12.1. R u l e v e r s u s Rule: C o m p a r i s o n A c r o s s P e r f o r m a n c e M e a s u r e s This method is called a dominance check. If one rule is better than or equal to some other rule with respect to all performance measures (should be better at least with respect to one PFM), the other rule is said to be dominated and can be eliminated. There was not any rule combination which dominated the others with respect to all the performance measures in this study. However, the rule combinations that could not be eliminated when each demand pattern is used are given in Table 4.

Table 4 The Rule Combinations not Eliminated

DP

Rule Combinations

1, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48 3

30, 37, 38, 41, 42, 43

4

25, 26, 27, 28, 29, 30, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48

The rule combinations 30, 37, 38, 41, 42 and 43 have not been dominated by any rule combination in four out of four cases. Therefore, the use of those rules is strongly recommended. The CL rule gave better results than the PM rule. The NFP rule performed very poorly in this case as well. When product selection rules are considered, the NCR rule outperformed the NFC rule. All the recommended rules use CP search priority. 12.2. R u l e v e r s u s Rule: C o m p a r i s o n C o n s i d e r i n g R o b u s t n e s s The rule combinations not eliminated in the previous section indicate that each rule combination was superior to another with respect to at least one PM. In other words, a rule combination may not be eliminated even though its performance is satisfactory with respect to only one measure and very poor with respect to the others. In this section, the robustness of each rule combination is measured considering the lowest rank it got during the simple ranking procedure

120 Table 5 Results of the Robustness Analysis

Rule Combination 42 37 43 30 25 48 41 38 26 29 44 47

Highest Rank

Lowest Rank

1 1 1 1 1 1 1 1 1 1 1 1

6 6 6 6 6 7 7 7 8 8 9 9

Average Rank 1.93 2.00 2.12 2.12 2.31 2.25 2.31 2.31 2.56 2.56 2.87 2.87

Standard Deviation 1.73 1.63 1.63 1.78 1.95 2.32 2.02 2.02 2.39 2.39 2.82 2.82

with respect to all the performance measures and demand patterns. The best rule combinations and relevant statistics are summarized in Table 5. In this case, NCR did better than NFC. However, CL and PM performed equally well. Cell priority rules outperformed product priority rules once again.

12.3. Rule v e r s u s Rule: Comparison Across Rules (Lexicography) The performance measures are ranked so that it is ascending for performance measures to minimize and descending for performance measures to maximize. For each performance measure, the rule combination which performs better is chosen. In case of a tie between two or more rules, break the tie selecting the rule t h a t performs better in the next performance measure or so on, until a single rule emerges or all the performance measures have been checked. If the tie cannot be broken, the set of rule combinations remains as the better set. Since there are four performance measures included in this study, there are twenty-four possible permutations of ordering the performance measures (n!). The analysis in this section is based on the assumption that the company is interested in minimizing nT, Tma x, TT and maximizing CU in the order of decreasing importance. The s u m m a r y of the results of the lexicographic analysis is given in Table 6. The rule combination 42 was the only best common rule combination in four out of four demand patterns. The details of the analysis are given in Appendix C.

121 Table 6 Results of Lexicographic Analysis

DP

Rule Combinations

1, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48 42/37/43/38, 41/30/25/26, 29/44, 47/48/

1, 6, 7, 9, 12, 13, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48

13. CONCLUSIONS There was no a single rule combination that performed very well with respect to all the performance measures. When performance measures were considered independently, it was observed that n T, TT and Tma x behaved similarly. The CP search gave the best results for those performance measures. The CL rule and the PM rule gave much better results than the NFP rule. NCR did slightly better than NFC. PM and CL performed equally well. However, the results varied significantly when CU was considered. Both CP search and PP search produced good results. Interestingly, none of the rule combinations recommended for CU were included in the best rule combinations for n T, TT and Tma x. When all the performance measures are considered simultaneously, the results of the robustness analysis might be used. Twelve rule combinations gave very close results out of which six have never been dominated in the experimentation carried out in this study, namely the rule combinations 30, 37, 38, 41, 42 and 43. The CP search proved to be superior to the PP search in this case as well. In general, choosing the best rule combination becomes more complicated if the performance measures do not have equal weight or if the user specifies a different preference among the performance measures. As mentioned before, experimentation has been performed under different demand patterns with many products and cells by using actual data. However, as in any other experimentation, it is still difficult to claim that the results obtained in this study are valid for other environments as well.

122 ACKNOWLEDGEMENT The authors are indebted to the current and former personnel of Avon Puerto Rico Operations, and particularly to Fernando Fernandez, Serafin Masol, Joe Quifiones, Dennis Roman and Florentino Quifiones for their support of our research activities and implementation efforts. The authors also would like to thank Ayse Siier and graduate student David Mir6 for their assistance in preparing the paper. REFERENCES [1]

[21 [3] [41

[5] [6] [71

[8] [9] [10]

[11]

[12] [131

Canada, J. R. and Sullivan, W. G., Economic and Multiattribute Evaluation of Advanced Manufacturing Systems, Prentice Hall, Englewood Cliffs, N.J., 1989. Chisman, J. A., "Manufacturing Cell: Analytical Setup Times and Part Sequencing," The International Journal of Advanced Manufacturing Technology, V 1, pp. 55-60, 1986. Conover, W. J., Practical Nonparametric Statistics, Second Edition, John Wiley and Sons, New York, NY, 1980. Dale, B. G. and Dewhurst, F., "Simulation of a Group Technology Cell," Engineering Costs and Production Economics, No. 8, pp. 45-54, 1984. Dillon, W. R. and Goldstein, M., Multivariate Analysis. Methods and Apolications, John Wiley and Sons, New York, NY, 1984. Espino, M., "Scheduling in Cellular Manufacturing with Interruptions in the Process Flow," Master's Thesis, University of Puerto Rico-Mayagiiez, 1991. Fry, T. D., Wilson, M. G. and Breen, M., "A Successful Implementation of Group Technology and Cell Manufacturing," Production and Inventory Management, Third Quarter, pp. 4-6, 1987. Gallagher, C. and Knight, W., Group Technology, Butterworth and Co., London, 1973. Greene, T. J. and Sadowski, R. P., "Cellular Manufacturing Control," Journal of Manufacturing Control, pp. 137-145, 1984. Greene, T. J. and Cleary, C.M., "Is Cellular Manufacturing Right for You?" Annual International Industrial Engineering Conference Proceedings, pp. 181-190, 1985. Guang-Xun, Y. and Li-Hang, Q., "A Production Scheduling Method for Group Technology Cell," Production Management Systems, pp. 121-133, 1987. Ham, I., "Introduction to Group Technology," SME Technical Report MMR76-03, Society of Manufacturing Engineers, Dearborn, MI, 1976. Mahmoodi, F., Tierney, E.J. and Mosier, C.T., "Dynamic Group Scheduling Heuristics in a Flow-through Cell Environment," Decision Sciences, Vol. 23, pp. 61-85, 1992.

123

[14] Miles, T. and Batra, A., "Scheduling of a Manufacturing Cell with Simulation," Proceedings of Winter Simulation Conference, pp. 668-676, 1986. [15] Mosier, C. T., Elvers, D. A. and Kelly, D., "Analysis of Group Technology Scheduling Heuristics," International Journal of Production Research, Volume 22, pp. 857-875, 1984. [16] Nelson, P. R., Design and Analysis of Experiments. Handbook of Statistical Methods for Engineers and Scientists, McGraw Hill, New York, NY, 1990. [17] Ranson, G. M., Group Technology, McGraw Hill, London, 1972. [18] Siier, G.A. and Gonzalez, W., "Synchronization in Manufacturing Cells: A Case Study," International Journal of Management and Systems, Vol.9, No.3, Sept.-Dec., 1993. [19] Siier, G.A and Saiz, M., "Cell Loading in Cellular Manufacturing Systems," Proceedings of the 15th Conference on Computer and Industrial Engineering, Cocoa Beach, Florida, March 8-10, 1993. [20] Sunduram, R. M., "Some Scheduling Algorithms for Group Technology Manufacturing Systems," Computer Applications in Production and Engineering, pp. 765-772, 1983. [21] Taylor, J. F. and Ham, I., "Group Scheduling by Shop Supervisory Personnel Using a Micro-Computer," SME Engineering Conference 1982, MS82-248, Society of Manufacturing Engineers, Dearborn, MI, 1982. [22] WemmerlSv, U. and Hyer, N., "Cellular Manufacturing in the US Industry: A Survey of Users," International Journal of Production Research, Volume 27, No. 9, pp. 1511-1530, 1989. [23] WemmerlSv, U., Production Planning and Control Procedures for Cellular Manufacturing. Concepts and Practices, The Library of Production, APICS, Falls Church, VA, 1988. [241 Wirth, G.T., Mahmoodi, F. and Mosier, C.T., "An Investigation of Scheduling Policies in a Dual-Constrained Manufacturing Cell," Decision Sciences, Vol. 24, No. 4, 1993.

124

Appendix A. Rule Combinations

No.

Rule Combination

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

PP/EDD/NFC/Min/CL/Min PP/EDD/NFC/Min/CL/Max PP/EDD/NFC/Min/NFP/Min PP/EDD/NFC/Min/NFP/Max PP/EDD/NFC/Min/PM/Min PP/EDD/NFC/Min/PM/Max PP/EDD/NFC/Max/CL/Min PP/EDD/NFC/Max/CL/Max PP/EDD/NFC/Max/NFP/Min PP/EDD/NFC/Max/NFP/Max PP/EDD/NFC/Max/PM/Min PP/EDD/NFC/Max/PM/Max PP/EDD/NCR/Min/CL/Min PP/EDD/NCR/Min/CL/Max PP/EDD/NCR/Min/NFP/Min PP/EDD/NCR/Min/NFP/Max PP/EDD/NCR/Min/PM/Min PP/EDD/NCR/Min/PM/Max PP/EDD/NCR/Max/CL/Min PP/EDD/NCR/Max/CL/Max PP/EDD/NCR/Max/NFP/Min PP/EDD/NCR/Max/NFP/Max PP/EDD/NCR/Max/PM/Min PP/EDD/NCR/Max/PM/Max

No. Rule Combination

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/ CP/

CL/M.in/EDD/NFC/Min CL/Max/EDD/NFC/Min NFP/Min/EDD/NFC/IV[in NFP/Max/EDD/NFC/Min PM/Min/EDD/NFC/Min PM/Max/EDD/NFC/Min CL/Min/EDD/NFC/Max CL/Max/EDD/NFC/Max NFP/Min/EDD/NFC/Max NFP/Max/EDD/NFC/Max PM/Min/EDD/NFC/Max PM/Max/EDD/NFC/Max CL/Min/EDD/NCR/Min CL/Max/EDD/NCR/Min NFP/Min/EDD/NCR/Min NFP/Max/EDD/NCR/Min PM/Min/EDD/NCR/Min PM/Max/EDD/NCR/Min CL/Min/EDD/NCR/Max CL/Max/EDD/NCR/Max NFP/Min/EDD/NCR/Max NFP/Max/EDD/NCR/Max PM/Min/EDD/NCR/Max PM/Max/EDD/NCR/Max

125 A p p e n d i x B. Results of Ranking Procedure

PM: n T DP

Rule Combinations 1, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 41, 42,43, 44, 45, 47, 48/46/3/40/14, 16, 17/20, 22, 23/2, 4, 518, 10, 11! 31, 32, 33, 34, 35, 36 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48/31, 32, 33, 34, 35, 36 42/37/43/38, 41/25, 26, 29, 30/44, 47/48/31, 32, 33, 34, 35, 36/27/1, 6, 9, 15, 39/13, 18, 28/2, 3, 4, 5, 7, 12, 40/8, 10, 11, 14, 16, 17/19, 21, 24, 45, 46/20, 22, 23 25, 26, 27, 28, 29, 30, 37, 38, 41, 42, 43, 44, 47, 48/31, 32, 33, 34, 35, 36/6/18/15, 39/9/45/21/24/12/1/13/7/40/46/8, 10, 11/20, 22, 23/3/2, 4, 5/14, 16, 40/46/14, 16, 17/3, 4, 5

PM: 'IT DP

Rule Combinations 1, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48/46/3/40/20, 22, 23/14, 16, 17/2, 4, 5/8, 10, 11/31, 32, 33, 34, 35, 36 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48/31, 32, 33, 34, 35, 36 42/37/43/38, 41/30/25/48/26, 29/44, 47/31, 32, 33, 34, 35, 36/27/18/12/15, 39/13/40/6/1, 9/28/14, 16, 17/7/2, 3, 4, 5/8, 10, 11/24/19/46! 21, 45/20, 22, 23 25, 26, 27, 28, 29, 30, 37, 38, 41, 42, 43, 44, 47, 48/31, 32, 33, 34, 35, 36/6/18/45/39/21/24/9, 15/12/13/1/19/7/40/46/8, 10, 11, 20, 22, 23/3/2, 4, 5/14, 16

126 PM: Tma x DP

Rule Combinations 1, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48/46/3/2, 4, 5, 14, 16, 17, 40/20, 22, 23! 8, 10, 11/31, 32, 33, 34, 35, 36 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48/31, 32, 33, 34, 35, 36 48/30/37/25, 43/42/26, 29! 38, 41! 44, 47/31, 32, 33, 34, 35, 36/1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 39, 40, 45, 46 25, 26, 27, 28, 29, 30, 37, 38, 41, 42, 43, 44, 47, 48/6/18/45/12, 24, 39/1, 19/7/13, 21/9, 15/31, 32, 33, 34, 35, 36/40/46/3/2, 4, 5, 14, 16! 8, 10, 11/20, 22, 23

PM: CUav DP

Rule Combinations 40/20, 22, 23/14, 16, 17/2, 4, 5/8, 10, 11/1, 3, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48/31, 32, 33, 34, 35, 36 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48/31, 32, 33, 34, 35, 36 48/25, 26, 29, 30, 38, 41/44, 47/42, 43/37/31, 32, 33, 34, 35, 36/27/46/1, 6, 9, 15, 19, 21, 24, 39, 45/13, 18/28, 40/7, 12/2, 3, 4, 5/8, 10, 11, 14, 16, 17, 20, 22, 23 46/40/20, 22, 23/8, 10, 11, 25, 26, 27, 28, 29, 30, 37, 38, 41, 42, 43, 44, 47, 48/6, 9, 12, 13,15, 18, 21, 24, 39, 45/1/19/7/14, 16! 3! 2, 4, 5! 31, 32, 33, 34, 35, 36

127 Appendix C. Results of Lexicographical Analysis

DP

Rule Combinations 1, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48/46/3/40/14, 16, 17/20, 22, 23/2, 4, 5/8, 10, 11/31, 32, 33, 34, 35, 36 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48/31, 32, 33, 34, 35, 36 42/37/43/38, 41/30/25/26, 29/44, 47/48/31, 32, 33, 34, 35, 36/27/15, 39/6/1, 9/18/13/28/12/40/7/2, 3, 4, 5/14, 16, 17/8, 10, 11/24/19/46/21, 45/20, 22, 23 25, 26, 27, 28, 29, 30, 37, 38, 41, 42, 43, 44, 47, 48/31, 32, 33, 34, 35, 36/6/18/39/15/9/45/21/24/12/1/13/19/7/40/46/8, 11/20, 22, 23/3/2, 4, 5/ 14, 16

10,