Assembly sequence planning for processes with

International Journal of Production Research

ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20

Assembly sequence planning for processes with heterogeneous reliabilities Shraga Shoval, Mahmoud Efatmaneshnik & Michael J. Ryan To cite this article: Shraga Shoval, Mahmoud Efatmaneshnik & Michael J. Ryan (2017) Assembly sequence planning for processes with heterogeneous reliabilities, International Journal of Production Research, 55:10, 2806-2828, DOI: 10.1080/00207543.2016.1213449 To link to this article: http://dx.doi.org/10.1080/00207543.2016.1213449

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Date: 11 April 2017, At: 22:37

International Journal of Production Research, 2017 Vol. 55, No. 10, 2806–2828, http://dx.doi.org/10.1080/00207543.2016.1213449

Assembly sequence planning for processes with heterogeneous reliabilities Shraga Shoval1

, Mahmoud Efatmaneshnik* and Michael J. Ryan

School of Engineering and IT (SEIT), UNSW Canberra at the Australian Defence Force Academy, Canberra, Australia (Received 27 October 2015; accepted 2 July 2016) Stochasticity in assembly processes is often associated with the processing time and availability of machinery, tools and manpower, however in this paper it is determined by probability of an assembly task successful completion which here is referred to as task reliability. We present a mathematical model for optimising the expected assembly cost, and consider two scenarios: the first a situation where a failure of one assembly task requires rework of that task alone; second a situation in which a failure in the midst of the process requires resumption of previously completed tasks. In the worst case scenario the assembly process must restart from the beginning. We show that the first scenario is insensitive to sequencing unless there are set-up costs. In the second scenario the process is sensitive to tasks’ sequence. We present a heuristic that argues for accomplishing more uncertain tasks (with less reliability) earlier in the process to decrease the expected cost of assembly, and show that in a mutually dependent assembly process, when tasks’ reliabilities are similar, the cheaper tasks should be executed earlier in the process. Keywords: assembly; reliability; sequencing; optimisation; uncertainty

Introduction Assembly is an important phase in the product lifecycle, in which components and subsystems are aggregated to generate a holistic unit that functions according to the product requirements. In a modern product life cycle, assembly is part of the Design for Manufacturing and Assembly (DFMA) methodology. Boothroyd, Dewhurst, and Knight (2010) list three incentives for performing DFMA: providing guidance for reducing manufacturing and assembly costs; providing a benchmark for studying competitors’ products and providing a tool for negotiating suppliers’ contracts. A crucial factor for successful DFMA is a reliable estimate of both the manufacturing and assembly costs at the early stages of the product lifecycle. General guidelines have been incorporated in the design lifecycle such as minimising the number of components/subassemblies by analysing the relative movements between the components, examining the materials they are made of, and determining the operational relations between them. In the past, the assembly process was believed to contribute only a small proportion to the total cost, while most efforts were to reduce manufacturing or procurement costs. However, technical and financial reports show that the benefits of Design for Assembly (DFA) are significant relative to the total system cost (Boothroyd 1982; Henchy 1988). Ford Corporation reported a saving of over 1 billion US$ in 1989 as a result of applying DFA (Burke and Carlson 1989). According to Boothroyd, Dewhurst, and Knight (2010) more than 800 enterprises in the US alone adopted DFA methodologies from 1990 to 1999, covering almost all types of industries and products. Other reports estimate that assembly consumes 50% of the production time and more than 20% of the production cost (Fan and Dong 2003). Assembly processes are typically characterised by the number of products’ models (single or mixed), the nature of the process (deterministic or stochastic) and the assembly flow (straight or U type). Many methods have been developed in order to improve the economics of the assembly process (Fan and Dong 2003) and to examine the effect of various factors on the total assembly cost (Yoosufani and Boothroyd 1978; Boothroyd 1979; Seth and Boothroyd 1979; Yoosufani, Ruddy, and Boothroyd 1983; Corbett and Crookall 1986). An important factor that affects the assembly cost is the ‘assembly efficiency’, which is affected by of the number of parts (components and subassemblies) in the process, the sequence of the assembly operation (assembly sequence planning [ASP]), the assignment of assembly tasks to workstations (assembly line balancing [ALB]), the assembly cycle time and the ease of the assembly tasks (Boothroyd, Dewhurst, and Knight 2010). Raj, Saravana Sankar, and Ponnambalam (2012) consider the assembly efficiency of products in which the components have probabilistic variations in the relevant features (e.g. dimensional and geometrical

*Corresponding author. Email: [email protected] 1 Department of Industrial Engineering and Management, Ariel University, Ariel, Israel © 2016 Informa UK Limited, trading as Taylor & Francis Group


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tolerances). They determine the assembly efficiency as the ratio between the accepted number of products and the batch size, and propose a method for classifying the components into assembly subgroups according to their tolerance variations. Boothroyd (2005) uses the downtime of the assembly machinery as a measure for assembly efficiency based on the probability for defective components that cause assembly failures. Combining the assembly cost of non-defective components and the cost of assembly failures and re-work for the given probability of defective components (component reliability), provides a good estimate for the expected assembly cost. Models for indexing assembly machines, free-transfer assembly machines and assembly robotic systems provide guidelines for selecting the optimal assembly configuration. Ohashi et al. (2002) propose the Extended Assemblability Evaluation Method (AEM) that was originally developed by Hitachi Inc. According to the AEM, all assembly operations are classified relative to basic assembly operations, and each operation is given a penalty score. Suzuki et al. (2003) suggest the Assembly Reliability Evaluation Method (AREM) for assembly quality evaluation. This method identifies components and assembly operations with low reliability values, and suggests improvements to reduce production time while maintaining high quality. Fei, Tang, and Bai (2014) describe the Distributed Collaborative Extremum Response Surface Method for dynamic assembly reliability analysis of complex systems, in particular gas turbines. Large and complex assemblies are partitioned into simpler models, then the extremum response of each model is constructed, and these responses are integrated for analysing the performance of the entire complex system. An important factor that affects both reliability and cost of the assembly process is the ASP. Complex products commonly consist of a mixture of heterogeneous components, which can be assembled in numerous configurations and sequences. These sequences form interoperable networks that comprise the architecture of the assembly process, which defines the assembly process based on the interconnections and relations between the various components. The assembly sequence affects a variety of issues such as resources assignments to tasks, process balancing, tooling, fixturing, testing and cost. Obviously, the number of valid assembly sequences increases staggeringly with the increase in the number of components and subassemblies. One method for determining possible assembly sequences is to analyse the disassembly process of the complete product as proposed by Lambert and Gupta (2004). Possible assembly sequences are determined from the disassembly graph that represents all possible sequences to disassemble the product. Henrioud, Relange, and Perrard (2003) present a method for generating precedence graphs based on a set of assembly sequences. These sequences use operative (hard) constraints and imposed subassemblies and grouping (soft) constraints for constructing a hypergraph. The hypergraph contains all possible precedence graphs that can be used for ALB and Assembly Line Design (ALD). Mascle (1998) models the functional liaisons between the product’s elements to distinguish between simple contacts and attachments, and between subsets and subassemblies. A functional liaison describes the link between two elements that consists of contacts and attachments. An assembly operation consists of positioning an element according to the geometric relations with other elements, and attaching it according to the functional liaison. The attachment provides functionality and stability between elements. According to Mascle’s model, only the external functional liaisons of a subassembly are relevant for connecting it to the rest of the product, and the difference between a subset and a subassembly is articulated by their stability as defined by the functional liaisons of their elements. Bourjault (1984) proposes a series of rules for generating all possible valid assembly sequences based on two questions: (1) is/are certain assembly liaison/liaisons cannot be done after other assembly liaison/liaisons is/are completed? (2) is/are certain assembly liaison/liaisons cannot be done if other liaison/liaisons have not been completed yet? A liaison network is constructed based on the answers to these questions that are repeated for all possible sequences. The network characterises the product architecture, where the nodes are the components, and the arcs/ links are the relations between them (physical, spatial, operational, etc.). De Fazio and Whitney (1987) modify Bourjault’s method by reducing the expected number of questions in order to cover all possible valid sequences. Karjalainen et al. (2007) present an algorithm for generating valid assembly sequences by constructing the adjacency matrix based on the assembly precedence constraints. Feasible sequences are determined using a backtracking algorithm based on three subassembly patterns: serial, parallel and loop. A contracted adjacency matrix presents the internal connections between the components, and an assembly tree is constructed for generating valid assembly sequences using a backtracking algorithm. Homem de Mello and Sanderson (1990) analyse several possible representations of assembly sequences (directed graphs, AND/OR graphs, establishment conditions and precedence relationships), and discuss the correctness and completeness of each representation. Wilson (1995) and Jones, Wilson, and Calton (1998) developed a Computer Aided Design (CAD) tool that identifies the interferences between parts based on their CAD model. Gottipolu and Ghosh (1995) propose an automatic assembly sequence-planning tool that is based on the contacts and interferences between the various components. Niu, Ding, and Xiong (2003) also propose a computerised tool for determining valid assembly sequences based on the Mating Relation Graph that is constructed from the CAD model of the product. Su (2007a) uses a Geometric Constraint Analysis (GCA) method, which is based on the assembly CAD model, and is suitable for 2D assembly tasks. Su and Lai (2010) present an extension of this method using the 3D-GCA method. The method generates a complete set of Assembly Precedence Relations (APR) and determines the optimal assembly sequence in terms of the assembly angle and assembly direction.

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Genetic Algorithm (GA) search method is another common technique to identify valid assembly sequences. Su (2007b) proposes an ASP using case-based reasoning and the GA method to enhance the efficiency and quality of the reasoning process. Lai et al. (2009) use a genetic algorithm for selecting the sequence of assembly operations in a multi-sequence modelling technique, in which a chromosome is a character string that represents a specific assembly sequence. A random population is initially selected, and using common GA features such as reproduction, crossover and mutation, new generations of chromosomes are generated. The most suitable chromosomes from each generation are selected for generating the next generation. When the averaged evaluation of a certain generation reaches a satisfactory value, the best chromosome from that generation is selected as the best assembly sequence. Xing and Wang (2012) also use the GA technique to identify and evaluate assembly sequences. They use a similar approach to Lai et al. (2009) by performing a tolerance analysis in order to evaluate the feasibility of the chromosomes. They propose a Hybrid Particle Swarm Optimisation (PSO) and Genetic Algorithm method that combines the advantages of the PSO technique, originally proposed by Kennedy and Eberhart (1999), and the genetic algorithm technique. As mentioned, the assembly process is often subject to uncertainties that affect its efficiency. The sources for such uncertainties are variations in the components, tools and fixtures, equipment malfunctions, human errors, etc. These uncertainties lead to variations in the processing times of the assembly tasks, resulting in possible disruption to the process flow. Variations in the completion time of the assembly tasks are common sources for disruptions in the assembly process, and several researchers propose methods for sequencing and balancing the assembly process under uncertainties (Erel, Sabuncuoglu, and Sekerci 2005; Kara, Paksoy, and Chang 2009; Cakir, Altiparmak, and Dengiz 2011; Özcan, Kellegöz, and Toklu 2011; Gurevsky, Battaïa, and Dolgui 2013; Hazır and Dolgui 2013; Dong et al. 2014; Pınarbaşı, Yüzükırmızı, and Toklu 2016). Cakir, Altiparmak, and Dengiz (2011) use a simulated annealing algorithm for multiobjective optimisation of line balancing in a single-mode stochastic system that minimises the design cost. Kara, Paksoy, and Chang (2009) propose a binary fuzzy programming approach for straight and U-shape lines for optimising the number of workstations and the cycle time of the assembly process. Gurevsky, Battaïa, and Dolgui (2013) present a stability measure for estimating the assembly cost, expressed by the weighted sum of the number of workstations, under possible variations in the processing times. Their solution combines robustness to variations and cost optimisation in assembly lines where tasks can be performed in parallel. Hazır and Dolgui (2013) propose two models for single model line balancing that considers the uncertainty in the operation times of the assembly tasks as well as the number of tasks in the workstations. These models do not require the accurate probabilistic distributions of the processing times, and can be extended for more complex systems such as multiple and mixed assembly lines. A genetic algorithm for mixed model U-shape balancing and sequencing assembly problem is proposed by Özcan, Kellegöz, and Toklu (2011). The algorithm considers stochastic task times and simultaneously solves the line balancing and model sequencing problems. Dong et al. (2014) also considers a mixed model U-shape assembly line with uncertain task times using a stochastic programming model to optimise the work overload time. They propose a simulated annealing algorithm for determining the near-optimal solution. Erel, Sabuncuoglu, and Sekerci (2005) use a beam search method to minimise the expected assembly cost of a single model U-shape stochastic assembly system. The expected cost is determined by the direct labour cost and the expected incompletion cost of tasks. An important observation from their model is that a good solution reduces the probability for incompletion of assembly tasks in the earlier stages of the assembly process. Pınarbaşı, Yüzükırmızı, and Toklu (2016) consider variation in the processing time of the assembly tasks, as well in the flow times between workstations. They use queuing networks and constraints programming in order to uniformly distribute the workload among all workstations in the process. Feasible solutions are generated using constraints programming, and the performance of each solution is determined by queuing networks approximation to determine the optimal ALD. In this paper we consider a different type of uncertainty in the assembly process: the probability of a task be completed successfully in a single trial. We investigate the ASP where each assembly operation (task) has a cost associated with it, as well as a reliability value (a probability of success). We focus on the additional costs related to the reliability of the assembly operations in case of failures. The model discussed in the paper refers to single model straight assembly system but then can be extended to other types such as multiple model and U-shape systems. In the next subsection we first look at the reliability of individual assembly tasks, then we provide a formulation of the assembly problem and present a practical and illustrative example. According to this formulation, assembly processes are categorised according to their sensitivity to failures, ranging from mutually independent to fully dependent processes. In mutually independent processes a failure of one assembly task requires rework of that task alone, while in mutually dependent processes a failure in the midst of a process requires resumption of previously completed tasks. We then tackle the optimisation problem for assemblies with mutual independent tasks that provides lower bound for the expected assembly cost. Next, we formulate the assembly cost for mutually dependent tasks, that expresses the upper bound of the expected cost, and present a heuristic based on K–L distance for optimal sequencing. Finally, we provide some concluding remarks and future research plans.


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Assembly task reliability There are many causes for failures during the assembly process, including: variations in components, faults in the feeding mechanism, inaccuracies of the assembly operations, external disruptions, human errors and more. The probability for success (or failure) in an assembly process can be evaluated based on the expected equipment’s performance, the probabilistic distribution of principal dimensions and geometric tolerances, and the specific assembly requirements. For example, consider the common robotic ‘peg in the hole’ assembly task shown in Figure 1 that consists of 5 tasks: (1) (2) (3) (4) (5)

Loading element A. Positioning element A in the assembly jig. Loading element B. Inserting element B into A. Removing the entire assembly.

Assuming normal distributions of the key dimensions (do and di) and robot repeatability (r), then the probability for i success of task 4 can be formulated by Ps ¼ P do d r , where Ps is the mean or expectation of the normal distribution 2 for success completion of the task in a single trial. Depending on the assembly setting (jigs, clamps, materials, etc.), a failure in a task may require repeating that task or returning to one of the previous tasks. For example, failure in inserting the peg (task 4) may require repositioning of element A (return to task 2), replacing one or both elements (return to task 1), or just repeating the insertion task (task 4). Let PAr be the probability that element A needs replacement after a failure in task 4, PBr the probability for replacing element B, PBi the probability for repeating the insertion task and PAp the probability of repositioning element A after a failure in Task 4. Referring to the state diagram shown in Figure 2 and defining P(m, n) as the probability of moving from task m to task n, then Pð4; 5Þ ¼ Ps , Pð4; 4Þ ¼ PBi , Pð4; 3Þ ¼ PBr , Pð4; 2Þ ¼ PAp and Pð4; 1Þ ¼ PAr . Formulating the problem as ‘Absorbing Markovian Chain’ (Task 5 is the absorbing state) and using its Fundamental Matrix, the expected cost for completing the entire assembly process can be determined. For example, let Pð1; 2Þ ¼ Pð2; 3Þ ¼ Pð3; 4Þ ¼ 1 (no risks in the first three tasks), Pð4; 5Þ¼ 0:6 (probability for success), Pð4; 4Þ¼ Pð4; 3Þ¼ Pð4; 2Þ¼ Pð4; 1Þ ¼ 0:1 and assume homogeneous costs for all tasks C ð1Þ¼ Cð2Þ ¼ C ð3Þ¼ Cð4Þ ¼ Cð5Þ ¼ 2, then the expected cost of the entire process is 13.33. If the nominal duration of each task is ‘1’, then the expected duration (cycle time) is 6.66. In the best-case scenario only one task needs to be repeated after a failure, resulting in Pð4; 4Þ ¼ 1 Ps ¼ 0:4, resulting in expected cost of 11.33. In the worst-case scenario (all tasks are repeated after a failure), (Pð4; 1Þ ¼ 1Ps ¼ 0:4), the expected cost is increased to 15.33. The best and worst case scenarios provide lower and upper limits to the expected assembly cost given the probabilities for success of the assembly tasks. Other than the direct rework, additional costs associated with an assembly failure are due to investigation of the causes for

Figure 1. A common ‘peg in the hole’ assembly with probabilistic distributions.

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Figure 2. State diagram for the ‘peg in the hole’ in a probabilistic assembly process.

Figure 3. Examples of different assembly sequences that affect mutual dependencies.

failure, production of replacement parts, queuing and other disruptions to the production schedules (Booker, Swift, and Brown 2005). These features have been studied by several researchers such as Pradhan and Damodaran (2009) that propose an analytical model for estimating the performance (e.g. flow time and work in process) of a manufacturing system with job circulation due to failures. Hulett and Damodaran (2011) consider random failures in manufacturing and service systems with mixed serial/parallel architecture (Pradhan and Damodaran 2009; Hulett and Damodaran 2011). In this paper we ignore these indirect costs and focus only on the direct costs associated with the assembly failures. Our motivation is to examine the effect of the assembly sequence on the process performance, and to develop methodologies for efficient ASP. The next example illustrates how modification of the assembly sequence can modify the process’s cost. Consider the simple product shown in Figure 3. The product consists of two plates that are fastened with two nuts and bolts through holes in each plate. Assuming that all components’ are required to be accurately aligned (notice the differences between the holes’ and bolts’ diameters), a possible assembly sequence (Figure 3(a)) consists of (1)-positioning plate B in jig F, (2)-positioning plate A on top of plate B, (3)-inserting the bolts through the holes and (4)-fastening the nuts. A failure during the positioning of plate A, the insertion of the bolts or fastening the nuts may affect previous tasks as illustrated in the state diagram (Figure 4(a)). Changing the assembly sequence as shown in Figure 3(b) ((1*)position the bolts in different jigs – G, (2*)-inserting plate B, (3*)-inserting plate A and (4*)-fastening the nuts) changes the mutual dependencies in case of failures as illustrated by the state diagram in Figure 4(b) (this sequence has an additional advantage as all assembly operations are performed from the top). Problem formulation Consider a set of n + 1 heterogeneous components that construct a product. The components are assembled in a serial sequence such that there are n assembly connections, where each connection is constructed in a single assembly task (a single model, straight flow system). The system is stochastic such that an assembly task i has a probability to be successful pi or to fail ð1 pi Þ. We refer to the probability of the assembly task to be successful as the assembly task reliability. An assembly task also has an associated cost, ci (cost can be measured in dollars or any other quantitative measure such as time, energy, etc.). We assume that the costs and reliabilities are static, and do not change over time – that is, if a task fails, the next attempt has the same cost and probability for success. Although these values can change over time due to variations in the components during manufacturing, learning curves, etc., their probabilistic distributions can be considered to be static for the same batch production. We also assume that there are several valid assembly sequences, and that the assembly constraints regarding the feasibility sequences are given. Finally, we assume that the reliabilities and costs of the assembly tasks are sequence-dependent, inferring that a certain task may have different cost and reliability values when performed after another assembly task. For example, an assembly task may require


(a)

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(b)

Figure 4. State diagrams for the different sequences shown in Figure 3.

additional resources (such as grippers, fixtures, clamps or tools) when executed after a different task. The cost and reliability values of tasks performed immediately after other tasks are given in the Cost C (n × n) and Reliability P (n × n) matrices, where ci,j and pi,j are the cost and reliability values of performing task j immediately after task i. All the diagonal values in P and C are set to ‘0’ and, since we assume that the connections between the components are sequencedependent, P and C are not necessarily symmetric. We define a Sequence Vector U = {ai}, i = 1 … n that indicates the position of task i in the serial assembly sequence. This vector will be used for registration of the assembly sequential constraints. We divide the tasks into three classes, and set the values of the reliability and expected cost matrices as follows: Class 0 – Assembly task j has no precedence constraints with task i. In this case the reliability value pi,j and the cost ci,j are not affected by the precedence constraints. Class 1 – Task j must be performed immediately after task i. In this case all cells in the ith row of the reliability matrix are set to ‘0’ and all cells in the ith row of the cost matrix are set to ‘∞’ except for ci,j and pi,j that remain unchanged. This guarantees that in any valid sequence, task j is performed immediately after task i. Class 2 – Task j must be performed after task i (but not necessarily immediately after). In this case the reliability and expected cost values maintain their original values, and the precedence constraint is expressed in terms of the Sequence Vector such that aj > ai. Given that the reliability value of a single attempt to complete task j immediately after task i is pi,j, the expected number of attempts required to complete that task is p1i;j , and the expected cost of completing task j immediately after task i – ^ci;j is given in (1) (for a proof see Efatmaneshnik and Ryan (2015)): ^ci;j ¼

ci;j pi;j

(1)

For a given sequence vector U, the expected cost of the entire process is given by:

^ njU ¼ C

n1 X

^cai ;aiþ1

(2)

i¼1

The objective is to find a sequence that conforms to APR and minimises the assembly cost: ^ njU min C

U U

(3)

Here U is the universe of all sequence vectors U that conform to APR. Assembly process characterisation and cost We define an assembly process A as a quadruple of the cost and reliability matrices, the Sequence Vector and a dependency parameter k: A(C, P, U, k). The dependency parameter k determines the maximum number of previous tasks that

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may need to be repeated after a failure has occurred. When k = 0 the assembly task is mutually independent as a failure requires repeating that task only (best case scenario). When 0 < k < n – 1 the task is considered partial mutual dependent, and when k = n – 1 the task is complete mutual dependent (worst case scenario). In this case a failure causes a cascade of failures that requires re-work of the failed task, as well as all previous tasks. Cascade failure is a result of tight interconnections between the components, and occurs when no ‘buffering’ mechanism that can isolate the cascading failures (such as mechanical, electrical, or chemical) exists. In practice, most assembly processes are mutually independent by the design of buffers and interfaces. Furthermore, in case it is impossible to create stable buffers, subassemblies or encapsulation techniques are used to lower k. For example Theroux, Baca, and Hamp (1995) introduce an improved encapsulation method for repair and replacement of defective components, thus allowing efficient rework and salvage of electronic assembly modules. They provide an electric housing with a structure that permits deformation with thermal expansion of the encapsulant, which results in reduced stresses on the electronic components. For a system that is constructed by n serial and heterogeneous assembly tasks, where the assembly cost and reliability values of task j when performed immediately after task i, are ci,j and pi,j (i = 1 … n), and for an assembly sequence, given by the Sequence Vector of U ¼fai g, i = (1, … , n), we have: ^ njU ¼ C

n1 X i¼1

cai ;aiþ1 Qminðn;iþkÞ pai ;aiþ1 j¼i

(4)

For a proof of (4) consider the sequence U1 (in which ai = i). From (2) we know that the expected cost for this sequence is: ^ njU ¼ C 1

n1 X i¼1

^cai ;aiþ1 ¼

n1 X ca ;a i

i¼1

iþ1

(5)

^pai ;aiþ1

^ pai1 ;ai is the conditional probability of successfully performing the aith assembly task immediately after task ai-1, assuming all previous tasks were successful. For n dependent assembly tasks with an order of a1 → a2 → a3 … → an, the probability of success in the aith task (i = 1 … n) depends on k subsequent tasks (all k subsequent processes must also be successful otherwise the entire assembly process begins from the first task) and therefore we have: ^ pai1 ;ai ¼ pai1 ;ai pai ;aiþ1 . . . paminðn1;iþk1Þ ;aminðn;iþkÞ

(6)

By replacing (6) into (5) we arrive at (4). For illustration consider the following sequences of a hypothetical process of size n = 10: Process 1: A(P, C, U1, k), U1 = (1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9 → 10) Process 2: A(P, C, U2, k), U2 = (10 → 9 → 8 → 7 → 6 → 5 → 4 → 3 → 2 → 1) Assume the cost and reliability values are not sequence dependent and symmetric, i.e. ci,j = cj,i and pi,j = pj,i, as given in Table 1. Figure 5 shows the expected cost for the two sequences according to (4). Starting with the mutual independent case (k = 0), the expected costs of the two sequences are identical. However for higher values of k, the expected costs grow differently, highlighting the sensitivity of sequencing for optimal assembly in higher values of k. Lower bound for mutually independent tasks Assuming all assembly tasks are mutually independent (k = 0), the assembly process can be visualised by a directional graph in which the nodes are the tasks, and the edges are the transitions between the tasks (Figure 6). Each edge is assigned with the expected cost of performing a task immediately after its predecessor using (1). We add two nodes to the graph: Initial (S) and Final (F) nodes. The edges from the initial node to the first task are assigned with the expected Table 1. Example of cost and reliabilities values. Costs(C) Reliabilities (P)

c1 = 0.96 p1 = 0.9

c2 = 0.97 p2 = 0.91

c3 = 0.98 p3 = 0.92

c4 = 0.99 p4 = 0.93

c5 = 1 p5 = 0.94

c6 = 1.01 p6 = 0.95

c7 = 1.02 p7 = 0.96

c8 = 1.03 p8 = 0.97

c9 = 1.04 p9 = 0.98

c10 = 1.05 p10 = 0.99


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15.5 A(C,P,U 1 ,k)

15

A(C,P,U 2 ,k)

Assembly cost $

14.5 14 13.5 13 12.5 12 11.5 11 10.5

0

1

2

3

4

5

6

7

8

9

k

Figure 5. The expected cost of assembly process vs. different dependency parameters k for two sequences U1 and U2.

Figure 6. Directed graph from the cost and reliability matrices.

costs of first tasks, and the edges to the final node have no cost associated with them (we do that in order to allow the assembly process to start and finish with any valid task). A valid sequence in this graph is a route from the initial node (S) to the final node (F) that visits all nodes that belong to a specific task exactly one time, and complies with the precedence constraints. Many factors can affect the selection of the optimal assembly sequence. Boothroyd (2005) suggests using the assembly spatial direction as a criterion for evaluation of the assembly sequence, while Pan and Smith (2006) use the re-orientation of components as the most significant parameter. Laperrière and ElMaraghy (1996) propose the stability and accessibility as features for selecting the optimal assembly sequence, and, as mentioned before, (Su and Lai 2010) use the assembly angle and direction as the objective function. As described in the previous section, our model uses the entire expected cost value as an objective function, derived from the cost and reliability values of a single attempt to complete the assembly tasks. The expected cost of the entire assembly process for mutually independent n heterogeneous and serial assembly tasks with Sequence Vector U ¼ fai g, i = 1 … n is given by: ^ njU ¼ C

n1 c X ai;a i¼1

iþ1

pai;aiþ1

(7)

Notice that (7) is a special case of (4) for k = 0. This problem can be considered as the Travel Salesperson Problem with Precedence Constraints (TSPPC), in which the constraints are the APR. The following subsection provides an illustrative example for determining the lower bound for the expected assembly cost using TSTPC solver (see Appendix 1).

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Illustrative example For illustration consider the three-pin plug, shown in Figure 7(a). The plug, used by Boothroyd (2005) in a feasibility study of assembly automation, consists of 14 components with 15 assembly tasks. The precedence diagram of the complete assembly is shown in Figure 7(b) (again, taken from Boothroyd (2005)). We choose this product as it is relatively cheap, manufactured in mass production, and the values of its components are relatively low compared with the assembly cost. The example consists of four subassemblies (as shown in the dashed closed lines in Figure 7(b)), four components and nine assembly tasks listed in Table 3. The liaison graph of the subassemblies is shown in Figure 8, where narrow lines show a regular precedence relation (Class 2) and a thick solid line indicates an immediate precedence (Class 1). In this simple example, there are 362,880 (9!) possible sequences, but considering the APR, many of them are excluded. Table 2 shows the APR in the form of the sequence vector. The construction of the reliability and cost matrices is based on the assembly methods to be used, the available data on the components and the assembly process. For an existing product, the required data can be determined using quality studies in which the performance is directly measured and analysed with probabilistic tools. For a new product, the matrices are constructed using indirect data such as the distribution of faults, equipment accuracy and reliability, environmental effects, etc. In this example we combine data provided in Boothroyd (2005) with motion analysis of a single assembly robot with a single arm. Table 3 lists the costs, reliability values and the expected costs of the individual tasks when performed independently. Based on the data shown in Table 3 and the precedence constraints, the Expected Cost matrix is constructed (Table 4). Assuming the robot uses a 2-fingers gripper for handling components with irregular shapes like the Base, Fuse clip and the Cover subassemblies, a collet gripper for handling components with cylindrical geometries such as the pins and the fuses, and a special third tool for handling the cover screw, every tool change involves a cost of 2. This cost is based on observation of a fast tool change mechanism commonly used in mass production robotic systems in which tools are mounted on a chain magazine and are presented to the robot at a fixed position (the tool change position). A quick swivelling mechanism delivers the new tool and removes the old tool. The cost is based on the change time relative to other assembly operations performed by the robot. For example, the expected cost of performing task 4

Figure 7. Assembly drawing of the three-pin plug (a) the precedence diagram of the complete assembly (b) and the precedence diagram of the subassemblies (Boothroyd 2005).


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Figure 8. The liaison graph for the three-pin plug (Boothroyd 2005). Table 2. Precedence constraints or APR rules. ai refers to the ith task of the assembly process. Rule # Rule

1

2

3

4

5

6

7

8

9

10

11

a2 > a1

a3 > a1

a6 > a1

a5 > a1

a7 > a6

a7 > a5

a4 > a2

a4 > a3

a7 > a4

a8 > a7

a9 > a8

Table 3. Expected task cost and reliability. #

Task

1 2 3 4 5 6 7 8 9

Base sub. Fuse clip sub. Live pin Fuse Ground pin sub. Neutral pin sub. Cover Cover screw Removal

Task cost

Task reliability

Expected cost

4.5 4.5 10 10 10 10 4.5 8 4.5

1 0.9762 0.998 1 0.983 0.997 1 0.978 1

4.5 4.609711 10.02004 10 10.17294 10.03009 4.5 8.179959 4.5

Table 4. Expected cost matrix. Task #

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

4.5 0 4.5 6.5 6.5 6.5 6.5 ∞ ∞ ∞ ∞

∞ 4.61 0 6.66 6.66 6.66 6.66 ∞ ∞ ∞ ∞

∞ 12.02 12.02 0 10.02 10.02 10.02 ∞ ∞ ∞ ∞

∞ 12 12 10 0 10 10 ∞ ∞ ∞ ∞

∞ 12.21 12.21 10.17 10.17 0 10.17 ∞ ∞ ∞ ∞

∞ 12.04 12.04 10.03 10.03 10.03 0 ∞ ∞ ∞ ∞

∞ 4.5 4.5 6.5 6.5 6.5 6.5 0 ∞ ∞ ∞

∞ 10.22 10.22 10.22 10.22 10.22 10.22 10.22 0 ∞ ∞

∞ 4.5 4.5 6.5 6.5 6.5 6.5 ∞ 6.5 0 ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 0 0

immediately after task 3 is 10 as there is no need for tool change. However, performing task 3 immediately after task 1 entails additional cost of 2 for the tool change resulting in an expected cost of 12/0.998 = 12.02. Using the Concorde TSPPC solver (Concorde 2011), we find that the optimal path sequence is given by U1 = (0 → 1 → 2 → 3 → 5 → 6 → 4 → 7 → 8 → 9 → 10) with a total expected cost of 74.56, which is the lower bound for the cost of this process. Upper bound for assembly cost of mutually dependent tasks In this section we analyse a system in which the assembly tasks are completely mutually dependent, (k = n – 1), and therefore a failure in the current task results in failures of all previous tasks. As discussed, this is the worst-case scenario

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of the more general case, where k < n − 1. Figure 9 shows the assembly costs of the three-pin plug used in the previous section as a function of k for the optimal sequence found for the mutual independent tasks (U1 in the figure). It also shows the costs for a different valid sequence of the same product – U2 = (1 → 2 → 4 → 5 → 3 → 6 → 7 → 8 → 9). As can be seen, the sequence is not optimal for all dependency amounts (k). The expected costs for all 40 valid sequences and for both sequence dependent and independent tasks are listed in Appendix 3. The complexity of the mutually dependent tasks problem is O(n2 × n!), which is NP-complete, and therefore we present heuristics that lead to smaller thus more manageable search. Consider an assembly process that consists of m tasks with no precedence constraints. Let H(m × m!) be a sequence matrix in which each row is a specific assembly sequence (a total of m! different sequence permutations). In the presence of sequential constraints, the length of H is reduced by removing all sequences that do not comply with the constraints. Let Hq, (q = 1 … m!), be a specific row in H that consists of m assembly tasks, each with a cost CHq ðiÞ and reliability level PHq ðiÞ (i = 1 … n). Note that CHq ðiÞ ¼ cHq ðiÞHq ðiþ1Þ; and PHq ðiÞ ¼ pHq ðiÞHq ðiþ1Þ . We have adopted this new notion to simplify the proof of the statements in this section. The expected assembly cost of assembling the entire system for a given sequence Hq is given by: m X CH ðiÞ ^ m Hq ¼ Qm q C j¼i PHq ðjÞ i¼1

(8)

Next, we wish to determine the upper and lower bounds for assembling the entire system. Assume H1 is an assembly sequence in which PH1 ð1Þ PH1 ð2Þ PH1 ð3Þ . . .: PH1 ðmÞ (tasks are ordered according to their probability for success in ascending order), and Hm! be an assembly sequence in which PHm !ð1Þ PHm !ð2Þ PHm !ð3Þ . . .: PHm !ðmÞ (descending order), assuming these sequences comply with the precedence constraints. All other valid sequences are the possible permutation between these two sequences. Let’s first assume that the assembly costs of all tasks are identical and equal ‘1’ and the setup costs are negligible compared with the tasks’ costs. Under homogeneous costs assumption, we can state that there are cost savings, if and only if one relatively difficult task (with relatively lower reliability) is performed earlier in the sequence. A proof of this is presented in Appendix 2. From this statement it follows that the lower bound for the expected assembly of mutually dependent cost homogenous tasks is given by the sequence H1, and the upper bound is given by Hm!. Also it is relatively trivial to see that, the lower bound for the expected assembly cost for mutually dependent tasks that have the same reliabilities (P1 ¼ P2 ¼ P3 . . . ¼ Pm for all sequences), is given by a sequence that sorts the costs in ascending order. To summarise, the following heuristics can be deduced regarding the cost homogeneity and reliability homogeneity of the tasks: Heuristic 1: When task costs are close in value, execute the most difficult tasks earlier in the process. Heuristic 2: When task reliabilities are close in value, execute the cheaper tasks earlier in the process. To analyse Heuristic 1, we can use Kullback–Leibler (K–L) divergence (Kullback and Leibler 1951). The K–L divergence measures the difference between two probability distributions and, although it does not provide a metric 77 U U

2 1

Expected cost

76.5

76

75.5

75

74.5

1

2

3

4

5

Dependency, k

Figure 9. Expected cost for partially and fully mutually dependent tasks.

6

7

8

9


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Figure 10. Expected assembly cost correlates well with K–L divergence of reliability from PH(1) = [0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99], shown in the lower left corner to PH(9!) = [0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91], shown in the upper right corner. All other sequences are permutations of PH(1).

Figure 11. Assembly cost for 40 valid sequences of the illustrative example. The sequence (2 → 8 → 5 → 6 → 3 → 1 → 4 → 7 → 9) has the lowest expected cost and the sequence (1 → 4 → 3 → 5 → 6 → 2 → 7 → 8 → 9) has the highest.

value (as it is asymmetric), it is a common tool for ranking a set of probability distributions. Let’s call H(q) the order of the assembly sequence (where H(q) is the qth row in H). Then, we rank the assembly sequence H(q) according to its distance from the lower bound given by H(1): m X PHq;i DðH ðqÞjjHð1ÞÞ ¼ PHq;i log (9) PH1;i i¼1 where H(1) is the sequence with the lowest assembly cost (ascending task difficulties). In Figure 10, the total assembly cost is almost a linear function of the K–L divergence, because of the low scatter in reliabilities. An important conclusion is that the assembly sequence in heterogeneous systems with mutual dependencies should consist of the tasks organised in ascending order of task reliability, if the assembly costs variation is negligible compared with the variation in task reliabilities. Figure 11 shows the expected costs for the valid sequences of the three-pin plug example. The K–L divergence is calculated relative to PH(1) = [0.9762 0.9780 0.9830 0.9970 0.9980 1 1 1 1]. Although the search space in this case is very small (40 sequences) the validity of the Heuristic 1 can be observed from this figure.

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Figure 12. Assembly cost for sequences of an assembly process with homogeneous reliabilities (p = 0.9) and heterogeneous costs where CH(1) = [1 2 3 … 9] is shown in the lower left corner and CH(9!) = [9 8 7 … 1] is shown in the upper right corner.

Figure 13. Algorithmic use of the heuristics for complete mutual assembly task dependency.


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To demonstrate Heuristic 2 we use the Euclidean (or algebraic) distance between two vectors, which is the square root of the sum of squared element-by-element differences given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X DðH ðk ÞjjHð1ÞÞ ¼ (10) ðCHk;i CH1;i Þ2 i¼1

This time the sequence vectors represent a different cost distribution in a sequential process characterised by homogeneous reliabilities. For example, assume that the reliabilities for a process of 9 tasks are all equal to 0.9, and the cost vector is CH(1) = [1 2 3 … 9]. Figure 12 shows the distances and corresponding costs of all possible sequences. Finally, Figure 13 shows a float-chart for determining the assembly sequence of complete mutually dependent tasks. As shown, the proposed heuristics can be used for sequence dependent, schedule independent tasks with similar costs and/or reliabilities, while a more exhaustive search is required for other cases. Conclusions This paper studies the effect of assembly task sequencing on the expected cost of the entire assembly process. In particular, it focuses on processes in which there are uncertainties regarding the success of the tasks comprising the assembly process. There are two extreme types of processes related to two principal assumptions. The first assumption is that a task failure does not halt the entire assembly process, and only the failed task is repeated until it is completed successfully. This type of process is called a mutually independent process. We show that in this case, the task sequencing does not affect the economy of the assembly process unless there are set-up costs that depend on immediate transition between successive tasks. These costs can be due to tool changes, re-positioning, recalibration, etc., and are given in a cost matrix. Similarly, the probability of successfully completing an assembly task immediately after the previous task is given in the probability matrix. Using the cost and probability matrices to determine the expected costs of all possible assembly sequences, the tasks are modelled as nodes in a directional graph, and the expected costs of successive tasks are modelled as the edges of the graph. Using a weighted graph methodology such as TSPPC solver, the optimal path denotes the sequence that provides the lower bound for the expected cost of the entire assembly process. The second principal assumption describes processes that are sensitive to failure at any stage during the assembly process. In such processes, the failure of an assembly task requires re-work of previously completed tasks, and in the extreme case restarting the process from the beginning. In other words, the entire assembly process needs to be completed by a series of successful tasks. These types of processes are called mutually dependent processes, and the assembly sequencing has a profound effect on the entire expected cost. The complexity of determining the optimal assembly sequence of such processes is shown to be O(n2 × n!), and therefore heuristics are proposed for determining the upper and lower bounds for the expected assembly costs. These heuristics lead to manageable search spaces, even for systems with a large number of heterogeneous tasks (in terms of reliabilities and costs). The paper proves that the lower bound for the expected assembly cost in which all assembly tasks are mutually dependent and have the same cost, is given by the sequence in which tasks are arranged in ascending order in terms of their reliabilities (the least reliable task is performed first and the most reliable task last). Similarly, the upper bound for the expected cost of the entire assembly process is given by the sequence in which tasks are arranged according to their reliability in descending order. Also, for a process that consists of tasks with identical reliability values and varied costs, the sequence in which the tasks are arranged in ascending cost order provides the lower bound, and the sequence with descending cost order provides the higher bound for the expected cost of the entire assembly process. Further sensitivity analysis of general assembly processes in which tasks vary in both their reliability and cost values leads to the following heuristics: • When tasks costs are close in value, execute the most difficult tasks earlier in the assembly process. • When tasks reliabilities are close in value, execute the cheaper tasks earlier in the assembly process. A numerical analysis of the general assembly process shows a close to linear dependency between the expected cost of the entire process and the heterogeneity of the tasks in terms of their costs and reliability values. The Kullback– Leibler (K–L) divergence is used as measure of heterogeneity between the different sequences in terms of the tasks’ reliability values, while the Euclidean distance is used for measuring the heterogeneity between the different sequences in terms of the tasks cost values. Although the results presented in the paper focus on a single model straight flow assembly processes with heterogeneous tasks, the proposed heuristics can be extended to multi-model and U- shape systems. Also, the model presented in the paper considers only the direct cost associated with rework of failed task, and does not consider additional costs related to queuing, latencies, etc. Indeed, failure in assembly tasks may have severe side effects on the entire process,

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particularly in terms of the ALB. Future work will concentrate on extending the model for more general assembly systems, and will consider the indirect costs due to failures. Also future model will consider processes that consist of tasks with simultaneous strong heterogeneity of reliability and cost values. Finally, future work will focus on the effect of adding buffers between tasks with mutual dependencies for reducing the expected assembly costs. For example, partitioning the entire assembly into subassemblies and modules can, on the one hand, reduce the effect of the mutual dependencies in case of failures, but may introduce additional demands for resources and assembly operations. Acknowledgement We would like to sincerely thank the reviewers of IJPR for their insightful inputs into this paper.

Disclosure statement No potential conflict of interest was reported by the authors.

ORCID Shraga Shoval

http://orcid.org/0000-0002-0582-4821

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Appendix 1. The following example illustrates the use of the proposed model for the case of mutually independent as well as dependent failures in a commercial high power transmission box shown in Figure A1. The product consists of 38 elements, listed in Table A1. The assembly process is fully automated using an assembly robot and automatic feeders for all elements. Table A2 lists all required tools, and Table A3 lists all 43-assembly tasks. Each task is assigned with a cost and reliability value. The costs are expressed in terms of the required time defined by the expected moving time of an assembly robotic arm. Each task includes movement of the arm to the feeder, picking the element, moving to the assembly jig and performing the required operation. All feeders are positioned within a close proximity around the assembly jig such that movements to/from the feeders are identical for all elements and involve a cost of ‘1’. The costs and probabilities of the assembly operations are proportional to the level of precision and to the required force, such that higher precision and force linearly increase the costs and reduce the reliability. Figure A2 shows the liaison graph of the assembly

Figure A1. High power transmission box and its elements (http://www.gearboxtransmission.com/demo26/pic/other/2015-07-16-0215-417.jpg).

Table A1. List of elements for the transmission box shown in Figure A1. Name 1 2 3 4 5 6 7 8 9 11 12 17 19

Motor Pinion Pinion #1 Gear Pinion #1 Shaft Pinion #2 Gear Pinion #2 Shaft Output Shaft Gear Output Shaft Output Shaft Ext. Key Output Shaft Seal Output Shaft Ext. Bearing Output Shaft Ext. Circlip Output Shaft Spacer Output Shaft Int. Key

Name 20 22 24 25 30 31 32 34 37 39 41 42 43

Breather Valve Gearcase Lifting Eyebolt Output Shaft Int. Bearing Pinion #2 Int. Bearing Pinion #2 Key Pinion #2 Spacer Pinion #2 Mid. Bearing Pinion #2 Ext. Bearing Pinion #2 Ext. Circlip Pinion #1 Ext. Circlip Pinion #1 Ext. Bearing Pinion #1 Key

Name 45 47 59 88 100 101 102 131 181 506 515 521

Pinion #1 Int. Bearing Pinion #1 Int. Circlip Screw Plugs Output Shaft Int. Circlip Gearcase Cover Gearcase Cover Hex Bolts Gearcase Cover Gasket Pinion #2 Cap Pinion #1 Cap Pinion #2 Shim Ring Pinion #1 Shim Ring Output Shaft Shim Ring


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Table A2. List of required tools.

1 2 3 4 5 6 7 8 9

Name

Operation

Two fingers gripper Three fingers hydraulic gripper Three fingers electric gripper Bearing insertion tool Bearing removal tool Hydraulic press Screwdriver Vacuum gripper Circlip gripper

For quadrilateral elements For high force cylindrical elements For regular cylindrical elements Inserting a bearing on a shaft Removing a bearing from a shaft For applying high power force For bolts and screws For flat soft elements For circlips

Table A3. Tasks details. Task #

Description

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 31 32 33 34 35 36 37 38 39 40 41 42 43

Position Gearcase in assembly fixture Position Motor Pinion Position Pinion #1 Shaft Position Pinion #1 Key Insert Pinion #1 Gear Insert Pinion #1 Int. Bearing Insert Pinion #1 Shim Ring Insert Pinion #1 Int. Circlip Insert Pinion #1 Ext. Bearing Insert Pinion #1 Ext. Circlip Press Pinion #1 Position Output Shaft Insert Output Shaft Seal Insert Output Shaft Ext. Circlip Insert Output Shaft Ext. Bearing Insert Output Shaft Spacer Insert Output Shaft Int. Key Insert Output Shaft Gear Insert Output Shaft Int. Bearing Insert Output Shaft Shim Ring Insert Output Shaft Int. Circlip Insert Output Shaft Ext. Key Press Output Shaft Position Pinion #2 Shaft Insert Pinion #2 Key Insert Pinion #2 Mid. Bearing Insert Pinion #2 Gear Insert Pinion #2 Spacer Insert Pinion #2 Int. Bearing Insert Pinion #2 Ext. Bearing Insert Pinion #2 Shim Ring Insert Pinion #2 Ext. Circlip Press Pinion #2 Position Gearcase Cover Gasket Position Gearcase Cover Lock Gearcase Cover Hex Bolts Lock Lifting Eyebolt Lock Screw Plugs Lock Breather Valve Insert Pinion #2 Cover Insert Pinion #1 Cover Insert Output Shaft Ext. Key Remove Gearbox

Elements

Tools

Cost

Probability

22 1 3 43 2 45 515 47 42 41 3 7 9 12 11 17 19 6 25 521 88 6 7 7 31 34 4 32 30 37 506 39 5 102 100 101 24 59 20 131 181 8 22

1 3 3 1 2 4 3 9 4 9 6 3 3 9 4 3 1 2 4 3 9 1 6 3 1 4 2 3 4 4 3 9 6 8 8 7 3 7 7 8 8 1 1

4 4 4 4 6 6 5 5 6 5 8 4 5 5 6 5 4 6 6 5 5 4 8 4 4 6 6 5 6 6 5 5 8 4 4 10 6 8 5 4 4 4 4

0.99 0.99 0.99 0.99 0.97 0.97 0.98 0.98 0.97 0.98 0.98 0.99 0.98 0.98 0.97 0.98 0.99 0.97 0.97 0.98 0.98 0.99 0.98 0.99 0.99 0.97 0.97 0.98 0.97 0.97 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.98 0.99 0.99 0.99 0.99

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Figure A2. Liaison graph for the gearbox. Tasks are identified in Table A3. tasks where, thick solid lines indicate Type 1 and regular lines represent Type 2 precedence. Finally, based on the lists of tasks and the liaison graph, the expected cost matrix is shown in Table A4. The costs in this table consist of the actual assembly operation, as listed in Table A3, and set-up costs. Each tool change involves a cost of ‘3’ that consists of moving to the tool changing mechanism, dispatching the old tool and loading the new tool. Assuming all tasks are mutually independent in case of a failure, the optimal sequence, determined by the TSP solver, is 1 → 3 → 2 → 4 → 5 → 6 → 7 → 8 → 9 → 10 → 11 → 12 → 13 → 14 → 15 → 16 → 17 → 18 → 19 → 20 → 21 → 22 → 23 → 24 → 25 → 26 → 28 → 27 → 29 → 30 → 31 → 32 → 33 → 34 → 35 → 36 → 37 → 38 → 39 → 40 → 42 → 41 → 43. The expected cost of this sequence is 332.5, compared with expected cost of 358.4 for the worst sequence and an average expected cost of 337.4 for all feasible random selected sequences. The optimal sequence reduces the number of tool changes during the process, and results in 7.78% saving compared with the worst case, and 1.47% savings compared with random selected sequences. Next, assume that all tasks are mutually dependent. Using the optimal sequence generated by a TSP solver, the total expected cost, as determined by (8), is 466.84 (increase of 40.4%). Reorganising the tasks in an ascending order by their probabilities (subject to the sequential constraints) results in total expect cost of 430.19 (increase of 29.3% compared with the optimal sequence of mutual independent tasks).

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 31 32 33 34 35 36 37 38 39 40 41 42 43

999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999

1

7.04 999 7.04 7.04 999 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

2

4.04 4.04 999 999 6.19 6.19 999 999 6.19 5.1 8.16 4.04 5.1 5.1 999 999 999 6.19 999 999 999 4.04 8.16 4.04 4.04 6.19 6.19 5.1 6.19 6.19 5.1 5.1 8.16 999 999 10.1 6.06 8.08 5.1 4.04 4.04 4.04 999

3

7.04 7.04 7.04 999 999 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

4

7.04 7.04 7.04 7.04 999 999 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

5

7.04 7.04 7.04 7.04 9.19 999 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

6

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

7

7.04 7.04 7.04 7.04 9.19 9.19 0 999 999 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

8

Table A4. Expected cost matrix.

7.04 7.04 7.04 7.04 9.19 9.19 999 0 999 999 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

9

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 999 999 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

10

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 999 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 999 7.04 999

11

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 999 999 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

12

4.04 4.04 4.04 4.04 6.19 6.19 999 999 6.19 5.1 8.16 4.04 999 999 999 999 999 6.19 999 999 999 4.04 8.16 4.04 4.04 6.19 6.19 5.1 6.19 6.19 5.1 5.1 8.16 999 999 10.1 6.06 8.08 5.1 4.04 4.04 4.04 999

13

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 999 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

14

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

15

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 0 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

16

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 0 999 999 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

17

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 0 999 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

18

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

19

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 0 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

20

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 0 999 999 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

21

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 0 999 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

22

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 999 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 999 999

23

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 999 999 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

24

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 999 999 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

25

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 999 999 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

26

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 999 999 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

27

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 999 999 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

28

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 999 999 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

29

4.04 4.04 4.04 4.04 6.19 6.19 999 999 6.19 5.1 8.16 4.04 5.1 5.1 999 999 999 6.19 999 999 999 4.04 8.16 4.04 4.04 6.19 6.19 5.1 6.19 999 999 5.1 8.16 999 999 10.1 6.06 8.08 5.1 4.04 4.04 4.04 999

30

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 999 999 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

31

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 999 999 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

32

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 999 999 999 13.1 9.06 11.1 8.1 999 7.04 7.04 999

33

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 7.04 999

34

4.04 4.04 4.04 4.04 6.19 6.19 999 999 6.19 5.1 8.16 4.04 5.1 5.1 999 999 999 6.19 999 999 999 4.04 8.16 4.04 4.04 6.19 6.19 5.1 6.19 6.19 5.1 5.1 8.16 0 999 999 6.06 8.08 5.1 4.04 4.04 4.04 999

35

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 0 999 999 11.1 8.1 7.04 7.04 7.04 999

36

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 999 11.1 8.1 7.04 7.04 7.04 999

37

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 999 8.1 7.04 7.04 7.04 999

38

4.04 4.04 4.04 4.04 6.19 6.19 999 999 6.19 5.1 8.16 4.04 5.1 5.1 999 999 999 6.19 999 999 999 4.04 8.16 4.04 4.04 6.19 6.19 5.1 6.19 6.19 5.1 5.1 8.16 999 999 10.1 6.06 8.08 999 4.04 4.04 4.04 999

39

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 999 7.04 7.04 999

40

4.04 4.04 4.04 4.04 6.19 6.19 999 999 6.19 5.1 8.16 4.04 5.1 5.1 999 999 999 6.19 999 999 999 4.04 8.16 4.04 4.04 6.19 6.19 5.1 6.19 6.19 5.1 5.1 8.16 999 999 10.1 6.06 8.08 5.1 4.04 999 4.04 999

41

7.04 7.04 7.04 7.04 9.19 9.19 999 999 9.19 8.1 11.2 7.04 8.1 8.1 999 999 999 9.19 999 999 999 7.04 11.2 7.04 7.04 9.19 9.19 8.1 9.19 9.19 8.1 8.1 11.2 999 999 13.1 9.06 11.1 8.1 7.04 7.04 999 999

42

4.04 4.04 4.04 4.04 6.19 6.19 999 999 6.19 5.1 8.16 4.04 5.1 5.1 999 999 999 6.19 999 999 999 4.04 8.16 4.04 4.04 6.19 6.19 5.1 6.19 6.19 5.1 5.1 8.16 999 999 10.1 6.06 8.08 5.1 4.04 4.04 4.04 999

43

International Journal of Production Research 2825

2826

S. Shoval et al.

Appendix 2. Assume two different sequences (say Ha and Hb) with n identical tasks except for the two tasks d and e (1 ≤ d < e ≤ n) such that PHa ðdÞ [ PHa ðeÞ , then we have: ^ n ðH b Þ [ C ^ n ðH a Þ C

(B1)

Proof: from the assumption for the two processes Ha, Hb we have: PHa ðiÞ ¼ PHb ðiÞ ; i 6¼ d; e PHa ðdÞ ¼ PHb ðeÞ ¼ p1 ; PHa ðeÞ ¼ PHb ðdÞ ¼ p2 ; p1 [ p2

(B2)

Because PHa ðiÞ ¼ PHb ðiÞ ; i 6¼ d; e and e > d, we can state that: n Y

n Y

PHa ð jÞ ¼

j¼i

PHb ð jÞ ; i [ e

(B3)

PHb ð jÞ ; i d

(B4)

j¼i

Since PHa ðdÞ ¼ PHb ðeÞ and PHa ðeÞ ¼ PHb ðdÞ , we can also state that: n Y

n Y

PHa ð jÞ ¼

j¼i

j¼i

It follows from (B3) and (B4) that: d n Y X i¼1

j¼i

n X

n Y

i¼eþ1

j¼i

!1 ¼

PHa ð jÞ

d n Y X i¼1

!1 PHa ð jÞ

¼

!1 PHb ð jÞ

(B5)

j¼i

n X

n Y

i¼eþ1

j¼i

!1 PHb ð jÞ

(B6)

Note that there is element-by-element equality between elements of the summations of the sides of above equalities. Because PHa ðeÞ [ PHb ðeÞ and PHa ðiÞ ¼ PHb ðiÞ ; for anyi 6¼ d; e we have PHa ðdþ1Þ . . .PHa ðeÞ . . .PHa ðnÞ [ PHb ðdþ1Þ . . .PHa ðeþ1Þ . . .PHa ðnÞ ; or

n Y j¼i

PHa ð jÞ [

n Y

PHb ð jÞ

(B7)

j¼i

So for d < i ≤ e we can state that:

Adding (B5), (B6) and (B7) results in (B8).

e X

n Y

i¼dþ1

j¼i

!1 PHa ð jÞ

\

e X

n Y

i¼dþ1

j¼i

!1 PHb ð jÞ

(B8)


2827

Appendix 3. Table C1. Schedule independent results. #

Sequence

k=0

k=1

k=2

k=3

k=4

k=5

k=6

k=7

k=8

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 31 32 33 34 35 36 37 38 39 40

1→5→4→6→3→2→7→8→9 1→5→4→6→2→3→7→8→9 1→5→4→3→6→2→7→8→9 1→5→4→3→2→6→7→8→9 1→5→4→2→3→6→7→8→9 1→5→4→2→6→3→7→8→9 1→5→3→4→6→2→7→8→9 1→5→3→4→2→6→7→8→9 1→5→3→2→4→6→7→8→9 1→5→2→4→6→3→7→8→9 1→5→2→4→3→6→7→8→9 1→5→2→3→4→6→7→8→9 1→4→5→6→3→2→7→8→9 1→4→5→6→2→3→7→8→9 1→4→5→3→6→2→7→8→9 1→4→5→3→2→6→7→8→9 1→4→5→2→3→6→7→8→9 1→4→5→2→6→3→7→8→9 1→4→3→5→6→2→7→8→9 1→4→3→5→2→6→7→8→9 1→4→3→2→5→6→7→8→9 1→4→2→5→6→3→7→8→9 1→4→2→5→3→6→7→8→9 1→4→2→3→5→6→7→8→9 1→3→5→4→6→2→7→8→9 1→3→5→4→2→6→7→8→9 1→3→5→2→4→6→7→8→9 1→3→4→5→6→2→7→8→9 1→3→4→5→2→6→7→8→9 1→3→4→2→5→6→7→8→9 1→3→2→5→4→6→7→8→9 1→3→2→4→5→6→7→8→9 1→2→5→4→6→3→7→8→9 1→2→5→4→3→6→7→8→9 1→2→5→3→4→6→7→8→9 1→2→4→5→6→3→7→8→9 1→2→4→5→3→6→7→8→9 1→2→4→3→5→6→7→8→9 1→2→3→5→4→6→7→8→9 1→2→3→4→5→6→7→8→9

74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56 74.56

75.20 75.14 75.15 75.20 75.15 75.16 75.19 75.19 75.11 75.06 75.06 75.18 75.12 75.06 75.08 75.12 75.07 75.08 75.08 75.08 75.11 75.06 75.06 75.07 75.17 75.17 75.07 75.19 75.20 75.18 75.19 75.12 75.02 75.02 75.05 74.94 74.94 74.95 75.03 75.06

75.65 75.76 75.65 75.78 75.72 75.76 75.72 75.90 75.80 75.64 75.58 75.61 75.51 75.61 75.51 75.63 75.57 75.62 75.47 75.64 75.63 75.50 75.44 75.45 75.74 75.87 75.78 75.66 75.78 75.81 75.64 75.68 75.40 75.36 75.40 75.33 75.28 75.29 75.49 75.41

76.20 76.13 76.14 76.14 76.12 76.12 76.33 76.33 76.32 75.95 75.95 76.03 76.05 75.99 75.99 75.99 75.98 75.98 76.01 76.01 76.00 75.75 75.75 75.71 76.30 76.35 76.34 76.15 76.20 76.19 76.10 76.02 75.73 75.72 75.79 75.59 75.58 75.54 75.81 75.73

76.69 76.50 76.66 76.53 76.24 76.25 76.81 76.68 76.50 76.20 76.19 76.39 76.54 76.35 76.52 76.38 76.09 76.10 76.49 76.35 76.11 75.99 75.98 76.00 76.83 76.69 76.52 76.68 76.54 76.31 76.41 76.26 75.97 75.96 76.15 75.82 75.81 75.83 76.12 75.97

77.03 76.74 77.01 76.76 76.47 76.49 77.21 76.96 76.79 76.32 76.30 76.49 76.88 76.59 76.86 76.61 76.32 76.34 76.88 76.63 76.39 76.11 76.09 76.10 77.18 76.93 76.76 77.03 76.78 76.54 76.52 76.37 76.21 76.19 76.38 76.06 76.04 76.06 76.40 76.25

77.27 76.98 77.25 77.00 76.71 76.73 77.44 77.19 77.02 76.56 76.53 76.73 77.12 76.83 77.10 76.85 76.56 76.58 77.11 76.86 76.63 76.34 76.32 76.34 77.46 77.21 77.04 77.31 77.06 76.83 76.80 76.65 76.31 76.29 76.49 76.17 76.15 76.16 76.50 76.36

77.37 77.08 77.35 77.10 76.81 76.83 77.55 77.30 77.13 76.66 76.64 76.83 77.22 76.93 77.20 76.95 76.66 76.68 77.22 76.97 76.73 76.45 76.43 76.44 77.57 77.32 77.14 77.42 77.17 76.93 76.90 76.76 76.42 76.40 76.59 76.27 76.25 76.27 76.61 76.46

77.37 77.08 77.35 77.10 76.81 76.83 77.55 77.30 77.13 76.66 76.64 76.83 77.22 76.93 77.20 76.95 76.66 76.68 77.22 76.97 76.73 76.45 76.43 76.44 77.57 77.32 77.14 77.42 77.17 76.93 76.90 76.76 76.42 76.40 76.59 76.27 76.25 76.27 76.61 76.46

2828

S. Shoval et al.

Table C2. Schedule dependent results. #

Sequence (U)

k=0

k=1

k=2

k=3

k=4

k=5

k=6

k=7

k=8

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 31 32 33 34 35 36 37 38 39 40

1→5→4→6→3→2→7→8→9 1→5→4→6→2→3→7→8→9 1→5→4→3→6→2→7→8→9 1→5→4→3→2→6→7→8→9 1→5→4→2→3→6→7→8→9 1→5→4→2→6→3→7→8→9 1→5→3→4→6→2→7→8→9 1→5→3→4→2→6→7→8→9 1→5→3→2→4→6→7→8→9 1→5→2→4→6→3→7→8→9 1→5→2→4→3→6→7→8→9 1→5→2→3→4→6→7→8→9 1→4→5→ 6→3→2→7→8→9 1→4→5→6→2→3→7→8→9 1→4→5→3→6→2→7→8→9 1→4→5→3→2→6→7→8→9 1→4→5→2→3→6→7→8→9 1→4→5→2→6→3→7→8→9 1→4→3→5→6→2→7→8→9 1→4→3→5→2→6→7→8→9 1→4→3→2→5→6→7→8→9 1→4→2→5→6→3→7→8→9 1→4→2→5→3→6→7→8→9 1→4→2→3→5→6→7→8→9 1→3→5→4→6→2→7→8→9 1→3→5→4→2→6→7→8→9 1→3→5→2→4→6→7→8→9 1→3→4→5→6→2→7→8→9 1→3→4→5→2→6→7→8→9 1→3→4→2→5→6→7→8→9 1→3→2→5→4→6→7→8→9 1→3→2→4→5→6→7→8→9 1→2→5→4→6→3→7→8→9 1→2→5→4→3→6→7→8→9 1→2→5→3→4→6→7→8→9 1→2→4→5→6→3→7→8→9 1→2→4→5→3→6→7→8→9 1→2→4→3→5→6→7→8→9 1→2→3→5→4→6→7→8→9 1→2→3→4→5→6→7→8→9

74.61 78.61 74.61 78.61 78.61 78.61 74.61 78.61 78.64 78.64 78.64 78.61 74.64 78.64 74.64 78.64 78.64 78.64 74.64 78.64 78.64 78.64 78.64 78.64 74.61 78.61 78.64 74.61 78.61 78.61 78.61 78.64 74.56 74.56 74.56 74.59 74.59 74.59 74.56 74.56

75.19 79.24 75.19 79.24 79.24 79.25 75.16 79.21 79.19 79.23 79.23 79.29 75.12 79.16 75.12 79.16 79.16 79.17 75.11 79.16 79.15 79.2 79.21 79.21 75.17 79.22 79.19 75.2 79.24 79.24 79.29 79.24 75.06 75.06 75.03 74.98 74.98 74.98 75.03 75.06

75.69 79.85 75.69 79.86 79.86 79.86 75.78 79.95 79.89 79.8 79.8 79.76 75.54 79.71 75.55 79.71 79.72 79.72 75.55 79.72 79.72 79.6 79.6 79.59 75.79 79.96 79.9 75.71 79.88 79.88 79.78 79.81 75.4 75.4 75.48 75.32 75.32 75.33 75.49 75.41

76.19 80.32 76.18 80.32 80.27 80.27 76.34 80.48 80.49 80.12 80.13 80.21 76.04 80.18 76.03 80.17 80.12 80.12 76.04 80.18 80.14 79.86 79.86 79.87 76.35 80.49 80.5 76.2 80.34 80.3 80.23 80.15 75.73 75.72 75.8 75.58 75.57 75.58 75.81 75.73

76.73 80.69 76.71 80.66 80.43 80.44 76.87 80.82 80.72 80.42 80.41 80.57 76.58 80.54 76.56 80.52 80.28 80.29 76.57 80.53 80.3 80.15 80.14 80.15 76.88 80.83 80.73 76.73 80.68 80.46 80.59 80.44 75.97 75.95 76.11 75.82 75.81 75.82 76.12 75.97

77.08 80.94 77.06 80.9 80.67 80.69 77.22 81.06 80.96 80.59 80.56 80.72 76.93 80.79 76.91 80.75 80.52 80.54 76.92 80.76 80.53 80.31 80.29 80.3 77.23 81.07 80.97 77.08 80.92 80.7 80.74 80.59 76.25 76.23 76.39 76.1 76.08 76.09 76.4 76.25

77.36 81.22 77.34 81.18 80.95 80.97 77.5 81.34 81.24 80.87 80.85 81 77.21 81.07 77.19 81.03 80.8 80.82 77.2 81.04 80.82 80.59 80.57 80.58 77.51 81.35 81.25 77.36 81.2 80.98 81.02 80.88 76.36 76.34 76.49 76.21 76.19 76.2 76.5 76.36

77.47 81.32 77.45 81.29 81.05 81.07 77.61 81.45 81.35 80.97 80.95 81.11 77.32 81.17 77.3 81.14 80.9 80.92 77.31 81.15 80.92 80.7 80.68 80.69 77.62 81.46 81.36 77.47 81.31 81.08 81.13 80.98 76.46 76.44 76.6 76.32 76.3 76.31 76.61 76.46

77.47 81.32 77.45 81.29 81.05 81.07 77.61 81.45 81.35 80.97 80.95 81.11 77.32 81.17 77.3 81.14 80.9 80.92 77.31 81.15 80.92 80.7 80.68 80.69 77.62 81.46 81.36 77.47 81.31 81.08 81.13 80.98 76.46 76.44 76.6 76.32 76.3 76.31 76.61 76.46

Assembly sequence planning for processes with

Assembly sequence planning for processes with

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