Lean Manufactury All Location

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International Journal of Production Research,

Vol. 46, No. 13, 1 July 2008, 34853502

Worker allocation in lean U-shaped production lines

JOHN P. SHEWCHUK*

Grado Department of Industrial and Systems Engineering,

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

(Revision received October 2006)

U-shaped lines are widely used in lean systems. In U-shaped production lines,each worker handles one or more machines on the line: the worker allocationproblem is to establish which machines are handled by which worker. This differsfrom the widely-investigated U-line assembly line balancing problem in that theassignment of tasks to line locations is fixed. This paper address the workerallocation problem for lean U-shaped production lines where the objectives are tominimize the quantity of workers and maximize full work: such allocationsprovide the opportunity to eliminate the least-utilized worker by improvingprocesses accordingly. A mathematical model is developed: the model allows forany allocation of machines to workers so long as workers do not cross paths.Walking times are considered, where workers follow circular paths and walkaround other worker(s) on the line if necessary. A heuristic algorithm for tacklingthe problem is developed, along with a procedure representing the traditionalapproach of constructing standard operations routines. Computational experi-ments considering three line sizes (up to 20 machines) and three takt time levels

are performed. The results show that the proposed algorithm both improves uponthe traditional approach and is more likely to provide optimal solutions.

Keywords: U-shaped line; Lean production; Worker allocation; Full work

1. Introduction

Lean manufacturing systems are in widespread use these days. One of the

cornerstones of lean manufacturing is the use of U-shaped lines and one-piece

flow, where the goal is to produce items in accordance with takt time (cycle time).When operations are manual, assembly line balancing is performed to determine how

many workers are needed and which tasks are done by which workers. Tasks are

grouped into stations, where each station has one worker assigned: each task is then

assigned a particular location on the line. When takt time changes, the line can

be rebalanced and a new assignment and grouping of tasks into stations obtained.

As takt time increases, the quantity of workers on the line can be decreased and

vice-versa. This is known as shojinka (Monden 1998).

In U-shaped production lines, where each task corresponds to a machine

performing an operation, the problem of dealing with changing demand is not

*Email: [email protected]


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as straightforward. The sequence of machines on the line is often fixed, making it

impossible to reallocate tasks to different line locations. Even when routing

alternatives are available, it may be difficult and/or uneconomic to move machines

whenever takt time changes. Thus, the assignment of tasks to line locations is fixed.

If a single operator can handle multiple machines (a concept known as jidoka;Monden 1998), however, the line balancing problem can be replaced with the

worker allocation problem, where the goal is to determine which machines are to be

handled by each worker. For a given takt time T, each worker will have a standard

operations routine which describes which machines will be visited and in what

sequence (figure 1). Whenever a worker arrives at a machine, he/she unloads the

completed item, loads the next item, performs any needed actions at the machine and

then walks to the next machine. Following the last machine, the worker returns to

the starting machine and waits until the start of the next cycle as necessary. Workers

can be reallocated when takt time changes, thus making shojinka possible on such

lines.While the worker allocation problem is somewhat similar to the U-line assembly

line balancing problem (UALBP), there are three important differences. The first is

that task assignments on the line are fixed, as previously described. The second is

that walking time can be significant on such lines and thus cannot be ignored

(Nakade and Ohno 1999). The third is that as there are no restrictions on what

machines can be assigned to a given worker, the possibility of workers crossing paths

exists. This should be avoided (e.g. Hyer and Wemmerlo v 2002) and thus only

assignments where workers do not cross paths should be allowed. A further

consideration is that while the objective in traditional line balancing is to balance

workload across operators, the objective in lean systems is often to maximize fullwork, i.e. have as many workers fully utilized as possible. The reason is that this

provides the opportunity to eliminate the least-utilized worker by improving

1

2

9

T

3

7

4

5

6

81 2 3 4

5

9 8 7 6

(a)

(b)

12

3

4

1:

3:

4:

2:

Figure 1. (a) Example 9-machine U-shaped production line, showing allocation of fourworkers. (b) Standard operations routines: solid line manual time; wavy line walking time;dashed line machine time.

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processes accordingly (Suzaki 1987, Dennis 2002). In figure 1, for example,

worker 2 handles machine 8 only, and is accordingly utilized only about 14% of

the time. This worker can be eliminated by either eliminating the process at

machine 8 entirely or improving processes at machines 13, 7, and/or 9 so that either

worker 1 or 3 can handle machine 8 as well. Being able to establish full workallocations such as this allows us to focus on where process improvement efforts on

the line should be directed.

The purpose of this paper is to:

. Develop a mathematical model of the worker allocation problem for lean

U-shaped production lines.

. Based upon the model, develop a suitable algorithm for tackling such

problems.

The objective is to minimize the quantity of workers on the line, then maximize

full work. A simple algorithm is desirable, such that it can quickly and easily be

implemented by lean practitioners. Relevant literature is reviewed in section 2,

followed by development of a mathematical model in section 3. Section 4 presents

both the traditional approach and a new algorithm for solving the worker allocation

problem; computational experiments to establish performance are described in

section 5. Concluding remarks are provided in section 6, along with a discussion

of where additional research is needed.

2. Literature review

The assembly line balancing problem (ALBP) has been widely studied over the pastseveral decades. Becker and Scholl (2006) provide a comprehensive review of

generalized assembly line balancing problems and their solution. Of relevance for

this paper is the U-line assembly line balancing problem (UALBP), where the

assembly line is arranged in a U-configuration and stations can be assigned tasks

from both sides of the line. In particular, the problem version UALBP-1 is concerned

with minimizing the number of stations (workers) for a given cycle (takt) time. The

UALBP was first introduced by Miltenburg and Wijngaard (1994) and tackled using

dynamic programming. Since then, researchers have addressed the problem using a

variety of approaches including commercial optimization packages (Urban 1998),

branch and bound (Scholl and Klein 1999, Aase et al. 2003), simulated annealing(Erel et al. 2001), and goal programming (Go kcen and Ag pak 2006). As previously

noted, however, the UALBP is fundamentally different from the worker allocation

problem in lean U-shaped production lines: the UALBP examines both the

assignment of tasks to locations on the line and assignment of workers to tasks,

while task (i.e. machine) locations are considered fixed in the worker allocation

problem. Even if machines are considered to be movable, the UALBP does not

consider walking times: hence they are ignored in such works. Miltenburg and

Wijngaard (1994), however, note the need to consider walking times as stations

sprawl across different parts of the U-line.

The closely-related problem of worker allocation in U-shaped production lineshas been largely ignored in the research community. This is surprising considering

that adding or removing operators from U-shaped lines often occurs frequently, with

Worker allocation in lean U-shaped production lines 3487


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moving of machines left for large changes in output rate (Miltenburg 2001).

The most well-known approach to the problem is presented by Monden (1998). He

provides a manual procedure for determining the standard operations routines, or

allocation of workers to machines, for a U-shaped line and product to be made

(figure 1). The procedure consists of selecting one operation (machine) at a time,from those available, and assigning to the current worker. This is continued until any

additional assignments would violate the takt time constraint: the implicit goal is to

provide each worker with full work. Walking time is considered, as is any time

needed for safety and/or quality checks. The solution is constructed graphically,

showing the relationships between manual time, machine time, walking time, and

takt time. While graphical methods such as this have become very popular in

industry, they suffer from the fact that they provide no guidance as to how to select

the next operation at each stage.

Ohno and Nakade (1997) develop a procedure for determining the minimum

cycle time for a given quantity of workers in U-shaped production lines. They

consider manual, machine, and walking times, as well as the waiting time which

results when a worker arrives at a machine and the item there is still in process.

Nakade and Ohno (1999) extend this work to develop an optimal procedure for

finding the minimum quantity of workers for a given takt (cycle) time and a worker

allocation which then balances workload on a U-shaped production line. They again

consider manual, machine, walking, and waiting times, and ensure that workers do

not cross paths in establishing stations. Lines have an even quantity of machines (i.e.

rectangular shape), and each worker must have an uninterrupted stretch of machines

when following a circular walking path around the line. The machine perimeter for

each worker thus forms either a point (single machine: worker 2 in figure 1), line (two

machines: worker 3), triangle (three machines: worker 1), rectangle (e.g. machines2, 3, 4, 6, 7, 8) or trapezoid (e.g. machines 1, 2, 7, 8, 9). The model developed in this

paper differs in that

(i) the objective is to maximize full work,

(ii) the quantity of machines on the line can be even or odd, and

(iii) any allocation of machines to workers is allowed so long as workers do not

cross paths.

In other words, there is no requirement for each worker to have an uninterrupted

set of machines when following a circular walking path: interrupted circular machine

assignments (e.g. machines 1, 3, 4, and 8, figure 1) are possible.

3. Mathematical model

As previously stated, our objective in allocating workers to lean U-shaped

production lines is to minimize the quantity of workers, then maximize full work.

To meet these objectives, we develop a mathematical model which maximizes full

work for a given quantity of workers W. Solution then involves starting with Wset at

its lower bound, then incrementing by 1 as necessary until the problem becomes

feasible. Only a few iterations at most should be required, as each additional workerprovides a substantial capacity increase and the quantity of workers is limited by the

number of machines.

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The model is as follows. The line consists of M machines in a U-shape

arrangement, as shown in figure 1 (Modd) or figure 2 (Meven). The manual time at

machine iis ti: takt time is T. Wworkers are to be allocated to the line. The followingassumptions are made:

1. Machines and workers are located via a grid arrangement, with

distance d between adjacent worker locations in the same row (e.g. locations

1 and 2, figure 2), and 2d between opposite worker locations (e.g. 1 and M).

When M is odd, the middle worker location is located at a distance offfiffiffiffiffiffiffiffi

2d2p

from the preceding and following locations.

2. Takt time is always greater than or equal to the longest operation time

(manual time machine time). This means that a feasible solution alwaysexists for a given problem. (If an operation time is larger than takt time in

practice, it must be reduced accordingly or other methods used to produce tothat takt time.)

3. Any worker can perform any manual task on the line.

4. Manual time at each machine and walking times between machines are both

deterministic and worker-independent.

5. Each worker follows a circular walking path inside the line, stopping at each

assigned machine in succession. If two successive machines are not adjacent

on the walking path (i.e. interrupted circular machine assignment, section 2),

the worker must walk around the other worker(s) in going from the first

machine to the second. Circular walking paths are used in Monden (1998) and

Nakade and Ohno (1999): in these works the approach minimizes totalwalking time as each path forms a convex polygon. While the approach may

not guarantee minimum walking times in this paper (walking paths become

non-convex when one worker must walk around another), it is still used as

it is common both in practice and research and to maintain tractability.

6. Direct walking time between worker locations (i.e. walking time when no

other workers on the line to walk around) is directly proportional to

Euclidean distance between locations.

7. To walk around one or more workers on a straight section of the line,

a worker travels to and along (if necessary) the major axis dividing the

U-shaped line, as shown in the examples of figure 2 (paths ( i 1) a (i 1),(i 1) a b (M/2)). This results in a constant additional walking distanceof 2

ffiffiffiffiffiffiffiffi2d 2

p d, regardless of how many workers must be walked around.

1 2 i i+1i1 M/2M/2

1

MM/2

+2

M/2

+1M1

M+2

i

M+1

iMi

a b

d

2d

Figure 2. Worker locations, even-numbered U-shaped line.



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8. Each worker can start with any machine in his/her assignment: starting

location does not affect the total work (manual walking) assigned to theworker. However, different starting locations can change the WIP (work in

process) required to prevent workers from starving. We assume the minimum

WIP required on the line such that starvation does not occur.

To establish the allocation of workers to machines, define

xik 1, machine iassigned to worker k,0, otherwise&

Given any solution {xik, i 1, . . . , M, k 1, . . . , W}, total walking time for eachworker is fixed, per assumption 5. To determine total walking time for a worker, we

first calculate the direct walking time (round-trip time when no other workers on

the line), then add any additional time resulting from walking around one or more

other workers. Define

qijk 1, worker k moves from machine worker location ito machine j,0, otherwise&

and

vijk 1, worker k walks around other worker

when going from machine ito machine j,

0, otherwise

8>>>>=>>>>>;

i 1, . . . , L2 2, j i 2, . . . , L2, k 1, . . . , Wi L1 2, . . . , M 2, j i 2, . . . , M, k 1, . . . , W

1724



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H2Fijk Pj1

ni1xnk !j i 1

H2Fijk

Pj1

ni1xnk5H2

j

i

1

Fjik Fijk

9>>>>>>>=>>>>>>>;

i 1, . . . , L2 2, j i 2, . . . , L2, k 1, . . . , Wi

L1

2, . . . , M

2, j

i

2, . . . , M, k

1, . . . ,W

2530

where L1 M/2 1 if M is even and (M 1)/2 if M is odd; L2 M/2, M even,(M 1)/2, M odd.

The objective function (1) maximizes the average utilization of all but the

least-utilized worker: we refer to this measure Z as the full work level for

the system. For each worker k, utilization is total time (manual directwalking walking around other workers) divided by takt time T: the least-utilizedworker has total time a. Constraints (2), (3) and (4) calculate a via parameter dk(dk 1 if worker k has total time a, 0 otherwise) and the big-M constant H1(H1! T). Constraint (5) ensures that takt time is not exceeded for any worker, whileeach machine must be assigned to only one worker via constraint (6). Constraint (7)

ensures that all machines assigned to each worker are adjacent. Given a U-shaped

line with an even number of machines (Meven) and any pair of opposing machines i

and j M 1 i(figure 2), a feasible assignment of machines to worker k requires atleast one of {i, j} to be assigned if one or more assigned machines are to the left of

{i,j} (i.e. Bik 0) and one or more assigned machines are to the right (Dik 0).Constraints (8)(11) then ensure that Bik and Dik are set accordingly, again via abig-M constant, H2 (H2! M). Note that when M is even, opposing pairs {i, j} endat i M/2 1: when Mis odd, they end at i (M 1)/2. This is taken care of via theparameter L1.

Constraint (12) ensures that workers do not cross paths. Consider, for example,

that on a line with an even number of machines, worker k is assigned two machines:

i and M 2 i (figure 2). Another worker n could then be assigned machines {i 1,M 1 i}, as these are adjacent. This would cause the workers to cross paths,however. Constraint (12) ensures this does not happen: given the possible

assignments of {i, M 2 i} to worker k and {i 1, M 1 i} to worker n, atmost three such assignments can be made. In similar manner to the adjacencyconstraints, the range for i depends upon whether M is even or odd: this is specified

via the parameter L2. Constraints (13)(15) calculate qijk values from xik values,

ensuring each worker moves in a circular path around the line. Constraint (16)

calculates the total additional time incurred by each worker k in walking around

other workers, Yk, based upon whether or not worker k walks around other workers

when going from machine i to machine j (vijk values) and constant time value s.

Variables vijk are established in constraints (17)(24), via Fijk (Fijk 1 if worker k isnot assigned all machines in-between machines iand j, 0 otherwize) and qijk. Finally,

variables Fijk are established via constraints (25)(30).

The above problem is a variation of the generalized assignment problem, which

is NP-hard.

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3.1 Calculation of lower bound on quantity of workers

To determine a lower bound on the quantity of workers, we use the approach of

Nakade and Ohno (1999). Given W workers, a lower bound on cycle time C for the

line, LB[C], can be found from the following:

LBC

PMi1

ti M W 1lminW

The numerator equals total operator time required, which consists of all manual time

plus all walking time. If we have W workers, walking time is minimized if the first

W 1 workers remain stationary and the remaining worker then coversM (W 1) M W 1 stations. Given that the minimum walking time betweenany two machines is lmin, total walking time is then at least (M W 1) lmin. Wecan then find LB[W] from

LBW fmin WjLBC ! Tgwhich results in

LBW

PMi1

ti M 1lminT lmin

2664

3775

This is equivalent to the expression provide by Nakade and Ohno (1999).

4. Algorithms

The goal of this paper is to develop an algorithm for allocating workers to lean

U-shaped production lines quickly and easily in practice. A simple, single-pass

procedure is as follows:

Step 0: Initialize. Set k worker 1, A set of unassigned machines {1, . . . , M}.Step 1: Prepare for next worker. Set Sk set of machines assigned to worker k .

Step 2: Establish feasible machines for worker k Fk. A machine m 2 A is feasibleiff:

it is adjacent to at least one other machine in Sk.

it does not result in worker k crossing paths with any other worker, and

it does not cause takt time to be violated, i.e. total time (manual total round-tripwalking time, circular walking path) for machines Sk m T.If Fk , go to Step 5.

Step 3: Select machine m from Fk for assignment to worker k.

Step 4: Assign machine m to worker k. Set Sk Sk m, A A m, Fk : go toStep 2.



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Step 5: Worker k completed. If A , done: set W k, calculate Z from Sk,k 1, . . . , W. Otherwize, set k k 1, go to Step 1.

Different approaches for Step 3 (machine selection) result in different algorithms

for the worker allocation problem. We first modify Step 3 to obtain the traditionalapproach; i.e. that employed when worker allocation is done by generating a

standard operations routine for each worker. An improved algorithm is then

presented.

4.1 Traditional approach

As outlined in section 2, Monden (1998) provides a basic, step-by-step procedure for

developing standard operations routines, without detailing exactly how machines are

to be selected. With no other information, it is reasonable to assume machines are

simply selected in order. This results in the following algorithm (procedure for Step 3above):

Step 3: Select machine m mini fki

, where fki ith machine in Fk.Figure 3 shows an example of the traditional approach applied to a seven-

machine problem, where walking times are assumed zero. An optimal solution is also

shown: comparing the two indicates improved performance is possible.

4.2 Proposed algorithm

The proposed algorithm begins with the idea of simply assigning machines in order

of largest total time (manual walking), as machines which result in large total timevalues become harder to fit as assignments are made. The algorithm is as follows:

Step 3:

(i) Calculate Vk {vki}, where vki total time (manual total round-trip walkingtime, circular walking path) for machines Skfki2 Fk.

(ii) Select machine m fki, where i argmaxi vkif g.Consider this initial algorithm applied to the problem in figure 3(a). Starting with

worker 1, machine 4 is selected first, following which F1 {3, 5} and V1 {14, 17}.Thus, machine 5 is selected next. Continuing on with the procedure in this mannerproduces the solution in figure 3(c), which is optimal. This simple approach,

however, has a drawback in that it can result in group splitting: the original group

1

3 3

2

8 8

3

2 2

4

7

4

6

7

5

5

4 12

4 7

3 8

3

2

4 12

4 7

5

5

1 2 3

7 6 5

1 2

6

T= 20

5

7

(a) (c)

W= 3, Z= 0.75 W= 3, Z= 0.95

(b)

12

Figure 3. (a) Example 7-machine problem. (b) Solution via traditional approach.(c) Solution via initial proposed algorithm (optimal).

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of feasible machines is split into two non-adjacent, disjoint sets. This can makesubsequent assignments less attractive as there are then fewer alternatives from

which to select in a given group. An example of this is shown in figure 4. Following

selection of the first machine (machine 8) for worker 1, F1 {1, 2, 7} and V1 {16,19, 17}, so machine 2 is selected, This selection splits the remaining machines into

two groups (this also occurs for worker 2). Continuing with the procedure produces

the solution of figure 4(b).

To address this shortcoming, we modify the above so that selections which result

in split groups are avoided. The resulting algorithm is as follows:

Step 3:

(i) Calculate Gk {gki}, where

gki grouping factor for ith feasible machine for worker k: 1, assignment ofith machine will result in all remaining machines

in Fk being adjacent i:e: single group remains: 2, assignment ofith machine will result in all remaining machines

in Fk being split into two disjoint, non-adjacent groups:

(ii) Determine the set of candidate machines for worker k, Ck, where

Ck fckig ffkig, gki 2, i 1, . . . , N, N jFkjffkijgki 1g, otherwise&

In other words, if at least one candidate machine does not result in split groups,

only such machines are considered.

(iii) Calculate Vk {vki}, where vki total time for machines Sk cki2 Ck.(iv) Select machine m cki, where i argmaxi vkif g.

Consider the use of the proposed algorithm for the problem of figure 4(a).Following selection of the first machine (machine 8) for worker 1, we have

F1 {1, 2, 7} and G1 {1, 2, 1}, so C1 {1, 7} and V1 {16, 17}. Hence, machine 7 is

1

4

2

7

3

6

8

12

7

5

6

9

4

10

5

2

1 2

8 7 6

4

5

12 5 9

10

2

T= 20

4 67 10

12 5 9 2

3 1

6

4

58 7

4 7 6

2 2

(a) (c)

W= 5, Z= 0.663 W= 3, Z= 0.95

(b)

Figure 4. (a) Example 8-machine problem, showing feasible machines (dashed lines)following selection of first machine for worker 1 using initial proposed algorithm.(b) Solution via initial proposed algorithm. (c) Solution via final proposed algorithm(optimal).



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selected next. Continuing with the procedure results in the solution of figure 4(c),

which is optimal.

5. Computational experiments

5.1 Experimental design

A set of computational experiments was performed to establish the performance of

both the traditional approach and the proposed algorithm. Three sizes of U-shaped

lines were investigated: small (59 machines), medium (1014 machines), and large

(1520 machines). Manual time was uniformly distributed between 530 seconds at

each machine. Following Nakade and Ohno (1999), walking time was 1 second for

adjacent machines (same row) and 2 seconds for opposite machines. For a given

quantity of machines Mand manual times {ti}, three different takt time valueslow,

medium, and highwere investigated. To determine appropriate values, we first

assume that manual time is no more than 50% of the total operation time at any

machine. As takt time must be greater than or equal to the largest operation time,

this results in a lower bound on takt time of LB[T] 2 max{ti}. At the other end,we note that a single worker is sufficient whenever there is enough time for the

worker to perform all manual operations and walk once around the line. This

provides an upper bound on takt time of UB[T] P ti (total round-trip walkingtime). We can then calculate the possible takt time range Trange UB[T] LB[T],and establish low, medium and high values based upon this range. The experimental

design thus consisted of two factors with three levels each as follows:

Line size: Small : M 59; medium : M 1014; large : M 1520Takt time: Low : T LBT; medium : T LBT 1=3 Trange;

high : T LBT 2=3 TrangeThis resulted in nine design points, or problem scenarios, for investigation. For each

machine quantity M in each scenario, 10 problems (i.e. sets of {ti} values) were

randomly generated. Thus, 50 problems were generated for each of the first six

problem scenarios (small lines, and medium lines, each with low, medium, and high

takt times) and 60 problems for each of the last three problem scenarios (large lines

with low, medium and high takt times), giving a total of 480 problems.Each problem was solved using both the traditional approach and the proposed

algorithm. These were coded using the C programming language and run on an IBM

RS/6000 Unix-based server: all 50 or 60 problems comprising a given scenario could

be solved, using either approach, in a few seconds. For comparison with optimal

values, the mathematical model of section 3 was coded using the AMPL

programming language and problems solved using CPLEX 9.0.0 on a 1.7 GHz

PC. Recall from section 3 that the optimal approach requires attempting to solve the

problem first with W LB[W], then incrementing W as required until the problembecomes feasible. It was found that for no problem were more than two solution

attempts required. Solution time increased as lines became larger and takt timesmaller (resulting in more workers): for low takt times, maximum solution time went

from 70 seconds for small lines (59 machines) to over sixteen hours with medium

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lines (1014 machines). For large lines (1520 machines) and low takt times, solution

time became so excessive that optimal solutions could not be established.

5.2 ResultsTo provide some overall idea of the range of values obtained for W and Z for the

different solution methods employed, table 1 presents these statistics for each of the

nine problem scenarios. A cursory examination of table 1 suggests that the proposed

algorithm performed better: values for W and Z are as good as or better than those

obtained with the traditional approach in almost every case. Figure 5 presents

6 10 22

6 5 5

W= 2, Z= 0.986

(Optimal)

(a) M= 6, T= 56 (high)

13

22

W= 3, Z= 0.899

(Optimal)

(b) M= 9, T= 90 (medium)

W= 5, Z= 0.927

(Optimal)

(c) M= 11, T= 54 (low)

W= 2, Z= 0.999 (Optimal)

(d) M= 12, T= 158 (high)

W= 3, Z= 0.962 (Optimal:W= 3, Z= 0.978)

(e) M= 16, T= 140 (med)

W= 3, Z= 0.970 (Optimal:W= 3, Z= 0.995)

(g) M= 20, T= 149 (medium)

W= 7, Z= 0.818 (LB[W] = 5)

(f) M= 17, T= 54 (low)

11 15 25

15 22 18

19

9 11 7 8

6

27 1730 30 30 20

1820

14 19 8 9

6586712109

27 12 24 2510

1015211026206

30262328

2726

19896

10581419

2727

201225

15 18 11 18

822

24

1610289710

241019122527

Figure 5. Example worker allocation problems and solutions, all solved using proposedalgorithm.

Table 1. Computational results (ranges of values obtained) for W and Z,each solution approach.

Quantity of workers, W Full work level, Z (%)

Machinequantity

Takttime Optimal

Traditionalapproach

Proposedalgorithm Optimal

Traditionalapproach

Proposedalgorithm

59 L 25 25 25 84100 6399 7210059 M 23 23 23 83100 65100 7810059 H 2 23 2 91100 54100 821001014 L 36 37 37 8598 7098 73981014 M 23 24 24 95100 54100 681001014 H 22 23 22 95100 57100 921001520 L 59 510 7295 70971520 M 3 34 34 9699 6199 70991520 H 2 24 2 98100 43100 90100



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example problems of varying size and their solution, all solved using the proposed

algorithm. The 16-machine example of figure 5(e) shows an optimal solution having

an interrupted circular machine assignment (section 2): the worker going from

machine 2 to machine 5 must walk around another worker attending machines 3

and 4. It was found that optimal solutions contained interrupted circular machineassignments 5%, 11%, and 31% of the time for small, medium, and large lines,

respectively.

The performance of the proposed algorithm relative to the traditional approach,

and of each of these relative to optimal, are summarized in table 2. Comparisons to

optimal are done for all scenarios except large lines (1520 machines) having low takt

times. We will refer to the eight scenarios where optimality was established as the

optimally-solved scenarios for convenience.

5.2.1 Quantity of workers. Table 2 shows that the traditional approach to worker

allocation on U-shaped production lines worked quite well for minimizing thequantity of workers: optimal solutions resulted at least half the time (and as high as

86% of the time) for each of the optimally-solved scenarios. The proposed algorithm

performed much better, however. In all eight optimally-solved scenarios, improve-

ment occurred more than half the time it was possible: in six of these, improvement

resulted 100% of the time. In three scenarios (medium-size lines with medium takt

times, large lines with both low and high takt times), this resulted in a reduction of

1.16 workers on average, indicating that for some problems two workers were

eliminated. Such manpower reductions would normally be considered quite

substantial in practice. Furthermore, the proposed algorithm provided worse

solutions than those obtained via the traditional approach much less than half thetime (7% on average), and always provided a larger quantity of optimal solutions.

5.2.2 Full work level. Table 2 shows that performance in maximizing full work level

Z is not as good as that found in minimizing W. This is not surprising: optimality is

more difficult to obtain for Z, as compared to W, as the former a real values and the

latter integer. At the same time, however, there is more opportunity for improvement

with Z. Using the traditional approach, optimal results were obtained 26% of the

time or less for the eight optimally-solved scenarios, with the average being about

13%. As with W, performance was better using the proposed algorithm.

Improvement again occurred at least half the time it was possible in all eightoptimally-solved scenarios: the level of improvement in Z averaged 14% across all

scenarios. In two scenarios (medium and large lines, high takt times), the

improvement was around 20%, a sizeable increase in full work for each worker.

The proposed algorithm again performed worse than the traditional approach less

than half the time (26%, on average), and provided a larger quantity of optimal

solutions in all but one scenario (medium-size lines, medium takt times).

5.3 Discussion of results

The computational results indicate that the proposed algorithm performed as goodas or better than the traditional approach for seven of the eight optimally-solved

scenarios. With medium-size lines having medium takt times, the traditional

3498 J. P. Shewchuk


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Table2.

ComparisonofcomputationalresultsforWandZ,eachsolutionapproach.

Quantityofworkers,W

Fullworklevel,Z(%)

Machine

quantity

Takt

time

Traditional

approach

Proposedalgorithm

Traditional

approach

Proposedalgorithm

%optimal

Improv

1

WI2

ws3

opt4

%optimal

Improv

1

ZI2

ws3

opt4

59

L

80

20(100)

1.0

0

4

96

26

56(76)

11.9

20

44

59

M

72

28(100)

1.0

0

2

98

24

60(79)

15.0

12

56

59

H

86

14(100)

1.0

0

0

100

20

48(60)

16.0

28

22

1014

L

62

22(58)

1.0

0

18

66

4

60(62)

7.6

36

10

1014

M

76

14(58)

1.2

9

2

88

12

58(66)

11.1

34

6

1014

H

66

34(100)

1.0

0

0

100

12

76(86)

18.5

12

32

1520

L

20()

1.0

8

23

60()

6.8

38

1520

M

85

15(100)

1.0

0

13

87

0

65(65)

10.7

27

3

1520

H

57

43(100)

1.1

2

0

100

10

70(78)

22.9

25

20

1

Percentageofsolutionsimprovedascomparedtotraditionalapproach(Percentageofsolutionsimprovedconsid

eringonlythosewhereimprovemen

tpossible).Bold

indicatesthatimprovementoccurredatlea

sthalfthetime.

2AverageimprovementinW(Z)overtraditionalapproach,

improvingsolution

sonly.

3Percen

tageofsolutionsworseascomparedtotraditionalapproach.

Bold

indicatesthatsolutionswereworseless

thanhalfthetime.

4Percen

tageofsolutionsfoundtobeoptim

al.Bold

indicatesalgorithmresulted

inoptimalatleastasoftenastrad

itionalapproach.



16/19

approach provided better results for Zonly, providing optimal solutions 12% of the

time versus 6% with the proposed algorithm. Overall, the proposed algorithm thus

appears to work better for allocating workers in U-shaped production lines for the

stated objectives. We also note the importance of allowing interrupted circular

machine assignments: these were found in a significant quantity (between 5 and 31%)of optimal solutions. Such solutions are not possible in related works such as Nakade

and Ohno (1999). One question which remains is how much additional improvement

in the proposed algorithm is possible, especially with respect to full work level, Z.

Further investigation seemed to indicate that additional improvement would require

a look-ahead feature and/or performing multiple passes through the data. It is

likely, however, that adding such methods would result in an algorithm too difficult

to use manually. Such further improvements were thus not explored.

It is also important to keep in mind that no matter how good we are at allocating

workers to maximize full work, the resulting solution may or may not be amenable to

worker elimination via process improvement (as discussed in section 1). Ideally, wewould like to see the least-utilized worker have a single machine and low workload,

such that the elimination of this worker through process improvement is a

reasonable proposition. In figure 5, for example, this occurs in examples (a), (b)

and (f), where the lowest worker utilization is 12% or less via a single machine.

Worker elimination via process improvement will be more difficult in examples (c)

and (d), where the least utilized worker has a utilization of about 33%. In any event,

the first step is to always to find the minimum quantity of workers which maximizes

full work: the results can then be used to target process improvement efforts to the

extent possible.

6. Conclusions

This paper has addressed the problem of worker allocation in lean U-shaped

production lines, where the objective is minimize the quantity of workers on the line,

W, then maximize full work level Z (average % utilization, all but least-utilized

worker). A mathematical model was developed: the model allowed for any allocation

of machines to workers as long as workers do not cross paths. Walking times were

considered, where workers followed circular paths and walked around other

worker(s) on the line if necessary. To solve worker allocation problems quickly andeasily in practice, a simple algorithm was developed. Computational results indicate

that the algorithm, which in selecting the next machine for a worker considers only

those machines which do not result in split groups and then selects the machine

having the largest total time (manual walking), performed better overall than thetraditional approach to the problem.

Regarding the mathematical model, it was found that optimal solutions to the

worker allocation problem in lean U-shaped production lines can be found in

reasonable time (less than one hour) for any size line up to 20 machines as long as

takt time is not too small. The quantity of workers required increases as takt

time decreases, and optimal solution time is extremely sensitive to the quantity of

3500 J. P. Shewchuk


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workers used. It was also found that the lower bound on the quantity of workers

proposed by Nakade and Ohno (1999) is very good: of the 420 problems solved

optimally in this paper, 372 had the minimum quantity of workers equal to the

bound and the remainder required only one more worker than the lower bound. This

implies that the bound can provide very good estimates of the quantity of workersneeded on U-shaped production lines in practice. A further conclusion based upon

optimal solution results is that it may be highly beneficial in lean U-shaped

production lines to allow interrupted circular machine assignments, i.e. those where

workers must walk around other worker(s) when following circular walking paths

around the line. The larger the line, the greater the likelihood the performance can be

improved by allowing such assignments.

Additional research is envisioned in three areas. First of all, the validity of the

mathematical model and algorithm can be improved. In real systems not every

worker can run every machine, some workers perform different tasks faster and/or

better than others, etc. Consideration of such factors leads to the worker assignmentproblem, where the goal is to assign tasks to specific workers. The mathematical

model and proposed algorithm presented here provide a natural starting point for

tackling this more-difficult operational problem. Second, this paper assumed circular

walking paths for workers, which may not necessarily result in minimum walking

times when interrupted circular machine assignments are allowed. Additional work is

required to establish whether this is indeed the case and, if so, how walking paths are

best established. Finally, this paper assumed sufficient WIP that workers are never

starved. In some cases, however, changing the starting location for workers on the

line can reduce the amount of WIP required. Work is needed to investigate how to

best select, for a given assignment of machines to workers, the starting locations for

the workers in order to minimize WIP.

References

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