On the robustness of the bowl phenomenon

EUROPEAN JOURNAL OFOPERATIONAL RESEARCH European Journal of Operational Research 89 (1996) 496-515

ELSEVIER

Theory and Methodology

On the robustness of the bowl phenomenon Frederick S. Hillier a,., Kut C. So

b

a Department of Operations Research, Stanford University, Stanford, CA 94305, USA b Graduate School of Management, University of California, Irvine, CA 92 717, USA

Received September 1993; revised October 1994

Abstract The bowl phenomenon provides a way of increasing the throughput of some production line systems with variable processing times by purposely unbalancing the line in a certain manner. However, achieving this increase in throughput depends on correctly identifying the values of the system parameters to estimate the optimal amount of unbalance and then actually being able to assign work to stations according to the optimal bowl allocation. In this paper we study the robustness of the bowl phenomenon by examining the effect of inaccurately estimating the optimal amount of unbalance and the effect of deviating from the optimal bowl allocation. Our results show that the bowl phenomenon is relatively robust in the sense that fairly large errors (even 50%) in the amount of unbalance still provide most of the potential improvement in throughput over a perfectly balanced line. Moreover, the throughput still exceeds that of a perfectly balanced line in most cases even when the work allocation to each station deviates from the optimal bowl allocation by as much as 10%. We also address the question of whether the optimal bowl allocation or the balanced line provides a more robust 'target' when assigning work to stations. When the deviations from these two targets are of the same magnitude, we found that the optimal bowl allocation target yields the larger throughput in most cases, where the average difference between their throughputs is roughly the same as the difference between the optimal throughput and the throughput of a balanced line. Furthermore, for the same magnitude of deviation, the throughput depends more heavily on the direction of the deviation from the balanced line than that from the optimal bowl allocation, so that the risk of a substantially reduced throughput is much larger when using the balanced line as the target. Therefore, the optimal bowl allocation provides a much more robust target than the balanced line. Keywords: Production line design; Manufacturing; Queueing

1. Introduction H i l l i e r a n d Boling (1966) d i s c o v e r e d t h e ' b o w l phenomenon' regarding the optimal allocation of

* Corresponding author.

w o r k in s o m e u n p a c e d p r o d u c t i o n line systems with v a r i a b l e p r o c e s s i n g times. I n p a r t i c u l a r , they f o u n d t h a t the t h r o u g h p u t can b e i n c r e a s e d by d e l i b e r a t e l y u n b a l a n c i n g t h e line in a c e r t a i n way, w h e r e a p l o t o f t h e o p t i m a l a l l o c a t i o n vs. station n u m b e r has t h e s h a p e of t h e cross-section o f the i n t e r i o r o f a bowl. H i l l i e r a n d Boling (1979) a n d H i l l i e r a n d So (1993) have o b t a i n e d n u m e r i c a l

0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(94)00287-8

F.S. Hillier, K~C. So/European Journal of Operational Research 89 (1996) 496-515

results on the optimal allocation for a variety of cases, and then gave guidelines for extrapolating these results to larger cases. These numerical results indicate that, for many cases, the improvement in throughput from using an optimal bowl allocation instead of a balanced allocation is approximately 1%, and perhaps can be as large as 2% or more. The improvement in throughput increases rapidly at first as the size of the line increases, approximately doubling when going from three stations to five and approximately tripling when going from three stations to ten. For example, when the line has no buffers and the processing times at all stations have a coefficient of variation of 1, the increase is 0.55% for three stations, 1.20% for five stations, and 1.59% for nine stations (the largest case for which exact results are available). These improvements decrease slowly when either the buffer sizes are increased or the coefficient of variation is decreased (but not both). For example, with about ten stations, the improvement is approximately 1% or more with a coefficient of variation of 1 for a range of (equal) buffer capacities up to three units each, or when there are no buffers for a range of values of the coefficient of variation down to 1/3. The improvement in throughput decreases more rapidly when both the buffer sizes are increased and the coefficient of variation is decreased (a somewhat unlikely combination given the current emphasis on decreasing work-in-process inventory). This improvement in throughput comes free, simply by allocating the required work more appropriately to the same N stations, with no increase in resource costs or capital costs. Given the high capital and operating costs associated with the typical production line, even a very modest 1% improvement in throughput can yield very substantial savings over the life of the line. Until recently, a major obstacle to achieving these substantial savings was the difficulty of finding an optimal bowl allocation. However, this task now has become straightforward for a wide variety of cases by referring to the numerical results and extrapolation guidelines provided by Hillier and So (1993). Two extrapolation procedures are given for estimating the optimal allocation of

497

work, where both are based upon obtaining a single constant from a table. The quick-and-dirty procedure then only requires plugging into two very simple formulas, whereas the more precise procedure requires some additional calculations. For the cases that could be checked exactly, the quick-and-dirty procedure consistently provided well over 95% of the potential increase in throughput over a balanced allocation, while the more precise procedure provided at least 99%. Although we do not expect results quite this good for other cases, these procedures now clearly provide easy and reliable methods for estimating the optimal bowl allocation. Following the early papers of HiUier and Boling, a considerable number of other researchers have explored certain aspects of the bowl phenomenon in a variety of contexts. Muth and Alkaff (1987) estimated that the bowl phenomenon had been the subject of 'nearly 25 papers' in recent years, and they provide a relatively comprehensive bibliography in this area. Nevertheless, a formal proof of the bowl phenomenon still has not been found. (For two-station tandem systems, Chao, Pinedo and Sigman (1989) showed that when the service time distributions are exponential and the sum of service rates is held constant, the throughput is maximized when the two rates are equal, which is consistent with the bowl phenomenon finding for N-station systems that the first and last stations should have equal workloads.) A related and well-studied problem is to determine the optimal arrangement of stations in a tandem queueing system with blocking; see the recent work of Ding and Greenberg (1989, 1991), Huang and Weiss (1990), Shanthikumar, Yamazaki and Sakasegawa (1991), Yamazaki, Sakasegawa and Shanthikumar (1992), Wan and Wolff (1993), and the references therein. In this problem, N fixed but distinct work allocations (service time distributions) have been provided to be assigned to the N stations on a one-to-one basis to maximize the throughput. Results from these studies are consistent with the bowl phenomenon in which the two highest workloads should be assigned to the first and last stations. Furthermore, Yamazaki, Sakasegawa and Shan-

498

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

thikumar (1992) found that while the ordering that conformed most closely to the bowl phenomenon (Rule 2 in their paper) is not always optimal, its throughput is always very close to the optimal throughput. Despite the potential impact, the bowl phenomenon remains a poorly understood and controversial theoretical curiosity that apparently has had little practical impact. Striving for as-balanced-as-possible workloads among stations remain the norm for designing production lines in practice. A commonly expressed reason for this is uncertainty about the robustness of the bowl phenomenon. Robustness is a key issue, because it generally is not possible in practice to achieve the precisely optimal bowl allocation for real production line systems. One reason for this lack of precision is that the mathematical model underlying the bowl phenomenon only provides a rough representation of the real problem. In addition, the computational limitations prevent solving exactly for the optimal bowl allocation with respect to this model for large production line systems (although the extrapolation guidelines now available largely solve this problem). Furthermore, even given an approximation of the optimal bowl allocation as the target, some deviations from this target inevitably will occur when assigning the various discrete tasks to the stations of the line. If the bowl phenomenon is not particularly robust, all these sources of error in striving for an optimal bowl allocation might actually lead to a smaller throughput than with a balanced line. Therefore, it can be argued that this approach is riskier than simply striving for a balanced line. This paper addresses the validity of this argument by studying the robustness of the bowl phenomenon and comparing the risk of error in striving for an optimal bowl allocation with the risk of error in striving for a balanced line. Some researchers have previously performed experiments to study the robustness of the bowl phenomenon. Smunt and Perkins (1985) presented a simulation study that purportedly shows that "perfectly balanced line designs are optimal for most cases in practice". However, Karwan and Philipoom (1989) and So (1989) (see also So, 1990) then independently argued that there were

major flaws in the design of the Smunt-Perkins study that invalidated its conclusions, as well as presenting their own simulation results that contradicted the Smunt-Perkins results. Smunt and Perkins (1989, 1990) then presented separate rejoinders to these two papers that defended the validity of their original study and provided additional simulation results on this subject. None of these papers, however, provided a comprehensive study of the robustness issue. We strive to do so here. Achieving the full increase of throughput from the bowl phenomenon depends on correctly identifying the optimal amount of unbalance and assigning workloads accordingly. Therefore, our study includes an examination of the effect of inaccurately estimating the optimal amount of unbalance. Determining just how much unbalance is optimal requires accurately identifying the system parameters, including especially the coefficient of variation of processing times. (We make the somewhat restrictive assumption in this study, discussed further at the end of this paper, that this coefficient of variation is the same at all stations.) The degree of unbalance in the optimal allocation of work increases as the coefficient of variation increases (until the coefficient reaches very large values). Therefore, misestimating the coefficient of variation can cause a substantial error in estimating the optimal allocation. This paper first studies (1) the effect of inaccuracies in estimating the optimal work allocation, (2) how to minimize this effect, and (3) how the throughput of the resulting work allocation compares with that for a balanced work allocation. Even when the optimal amount of unbalance is determined accurately, another source of error mentioned earlier is missing the optimal bowl allocation because of the necessity of assigning discrete microelements of work to the stations of the line. The optimal bowl allocation can only be regarded as a target for assigning work to stations. The latter part of this paper studies the effect of incurring deviations from this target. We also compare this effect with that when a work allocation deviates from a balanced line by about the same magnitude to determine which alloca-

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

tion provides a better target when assigning workloads to stations. Section 2 summarizes the basic model that yields the bowl phenomenon, without yet considering robustness issues. Section 3 studies the effect of inaccurately estimating the optimal amount of unbalance. Section 4 then focuses on the effect of incurring deviations from the targeted work allocation (either the optimal bowl allocation or a perfectly balanced line). Finally, Section 5 summarizes the resulting conclusions.

2. Formulation As described in detail by Hillier and So (1993), the model used to represent a production line system is the classical queueing model with finite queues in series. The model assumes that there always is a unit (customer) available to begin being processed at the first station. We also assume a single processing unit (server) at each station. We use the following notation: N = N u m b e r of stations. q = Capacity of buffers between stations (assumed to be the same for all such buffers). wj = Expected processing time at station j. W

=

(Wl,

W2,...

,

WN).

c = Coefficient of variation of processing times (assumed to be the same at all stations). R ( w ) = Throughput (production rate) with w. w ° = Optimal bowl allocation. R* wb ...

Rb

= R(w*).

= Balanced work allocation, i.e., w 1 = w E = ---~W N -

= g(wb).

For convenience, we use a unit of time such that the total expected processing time per unit is

499

Erlang when c < 1, exponential when c = 1, and two-stage Coxian when c > 1. The bowl p h e n o m e n o n observes that the optimal allocation of work w ° follows the exact shape of a bowl under the assumptions that q is the same for all buffers between stations and that c is the same at all stations. We do not consider here the more general case where q and c may have different values at different buffers or stations. The design problem under consideration is one where N, q, and c are fixed whereas the wi are continuous decision variables and the objective is to maximize R ( w ) . For this problem, Hillier and Boling (1966, 1979) found that the optimal allocation has the form for j = 1, 2 . . . . , N

w/*=w~v+l_ /

(the symmetrical allocation property) and *

*

*

1

w~ > w/+ l ( w N + l _ s > wPv_/) for j < ~N

(the monotonicity property). Furthermore, this bowl shape is fairly fiat in the sense that w 2 , w~ . . . . . WN-1 are fairly close in value, whereas w~* = w~, is considerably larger. Hillier and So (1993) use this observation about the flatness of the bowl to develop relatively precise extrapolation guidelines that are based (except for an optional final adjustment) on sett i n g w 2 = w 3 = . . . = w N _ 1 as well a s w l = w N. For example, when N = 3, q = 0, c = 1, then w * = (1.086, 0.828, 1.086), which yields R * = 0.56718 vs. R b = 0.56410, so this optimal 'bowl allocation' yields a 0.55% improvement in throughput over a perfectly balanced line. For considerably larger values of N, this improvement in throughput increases to approximately 2% or more.

w~ + w2 + . . . + w N = N ,

so that R ( w ) < 1 and wb = (1, 1 . . . . . 1).

See Hillier and So (1992) for information about how to obtain R ( w ) and w*. The probability distribution of processing times is assumed to be

3. The effect of inaccurately estimating the optimal amount of unbalance In this section we study how R ( w ) changes (and how it compares with R b) a s changes are

500

ES. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

m a d e in the degree of unbalance in w when trying to estimate w *. First, let N

u*=

Table 2 V a l u e s of P(w) w h e n w = ~,(u) for N = 3, q = 0

U/U *

Y'. Iw~.*- 11 /=1

denote the (optimal) degree of unbalance for the optimal work allocation w *. Then, for any given degree of unbalance u, we associate this degree of unbalance u with the following work allocation w: u

i f ( u ) = (1, 1 . . . . . 1) + - - [w* - (1, 1 . . . . 1)] U*

'

'

Note that i f ( u ) = w* when u = u*, whereas a perfectly balanced allocation is obtained when u = 0. Thus, when u > 0, a true bowl shape is obtained, but with less unbalance than is optimal when u / u * < 1 and more unbalance than is optimal when u / u * > 1. W h e n u < 0, an inverted bowl shape is obtained. For various w, we will tabulate both R(w) and P(w) =

R(w) - R b R* - R b

* 100%,

where P(w) is interpreted as the percentage of the Potential improvement, R*--Rb, that is achieved by w. Note that P(w °) = 100%, and P(w) < 0 when R(w) < R b.

Table 1 V a l u e s o f R(w) w h e n w = ~ ( u ) for N = 3, q = 0

///U*

C 0.5

1

2

-2 - 1 -0.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.5 3 4

0.69069 0.70510 0.71028 0.71400 0.71534 0.71622 0.71679 0.71695 0.71674 0,71621 0.71526 0.71405 0.71058 0.70591 0.69350

0.54134 0.55521 0.56033 0.56410 0.56544 0.56641 0.56699 0.56718 0.56697 0.56636 0.56535 0.56394 0.55992 0.55434 0.53893

0.45963 0.46825 0.47143 0.47378 0.47462 0.47523 0.47560 0.47573 0.47560 0.47521 0.47456 0.47362 0.47090 0.46695 0.45468

C

0.5

1

2

-2

-790.16

- 738.92

- 725.50

-1 - 0.5 0 0.25 0.5

-301.69 - 126.10 0 45.42 75.25

- 288.80 - 122.27

- 283.50 - 120.36

0 43.66

0 43.11

0.75 1

94.58 100.00

1.25

92.88

1.5 1.75

74.92 42.71

75.03 93.91 100.00 93.30 73.51 40.65

74.39 93.43 100.00 93.30 73.40 39.77

- 5.27 - 135.88

- 7.99 - 146.67

- 316.75 - 817.25

- 350.07 - 979.46

2 2.5 3 4

1.69 -115.93 -274.24 -694.91

Table 1 tabulates R(w) when w = i f ( u ) for various u / u * and three values of c when N = 3 and q = 0. Table 2 gives the corresponding P(w). Tables 3 and 4 then give R(w) and P(w) for various other values of (N, q) with c = 1. To show the form of these functions graphically, Figs. 1-3 plot R(w) from Tables 1 and 3 for a sampling of three values of (N, q), and then Figs. 4 - 6 plot P(w) for the corresponding cases. Note especially from Tables 1 - 4 that the R(w) and P(w) functions are (1) roughly symmetrical around u / u * = 1 (corresponding to w = w * ), (2) relatively flat near u / u * - - 1 , and (3) relatively steep for u / u * < 0 (unbalanced in the wrong direction) and u / u * > 2 (more than double the optimal degree of unbalance). Furthermore, our numerical results indicate that the bowl p h e n o m e n o n is relatively robust with respect to the degree of unbalance in work allocations. One can be off the optimal amount of unbalance by 25% in either direction and still achieve nearly 95% of the potential improvement in throughput over that for a perfectly balanced work allocation. Similarly, missing by 50% in either direction still yields about 75% of the potential improvement. On the other hand, relatively large losses in throughput result when the line is unbalanced in

F.S. Hillier, K.C. So / European Journal of Operational Research 89 (1996) 496-515

501

R(w)

R(w) o.57 i

0.808

0.585 .56

0.802 0.8 080il 0.798 0.798 0.794 0,792

'J]i"

0.555 0.55

0.54 -2

-,

;

o

~

&

u_ Oat

Fig. 1. G r a p h o f N=3, q=0. Table 3 Values of

u/u* - 2 - 1 - 0.5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.5 3 4

Table 4 Values of

u/u * - 2 - 1 - 0.5 0 0.25 0.5 0.75 1

1.25 1.5 1.75 2 2.5 3 4

R(w)

R(w)

vs.

u/u*

when

w=ff(u)

for c=l,

i

.7

2Fig. 2. G r a p h o f N=3, q=4.

R(w)

vs.

u/u*

-~

¢

u.A_

u*

when w=~(u)

for c=1,

when w = ~(u) for c = 1

(N, q)

(3, 2)

(3, 4)

(4, 0)

(4, 2)

(5, 0)

(6, 0)

(7, 0)

0.71160 0.72541 0.73041 0.73402 0.73529 0.73619 0.73674 0.73692 0.73675 0.73622 0.73535 0.73414 0.73076 0.72617 0.71389

0.79075 0.80113 0.80483 0.80749 0.80842 0.80908 0.80949 0.80963 0.80952 0.80916 0.80855 0.80771 0.80534 0.80212 0.79346

0.47880 0.50060 0.50878 0.51478 0.51690 0.51842 0.51933 0.51963 0.51932 0.51841 0.51692 0.51486 0.50917 0.50158 0.48190

0.66825 0.68792 0.69529 0.70071 0.70264 0.70402 0.70485 0.70513 0.70486 0.70405 0.70272 0.70088 0.69579 0.68899 0.67131

0.44430 0.46906 0.47866 0.48580 0.48835 0.49018 0.49128 0.49165 0.49130 0.49024 0.48850 0.48611 0.47959 0.47105 0.44961

0.42349 0.44883 0.45900 0.46671 0.46949 0.47150 0.47271 0.47312 0.47273 0.47157 0.46966 0.46706 0.45998 0.45082 0.42829

0.41010 0.43495 0.44525 0.45318 0.45606 0.45818 0.45941 0.45982 0.45938 0.45811 0.45605 0.45326 0.44573 0.43607 0.41269

P(w) w h e n

w = ff,(u) f o r c = 1

(N, q) (3, 2)

(3, 4)

(4, 0)

(4, 2)

(5, 0)

(6, 0)

(7, 0)

- 773.17 - 296.73 - 124.54 0 43.65 74.95 93.76

- 782.27 - 297.02 - 124.22 0 43.42 74.48 93.39

- 741.80 - 292.31 - 123.79 0 43.73 75.11 93.89

- 734.32 - 289.48 - 122.57 0 43.59 74.82 93.61

- 709.35 - 286.11 - 122.11 0 43.54 74.86 93.70

- 674.29 - 278.96 - 120.21 0 43.42 74.74 93.65

- 648.77 - 274.57 - 119.48 0 43.45 74.91 93.84

100.00

94.12 75.96 45.88 4.17 - 112.58 - 270.76 - 694.12

100.00

94.97 78.05 49.70 10.22 - 100.58 - 251.01 - 655.53

100.00

93.60 74.84 44.05 1.67 - 115.69 - 272.12 - 677.91

100.00

93.82 75.55 45.43 3.88 - 111.32 - 265.09 - 665.14

100.00

93.96 75.82 46.08 5.34 - 106.20 - 252.18 - 618.64

100.00

93.94 75.77 46.02 5.40 - 105.02 - 274.9 - 599.33

100.00

93.32 74.28 43.29 1.26 - 112.22 - 257.66 - 609.71

F.S. Hillier, K.C. So/European Journalof Operational Research 89 (1996) 496-515

502

P(w)

R(w)

O.46 0.455 0.45"

loo

5o _L

, -1

0A4 O.435 0.43 25

/

C

-1

1

2

3

u

--~

R(w) vs. u/u*

w h e n w = ~,(u) for c = 1,

4B

U~

- / i -200 50 -250 O

4

LI =

Fig. 3. Graph of

= 3

-100

0,415 .41

i

-2

o

-~o

Fig. 5. Graph of P ( w ) vs.

u/u*

when w = ~,(u) for c = 1,

N=3, q=4.

N=7, q=0.

the 'wrong' direction (u < 0) or w h e n it is unbalanced in the 'right' direction but by more than double the optimal amount (u/u * > 2). To avoid the latter possibility, it is better to err on the small side w h e n in doubt about the optimal amount of unbalance. Because of the extrapolation guidelines mentioned in the preceding section, we also consider a set of values of w that correspond to using these guidelines. In particular, let =w

with W1 =

WN~

...

W2=W3=

=WN_

1

and 2(w,-

1) + ( N -

2)(1 - w2) = u .

With this flat bowl shape, if(u) approximates w ° when u = u *, whereas it gives a perfectly bal-

anced allocation when u = 0. N o t e that # ( u ) differs from if(u) only when N > 5. Table 5 tabulates R(w) when w = ff,(u) for various u/u* and N = 5 , 6 , or 7 when q = 0 and c = 1. Table 6 gives the corresponding P(w). N o t e how close the values in Tables 5 and 6 are to the corresponding values in the last three columns of Tables 3 and 4. In other words, using a perfectly fiat bowl (w 2 = w 3 = ... = WN_1) provides essentially the same throughput as the optimal slightly rounded bowl, even when the optimal amount of unbalance is misestimated. With the correct amount of unbalance, the u/u * = 1 row of Table 6 indicates that the perfectly fiat bowl achieves about 99% of the potential improvement in throughput given by the optimal bowl. (However, we would expect to drop a little below 99% for somewhat larger values of N, since the optimal bowl becomes a little more rounded for larger N.) These results help to support a high degree of precision in the extrapolation guidelines of Hillier

P(w)

P(w)

lOO

100

50 -2

-~

3

___U

r

=

o

i

-2

-1

- 50

3

:~ --U

4 -

U*

U* -100

-/350 -200

-200 -7350 -250 0 Fig. 4. Graph of

N=3, q=0.

P(w) vs. u/u*

-250 O when w=~(u)

for c = l ,

Fig. 6. Graph of

N=7, q=0.

P(w) vs. u/u*

when w = ~ ( u )

for c = 1 ,

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515 Table 5 V a l u e s of

u/u* - 2 - 1 -0.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.5 3 4

R(w) w h e n

Table 6 V a l u e s of

w = ~,(u) f o r c = 1

u/u*

(N,q) (5, 0)

(6, 0)

(7, 0)

0.44513 0.46940 0.47880 0.48580 0.48831 0.49012 0.49122 0.49162 0.49131 0.49031 0.48865 0.48637 0.48007 0.47177 0.45076

0.42526 0.44948 0.45931 0.46671 0.46940 0.47137 0.47258 0.47304 0.47275 0.47173 0.47000 0.46761 0.46100 0.45233 0.43065

0.41265 0.43603 0.44569 0.45318 0.45594 0.45798 0.45926 0.45977 0.45950 0.45847 0.45671 0.45425 0.44746 0.43856 0.41645

- 2 - 1 -0.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.5 3 4

and So (1993), since they are based in part on using a perfectly flat bowl. (For greater precision, these guidelines also provide an optional final adjustment that involve readjusting w 2 = WN_ ~ a little upward and w 3 = w4 = ... = W N _ 2 a little downward.) Therefore, our results suggest that very little is lost by the a p p r o x i m a t i o n o f e q u a t i n g

P(w) w h e n

503

w = ~,(u) f o r c = 1

(N, q) (5, 0)

(6, 0)

(7, 0)

- 695.13 - 280.28 - 119.71 0 42.84 73.82 92.70 99.44 94.15 77.13 48.79 9.70 - 98.02 - 239.88 - 599.01

- 646.66 - 267.31 - 115.38 0 41.99 72.63 91.61 98.82 94.30 78.34 51.37 13.97 - 89.13 - 224.34 - 562.54

- 610.46 - 258.24 - 112.74 0 41.54 72.23 91.53 99.19 95.17 79.69 53.15 16.16 - 86.12 - 220.11 - 553.12

W2, W 3 , . . . , W N _ 1. It also implies that the basic observations and the three properties cited above for the R ( w ) and P ( w ) functions when w = if(u) also hold when w = ff,(u). The last set of numerical results in this section directly examines the effect of inaccurately estimating the coefficient of variation c and then using w that is optimal for this inaccurate c. In

Table 7 The effect of misestimating that c = 1 f o r N = 3, q = 0

c

w~ /w~

/w~

0.333 0.5 0.707 1 1.5 2 2.5

1.034/0.932/1.034 1.050/0.900/1.050 1.067/0.866/1.067 1.086/0.828/1.086 1.101/0.798/1.101 1.105/0.790/1.105 1.103/0.794/1.103

R*

gb

R(w(c = 1))

P(w(c = 1))

0.79164 0.71695 0.64360 0.56718 0.50723 0.47573 0.45807

0.78921 0.71400 0.64041 0.56410 0.50478 0.47378 0.45645

0.78653 0.71543 0.64333 0.56718 0.50718 0.47566 0.45802

- 110.29 48.47 91.54 100.00 97.96 96.41 96.91

R*

Rb

R(w(c = 1))

P(w(c = 1))

0.89129 0.82387 0.73692 0.64096 0.58272 0.54691

0.89012 0.82189 0.73402 0.63810 0.58035 0.54501

0.88641 0.82290 0.73692 0.64069 0.58237 0.54663

- 317.09 51.01 100.00 90.56 85.23 85.26

Table 8 The effect of misestimating that c = 1 f o r N = 3, q = 2

c

w{

0.5 0.707 1 1.5 2 2.5

1.014/0.972/1.014 1.026/0.948/1.026 1.044/0.912/1.044 1.063/0.874/1.063 1.071/0.858/1.071 1.071/0.858/1.071

/w~

/w~

504

F.S. Hillier, IC C. So / European Journal of Operational Research 89 (1996) 496-515

Table 9 The effect of misestimating that c = 1 for N = 4, q = 0

c 0.5 0.707 1 1.5 2 2.5

w{

=

w,~/w~

=

w~

1.079/0.921 1.106/0.894 1.137/0.863 1.168/0.832 1.178/0.822 1.177/0.823

R*

Rb

R(w(c = 1))

P(w(c = 1))

0.68284 0.60270 0.51963 0.44768 0.40909 0.38699

0.67823 0.59771 0.51478 0.44386 0.40611 0.38460

0.68037 0.60226 0.51963 0.44755 0.40892 0.38687

46.62 91.18 100.00 96.60 94.30 94.98

particular, we consider the sample case where the estimated, but inaccurate, value of c is c = 1. Let w(c = 1) denote the w that is optimal for c = 1. Tables 7 - 9 show the effect of this inaccuracy for various true values of c when (N, q) = (3, 0), (3, 2), and (4, 0), respectively. In each case, the second column identifies w * for the true c given in the first column, and then the corresponding R * and R b are given in the third and fourth columns. The fifth and sixth columns provide R(w(c = 1)) and P(w(c = 1)), where w(c = 1) is given by w * in the c = 1 row. Fig. 7 also plots the P(w(c--1)) column from the three tables as a function of the true value of c. The results in Tables 7 - 9 provide some interesting information about the effect of misestimating the coefficient of variation of processing times c, especially when the estimated value is in the vicinity of 1. First, note how P(w(c = 1)) barely

decreases as c increases above the estimated value of 1, whereas it decreases fairly rapidly as c decreases below the estimated value. When the true value is somewhat larger than the estimated value (so that the optimal amount of unbalance in the work allocation is a little larger than the estimated amount), most of the potential improvement in throughput still is achieved. However, when the true value is somewhat smaller than the estimated value (so that the optimal amount of unbalance is somewhat smaller than the estimated amount), the throughput drops off considerably more. Therefore, when in doubt about the true value of c(in the vicinity of 1), it is better to underestimate somewhat than to overestimate somewhat. It is especially important to avoid overestimating so much that the estimated amount of unbalance in the work allocation is close to (or even more than) double the optimal

P(w(c=I)) ...-"~ .......... ~ : - - --:-:.";~_7- - - - : ~

~oo

50 175 ol

0.33

~

,,..-" i . ]o17,

,

/ o.5o

.

. vDo

.

. . ,.5o

2.oo

2.50

-50

-75 t /

-looyl/

-125J-

/i

"

i

• (..q):c~.o~

I

I

I (N.q)=(4.0)I

Fig. 7. Graph of P(w(c = 1)) vs. c for (N,q) = (3,0), (3,2), (4,0).

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

amount of unbalance, since this is where the throughput starts dropping off rapidly. (This relates to the earlier conclusion that it is better to err on the small side when in doubt about the optimal amount of unbalance.)

4. The effect of deviating from the target allocations w* and w b In this section we analyze the effect on the throughput of a line when an actual work allocation deviates from a specific target allocation w t due to, for example, the necessity of assigning d i s c r e t e microelements of work to the stations. We shall compare the effect for the two cases where the optimal bowl allocation w * and the balanced work allocation w b are used as the target allocation. We first focus on the N = 3 and q = 0 case. Results for the q > 0 case and longer lines will then be presented. For N = 3, let

(W- Wt) =

(dl,

d2, d3)

denote the deviation of the actual work allocation w from the target allocation wt. Therefore, a positive (negative) d i represents a higher (lower) than targeted workload being assigned to station i, whereas d 1 = d 2 = d 3 = 0 represents the situation where the target allocation is achieved. Note that since the total workload is constant, d 1 + d 2 + d 3 = O.

505

Let

A = max{I dl ), Id21, )d31} represent the largest (in absolute value) difference between the actual and target work allocation at all stations. Since we have normalized the average workload per station to one in our model, A can then be interpreted as the maximum deviation as a fraction of the average workload per station. To provide a systematic analysis regarding the effect on throughput when the work allocation deviates from a specific target allocation, we con1 sider all possible cases where d i ~ {0, + 7A, + A}, i = 1, 2, 3 (with I d i l = A for at least one i) for different values of A > 0. This then corresponds to a total of 12 possible cases for w - w t, as listed in Exhibit 1. Exhibit 1 (½A

½A

- A)

2. (½a

1.

-a

½A)

3.

(-½A

--1d

A)

4. 5. 6. 7.

(-½A (A (A (0

A -A 0 A

-½A) 0) -A) -A)

8.

(-A

½A

IA)

9.

(A

-½A

-½A)

10. 11. 12.

(0 (-a (-A

-A 0 A

A) A) 0)

Note that any two cases given in the same row represent a 'mirror image' of each other. The reversibility p r o p e r t y (see, e.g. Y a m a z a k i , Kawashima and Sakasegawa, 1985) implies that R ( w ) is the same for each pair of mirror images, so computations are needed for only the first 7 cases in order to consider all 12.

Table 10 Effect of deviating from the optimal bowl allocation for N = 3 and c = 0.5

w - w*

Q(w)

(½a

1 7A

(½A ( - ½A ( - ½A (A (A (0

-- A -- 1A A -A 0 A

-- A) ½A) A) -- ½A) 0) --A) --k)

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

100.41 100.41 100.41 100.41 100.41 100.41 100.41

100.37 100.39 100.37 100.39 100.37 100.35 100.37

100.26 100.31 100.25 100.31 100.25 100.18 100.25

99.79 100.01

98.07 98.87 97.89 98.75 97.89 96.90 97.94

99.77 100.00 99.76 99.48 99.77

F.S. Hillier, K.C. So /European Journal of Operational Research 89 (1996) 496-515

506

Table 11 Effect of deviating from the balanced allocation for N = 3 and c = 0.5 W -- Wb

Q(w)

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

(½A

½A

- A)

100.00

99.86

99.64

99.00

96.98

(½A

-- A

½a)

100.00

100.18

100.31

100.41

100.01

( - ½A

- ½A

A)

100.00

100.07

100.05

99.80

98.37

( - ½A (A (a (0

A -A 0 ,5

- ½A) 0) -A) -A)

100.00 100.00 100.00 100.00

99.76 100.17 99.94 99.75

99.48 100.25 99.78 99.42

98.75 100.18 99.12 98.55

96.74 99.04 96.69 96.01

F o r each target ( w t = w b or w t = w * ) and each A = 0.025, 0.05, 0.1, 0.2, we c o m p u t e d R ( w ) for the c o r r e s p o n d i n g work allocation w in each of the first 7 cases. In each case, we also tabulated

robustness of using w* and w b as the target work allocation w r F o r example, for c = 0.5 with w t = w* = (1.05, 0.90, 1.05),

w - w t = ( - ~1A ,

R(w) Q(w)

- Rb

1 ~A, -A),

* 100% and d = 0.1, we have w = (1.10, 0.95, 0.95) and with Q ( w ) = 99.79% ( R b = 0.71400). Table 10 presents Q ( w ) for the first seven cases with the four different values of A when the

w h e r e Q ( w ) is then the t h r o u g h p u t u n d e r work allocation w r e p r e s e n t e d as a p e r c e n t a g e of the t h r o u g h p u t o f the b a l a n c e d line. W e shall use Q ( w ) as our basic m e a s u r e in c o m p a r i n g the

R ( w ) = 0.71253

Table 12 Mean and standard deviation of Q(w) vs. A for N = 3 and c = 0.5 Target wt

~)(w) and (in parentheses) s

Bowl - w* Balanced - wb

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

100.41 (0) 100.00 (0)

100.37 (0.01) 99.96 (0.16)

100.25 (0.04) 99.84 (0.32)

99.76 (0.15) 99.37 (0.64)

97.92 (0.56) 97.58 (1.27)

Table 13 Effect of deviating from the optimal bowl allocation for N = 3 and c = 1 w -

w °

Q(w)

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

- d)

100.55

100.53

100.48

100.28

99.49 99.77

(½d

½A

(½A

-- ,5

½A)

100.55

100.53

100.50

100.35

(-- ½A

-- ½A

A)

100.55

100.53

100.48

100.28

99.51

( - ½A (A (A (0

A --A 0 A

-- ½A) 0) --A) -- A)

100.55 100.55 100.55 100.55

100.54 100.53 100.52 100.53

100.50 100.48 100.45 100.48

100.36 100.26 100.16 100.26

99.81 99.41 99.03 99.41

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

507

Table 14 Effect of deviating from the balanced allocation for N = 3 and c = 1 w -

w b

Q(w)

A = 0.05

A = 0.1

(~A

½A

-a)

A= 0 100.00

A = 0.025 99.91

99.78

99.42

98.33

(½A

-A

½A)

100.00

100.15

100.27

100.45

100.53

( - ½A

- ½A

A)

100.00

100.06

100.09

100.05

99.58

( - ½A (A (A (0

A -A 0 A

- ½A) 0) -A) -A)

100.00 100.00 100.00 100.00

99.83 100.14 99.98 99.83

99.64 100.25 99.90 99.62

99.20 100.35 99.61 99.10

98.08 100.15 98.46 97.69

optimal bowl allocation w * = (1.05, 0.90, 1.05) is used as the target work allocation w t and the coefficient of variation c = 0.5. For comparison purposes, we also include the results for A = 0, which corresponds to the situation where the target allocation w t = w * is achieved (which gives a throughput of R * = 0.71685, representing 100.41% of R b = 0.71400). Table 11 presents the corresponding results when the balanced allocation w b is used as the target allocation w t. To further provide a single consistent measure for the effect when using a specific target allocation, we computed the average value of Q ( w )

A = 0.2

over the 12 possible cases, denoted by ~)(w), for each value of A. In other words, Q ( w ) can be interpreted as the average throughput (expressed as a percentage of the throughput of the balanced line) when the work allocation deviates from the specific target allocation by at most A at any stations of the line, assuming that the direction of the deviation is equally likely among the 12 cases. We also computed the standard deviation of Q ( w ) over the 12 cases, denoted by s. Table 12, which is derived from Tables 10 and 11, gives the values of ~)(w) and (in parentheses) s vs. A for the two targets, w* and w b.

Table 15 Mean and standard deviation of Q(w) vs. A for N = 3 and c = 1 Target wt

Q(w) and (in parentheses) s

Bowl - w * Balanced - wb

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

100.55 (0) 100.00 (0)

100.53 (0.00) 99.99 (0.12)

100.48 (0.01) 99.93 (0.24)

100.27 (0.06) 99.73 (0.48)

99.44 (0.22) 98.92 (0.97)

Table 16 Effect of deviating from the optimal bowl allocation for N = 3 and c = 2 w -

w *

Q(w)

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

(~a

½a

- a)

100.41

100.40

100.38

100.29

99.91

(½A

- A

½A)

100.41

100.41

100.39

100.31

100.01

A)

100.41

100.40

100.38

100.29

99.93

-- ½A) 0) -A) -A)

100.41 100.41 100.41 100.41

100.41 100.40 100.40 100.40

100.39 100.38 100.37 100.38

100.32 100.27 100.24 100.27

100.04 99.85 99.72 99.85

(_ i ( - ½A (A (A (0

1 A -A 0 A

F.S. Hillier, K.C. So ~European Journal of Operational Research 89 (1996) 496-515

508

Table 17 Effect of deviating from the balanced allocation for N = 3 and c = 2 w - wb

Q(w)

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

(½a

~A

-A)

100.00

99.95

99.87

99.68

99.11

(½A

-A

½A)

100.00

100.09

100.17

100.30

100.41

( - ½A

- ½A

A)

100.00

100.04

100.07

100.07

99.89

( - ½A (A (A (0

A -A 0 A

- ½A) 0) -A) -A)

100.00 100.00 100.00 100.00

99.90 100.09 99.99 99.90

99.79 100.16 99.95 99.78

99.53 100.25 99.82 99.48

98.90 100.23 99.27 98.71

Table 18 Mean and standard deviation of Q(w) vs. A for N = 3 and c = 2 Target wt

~)(w) and (in parentheses) s

Bowl - w * Balanced - wa

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

100.41 (0) 100.00 (0)

100.40 (0.00) 99.99 (0.07)

100.38 (0.01) 99.97 (0.17)

100.28 (0.02) 99.87 (0.30)

99.88 (0.09) 99.48 (0.59)

Table 19 Effect of deviating from the optimal bowl allocation for N = 3, c = 1 and q = 2 w - w *

Q(w)

A= 0

A = 0.025

d = 0.05

A = 0.1

A = 0.2

(½A

½A

-A)

100.40

100.35

100.21

99.68

97.72

(½A

-A

½A)

100.40

100.36

100.27

99.91

98.53

( - ½A

- ½A

A)

100.40

100.35

100.21

99.64

97.46

( - ½A (A (A (0

A -A 0 A

- ½A) 0) -A) -A)

100.40 100.40 100.40 100.40

100.36 100.35 100.33 100.35

100.27 100.20 100.12 100.20

99.88 99.61 99.32 99.64

98.38 97.37 96.40 97.51

A = 0.1

d = 0.2

Table 20 Effect of deviating from the balanced allocation for N = 3, c = 1 and q = 2 w-w~

Q(w)

A= 0

A = 0.025

A = 0.05

(½a

½A

-- A)

100.00

99.84

99.60

98.88

96.65

(½d ( 1 ~A

- A 1 -- ~A

½A)

100.00

100.19

100.32

100.39

99.79

A)

100.00

100.07

100.04

99.72

98.03

(-- ½A (A (A (0

A --A 0 A

-- ½A) 0) --A) --A)

100.00 100.00 100.00 100.00

99.74 100.18 99.93 99.73

99.43 100.25 99.74 99.37

98.61 100.11 98.99 98.38

96.32 98.66 96.26 95.55

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

509

Table 21 Mean and standard deviation of Q(w) vs. A for N = 3, c = 1 and q = 2 ~)(w) and (in parentheses) s

Target wt

Bowl - w * Balanced - wb

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

100.40 (0) 100.00 (0)

100.35 (0.01) 99.95 (0.17)

100.20 (0.04) 99.81 (0.34)

99.63 (0.17) 99.26 (0.68)

97.49 (0.60) 97.20 (1.32)

Table 22 Effect of deviating from the optimal bowl allocation for N = 4 and c = 1 w -

w °

(-A ( -A (-A (-A

Q(w)

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

A) 0) 0) 0)

100.94 100.94 100.94 100.94

100.92 100.92 100.93 100.93

100.85 100.86 100.88 100.89

100.56 100.61 100.70 100.72

99.43 99.62 99.99 100.06

0 0 A ½A

0 A 0 ½A

(-A

½A

0

½A)

100.94

100.93

100.89

100.71

100.04

(-A (0 (0 (A (½A

0 -A -A -A -A

½A 0 A 0 ½A

½A) A) 0) 0) 0)

100.94 100.94 100.94 100.94 100.94

100.92 100.92 100.93 100.93 100.93

100.87 100.86 100.87 100.88 100.90

100.64 100.60 100.67 100.70 100.77

99.76 99.60 99.87 99.98 100.25

(½A

-A

0

½A)

100.94

100.93

100.89

100.75

100.17

(0 (- A ( - ½A

-A - A - A

½A A ½A

½A) A) A)

100.94 100.94 100.94

100.93 100.89 100.91

100.88 100.73 100.81

100.70 100.09 100.43

99.96 97.69 98.94

( - ½A

- A

A

½A)

100.94

100.91

100.83

100.48

99.12

(-A

- ½A

½A

A)

100.94

100.91

100.80

100.39

98.80

(-A

- ½A

A

½A)

100.94

100.91

100.81

100.43

98.96

( - ½A

- ½A

A

0)

100.94

100.93

100.88

100.70

99.98

( - ½A (-A ( - ½A

- ½A A ½A

0 - A - A

A) A) A)

100.94 100.94 100.94

100.92 100.91 100.92

100.87 100.83 100.87

100.64 100.48 100.65

99.75 99.14 99.79

( - ~A

A

- A

½A)

100.94

100.93

100.88

100.67

99.88

(--A

½A

-- ½A

A)

100.94

100.92

100.85

100.59

99.55

( -- A ( - ½A

A ½A

- 5A1 -- A

½A) A)

100.94 100.94

100.92 100.93

100.87 100.90

100.65 100.77

99.80 100.27

( - ½A (- A ( - ½A

A A ½A

- A A A

½A) -- A) -- A)

100.94 100.94 100.94

100.93 100.91 100.92

100.88 100.82 100.87

100.71 100.44 100.63

100.03 98.94 99.70

( - ~A

A

½A

- A)

100.94

100.92

100.86

100.61

99.61

( - ½A (A

(½,a

A --A - ~A

0 -- A - A

- IA) A) A)

100.94 100.94 100.94

100.93 100.91 100.92

100.90 100.81 100.86

100.75 100.44 100.63

100.18 99.01 99.72

(A

- ½A

- A

½A)

100.94

100.92

100.86

100.61

99.63

(A

- ½A

- ½A

0)

100.94

100.93

100.89

100.72

100.06

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

510

Tables 13-15 give the corresponding results for c--- 1. In this case, w * = (1.086, 0.828, 1.086) with R * = 0.56718, representing 100.55% of R b = 0.56410. Tables 16-18 give the corresponding results for c = 2. In this case, w* = (1.105, 0.790, 1.105) with R* = 0.47573, representing 100.41% of R b = 0.47378.

There are three major observations from the results given in Tables 10-18. First, in most cases when wt = w*, the throughput of a line still exceeds that of a balanced line - i.e., Q ( w ) > 100% even when the workload allocation deviates from the optimal bowl allocation by as much as 10% (A = 0.1). Second, the optimal bowl alloca-

Table 23 Effect of deviating from the balanced allocation for N = 4 and c = 1 W -

Wb

Q(w)

,4 = 0.05

,4 = 0.1

,4 = 0.2

(-'4 ( -,4 ( -,4 ( -,4

0 0 ,4 ½,4

0 ,4 0 ½,4

,4) 0) 0) 0)

,4 = 0 100.00 100.00 100.00 100.00

99.98 99.81 99.81 99.81

99.91 99.58 99.60 99.60

99.63 99.00 99.08 99.10

(_,4

(-,4 (0 (0 (,4 (½,4 (½,4

~,4 1 0 -,4 -,4 -,4 - ,4 - A

0 ½,4 0 ,4 0 ½,4 0

1,4) ½,4) ,4) 0) 0) O) ½,4)

100.00 100.00 100.00 100.00 100.00 100.00 100.00

99.90 99.89 100.15 99.98 100.16 100.07 100.16

99.77 99.75 100.26 99.94 100.28 100.13 100.30

99.44 99.37 100.35 99.75 100.45 100.18 100.50

98.54 97.41 97.74 97.80 98.45 98.20 100.03 99.00 100.42 100.03 100.61

(0 ( -'4 ( - ½,4

--A - A - ,4

½A '4 ½'4

½a) ,4) ,4)

100.00 100.00 100.00

100.07 99.95 100.05

100.11 99.80 100.05

( - ½,4 (_,4

- ,4 _ ½,4

,4 1,4

½,4) ,4)

100.00 100.00

99.97 99.97

99.89 99.87

100.11 99.20 99.85 99.56 99.47

(-,4 (__1i'4 ( - ½'4 ( -,4 ( - ½,4 ( - ½,4

- ½,4 1 - i'4 - ½,4 A ½,4 A

,4 "4 0 -- "4 - ,4 - "4

½,4) 0) ,4) ,4) ,4) ½"4)

100.00 100.00 100.00 100.00 100.00 100.00

99.88 99.90 100.07 99.97 100.07 99.98

99.71 99.77 100.10 99.89 100.10 99.94

99.18 99.43 100.05 99.56 100.06 99.75

(--A (-d ( - ½A ( - ½d (- A ( _ 1~A ( - ½A ( - ½a (A (½A (A

½A A ½A A A ½A A "4 --a - ½a -- ½A

-- ½'4 -- ½A -- '4 - A A A ½A 0 -- A -- a -- A

A) ½A) A) ½A) -- A) -- A) -- A) - ½A) A) A) 1A)

100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00

99.98 99.90 99.90 100.07 99.63 99.72 99.72 99.82 100.31 100.24 100.24

½"4

- ½"4

0)

100.00

100.16

99.91 99.76 99.79 100.11 99.19 99.41 99.40 99.61 100.56 100.44 100.43 100.29

99.65 99.38 99.50 100.12 98.15 98.67 98.65 99.13 100.87 100.72 100.70 100.47

(A

-

,4 = 0.025

99.74 96.91 98.73 98.28 97.95 97.48 98.43 99.51 98.27 99.56 98.99 98.65 98.23 98.69 99.79 95.45 96.80 96.72 97.92 100.74 100.82 100.72 100.49

511

E S. Hillier, K.C. So / European Journal of Operational Research 89 (1996) 496-515

Table 24 Mean and standard deviation of Q(w) vs. A for N = 4 and c = 1 Q(w) and (in parentheses) s

Target wt Bowl - w * Balanced - w b

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

100.94 (0) 100.00 (0)

100.92 (0.01) 99.98 (0.15)

100.86 (0.04) 99.92 (0.29)

100.61 (0.14) 99.67 (0.59)

99.65 (0.52) 98.76 (1.24)

tion provides a b e t t e r target t h a n the b a l a n c e d allocation in the sense that w h e n the actual work allocation deviates from e i t h e r target by the same m a g n i t u d e (as r e p r e s e n t e d by A), a n average higher t h r o u g h p u t (as r e p r e s e n t e d by Q(w)) is achieved w h e n the target is the o p t i m a l bowl allocation. F u r t h e r m o r e , the difference b e t w e e n these two average t h r o u g h p u t s r e m a i n s roughly the same for the four different values of A, 0.4% for c = 0.5, 0.5% for c = 1 a n d 0.4% for c = 2. T h e third o b s e r v a t i o n is that, for any given A, the t h r o u g h p u t of the line d e p e n d s less o n the direction of the deviation w h e n the o p t i m a l bowl allocation is used as the target t h a n w h e n the bala n c e d line is the target. T h e r e f o r e , n o t only does the o p t i m a l bowl allocation provide a higher average t h r o u g h p u t t h a n the b a l a n c e d allocation, t h e r e is also a m u c h s m a l l e r 'risk' associated with the bowl allocation w h e n the actual work allocation deviates from this target. F o r example, for c = 1 a n d A = 0.2, P ( w ) c a n be b e t w e e n 97.69% a n d 100.53% - a r a n g e of 2.84% - w h e n w t = w b, d e p e n d i n g o n the d i r e c t i o n of the deviation, while Q ( w ) only varies from 99.03% to 99.81% - a r a n g e of 0.78% - w h e n wt = w*. This very substantial difference in risk is reflected by the very large difference in s t a n d a r d deviations for the two targets shown in T a b l e s 12, 15, a n d 18. T a b l e s 19-21 p r e s e n t the c o r r e s p o n d i n g re-

Suits for the N = 3 a n d c = 1 case with the buffer capacity q = 2. I n this case, w * = (1.044, 0.912, 1.044) with R * = 0.73692, r e p r e s e n t i n g 100.40% of R b = 0.73402. F r o m T a b l e 12, w h e n w t = w * , Q ( w ) exceeds 100% for A = 0.05 b u t is less t h a n 100% for A = 0.1. T h e second a n d third o b s e r v a t i o n s r e m a i n valid for the q = 2 case. W e next p r e s e n t the results for the N = 4 case. Let W

--

W t :

(dl, d2, d3, d4)

with d 1+ d 2 + d 3 + d4 = 0 and A = m a x { I d l [, I d 2 l , I d 3 l ,

[d4] }.

As before, we c o n s i d e r all possible cases w h e r e + ~ 1A , _+A}, i = 1 , 2 , 3 , 4 (with I d i l = A for at least o n e i) for different values of A > 0. This c o r r e s p o n d s to a total of 66 possible cases for w - w t, as listed in the A p p e n d i x . (Again, any two cases given in the same row r e p r e s e n t a m i r r o r image of each o t h e r for which the reversibility p r o p e r t y holds, so c o m p u t a t i o n s are n e e d e d for only the first 34 cases in o r d e r to c o n s i d e r all 66.) F o r each target ( w t = w b or wt = w*) a n d each d iE{O,

Table 25 Effect of deviating from the optimal bowl allocation for N = 7 and c = 1 w - w*

(A (A (_ J (½Z

Q(w)

A -A l

- ½A

A A

0 0

- A - A

- A A l

-

-A

½A

- ~A'

½A)

l

*A_

- A) - A) ½A)

A= 0

A = 0.025

A = 0.05

A = 0.1

A = 0.2

101.48 101.48 101.48 101.48

101.36 101.43 101.47 101.47

101.06 101.33 101.43 101.43

99.90 100.94 101.27 101.28

95.85 99.44 100.65 100.68

F. S. Hillier, K. C. So/European Journal of Operational Research 89 (1996) 496-515

512

za = 0.025, 0.05, 0.1, 0.2, we computed R ( w ) and Q ( w ) for the corresponding work allocation w in each of the first 34 cases. Table 22 presents Q ( w ) for these 34 cases with the four different values of A when c = 1 and the optimal bowl allocation w* = (1.137, 0.863, 0.863, 1.137) is used as the target work allocation wr For comparison purposes, we again include the results for A = 0. Table 23 presents the corresponding results when the balanced allocation w b is used as the target allocation w r In this case, R * = 0.51963 and R b = 0.51478, so the optimal bowl allocation w* provides a throughput R * that equals 100.94% of R b. Table 24, which is derived from Tables 22 and 23, gives the mean and standard deviation (in parentheses) of Q ( w ) over the 66 possible cases vs. A for the two targets, w* and w b. Observe that for all cases where w t = w *, Q ( w ) well exceeds 100% even for A = 0.1. The second and third observations given for the N = 3 case also remain valid in this case. Finally, in Tables 25 and 26, we sampled some results when the work allocation deviates from the optimal bowl allocation and the balanced allocation for the case where N = 7, c = 1 and q = 0. In this case, w°=(1.210, 0.942, 0.901, 0.895, 0.901, 0.942, 1.210) with R * = 0.45988, representing 101.48% of R b = 0.45318. Again, the three conclusions hold for these results as well. Furthermore, the increase in the n u m b e r of stations (from N = 3 or 4 in the preceding tables to N = 7 here) substantially increases the improvement in throughput (to roughly 1.5%) from using w* as the target rather than w b.

5. Conclusions Earlier studies have shown that an optimal bowl work allocation can increase the throughput of an unpaced production line over that of a balanced line. However, achieving this increase in throughput requires correctly estimating the system parameters (especially the coefficient of variation of the processing times) and then actually being able to assign work to stations according to the optimal bowl allocation. In this p a p e r we provide a relatively comprehensive study of the effect on the throughput when we misestimate the coefficient of variation of the processing times or deviate from the optimal bowl allocation because of the necessity of assigning discrete microelements of work to the stations of the line. We found that the optimal bowl allocation is relatively robust when there are inaccuracies in estimating the coefficient of variation of the processing times to determine the optimal amount of unbalance. Our numerical results show that even fairly large errors in the amount of unbalance still provide most of the potential improvement in throughput over a perfectly balanced line. For example, errors of + 5 0 % from the optimal amount of unbalance still provide about 75% of the potential improvement in throughput. The optimal bowl allocation usually cannot be achieved exactly in practice because of the necessity of assigning d i s c r e t e microelements of work to the stations of the line. Consequently, it has been argued that a balanced line should remain as the objective for assigning work to stations for all practical purposes. However, for the same reason, it ordinarily is not feasible to obtain a perfectly balanced line. Either the optimal bowl allocation or the balanced allocation can only be

T a b l e 26 E f f e c t o f d e v i a t i n g f r o m the b a l a n c e d allocation for N = 7 a n d c = 1

w - wb

Q(w) zl = 0

(A (A

A -A

A A

0 0

( - ½zi (½A

- A - A

- A A

½A

- ½A

- ~1A

~, A

d

- ½A

-A

~1A

A = 0.025

A = 0.05

A = 0.1

A = 0.2

99.64 99.88

98.62 99.52

94.98 98.13

- A) - A)

100.00 100.00

½A

- ½A)

100.00

99.85

99.67

99.24

98.12

' - ~zi

½A)

100.00

100.13

100.23

100.36

100.32

99.91 99.97

F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

considered as a target when assigning work to the stations of a line. Therefore, our study also addresses the question of which one of these two allocations provides a better target. We provide a rather comprehensive analysis of the effect on the throughput of a line when the work allocation deviates from each of these two specific targets. Our numerical results suggest that when using the optimal bowl allocation as the target, the throughput of the line still exceeds the throughput of a perfectly balanced line in most cases even when the work allocation deviates from the target by as much as 10% of the average workload per station. We also compare the optimal bowl allocation and the balanced line as targets when assigning work to stations. When the actual work allocations deviate from these two targets by the same magnitude, the difference between their resulting throughputs tends on the average to be roughly the same as the difference between the optimal throughput and the throughput of a balanced line. Probably more important, given the magnitude of the deviation, the actual throughput of a line depends more heavily on the direction in which the work allocation deviates from the target allocation when the perfectly balanced line is the target than when the optimal bowl allocation is the target. In other words, the balanced allocation is a 'riskier' target while the optimal bowl allocation is a much more robust target. Therefore, our results show that in allocating work to stations, the optimal bowl allocation provides a better target than the balanced line whenever the model presented in Section 2 provides a reasonable representation of the real system. Furthermore, the resulting benefit in increased throughput grows substantially as the number of stations in the line increases. The results of this paper clearly suggest that the optimal bowl allocation is quite robust in providing the appropriate 'target' when assigning work to stations in many production line systems. It is important to use this target, since even a rough approximation of the theoretically optimal allocation yields most of the potential improvement in throughput. Using the perfectly balanced line as a target instead not only would sacrifice this small improvement, it also would risk incur-

513

ring an even larger loss in throughput by ending up with a significant unbalance in the wrong direction. Earlier work by Hillier and Boling (1966, 1979) has provided exact numerical results for the optimal bowl allocation for many cases. Recent work by Hillier and So (1993) further extended these results and then developed extrapolation guidelines for determining near-optimal bowl allocations for even larger production line systems. These extrapolation guidelines enable estimating the optimal amount of unbalance to a high degree of precision (perhaps even within 5% of the true value) for fairly large values of N, provided that the coefficient of variation of processing times is estimated accurately. Thus, identifying a close approximation of the optimal bowl target now is reasonably straightforward for numerous cases. Therefore, it now is frequently feasible to obtain a small but significant increase in throughput by using this target instead of the less advantageous target of a perfectly balanced line. The results obtained in this study are based on the assumptions that the buffer size between each pair of stations is identical and the coefficient of variation of the processing times at each station is the same. This experimental design allows us to isolate the impact of unbalancing the workload such that the potential benefits of the bowl phenomenon (which makes these same assumptions) can be accurately evaluated. On the other hand, previous studies by Hillier and So (1991), Hillier, So and Boling (1993), Rao (1975) and Lau (1992) have analyzed the impact of unequal buffer sizes and unbalanced variances of processing times in unpaced lines, and showed that either of these two factors in isolation can also significantly affect the performance of a production line. In general, the workload, buffer size, and variance of processing times can be simultaneously unbalanced to achieve the optimal performance when there is a high degree of variability in a production line. Evidently, the interrelations between these three factors are very complex, and a comprehensive study is needed to analyze their combined impact on the performance of a production line system. Ideally, such a study should precede an investigation of the robustness of the resulting

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F.S. Hillier, K.C. So/European Journal of Operational Research 89 (1996) 496-515

References Chap, X., Pinedo, M., and Sigman, K. (1989), "On the interchangeability and stochastic ordering of exponential queues in tandem with blocking", Probability of the Engineering and Informational Sciences, 3, 223-236. Ding, J., and Greenberg, B.S. (1989), "Optimal order for server in series with no queue capacity", Technical Report, University of Texas, Austin. Ding, J., and Greenberg, B.S. (1991), "Bowl shapes are better with buffers - sometimes", Probability of the Engineering and Informational Sciences 5, 159-169. Hillier, F.S., and Boling, R.W. (1966), "The effect of some design factors on the efficiency of production lines with variable operations times", Journal of Industrial Engineering 17, 651-858. Hillier, F.S., and Boling, R.W. (1979), "On the optimal allocation of work in symmetrically unbalanced production line systems with variable operations times", Management Science 25, 721-728. Hillier, F.S., and So, K.C. (1991), "The effect of the coefficient of variation of operation times on the allocation of storage space in production line systems", liE Transactions 23, 198-206. Hillier, F.S., and So, K.C. (1993), "Some data for applying the bowl phenomenon to large production line systems", International Journal of Production Research, 31, 811-822. Hillier, F.S., So, K.C., and Boling, R.W. (1993), "Towards characterizing the optimal allocation of storage space in production line systems with variable processing times", Management Science 39, 126-133. Huang, C.C., and Weiss, G. (1990), "On the optimal order of m machines in tandem", Operations Research Letters 9, 299-303. Karwan, K.R., and Philipoom, P.R. (1989), "A note on stochastic unpaced line design: Review and further experimental results", Journal of Operations Management 8, 4854.

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Lau, H.S. (1992), "On balancing variances of station processing times in unpaced lines", European Journal of Operational Research 56, 345-356. Muth, E.J., and Alkaff, A. (1987), "The bowl phenomenon revisited", InternationalJournal of Production Research 25, 161-173. Rap, P. (1975), "A generalization of the 'bowl phenomenon' in series production systems", International Journal of Production Research 14, 437-443. Shanthikumar, J.G., Yamazaki, G., and Sakasegawa, H. (1991), "Characterization of optimal order of servers in tandem queues with blocking", Operations Research Letters 10, 17-22. Smunt, T.L., and Perkins, W.C. (1985), "Stochastic unpaced line design: Review and further experimental results", Journal of Operations Management 5, 351-373. Smunt, T.L., and Perkins, W.C. (1989), "Stochastic unpaced line design: A reply", Journal of Operations Management 8, 55-62. Smunt, T.L., and Perkins, W.C. (1990), "On the efficiency of unbalancing production lines: An alternative interpretation", International Journal of Production Research 28, 1219-1220. So, K.C. (1989), "On the efficiency of unbalancing production lines", International Journal of Production Research 27, 717-729. So, K.C. (1990), "On the efficiency of unbalancing production lines: A reply to an alternative interpretation", International Journal of Production Research 28, 1221-1222. Wan, Y.-W., and Wolff, R.W. (1993), "Bounds for different arrangements of tandem queues with nonoverlapping service times", Management Science 39, 1173-1178. Yamazaki, G., Kawashima, T., and Sakasegawa, H. (1985), "Reversibility of tandem blocking queueing systems", Management Science 31, 78-83. Yamazaki, G., Sakasegawa, H., and Shanthikumar, J.G. (1992), "On optimal arrangement of stations in a tandem queueing system with blocking", Management Science 38, 137153.