Robust sequencing of heterogeneous items in Bucket Brigade

systems

Yossi Bukchin∗ Eran Hanany∗ Eugene Khmelnitsky∗

February 2021

Abstract

Bucket Brigade is a common approach for dynamic work-sharing in serial lines. Differently

from typical analysis, we assume item heterogeneity, which creates blockages and reduced ef-

ficiency. The problem is how to sequence entering items in order to minimize this potential

inefficiency. The framework models real-world processes such as mixed-model assembly lines in-

volving the simultaneous assembly of multiple product variants, and order picking with workload

distributed along the picking aisle according to the number of items to be picked. To analyze

such systems, we propose a measure for robustly quantifying the generated blockage inefficiency

(BI) as a proxy for the makespan. As the BI depends on the sequence of items, several strate-

gies are proposed to identify robust sequences with no-blockage or with minimal BI. We provide

several practical sequencing policies requiring an increasing order of detail about the items. Se-

quencing based on no-blockage notions and steady-state hand-off positions is proved useful, and

no-blockage in a robust sense is implied by first-order distributional dominance sequencing. A

simple policy ensures low BI for large sets of items, for which we show an asymptotic result: the

robust BI of any efficient sequence approaches zero in the limit as the sequence length tends to

infinity. Hamiltonian path and TSP modeling with corresponding MILP formulations are pro-

posed as exact and approximate computational methods of robust, item specific sequences with

no-blockage or minimal BI. In general, sequencing items is a practically relevant and effective

managerial strategy, as it typically substantially reduces the robust BI, and often eliminates it

entirely. The latter is relevant for any number of items.

Key words: Work sharing, robust optimization, sequencing.

1 Introduction

Modern environments of serial production and assembly apply work-sharing methods, whereby

multiple cross-trained workers share the sequential execution of multiple operations on a given item

in order to finish a product or service. Such environments become quite complex when demand

is customer specific and items are heterogeneous, as the system control becomes more challenging

∗Department of Industrial Engineering, Tel Aviv University. E-mail: bukchin@tau.ac.il, hananye@tau.ac.il,xmel@tau.ac.il

1

and operational performance may deteriorate. Possibly the most commonly analyzed work-sharing

approach in serial systems is the Bucket-Brigade (BB) proposed by Bartholdi and Eisenstein (1996).

According to BB, whenever the last downstream worker completes an item, the worker moves back

and takes over the item from the immediate upstream worker, this preempted worker continues

similarly with the next upstream worker, and the procedure continues until the first upstream,

preempted worker starts processing a new item.

In general, serial lines suffer from throughput loss due to starvation and blockage. The former

does not exist in BB, while the latter occurs when workers are not allowed to overtake each other.

Most of the relevant literature, as well as the current paper, assume that overtaking is not allowed.

This stems from the fact that overtaking is very difficult to apply in real-world manufacturing

systems and order picking in forward storage areas due to technological reasons and cost of footprint.

Bartholdi and Eisenstein (1996) suggested conditions ensuring a self-balancing BB line under the

assumptions of uniform items, full cross-training of the workers, no restriction on the hand-off point

(the point in which a job is handed from one worker to another), and a completely deterministic

nature of the system. They show that when the workers are located from slowest to fastest,

the line does not suffer from blockages, thus achieving the maximal theoretical throughput rate.

Additionally, they show that under the above assumptions the hand-off positions converge to a

constant steady-state, consequently each worker specializes on executing their own work segment

for each item.

One of the common assumptions in the BB literature is that of homogeneous items, namely

identical items with workload uniformly distributed along the line. In this paper we relax this

assumption and consider heterogeneous items, i.e. items that differ from one another in their total

workload or their distribution along the line. Missing from the literature is a general understanding

of simple and efficient operational policies for BB systems with heterogeneous items. This is the

contribution of our paper.

Our proposed modeling approach fits common real-world situations. For example, mixed-model

assembly lines involve the simultaneous assembly of multiple model types (or product variants), for

which the main components are typically common and some sub-components are different. These

models may have varying assembly times along the line, namely different workload distributions (see

Figure 1.1(a)). This variation may generate many item types, thus leading to potential blockage and

starvation phenomena between the stations, which are significantly affected by the arrival sequence

of the models into the line (see for example, Scholl 1999, Grzechca 2011, Thomopoulos 2014).

Another example is an order picking process, where each order is considered an item that may

involve different picking requirements. As demonstrated in Figure 1.1(b), two pickers are working

by an aisle with eight pick faces, and the workload is distributed along the picking aisle according

to the number of physical objects to be picked at each face. Justified by the demonstration in the

figure, we will refer to the type of such items as piecewise linear. Therefore, one may consider a

continuous or discontinuous workload distribution.

Applying BB dynamics to such systems, we show that blockages may occur due to item hetero-

2

(a) (b)

Figure 1.1: Applications with heterogeneous items: (a) mixed-model assembly, (b) order pickingprocess

geneity. The amount of blockage is affected by the particular set of items and the exact sequence

in which they enter the line. Two common operational approaches for reducing blockages include

item batching and sequencing. The former determines the set of items and is applicable for order

picking, whereas the latter determines the item order and is applicable in general. In this paper we

concentrate on sequencing as the operational strategy given the set of items, and our contribution

is to show that this is very effective in handling the blockage problems. To this end, we propose a

measure of Blockage Inefficiency (BI), defined for a given sequence of items as the work capacity

loss due to blockage divided by the total workload for all items. Moreover, taking concern with pos-

sible unavoidable disturbances in the work dynamics, we adopt a robust approach for analyzing the

system, according to which the worst-case BI is evaluated within an interval of work rate scenarios.

As this robust BI might be high or low depending on the chosen sequence of items, we propose to

solve a static sequencing problem for finding efficient sequences that minimize the robust BI over

a given interval of work rates. This objective may be seen as a proxy for the robust minimization

of the production makespan.

We demonstrate with simple examples the potentially severe inefficiency that might obtain,

depending on the work rates and the sequence of items. Nevertheless, our general conclusion is

that the sequencing problem we formulate is practically very relevant, as it typically substantially

reduces this inefficiency and often eliminates it entirely. This is relevant when the number of

items is either small or large. Although our approach is applicable to any number of workers, we

concentrate throughout the paper on systems with two workers and discuss the extension to more

workers in the conclusions section.

To establish the above general conclusion, we propose several sequencing policies requiring an

increasing order of detail about the items. In order to achieve simple and effective operations, we

suggest to apply any sequencing policy only in case the previous policies in the described order

were not sufficiently useful. As preliminary analysis, we provide necessary and sufficient conditions

for no-blockage and for blockage during a cycle using a cumulative workload difference function

that also takes into account the work rates.

The first sequencing policy concerns non-uniform identical items, for which we show that no-

blockage in any cycle is guaranteed as long as the workers are positioned slowest to fastest. This

3

analysis leads to a generalization of Bartholdi and Eisenstein’s (1996) steady-state hand-off positions

to items with non-uniform distributions.

Two of the sequencing policies we propose are based on particular, nested notions of no-blockage,

each ensuring the existence of a robust no-blockage sequence. The first, most demanding notion is

the basis of the second sequencing policy. It involves a set of item types with universal no-blockage,

in the sense that for any sequence of item types selected from the set, no blockage occurs in any

cycle. Moreover, this set of items is maximal in the sense that including in the set any additional

item type from outside the set necessarily violates the universality requirement. We provide an

algorithm for determining the inclusion in such a set of item types, which therefore ensures that

any sequence has no-blockage.

The third sequencing policy concerns items with identical total workload but possibly different

distributions. For such items we show that sequencing the items in a non-increasing order of their

steady-state hand-off positions can ensure no-blockage.

The fourth sequencing policy is based on a second, less demanding notion of no-blockage. It

involves a sequence of items with strong no-blockage, in the sense that no blockage would occur

when each consecutive pair of items in the sequence were processed alone. We show that first-order

distributional dominance is sufficient for strong no-blockage, thus there is a wide opportunity for

strong no-blockage.

The fifth sequencing policy generates low robust BI when the number of items to be sequenced

is relatively large. The policy involves a sequence in which groups of identical items are ordered

according to increasing total workload. Such sequences are first used for items with quadratic

distributions, i.e. with weakly linearly increasing or decreasing work content along the line, for

which we demonstrate substantial reductions in the BI already for sequences with at least 10 items.

This computation uses exact expressions we develop for the BI for item types with quadratic

distributions. Then we use the sequencing policy for piecewise linear items, which are relevant e.g.

in order picking processes (as in Figure 1.1b), for which we show an asymptotic no-blockage result:

under uniform, i.i.d. sampling of items from a set of piecewise linear item types with identical

workers, the probability that there exists a no-blockage sequence approaches 1 in the limit as the

sequence length tends to infinity. This result is important also because piecewise linear item types

are a good approximation for the domain of all item types when the number of break points is

large. We then show a more general asymptotic result: for piecewise linear item types with any

given approximation level and slowest to fastest assignment of the workers, the robust BI of any

efficient sequence approaches zero in the limit as the sequence length tends to infinity.

The sixth and last policy involves item specific sequences, generated using the following three

increasingly detailed approaches we propose: (1) a Hamiltonian path based MILP formulation

for finding strong no-blockage sequences in general, (2) a type of Hamiltonian path based MILP

formulation for finding a no-blockage sequence for identical workers and (3) relying on discrete ap-

proximations, a robust Traveling Salesman Problem (TSP) formulation for finding sequences with

approximately efficient robust BI for any given interval of work rate scenarios. We accompany the

4

description of each approach with demonstrating examples, and the TSP formulation is solved nu-

merically to demonstrate that under uniform, i.i.d. sampling of items with quadratic distributions,

the robust BI decreases as the number of items to be sequenced increases.

In the remainder of the Introduction we review the relevant literature on work sharing systems

and BB in particular. Systems that adopt work sharing differ by the level of worker cross-training,

ranging from full cross-training where all workers are capable of performing the whole work, to

partial cross-training where there is only some overlapping between the capabilities of the workers.

On the operational side, Ostolaza et al. (1990) use the term Dynamic Line Balancing (DLB) to

describe predefined tasks that are shifted dynamically between adjacent stations/workers based on

the state of the system. Hopp et al. (2004) investigated these issues in the context of achieving

a low ratio of work-in-process (WIP) to throughput. Besides the level of cross-training, they

investigated the line topology by studying D-Skilled Chaining (DSC), which was initially coined

by Jordan et al. (2004), and Cherry Picking (CP). In CP, only the bottleneck worker is assisted

by the other workers, while the cross-training under DSC is symmetric among the workers, as each

worker can help the adjacent upstream/downstream workers. The literature has also addressed

other variations of work-sharing systems, such as preemption by which a task may be split between

workers (McClain et al. 2000), and processes in which machines are involved (Zavadlav et al. 1996).

Following Bartholdi and Eisenstein (1996), several extensions have been suggested (see Bratcu

and Dolgui 2005 for a review). In the deterministic domain with homogeneous items, the BB system

dynamics for two and three workers was analyzed in Bartholdi et al. (1999). The chaotic behavior

of the hand-off point when the convergence condition does not hold was studied in Bartholdi et al.

(2009). Armbruster and Gel (2006) relaxed the dominance assumption, and studied a 2-worker line

in which one worker may be faster than his neighbor in some part of the line, and slower in the

other part. They provided insights and operation principles for various scenarios. Gurumoorthy

et al. (2009) analyzed the dynamics of a line with two workers, each having an arbitrary, constant

speed at each station. When considering discrete workstations, the results of the basic continuous

model approximately hold for large number of stations, however a different analysis is needed when

the number of stations is small. This issue was considered in Lim and Yang (2009), who found

conditions under which 2- and 3-station lines maximize throughput for a given number of stations.

Bartholdi et al. (2006) applied the BB principles in an in-tree network of sub-assembly lines.

Bratcu and Dolgui (2009) and Lim (2011) relaxed the common assumption of infinite worker walk

back speed, where the latter proposed a cellular configuration model. Lim and Wu (2013) studied

a U-line BB system with discrete stations. In another direction, Armbruster et al. (2007) and

Montano et al. (2007) studied the effect of learning in BB systems, where the latter suggested

an alternative control rule named modified-work-sharing. Some industrial case studies of BB have

been studied in Bartholdi and Eisenstein (2005), Lim (2005) and De Carlo et al. (2013).

In the stochastic domain with homogeneous items, Bartholdi et al. (2001) studied the perfor-

mance of BB under the assumption of stochastic processing times, and showed that the throughput

rate converges to its optimal value as the number of stations increases. Buzacott (2002) considered

5

4-station and 2-worker stochastic lines with or without preemption and derived the optimal policy,

which modifies the non-preemptive BB rule. Bratcu and Dolgui (2009) studied a BB system with

stochastic performance rates via simulations, assuming that both working and walk back speeds

are normally distributed. Hong (2014) derived an analytic expression for the 2-worker blocking

congestion in a circular-passage system under the assumptions of constant worker speeds and prob-

abilistic picks. In a later paper, Hong et al. (2015) provided a closed-form expression to the level

of blocking for two extreme walk speed cases. Bukchin et al. (2018) studied three variants of BB

production systems under the assumption of stochastic worker speeds: the traditional BB system,

BB with overtaking allowed (BBO), and a benchmark system of parallel workers. They showed

that BB systems may perform better than a comparable system with parallel workers, and that

the best configuration is BB with overtaking. Additionally, they showed that slowest to fastest

is not always optimal when speeds are stochastic. The general conclusion that BB systems are

immune to stochastic workloads even without sequencing is a direct consequence of the assumption

of homogeneous items. In contrast, our contribution is to show that sequencing policies are very

much required when items are heterogeneous.

Hong et al. (2016) considered heterogeneous items in deterministic environments. They devel-

oped a batching and sequencing MIP formulation to reduce blocking delays in BB order picking

systems with work content dependent picking times. Integrating a ’rolling horizon’ implementa-

tion of this MIP, they considered at each step a small number of orders within a simulation with

stochastic picking times and compared the results with a random policy. This formulation was later

modified in Fibrianto et al. (2017) to consider on-line order arrival. Our contribution compared

to these papers is to offer simple and robustly efficient sequencing policies that typically eliminate

the blockage inefficiency almost entirely. Such policies may be used as substitutes or in addition to

batching operations.

The rest of the paper is organized as follows. In Section 2 we present the model and propose our

robust measure of blockage inefficiency (BI) for a BB system with heterogeneous items. In Section

3 we propose several sequencing policies for reducing the BI, and provide an evaluation of these

policies using formal results and examples. Section 4 discusses the generalization to more than two

workers and concludes. A preliminary analysis of no-blockage and blockage sequences is included

in Appendix A, and auxiliary results and all proofs are collected in Appendices B, C and D.

2 Model

We study a Bucket Brigade (BB) system (Bartholdi and Eisenstein, 1996) for coordinating the ef-

forts of several workers along a production line. The protocol of BB includes forward (downstream)

and backward (upstream) movements of each worker along the line. During a forward movement,

the worker is involved in production of an item. Moving forward ends for the most downstream

worker upon reaching the end of the line with a finished item, and ends for any other worker upon

meeting their immediate downstream worker who is moving backward. When moving backward,

6

the worker does not hold any item. Moving backward ends for the most upstream worker upon

reaching the start of the line, and ends for any other worker upon meeting their immediate up-

stream worker who is moving forward. When such a meeting occurs, the downstream worker takes

the item from the upstream worker and starts a forward movement to continue processing it, and

the upstream worker starts a backward movement to take a different item from an upstream worker

or from the start of the line . Therefore the completion of an item by the most downstream worker

initiates a sequence of such meetings, which are together called a hand-off event.

The work rate of each worker k, i.e. the amount of work that the worker can process in a unit

of time, is fixed at rk > 0. In line with most of the literature, we assume that the time required to

return upstream is insignificant compared with the time required to work downstream. The order

of the workers along the line does not change over time, and we mostly (but not always) assume

that the workers are located in a non-decreasing order of their work rate. As a result, each worker

either proceeds at their own work rate or at a reduced pace when blocked by the next downstream

worker (recall that overtaking is not allowed). Without loss of generality, we identify the line

with the interval [0, 1]. At time 0, a given set of heterogeneous items is ready to be processed at

the start of the line. Item j = 1, 2, ..., J is identified by an item type, i.e. a bounded workload

density function wj : [0, 1] → R+, for which the corresponding cumulative workload distributionfunction, Wj(x), is assumed to be well defined for any x ∈ [0, 1] as the integral of wj(x) over[0, x]. Differently from probability distributions, the total workload Wj(1) may be above, equal

or below 1. The permutation sequence of the items, denoted by s = (s1, ..., sJ), is given before

production starts. We aim at determining the appropriate sequence of items s, thus a set of items

W = {Wj , j = 1, ..., J} together with a vector of work rates r = (rk, k = 1, ...,K) will be referredto as a sequencing problem (W, r).

For clarity, our analysis will be presented for two workers, K = 2, with the generalization to

any number of workers discussed in the conclusions section. Fixing the order of the two workers

as 1 → 2, denote their work rate ratio by r̄ = r1r2 . Given a sequencing problem (W, r), for anysequence s, the nth state of the system, 0 ≤ xn(W, r, s) ≤ 1, defines the position along the line justbefore hand-off n = 1, 2, ..., J − 1, i.e., the position of the upstream worker 1 with item sn+1 whenthe downstream worker 2 with item sn has reached the end of the line. We assume x0(W, r, s) = 0

for any sequence s. Since worker 2 cannot be blocked and always reaches the end of the line, the

cycle time, i.e. the time elapsed between hand-off n− 1 and hand-off n, is always

CTn(W, r, s) =Wsn(1)−Wsn(xn−1(W, r, s))

r2. (2.1)

Accordingly, we next define the system makespan.

Definition 2.1. The system makespan for sequence s is the total time to complete processing,

J∑n=1

CTn(W, r, s). (2.2)

7

Blockage may occur over any partial interval of the line during any cycle n < J , depending on

the work rates and the type of the item held by each worker. Such blockages create inefficiency

due to the reduced pace of the upstream worker, which in turn generates longer cycle times, and

consequently a higher makespan. We next define a measure to quantify this blockage inefficiency.

Minimizing this measure is then shown to be a proxy for minimizing the makespan.

Definition 2.2. The makespan work capacity for sequence s is the maximal amount of work

potentially accomplished by the workers during the makespan, i.e. if there were no blockages,

MC(W, r, s) =J−1∑n=1

CTn(W, r, s)(r1 + r2) + CTJ(W, r, s)r2. (2.3)

Note that the downstream worker is busy in all cycles, whereas the upstream worker is busy in

all but the last, J th cycle. Now, defining the total workload of all items by

TW (W ) =J∑j=1

Wj(1), (2.4)

the difference

BL(W, r, s) ≡MC(W, r, s)− TW (W ) (2.5)

will be referred to as the work capacity loss due to blockage for sequence s. Using these notions,

we may define our measure of inefficiency.

Definition 2.3. The Blockage Inefficiency (BI) for sequence s is

BI(W, r, s) =BL(W, r, s)

TW (W )=MC(W, r, s)− TW (W )

TW (W ). (2.6)

We are interested in sequences s that guarantee low values of our inefficiency measure, BI(W, r, s).

It is worth noting that, using Equations (2.3) and (2.5) and Definition 2.3, for any sequencing prob-

lem (W, r),

(BI(W, r, s) + 1)TW (W )

r1 + r2=

J−1∑n=1

CTn(W, r, s) + CTJ(W, r, s)r2

r1 + r2, (2.7)

which approaches the system makespan, in the relative sense, as the number of items increases.

Therefore, when the number of items is large, since TW and r are independent of the sequence of

items, minimizing the BI over all sequences is a proxy to minimizing the system makespan over all

sequences.

Blockage inefficiency has a simple tight upper bound: the ratio of work rates, r̄. This is stated

in Proposition 2.1. The proof of the proposition shows that high inefficiency occurs in particular

when an item with large total workload is followed by an item with a small total workload.

8

Proposition 2.1. For any sequencing problem (W, r) and any sequence s, an upper bound for

BI(W, r, s) is r̄, and this bound is tight.

A higher work rate ratio r̄ may therefore generate higher inefficiency. Nevertheless, due to

item heterogeneity, blockages in early cycles may alleviate blockages in later cycles. As the next

proposition shows, BI(W, r, s) is not always increasing with the work rate ratio r̄, and may even

decrease from a positive value to zero.

Proposition 2.2. There exists a set of items W , a sequence s and r1, r2 with r̄1 < r̄2 such that

BI(W, r1, s) > BI(W, r2, s) = 0.

A main subject of interest in our analysis will be sequences with no-blockage.

Definition 2.4. A sequence s has no-blockage for (W, r) if BI(W, r, s) = 0, i.e., for any cycle n,

worker 1 is not blocked by worker 2 throughout the cycle.

Finally for this section, our model makes assumptions about exact and deterministic item work-

load distributions and work rates. Since realistic environments are likely to suffer from uncontrolled

disturbances that affect these parameters, when studying the system performance we rely on se-

quence robustness with respect to the work rate ratio, r̄. Consider an interval I(r̄, ε) of radius εaround a given rate ratio r̄. For a given sequence s, larger intervals may lead to weakly larger worst

case BI values. Say that a sequence s is (ε, δ)-robust efficient if among all sequences it achieves the

minimum worst case BI, which equals δ, when considering any work rate ratio r̂ within the interval

I(r̄, ε). Additionally, ε is required to generate the maximal sized interval for which δ is attained.Formally, we define robust efficiency as follows:

Definition 2.5. Given a sequencing problem (W, r) and ε, δ ≥ 0, say that a sequence s is (ε, δ)-robust efficient for (W, r) if

δ = BIε(W, r, s) = minŝBIε(W, r, ŝ)

and

ε = maxŝ,ε̂|ε̂≥0,BIε̂(W,r,ŝ)≤δ

ε̂,

where for every ŝ and ε̂ ≥ 0,

BIε̂(W, r, ŝ) ≡ maxr̂∈I(r̄,ε̂)

BI(W, r̂, ŝ)

and

I(r̄, ε̂) ≡ {r̂ = (r̂1, r̂2) |r̄

1 + ε̂≤ r̂1r̂2≤ (1 + ε̂)r̄}.

Say that s is δ-robust efficient if it is (ε, δ)-robust efficient for some ε > 0. Say that s is no-blockage

robust efficient if it is 0-robust efficient.

9

The following example demonstrates Definition 2.5 (the calculation details for the example rely

on the preliminary analysis provided in Appendix A and are omitted here).

Example 2.1. Consider a BB sequencing problem (W, r) with three items to be processed,

W1(x) = x (2− 0.95x) , W2(x) = 1.6x2, W3(x) = x(4− x).

As shown in the left graph of Figure 2.1, the BI can be zero or significantly high, depending on the

work rate ratio r̄ and sequence of the items. Consider the case where the work rates are r = (1, 2),

thus r̄ = 0.5. The right graph of the figure depicts the resulting (ε, δ)-robust efficient curve for

(W, r) with this r̄. Each point on the efficiency curve corresponds to a sequence s with minimal

worst-case BI, BIε(W, r, s), within the interval I(r̄, ε) of radius ε around r̄, thus generating thecorresponding value of δ. For low values of ε, the robust efficient sequence is s = (2, 1, 3), as the

BI curve of this sequence is the lowest among the BI curves of all 3! possible sequences. For a

sufficiently low value of ε, s = (2, 1, 3) is a no-blockage robust efficient sequence with zero worst-

case BI, i.e. δ = 0, whereas for higher values of ε this sequence is still robust efficient, but with

positive and increasing value of δ. The robust efficient sequence switches to s = (1, 2, 3) at the ε

corresponding to the intersection point with the BI curve of this sequence, resulting in an increase

of δ with a lower slope. Finally, the robust efficient sequence switches to s = (2, 3, 1) when the

BI curve of s = {1, 2, 3} reaches the maximal value of the BI curve of s = (2, 3, 1), as from thispoint onward the minimal worst-case BI is achieved by s = (2, 3, 1) with δ constant and equal to

this maximal value. Note that in general the efficiency curve is increasing, and is not necessarily

piecewise linear.

Figure 2.1: (ε, δ)-robust efficiency for r̄ = 0.5

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3 Sequencing policies

Our approach leads to several sequencing policies, described and analyzed in the subsections below.

These policies require an increasing order of detail about the items. Each of these policies may lead

to robust efficient sequences. Consequently, in order to achieve simple and effective operations, any

sequencing policy should be applied only in case the previous policies in the described order were

not sufficiently useful. Additionally, as the following result shows, whenever a sequencing policy

ensures no-blockage, it typically also has the advantage of ensuring the existence of a no-blockage

robust efficient sequence.

Proposition 3.1. If there exists a no-blockage sequence for a generic (W, r), then there exists a

no-blockage robust efficient sequence for (W, r).

3.1 Non-uniform identical item sequences

The simplest case in our analysis is where W = (W̄ , ..., W̄ ), i.e. all items are identical, say with

type W̄ , but the workload may not be distributed uniformly across the line. Intuitively, for this

case, since at any position x the instantaneous amount of work required by each worker is the

same, if the workers are ordered slowest-to-fastest, blockage cannot occur at any position x. This

intuition is confirmed by the following no-blockage result.

Proposition 3.2. For any sequencing problem (W̄ , r) with work rate ratio r̄ ≤ 1, any sequence sis a no-blockage sequence for (W̄ , r).

In steady-state, the hand-off position remains unchanged across cycles. This leads to the follow-

ing definition, which generalizes Bartholdi and Eisenstein (1996)’s definition for uniform, identical

items.

Definition 3.1. The steady-state hand-off position of type W̄ is1 x∗ = max{x | W̄ (x) = W̄ (1) r̄1+r̄}.

The following proposition shows that the no-blockage property of non-uniform identical item

sequences guarantees the convergence to steady-state.

Proposition 3.3. For any sequencing problem (W̄ , r) with work rate ratio r̄ ≤ 1 and any sequences, hand-off position n, xn(W̄ , r, s), converges to the steady-state hand-off position x

∗ as the number

of items J approaches infinity, and the convergence rate is exponential.

Proposition 3.3 leads to the following important insight: each worker specializes on executing

their own work segment for each item. This implication is due to the items being identical rather

than being uniform.

1The max is relevant when W̄ (x) is constant because the workload density is zero.

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3.2 Maximal universal no-blockage sequences

Two of the sequencing policies we propose are based on particular, nested notions of no-blockage.

The following policy is based on the first, most demanding no-blockage notion, which therefore

has the advantage of ensuring no-blockage in a very wide set of scenarios. Recall the definition

of item types in the second paragraph of Section 2. A consequence of our proposed policy is that

if a system designer can affect which item types may enter the line, for example by batching in

an order picking system, the approach described here suggests a way to ensure efficiency without

the need to worry about item sequencing. Specifically, we investigate sets of item types for which

no blockage occurs in any sequence, thus generalizing the case of identical items studied in the

previous subsection. This is formalized in the following definition.

Definition 3.2. Given work rates r, a set of item types satisfies maximal universal no-blockage if:

(a) Universality: for any sequence of items with item types selected from the set, no blockage occurs

in any cycle; and

(b) Maximality: including in the set any additional item type from outside the set necessarily

violates the universality requirement (a), i.e., there would exist a sequence of items with item types

selected from the set including the additional item type, for which blockage occurs in some cycle.

Our next result characterizes sets of item types satisfying maximal universal no-blockage. Given

any work rate ratio r̄ and item type W 0, denote by Hr̄,W 0 the set of all item types W̄ such thatW̄ (x) ∈

[r̄W 0(x),W 0(x)

]for all x ∈ [0, 1]. Note that Hr̄,W 0 is non-empty if and only if r̄ ≤ 1, i.e. a

slowest to fastest configuration of workers. There are infinitely many such sets, and one of them is

illustrated in Figure 3.1 for a linear (uniformly disributed) item type W 0, together with two possible

non-linear item types within this set. As shown in the figure, Hr̄,W 0 is a strict super-set of the setof all item types with bounded workload density functions, i.e., such that w̄(x) ∈

[r̄w0(x), w0(x)

]for all x ∈ [0, 1].

Figure 3.1: Set of item types satisfying maximal universal no blockage

Proposition 3.4. For any work rate ratio r̄ ≤ 1 and item type W 0, the set of item types Hr̄,W 0satisfies maximal universal no-blockage.

The simple case described in Subsection 3.1 of non-uniform identical items is always a subset of some

maximal universal no-blockage set Hr̄,W 0 . In particular, when the workers identical, i.e. r1 = r2,

12

the maximal universal no-blockage set Hr̄,W 0 includes the single item type W 0. Additionally,Proposition 3.4 directly implies the following natural monotonicity property of maximal universal

no-blockage with respect to work rate ratios.

Corollary 3.1. Given an item type W 0, a pair of work rates r̂ with r̂1r̂2 < r̄ and a set of items W

with Wj ∈ Hr̄,W 0 for each j, any s is a no-blockage sequence for (W, r̂).

The following algorithm may be applied in order to check whether a set of items W is a subset

of some maximal universal no blockage set: Defining W 0 as the pointwise maximum of the CDFs

of all items j in the set W , i.e. let W 0(x) = max1≤j≤JWj(x) for each x ∈ [0, 1], check whethereach Wj is in the set Hr̄,W 0 , i.e. check if Wj(x) ∈

[r̄W 0(x),W 0(x)

]for all x ∈ [0, 1] and each item

j.

Proposition 3.5. The decision given by the algorithm always identifies correctly whether a set of

items W is a subset of some maximal universal no blockage set.

In sum, we find that maximal universal no-blockage is easily identified, and expands when the

gap between the work rates increases, thus increasing the no-blockage opportunities.

3.3 Identical total workload sequences

General items with identical total workload allow a simple no-blockage sequencing policy based on

the the steady-state hand-off position introduced in Definition 3.1.

Definition 3.3. Let s∗ be a sequence in which the items are in a non-increasing order of their

steady-state hand-off position, i.e. x∗s∗1≥ ... ≥ x∗s∗J .

The policy partly relies on the ability to initialize the process by letting worker 2 work alone

up to the first hand-off position before the first cycle starts:

Proposition 3.6. Consider a sequencing problem (W, r) where all items have identical total work-

load denoted by a, i.e. Wj(1) = a for all j. If the starting position of worker 2 in the first cycle is

the steady-state hand-off position of the first item, i.e. x0(W, r, s∗) = x∗s∗1

, then s∗ has no blockage.

The condition stated in Proposition 3.6 ensures that each cycle starts at the steady-state hand

position of the item currently processed by worker 2, which guarantees a no-blockage sequence.

This holds regardless of the shape of the cumulative workload distribution functions, and for any

work rate ratio r̄. The following proposition shows that s∗ can be useful even without initiating

the process at the steady-state hand-off position.

Proposition 3.7. Consider a sequencing problem (W, r) where all items have identical total work-

load denoted by a, i.e. Wj(1) = a for all j. If the starting position of worker 2 in the first cycle is

zero, i.e. x0(W, r, s∗) = 0 as assumed throughout our analysis, then a sufficient condition for s∗ to

be a no-blockage sequence is that max0≤x≤1 r̄Ws∗1(x)−Ws∗2(x) ≤ 0 for the first cycle, n = 1, and

W−1s∗n+1

(a

r̄

1 + r̄(1− (−r̄)n)

)≤W−1s∗n

(a

r̄

1 + r̄

(1− (−r̄)n−1

))(3.1)

13

for all odd cycles n ≥ 3, where for any j, W−1j is defined by W−1j (α) = sup{x |Wj(x) = α}.

Example 3.1. Consider a BB sequencing problem (W, r), with r = ( 916 , 1) and with four items to

be processed with identical total workload a = 1 such that

W1(x) = x2, W2(x) = 1.2x

2 + 1.6x3 − 1.8x4, W3(x) = x, W4(x) = 2x− x2.

By Definition 3.1, the respective steady-state hand-off positions are x∗1 = 0.6, x∗2 ≈ 0.481, x∗3 = 0.36,

x∗4 = 0.2 (see Figure 3.2). Then, according to Definition 3.3 and Proposition 3.6, if the items are

ordered with decreasing x∗j , i.e. s∗ = (1, 2, 3, 4), and the starting position of worker 2 in the first

cycle is x0(W, r, s∗) = x∗1, then s

∗ has no blockage. Note that for this example, the sequence s∗

has no blockage also if the starting position is zero, x0(W, r, s∗) = 0. To see this, first note that

916W1(x) −W2(x) ≤ 0 for all 0 ≤ x ≤ 1, and Condition (3.1) holds for all odd n ≥ 3, i.e. only forn = 3, since W−14

(a r̄1+r̄

(1− (−r̄)3

))= 0.2411 is smaller than W−13

(a r̄1+r̄

(1− (−r̄)2

))= 0.2461.

Figure 3.2: 4-item example with identical total workload

3.4 Strong no-blockage sequences

The no-blockage requirement of Subsection 3.2, despite being useful, is very demanding as it requires

no-blockage simultaneously for all possible sequences of a large set of item types. In this subsection

we study a weaker requirement based on a robust optimization principle with respect to the starting

position of worker 2. It is based on the idea that the zero starting position x0(W, r, s) = 0,

consequently the first cycle in a sequence, is a blockage worst case.

Definition 3.4. An ordered pair of items has strong pairwise no-blockage given r if no blockage

would occur in any cycle of a sequence consisting only of this ordered pair. A sequence s has strong

no-blockage for (W, r) if each consecutive pair in s satisfies strong pairwise no-blockage given r.

Strong no-blockage is also monotonic with respect to work rate ratios, as shown next.

Proposition 3.8. Given a pair of work rates r̂ with r̂1r̂2 < r̄ and a set of items W , if a sequence s

has strong no-blockage for (W, r) then the same holds also for (W, r̂).

14

3.4.1 First-order distributional dominance sequences

Strong no-blockage allows us to propose a sequencing policy based on a dominance relation between

item types.

Definition 3.5. For any set of items W , say that s is a first-order distributional dominance

sequence if Wsn+1(x) ≥Wsn(x) for each n and x ∈ [0, 1].

Since first-order distributional dominance is a relatively weak requirement, the following propo-

sition shows that many sequencing problems are guaranteed to have strong no-blockage.

Proposition 3.9. For any sequencing problem (W, r) with work rate ratio r̄ ≤ 1, if s is a first-order distributional dominance sequence then s has strong no-blockage for (W, r). The converse

holds when r̄ = 1.

3.5 Sequences for large sets of items

In this subsection we propose a sequencing policy that generates low robust BI when the number

of items to be sequenced is relatively large.

Definition 3.6. Let stot be a sequence in which groups of identical items are ordered according to

increasing total workload.

Intuitively, the increasing total workload sequencing policy stot has the advantage of no-blockage

sequencing of consecutive identical items, as explained in Section 3.1. On top of that, it leads to

higher total workload for worker 1 as compared to worker 2, which tends to reduce the overall

physical progress of worker 1 along the line relative to worker 2. This holds because the difference

between the work accomplished by worker 1 and the work worker 1 would accomplish were worker

1 to move together with worker 2 tends to decrease, thus reducing the blockage opportunities. This

approach is demonstrated for two general classes of item types we define: quadratic items and

piecewise linear items.

3.5.1 Item types with quadratic distributions

Consider item types with quadratic distributions, i.e. each item j has a quadratic cumulative

workload distribution function Wj (as in Example 2.1 at the end of Section 2). Such item types

represent realistic settings in which the workload density function wj is linearly increasing or

decreasing (or constant) along the line. Moreover, as a simple parametric family of item types,

it allows an exact analytic expression for the BI, which is useful in drawing general conclusions for

many item types. Since Wj(0) = 0 and wj(x) ≥ 0 for all x ∈ [0, 1], quadratic item types mustbe of the form Wj(x) = x[bj + (aj − bj)x] for all x ∈ [0, 1], for some constants aj , bj satisfying0 ≤ bj ≤ 2aj .

We now demonstrate the sequencing policy stot with quadratic items types. Figure 3.3 depicts

the BI values of randomly sampled sequencing problems with increasing number of items J , where

15

each item is uniformly and independently drawn with a quadratic cumulative workload distribution

function where the parameters are 0 ≤ aj ≤ 1 and 0 ≤ bj ≤ 2aj . Using the sequencing policy stot

leads to decreasing BI values, which are relatively low already for 10 items. For comparison, a

random sequencing policy generates substantially higher BI values.

Figure 3.3: BI for a random sequence (left) and for stot (right)

3.5.2 Piecewise linear item types

Consider a general, parametric domain of distributions, motivated by the order picking application

described in the Introduction (see Figure 1.1). For a given finite ordered set X = {yp}Pp=0 withan integer P ≥ 2 of increasing positions 0 = y0 < y1 < y2 < ... < yP = 1, let WX be the set ofall item types W̄ such that W̄ is piecewise linear with X being its set of break points, and whereW̄ (yp) ∈ [0, 1] for 1 ≤ p ≤ P . When the number of breakpoints P is large, this domain is a goodapproximation for the domain of all item types normalized to have total workload of at most 1.

The number of breakpoints P is therefore referred to as a positional approximation level. Each

W̄ ∈ WX may be represented by the vector A = [a1, ..., aP ] with ap = W̄ (yp) for each 1 ≤ p ≤ P ,thus 0 ≤ a1 ≤ a2 ≤ ... ≤ aP ≤ 1.

Lemma 3.1. Given a sequencing problem (W, r) with item types in the domain WX of piecewise lin-ear item types with corresponding representing vectors {Aj = [aj,1, ..., aj,1], j = 1, ..., J}, a sequences has no blockage if and only if for all cycles n = 1, 2, ..., J − 1,

max1≤p≤P

(r̄asn,p − asn+1,p

)≤

n−1∑i=1

(−r̄)i+1Wsn−i(1). (3.2)

We concentrate on a particular subset of piecewise linear item types, which will be then used to

provide asymptotically robust efficient sequencing policies. Specifically, for two integers L ≥ 1 andQ ≥ 2, where L is a workload approximation level and P = QL is the positional approximationlevel, consider item types such that each ap ∈ {0, 1L ,

2L , ..., 1}, and each of these values up to aP

is attained for some p. Figure 3.4 illustrates such item types for two approximation specifications

alongside an approximated item type with smooth cumulative workload distribution function. Note

that we may partition any such set of item types to groups l = 0, 1, ..., L, where all items in group

16

l have identical total workload of lL , and the number of item types in group l is Cl =P !

l!(P−l)! , thus

the total number of all item types is C =∑L

l=0Cl.

Figure 3.4: Approximation using piecewise linear distributions for L = 2, P = 4 (left), and L =6, P = 24 (right)

We first investigate the likelihood that a no-blockage sequence exists for piecewise linear item

types and identical workers. To this end, consider a uniform probability distribution over the sub-

set of piecewise linear item types described in the previous paragraph, i.e. the probability of each

item type is 1C . Assuming identical workers, under i.i.d sampling of J items with this uniform

distribution we may evaluate the probability PL,P,J that the corresponding sequencing problemhas a no-blockage sequence. Consider for example L = 2, P = 4 and J = 2. In this case, each of

the two items is independently sampled as one of the 1 + 4 + (4!)(2!)2

= 11 item types with represent-

ing vectors [0, 0, 0, 0], [0, 0, 0, 12 ], [0, 0,12 ,

12 ], [0,

12 ,

12 ,

12 ], [

12 ,

12 ,

12 ,

12 ], [0, 0,

12 , 1], [0,

12 ,

12 , 1], [

12 ,

12 ,

12 , 1],

[0, 12 , 1, 1], [12 ,

12 , 1, 1], [

12 , 1, 1, 1] with equal probability. Since any pair of items except for the

pairs W = {{0, 0, 12 , 1}, {0,12 ,

12 ,

12}}, W = {{0, 0,

12 , 1}, {

12 ,

12 ,

12 ,

12}}, W = {{0,

12 ,

12 , 1}, {

12 ,

12 ,

12 ,

12}},

W = {{0, 12 , 1, 1}, {12 ,

12 ,

12 ,

12}}, W = {{0,

12 , 1, 1}, {

12 ,

12 ,

12 , 1}} (and the same pairs in reverse or-

der) can be ordered according to first-order distributional dominance, Proposition 3.9 and Lemma

3.1 imply that a no-blockage sequence exists for all but 10 of the 112 possible pairs, therefore

PL,P,J = 1− 10112 =111121 ≈ 0.9174.

Proposition 3.10. For any sequencing problem (W, r) with piecewise linear W under some work-

load approximation level L and positional approximation level P and with identical workers, using

(a variant of) the sequencing policy stot, as the number of items J approaches +∞, the no-blockageprobability PL,P,J approaches 1.

Figure 3.5 demonstrates the fast rate of convergence characterizing the asymptotic result in

Proposition 3.10. The figure plots the estimated no-blockage probability based on simulation with

2000 sized samples. For fixed values of workload approximation level L and positional approxima-

tion level P , the estimated probability tends to increase with the number of items J , and despite

possibly starting relatively low it reaches values close to 1 for relatively small values of J .

17

Figure 3.5: No-blockage probability

Finally for this subsection, the asymptotic no-blockage sequence result of Proposition 3.10

is next extended to a much more general conclusion: optimal sequencing always eliminates any

blockage inefficiency for large problems.

Proposition 3.11. For any sequencing problem (W, r) with piecewise linear W under some work-

load approximation level L and positional approximation level P , using the sequencing policy stot, as

the number of items J approaches +∞, all (ε, δ)-robust efficient sequences for (W, r) with ε ≤ 1r̄ −1have δ approaching 0.

3.6 Item specific sequences

In previous subsections we presented sequencing policies that together apply in many, though not

all circumstances. This subsection establishes several sequencing policies which apply generally by

utilizing the specific properties of the items at hand. We define three formulations to be applied

in the following order: (1) Hamiltonian path based ILP formulation to find a strong no-blockage

sequence; (2) a type of Hamiltonian path based ILP formulation to find a no-blockage sequence

for identical workers; and (3) TSP based MILP formulations to approximate a robustly efficient

sequence, taking as input some method for computing the contribution of a single cycle to the BI.

3.6.1 Hamiltonian paths

We now link the problem of finding a strong no-blockage sequence for (W, r) to the problem of

finding a Hamiltonian Path (see e.g. Garey and Johnson, 1983) in an appropriately defined graph.

Given (W, r), consider a directed graph where the set of nodes is the set of items W , and the set of

arcs is defined such that an arc connecting item j to item j′ exists if and only if this ordered pair

has strong pairwise no-blockage given r. Then the set of Hamiltonian paths in this graph is exactly

the set of strong no-blockage sequences for (W, r). It follows by Proposition 3.8 that a sequence

s is no-blockage robust efficient for (W, r) if a corresponding Hamiltonian path exists for some r̂

with r̂1r̂2 > r̄. A solution approach based on an ILP formulation is detailed as a special case of the

analysis in Section 3.6.3.

18

Example 3.2. Figure 3.6 demonstrates the Hamiltonian paths approach with an example including

5 items, of which the cumulative workload distribution and density functions are depicted in the

left and right upper graphs, respectively. The four directed graphs shown in the figure correspond

to the work rate ratios r̄ = 0.1, 0.3, 0.5, 0.7, respectively. The sequence s = (3, 2, 4, 5, 1) has strong

no-blockage for (W, r) with each of these four work rate ratios, as it corresponds to a Hamiltonian

path in each of the four directed graphs. Moreover, s is the unique such sequence in the right-most

graph, which corresponds to the highest of the four work rate ratios. Calculation shows that s has

strong no-blockage for all r̄ up to 0.75, thus, by Proposition 3.8, s is a no-blockage robust efficient

sequence for any r̄ < 0.75.

Figure 3.6: 5-item strong no-blockage example; directed graphs for r̄ = 0.1, 0.3, 0.5, 0.7, respectively;no-blockage robust efficient sequence (3,2,4,5,1)

3.6.2 A type of Hamiltonian path based ILP formulation

The following approach may be taken in order to link the problem of finding a no-blockage sequence

with identical workers to a version of the problem of finding a Hamiltonian Path in an appropriately

defined graph. Given a sequencing problem with piecewise linear items having break points X ={yp}Pp=0 and cumulative workloads (aj,p̄)Pp=0 for each item j, extend the directed graph definedin Section 3.6.1 so that each node is a pair (j, yp) consisting of an item and a position for some

p = 0, ..., P of worker 2 at the beginning of a cycle, and an arc connecting (j, yp) to (j′, yp′) exists

if and only if max1≤p̄≤P(aj,p̄ − aj′,p̄

)≤ aj,p and p′ = max{p̄ | aj′,p̄ = aj,P − aj,p̄}. Then any path

initiating at y0 = 0 and traversing each item j exactly once for some p corresponds to a no-blockage

sequence. A solution approach based on an ILP formulation is detailed as a special case of the

analysis in Section 3.6.3. Figure 3.7 demonstrates the directed graph generated for 3 items with

19

representing vectors A1 = [0, 0.5, 0.5, 0.5], A2 = [0.5, 0.5, 0.5, 0.5] and A3 = [0, 0, 0.5, 1], where the

unique feasible path is (1, 0), (2, 1), (3, 12) and the identified no-blockage sequence is (1, 2, 3).

Figure 3.7: Directed graph for a 3-item example with L = 2, P = 4; no-blockage sequence s =(1, 2, 3)

3.6.3 Approximating (ε, δ)-robust efficiency using TSP based MILP formulations

We propose to utilize two MILP, robust optimization formulations of the Traveling Salesman Prob-

lem (TSP) for approximating (ε, δ)-robust efficient sequences for a sequencing problem (W, r) with

workers assigned according to slowest to fastest. Here we describe the main idea of these formula-

tions. For three integers P ≥ 1, R ≥ 1 and Re ≥ 1, where P is the positional approximation level, Ris the work rate approximation level and Re is the entire work rate approximation level, consider the

approximation according to which the BB line is discretized to a finite sequence X ≡ {yp = pP }Pp=0

of additively, equally spaced, increasing positions, and the ε-neighborhood of r̄ ≤ 1 is discretizedto a finite sequence Iε ≡ {r̂ = (r̄(1 + ε)

mR , 1)}Rm=−R of multiplicatively, equally spaced, increasing

work rate ratio values. Relying on a directed graph as described in Section 3.6.2, the first MILP

formulation takes as given an initial lower bound 0 ≤ ε ≤ 1r̄ − 1 on the maximal work rate radiusε∗, and finds a corresponding δ∗ as the robustly minimal TSP cost. Then the second MILP takes

as given the δ∗ computed using the first formulation and approximates the actual ε∗ together with

an (ε∗, δ∗)-robust efficient sequence s∗. The entire work rate approximation level Re is only used

in the second MILP formulation while identifying ε∗. Further details are provided in Appendix C.

The approach described in Section 3.6.1 may be implemented with the first MILP formulation

in the special case of a degenerate interval, ε = 0, in which case the formulation is equivalent to an

ILP, and with the whole line as a single interval, P = 1. In this case, zero cost arcs connecting any

pair of items represent strong pairwise no-blockage, the cost of any feasible tour is an upper bound

to BI(W, r, s) of the corresponding sequence s, any zero cost sequence has strong no-blockage for

(W, r), and there exists a sequence s that has strong no-blockage if and only if δ∗ = 0.

Similarly, the approach described in Section 3.6.2 is the special case of the first MILP formulation

20

in which ε = 0 (thus again it is equivalent to an ILP formulation), W includes piecewise linear

items with common break points given by the set X , and the workers are identical. In this case,the zero cost arcs connecting any pair of items exactly represent cycles with no-blockage, any zero

cost sequence has no-blockage for (W, r), and there exists a sequence with no-blockage if and only

if δ∗ = 0.

Finally, using Proposition B.1 as input for the above two MILP formulations, we next show the

computational results of (ε, δ)-robust efficiency values for randomly sampled sequencing problems,

where each item has a quadratic cumulative workload distribution function that is uniformly and

independently drawn with parameters 0 ≤ aj ≤ 10 and 0 ≤ bj ≤ 2aj . Figure 3.8 provides a scatterplot of 500 instances with J = 2 and J = 8 items under the approximation levels P = 4, R = 10

and Re = 100. Out of the entire sample, the subset of instances resulting with δ ≤ 0.01 were 92.6%when J = 2, which increased to 94.8% when J = 8. Additionally, for all other instances, δ averaged

to 0.039 when J = 2, which decreased to 0.021 when J = 8. Overall, the results demonstrate that

the BI values tend to decrease when the number of items increases.

Figure 3.8: (ε, δ)-robust efficiency scatter plots

4 Concluding remarks

Item heterogeneity in work sharing systems may potentially reduce the resulting throughput due

to blockages. We provide methods to robustly quantify this inefficiency in BB systems and sub-

stantially reduce it by allowing item sequencing. Several sequencing policies are proposed and ex-

plored, ranging from ensuring no-blockage in any sequence through finding specific robust efficient

sequences. In order to achieve simple and effective operations, the policies should be implemented

in increasing order of detail about the items. The simplest policies involve sequencing according to

first-order distributional dominance of items and according to their steady-state hand-off positions.

Such policies require almost no computation and are often sufficient. The most involved policies we

propose are based on MILP formulations of Hamiltonian path and TSP. Our general conclusion is

that due to the useful self-balancing property of BB systems with slowest-to-fastest ordering of the

21

workers, we are able to identify sequencing policies which effectively solve the blockage inefficiency

when the number of items to be sequenced is either small or large.

For clarity of the presentation, our analysis concentrated on two-worker BB systems. Never-

theless, the approach is readily generalized to any number K > 2 of workers without encountering

conceptual complications. In particular, our conclusions would continue to hold without any change

with regard to non-uniform identical items, strong no-blockage (generalized so that each consecu-

tive K items satisfy strong K-wise no-blockage), first-order distributional dominance sequences and

sequences for large sets of items. Universal no-blockage would be obtained using the maximal work-

rate ratio among all adjacent pairs of workers along the line. Identical total workload sequences

would hold provided that the same non-increasing order of steady-state hand-off positions applies

to all workers. Finally, the proposed Hamiltonian path and TSP formulations would continue to

quantify correctly the robust blockage inefficiency.

Another direction in which our approach may be generalized is to strengthen the robustness

consideration to allow for varying work rates across the process cycles. The characterization of

no-blockage and blockage conditions would be adapted accordingly, and the definition of robust

efficient sequences would be modified similarly using a worst-case BI over any combination of work

rate scenarios for each cycle. Our proposed methods for evaluating the robust BI, and the conclusion

that sequencing is effective in reducing blockages would remain intact.

A Appendix: preliminary analysis

In this section we take the first, technical steps in our analysis, providing a necessary and sufficient

condition for no blockage in any cycle, as in Definition 2.4, and for blockage in any cycle.

A useful way of understanding our BI measure of inefficiency is through its numeratorBL(W, r, s),

the work capacity loss due to blockage. First, note that, using Equation (2.1),

BL(W, r, s) =

J−1∑n=1

(r̄[Wsn(1)−Wsn(xn−1(W, r, s))]−Wsn+1(xn(W, r, s))

),

which, since worker 2 is never blocked, may be interpreted as the sum, over all busy cycles for

worker 1, of the difference between the amount of work, r̄[Wsn(1) −Wsn(xn−1(W, r, s))], that canpotentially be accomplished by worker 1 during cycle n, and the amount of work, Wsn+1(xn(W, r, s)),

actually accomplished by worker 1 due to possible blockages during this cycle. In particular, for

cycle n with xn−1(W, r, s) = 0, e.g. n = 1, the summand simplifies to

r̄Wsn(1)−Wsn+1(xn(W, r, s)).

Now, for any item j immediately followed by item j′ and all x ∈ [0, 1], define the cumulativeworkload difference function

dj′,j(x) ≡ r̄Wj(x)−Wj′(x).

22

To interpret this difference, consider some cycle n with xn−1(W, r, s) = 0, where worker 2 holds item

j = sn and worker 1 holds item j′ = sn+1. Suppose first that no blockage occurs for any position of

worker 2 within the interval I = (0, x) for some x ∈ (0, 1]. Then dj′,j(x) is the difference betweenthe work accomplished by worker 1 throughout interval I of worker 2, and the work worker 1 would

accomplish were worker 1 to reach position x together with worker 2. In this case, dj′,j(x) ≤ 0, andinterval I has zero contribution to the work capacity loss due to blockage.

Formally, to study no-blockage sequences, the following lemma is key.

Lemma A.1. Given a sequencing problem (W, r), a sequence s has no-blockage if and only if for

all cycles n = 1, 2, ..., J − 1 and x ∈ [xn−1(W, r, s), 1],

dsn+1,sn(x) ≤ r̄Wsn(xn−1(W, r, s)), (A.1)

or equivalently, for all such n,

max0≤x≤1

dsn+1,sn(x) ≤n−1∑i=1

(−r̄)i+1Wsn−i(1). (A.2)

Note that Lemma A.1 implies that if no blockage occurs throughout cycle n when the starting

position of worker 2 is xn−1(W, r, s), then no blockage would continue to hold throughout the cycle

if the starting position was larger. Consequently, the zero starting position of worker 2 is the worst

case in terms of blockage. This implies that the first cycle in a sequence, for which we always

assume x0(W, r, s) = 0, is in particular a blockage worst case.

In cycle n of a no-blockage sequence s, the position of hand-off n satisfies the recursive relation

xn(W, r, s) = max{x |Wsn+1(x) = r̄(Wsn(1)−Wsn(xn−1(W, r, s)))}. (A.3)

In this recursion, the max is relevant when the condition Wsn+1(x) = r̄(Wsn(1)−Wsn(xn−1(W, r, s)))holds over a closed interval of x values, which happens when Wsn+1(x) is constant there because

the workload density is zero.

Suppose now that worker 1 is blocked by worker 2 for the entire position interval I. Then

dj′,j(x) is the difference between the amount of work, r̄Wj(x), that can potentially be accomplished

by worker 1 within interval I, and the amount of work, Wj′(x), actually accomplished by worker 1

due to this blockage. In this case, dj′,j(x) > 0, and is exactly the positive contribution of interval

I to the work capacity loss due to blockage.

To understand the possibility of blockages in general, we may consider intervals I = (y1, y2) ⊆[xn−1(W, r, s), 1] for xn−1(W, r, s) ≥ 0, where y1 is the largest joint position for both workers in theinterval [xn−1(W, r, s), y2] if one exists otherwise y1 = xn−1(W, r, s), and let y0 = y1 in the former

case and y0 = 0 in the latter. Then, extending the function dj′,j(x) to such intervals I by defining

dj′,j(I) = dj′,j(y2)−dj′,j(y1), no blockage occurs for any position of worker 2 within the interval I ifand only if dj′,j(I

′) ≤ r̄[Wj(y1)−Wj(y0)] for all sub-intervals I ′ = (y1, x) ⊆ I, in which case interval

23

I has zero contribution to the work capacity loss due to blockage. Additionally, blockage occurs

throughout I if and only if dj′,j(I) > r̄[Wj(y1)−Wj(y0)] for all such sub-intervals I ′, in which casedj′,j(I) is exactly the contribution of the interval I to the work capacity loss due to blockage. In the

latter case, dj′,j(I) is the difference between the amount of work, r̄(Wj(y2)−Wj(y1)), that can bepotentially accomplished by worker 1 during interval I, and the amount of work, Wj′(y2)−Wj′(y1),actually accomplished by worker 1 due to this blockage. It follows that BL(W, r, s), the work

capacity loss due to blockage and the numerator of BI(W, r, s), is equal to the sum of dj′,j(I) over

all disjoint blockage intervals in all cycles.

Formally, to study blockage sequences, the following lemma is key.

Lemma A.2. Given a sequencing problem (W, r), a sequence s and a cycle n = 1, 2, ..., J − 1,worker 1 is blocked by worker 2 at position x ∈ (xn−1(W, r, s), 1) if and only if

dsn+1,sn(x)− dsn+1,sn(l1(x)) > r̄[Wsn(l2(x))−Wsn(l1(x))] (A.4)

and

d′sn+1,sn(x

−) > 0, (A.5)

where d′sn+1,sn(x

−) is the left derivative of dsn+1,sn(x) at x, and either l1(x) = l2(x) is the joint

position of the two workers during cycle n at the end of the previous blockage interval, or, in case

there is no such previous blockage interval, l1(x) = 0 and l2(x) = xn−1(W, r, s) are the initial

positions of worker 1 and worker 2, respectively, at the beginning of cycle n.

Example A.1. Consider a BB sequencing problem (W, r), with identical work rates that are equal

to 1 for the two workers, i.e. r = (1, 1) and r̄ = r1r2 = 1, and with two piecewise linear items

to be processed, W = (Wj)j=1,2, with joint break points X = {13 ,23 , 1} and representing vectors

A1 = [13 ,

12 , 1] and A2 = [

16 ,

12 ,

23 ]. Consider the sequence s = {1, 2}, thus j = 1 and j

′ = 2. The

following table presents the functions used in Lemma A.2 in order to characterize the blockage

obtained. Since the first cycle ends with blockage, the entire work is completed in a single cycle.

As shown in the table, several intervals are distinguished, depending on whether Conditions (A.4)

or (A.5) are met, which in turn determine two blockage intervals during the single cycle, one interval

at the beginning and the other at the end of the cycle. The contribution of each blockage interval

I to BI(W, r, s) isd1,2(I)TW (W )=

1/61+2/3 =

110 .

24

interval x ∈ (0, 13) x ∈ [13 ,

23) x ∈ [

23 ,

56 ] x ∈ (

56 , 1]

Wj(x) xx2 +

16

3x2 −

12

3x2 −

12

Wj′(x)x2 x−

16

x2 +

16

x2 +

16

dj′,j(x)x2 −

x2 +

13 x−

23 x−

23

l1(x) = l2(x) 013

13

13

Cond. (A.4) True False False True

Cond. (A.5) True False True True

Blockage True False False True

Contribution to BI 110 0 0110

The following example summarizes the preliminary analysis of this section.

Example A.2. Consider a BB sequencing problem (W, r), with identical work rates that are equal

to 1 for the two workers, i.e. r = (1, 1) and r̄ = r1r2 = 1, and with three items to be processed,

W = (Wj)j=1,2,3, such that

W1(x) = x(

1− x2

), W2(x) = x

(1 + x

2

), W3(x) =

x

2,

as shown in the left graph of Figure A.1.

Figure A.1: 3-item blockage example

Suppose that the chosen processing sequence for these items is 1 → 2 → 3, i.e. s = (1, 2, 3).Then, in the first cycle, worker 2 processes item 1 and worker 1 processes item 2, and both start

at position x = 0, i.e. x0(W, r, s) = 0. Cycle 1 has cycle time of CT1 =W1(1)−W1(0)

r2= 12 . For this

cycle, the cumulative workload difference function d2,1(x) = r̄W1(x)−W2(x) = x(12 − x), as shownin Figure A.1, is concave and achieves its maximal value of dmax2,1 =

116 > 0 at x

max = 14 . Therefore

worker 1 has their first blockage interval I1 = (0, xmax) starting immediately at the beginning of

the cycle and ending at position xmax, with no blockage occurring from that point until the end

of the cycle, and the contribution of this blockage interval to the work capacity loss BL(W, r, s) is

∆1 = d2,1(I1) = dmax2,1 − d2,1(0) = 116 (the details for this calculation are given in Proposition B.1

25

in the Appendix). Worker 1 ends the cycle at the position x for which their cumulative workload

distribution function W2(x) is equal to ω1 ≡ r̄W1(1)− dmax2,1 = 716 , i.e. x =3√

2−24 ≈ 0.561. So this

is the hand-off position x1(W, r, s) at which worker 2 takes item 2 from worker 1, worker 1 takes

item 3 at position x = 0, and cycle 2 initiates. It follows that cycle 2 has cycle time of CT2 =W2(1)−W2(x1(W,r,s))

r2= W2(1)−ω1r2 =

916 . For this cycle, the cumulative workload difference function

d3,2(x) = r̄W2(x) −W3(x) = x2

2 is convex with d3,2(1) > r̄ω1 =716 and d3,2(

√78) =

716 , therefore

worker 1 has their second blockage interval I2 = (√

78 , 1) starting from position x =

√78 ≈ 0.935

and ending at position x = 1 at the end of the cycle, thus the contribution of this blockage interval

to the work capacity loss BL(W, r, s) is ∆2 = d3,2(I2) = d3,2(1)−d3,2(√

78) =

116 (again, this follows

from Proposition B.1). Consequently the second hand-off position is x2(W, r, s) = 1, at which

point both workers complete processing their respective items. Cycle 3 therefore has cycle time

CT3 =W3(1)−W3(1)

1 = 0.

To summarize the example, the makespan for this sequence is∑3

n=1CTn = 1116 , the total

workload of the three items is TW (W ) =∑3

j=1Wj(1) = 2, and the makespan work capacity is

MC(W, r, s) = (12 +916)(1 + 1) + 0 · 1 = 2

18 , therefore the work capacity loss due to blockage is

BL(W, r, s) = MC(W, r, s) − TW (W ) = 18 . Note that the two blockage intervals I1, I2 contributeequally to BL(W, r, s). Finally, the blockage inefficiency for this sequence is BI = 116 .

B Appendix: Auxiliary results

The following corollary is a direct consequence of Lemma A.1.

Corollary B.1. A sequence s has strong no-blockage for (W, r) if and only if for all cycles n =

1, 2, ..., J − 1,max

0≤x≤1dsn+1,sn(x) ≤ 0.

The following result provides an exact analytic expression for the work capacity loss due to

blockage, thus for the BI, and for the hand-off positions for each cycle n.

Proposition B.1. For a sequencing problem (W, r) with quadratic Wj and a sequence s, the work

capacity loss due to blockage BL(W, r, s) =∑J−1

n=1 ∆n, where ∆n is recursively given, together with

the workload ωn accomplished by worker 1 during cycle n = 1, 2, ..., J − 1, by

(∆n, ωn) =

(dmaxj′,j − r̄ωn−1, r̄Wj(1)− dmaxj′,j ), 0 < d

′j′,j(0) < −d

′′j′,j(0), d

maxj′,j > r̄ωn−1

(dj′,j(1)− r̄ωn−1,Wj′(1)), ¬(0 < d′j′,j(0) < −d

′′j′,j(0), d

maxj′,j > r̄ωn−1), dj′,j(1) > r̄ωn−1

(0, r̄(Wj(1)− ωn−1)), otherwise,

for j = sn, j′ = sn+1, d

maxj′,j ≡ dj′,j(

d′j′,j(0)

−d′′j′,j(0)

) and ω0 = 0.

Additionally, the hand-off positions xn(W, r, s) satisfy ωn = Wj′ [xn(W, r, s)], and whenever

∆n > 0, the unique blockage interval In = (yn1, yn2) during cycle n satisfies ∆n = dj′,j(In) =

26

dj′,j(yn2)− dj′,j(yn1) for yn1 satisfying dj′,j(yn1) = r̄ωn−1 and

yn2 =

d′j′,j(0)

−d′′j′,j(0)

, 0 < d′j′,j(0) < −d

′′j′,j(0), d

maxj′,j > r̄ωn−1

1, otherwise.

C Appendix: TSP based MILP formulations

In this appendix we describe the TSP based MILP formulations proposed in Section 3.6.3. Start

with the first MILP formulation. Consider a multi-cost TSP complete directed graph, where the

set of nodes is the set of item-position pairs (W ∪ {W0,WJ+1})×X , including two dummy items,a source item 0 and a sink item J + 1. Take as given an initial lower bound ε ≥ 0 on ε∗. Foreach r̂ ∈ Iε, letting Ŵ = ( Wj(yp)r̂1Wj(1)Wj ,Wj ,Wj′) for 1 ≤ j, j

′ ≤ J , j 6= j′ and ŝ = (1, 2, 3), the

r̂-cost, cr̂jpj′p′ , assigned to an arc connecting node (j, yp) to node (j′, yp′) is equal to

BL(Ŵ ,r̂,ŝ)TW (W ) if

x2(Ŵ , r̂, ŝ) ∈ [yp′ , yp′ + 1P ) and yp′ < 1 or x2(Ŵ , r̂, ŝ) = yp′ = 1; for any 1 ≤ j, j′ ≤ J and any p, p′,

this cost is equal to zero from (0, 0) to (j′, 0) or from (j, yp) to (J + 1, yp′) or from (J + 1, yp) to

(0, yp′), and is equal to +∞ from (0, yp) to (j′, yp′) with min{yp, yp′} > 0 or from (j, yp) to (0, yp′)or from (J+1, yp) to (j

′, yp′) or from (0, yp) to (J+1, yp′) or for any 0 ≤ j = j′ ≤ J+1. The valuesfor BL(Ŵ , r̂, ŝ) may be obtained using closed form formulas such as those provided in Proposition

B.1. The first MILP formulation is defined as follows.

δ∗ = min δ

s.t.∑

j,p,j′,p′

cr̂jpj′p′zr̂jpj′p′ ≤ δ , ∀r̂ ∈ Iε∑p,p′

(zr̂jpj′p′ − zr̂′jpj′p′

)= 0 , ∀r̂, r̂′ ∈ Iε, ∀j, j′∑

p,j′,p′

zr̂jpj′p′ = 1 , ∀r̂ ∈ Iε, ∀j∑j,p,p′

zr̂jpj′p′ = 1 , ∀r̂ ∈ Iε, ∀j′∑j′,p′

(zr̂jpj′p′ − zr̂j′p′jp

)= 0 , ∀r̂ ∈ Iε,∀j, p

ur̂j − ur̂j′ + (J + 2)∑p,p′

zr̂jpj′p′ ≤ J + 1 , ∀r̂ ∈ Iε, j 6= 0, j′ 6= 0

ur̂j ∈ Z , ∀r̂ ∈ Iε, ∀j 6= 0

zr̂jpj′p′ ∈ {0, 1} , ∀r̂ ∈ Iε,∀j, p, j′, p′.

In this formulation, there are binary variables zr̂jpj′p′ that equal 1 if and only if the TSP tour

for work rate ratio r̂ includes the arc from (j, yp) to (j′, yp′). The objective function and the first

constraint together ensure that δ is a worst-case BI among all work ratios r̂ in Iε, the second

constraint requires that the TSP tours of all these r̂ imply the same sequence s, the next three

27

constraints ensure traversing each item j exactly once for some p, and the remaining constraints

eliminate sub-tours based on the Miller-Tucker-Zemlin (Miller et al. 1960) approach.

Next consider the second MILP formulation. Here, the entire work rate ratio interval [0, 1] is

discretized to a finite number of values, I ≡ Iε ∪ {r̂ = ( mRe , 1)}Rem=1. The value of δ

∗, which was

determined by the first MILP formulation, is given as a parameter for this formulation:

ε∗ = max ε

s.t. ε ≤ max{ r̂r̄,r̄

r̂} − 1 +Byr̂ , ∀r̂ ∈ I

yr̂ ≤ 1−

∑j,p,j′,p′

cr̂jpj′p′zr̂jpj′p′ − δ∗ , ∀r̂ ∈ I

∑p,p′

(zr̂jpj′p′ − zr̂′jpj′p′

)= 0 , ∀r̂, r̂′ ∈ I, ∀j, j′

∑p,j′,p′

zr̂jpj′p′ = 1 , ∀r̂ ∈ I, ∀j∑j,p,p′

zr̂jpj′p′ = 1 , ∀r̂ ∈ I, ∀j′∑j′,p′

(zr̂jpj′p′ − zr̂j′p′jp

)= 0 , ∀r̂ ∈ I, ∀j, p

ur̂j − ur̂j′ + (J + 2)∑p,p′

zr̂jpj′p′ ≤ J + 1 , ∀r̂ ∈ I, j 6= 0, j′ 6= 0

ur̂j ∈ Z , ∀r̂ ∈ I, ∀j 6= 0

zr̂jpj′p′ ∈ {0, 1} , ∀r̂ ∈ I, ∀j, p, j′, p′

yr̂ ∈ {0, 1} , ∀r̂ ∈ I,

where B is a sufficiently large number. Compared to the first MILP formulation there are additional

binary variables yr̂ that equal 1 if and only if there exists a TSP tour for work rate ratio r̂ with

total cost at most δ∗. Note that the parenthesized difference in the right-hand side of the second

constraint is at most 1 by Proposition 2.1 because r̂1 ≤ 1 for all r̂ ∈ I. Examining each r̂ ∈ I,the objective function and the first two constraints together achieve the maximal ε∗ for which

the corresponding interval I(r̄, ε∗) includes all r̂ with total cost at most δ∗, where the remainingvariables and constraints are the same as in the first MILP formulation but with I instead of Iε.

D Appendix: proofs

Proof of Proposition 2.1. By Equation (2.1), since CTn(W, r, s) ≤ Wsn (1)r2 for any cycle n, Equation(2.3) implies

MC(W, r, s) ≤J∑n=1

Wsn(1)

r2(r1 + r2) = (r̄ + 1)TW (W ),

28

thus BI(W, r, s) ≤ r̄. To see that this bound is tight, consider the two-item sequencing problemwhere W1(x) has W1(1) > 0 and W2(x) = 0 for x ∈ [0, 1] and the chosen processing sequenceis 1 → 2, i.e. s = (1, 2). Then, in the first cycle worker 2 processes item 1 and determinesCT1(W, r, s) =

W1(1)−0r2

= W1(1)r2 , worker 1 processes item 2 and, being blocked throughout the

cycle, ends at x1(W, r, s) = 1, and the second cycle has CT2(W, r, s) =0−0r2

= 0. Therefore,

BI(W, r, s) =

W1(1)r2· (r1 + r2) + 0 · r2W1(1) + 0

− 1 = r̄.

Proof of Proposition 2.2. Consider three items to be processed, W = (Wj)j=1,2,3, such that

W1(x) = 2.2x2, W2(x) = x (7− 3.5x) , W3(x) = x (2− 0.6x) .

For this system with the sequence s = (1, 2, 3), the correspond BI as a function of r̄ is

−75

+7

2r̄ − 11

5r̄2 +

8

5(−6 + 35r̄)

when positive, i.e. approximately for 0.578 ≤ r̄ ≤ 0.947, and is equal to zero otherwise. This BI isdepicted in Figure D.1. Any r̄ in this interval has positive BI, whereas higher values of r̄ generate

zero BI.

Figure D.1: BI decreasing to zero

Proof of Lemma A.1. Consider any cycle n = 1, 2, ..., J−1, and let j = sn and j′ = sn+1. Condition(A.1) is equivalent to r̄ (Wj(x)−Wj(xn−1(W, r, s))) ≤Wj′(x), which means that while the workersare moving to position x ∈ [xn−1(W, r, s), 1] from their respective starting positions, namely 0 forworker 1 and xn−1(W, r, s) for worker 2, the amount of work, r̄ (Wj(x)−Wj(xn−1(W, r, s))), that

29

can be potentially accomplished by worker 1 is at most the required amount of work, Wj′(x), i.e.,

worker 1 is not blocked by worker 2 up to position x. Since this holds for all x ∈ [xn−1(W, r, s), 1],no blockage occurs throughout the cycle.

To verify Condition (A.2) note that by definition of the difference function dj′,j and monotonicity

of Wj together with Wj′ ≥ 0, if x < xn−1(W, r, s) then dj′,j(x) ≤ r̄Wj(x) ≤ r̄Wj(xn−1(W, r, s)).Thus the no-blockage Condition (A.1) is equivalent to max0≤x≤1 dj′,j(x) ≤ r̄Wj(xn−1(W, r, s)).The conclusion follows from recursive substitution of Equation (A.3) into the right-hand side of

Condition (A.1).

Proof of Lemma A.2. Let j = sn and j′ = sn+1. Condition (A.4) is equivalent to r̄ (Wj(x)−Wj(l2(x))) >

Wj′(x)−Wj′(l1(x)), which means that while the workers are moving to position x ∈ (xn−1(W, r, s), 1)from their respective starting positions, lk(x) for worker k, the amount of work, r̄ (Wj(x)−Wj(l2(x))),that can be potentially accomplished by worker 1 is larger than the required amount of work,

Wj′(x) − Wj′(l1(x)). This is necessary for blockage to occur at position x, as otherwise worker1 will not be able to reach the position of worker 2. Adding Condition (A.5), both conditions

are together sufficient for blockage at position x because Condition (A.5) means that this positive

work difference is increasing also at position x. Note that even if Condition (A.4) holds, violation of

Condition (A.5) implies that blockage does not occur at x. This is the case because Condition (A.4)

implies that the two workers are positioned at x simultaneously, but violation of Condition (A.5)

implies that such blockage will no longer hold when the two workers infinitesimally proceed with

their work. This shows that the two conditions are together necessary and sufficient for blockage

to hold at position x.

Proof of Proposition 3.1. Let s be a no-blockage sequence for (W, r), and let j = sn, j′ = sn+1 and

R ={r̂1r̂2| max

1≤n≤J−1

(max

xn−1(W,r̂,s) 0,

I(r̄, ε) ⊆ R, and by Lemma A.1, for any r̂ ∈ I(r̄, ε), s is a no-blockage sequence and BI(W, r̂, s) =0. Consequently, BIε(W, r, s) = 0. Therefore ε̄ = maxŝ,ε̂|ε̂≥0,BIε̂(W,r,ŝ)=0 ε̂ exists and is positive.

Letting s̄ ∈ arg minŝBIε̄(W, r, ŝ), s̄ is (ε̄, 0)-robust efficient, thus s̄ is no-blockage robust efficientfor (W, r).

30

Proof of Proposition 3.2. Consider any sequence s. Since for each cycle n = 1, 2, ..., J − 1 andj = sn and j

′ = sn+1,

max0≤x≤1

dj′,j(x) = max0≤x≤1

(r̄W̄j(x)− W̄j′(x)

)≤ max

0≤x≤1

(W̄ (x)− W̄ (x)

)= 0

≤ r̄W̄ (xn−1(W̄ , r, s)),

the conclusion follows from Lemma A.1.

Proof of Proposition 3.3. Define the function f : [0, W̄ (1)]→ [0, W̄ (1)] by f(y) = r̄(W̄ (1)− y). ByEquation (A.3), the function f maps the cumulative workload at hand-off n− 1 to the cumulativeworkload at hand-off n, i.e. W̄ [xn(W̄ , r, s)] = f [W̄ [xn−1(W̄ , r, s)]]. Note that for r̄ < 1, f(y) is

a contraction mapping because |f(y1)− f(y2)| = r̄ |y1 − y2|. Applying a fixed-point theorem, thehand-off position xn(W̄ , r, s) converges to the steady-state hand-off position x

∗ as the number of

items J approaches infinity, and the convergence rate is exponential.

Proof of Proposition 3.4. Fix a sequence s. To see (a), note that for any pair of items processed in

cycle n with item types Wj ,Wj′ ∈ Hr̄,W 0 for j = sn and j′ = sn+1,

maxxn−1(W,r,s)≤x≤1

dj′,j(x) = maxxn−1(W,r,s)≤x≤1

(r̄Wj(x)−Wj′(x)

)≤ max

xn−1(W,r,s)≤x≤1

(r̄W 0(x)− r̄W 0(x)

)= 0 ≤ r̄Wj(xn−1(W, r, s)),

which implies by Lemma A.1 that no blockage occurs during cycle n.

To see (b), first note that for any item type W̄ /∈ Hr̄,W 0 , there exists 0 < x′ ≤ 1 such thateither (1) W̄ (x′) > W 0(x′), or (2) W̄ (x′) < r̄W 0(x′). In case (1), considering W = (W̄ , r̄W 0) with

j = s1 = 1 and j′ = s2 = 2,

max0≤x≤1

dj′,j(x) ≥ dj′,j(x′) = r̄W̄ (x′)− r̄W 0(x′) > r̄W 0(x′)− r̄W 0(x′) = 0,

thus, since x0(W, r, s) = 0, by Lemma A.1, there is blockage during the first cycle. In case (2),

considering W = (W 0, W̄ ) with j = s1 = 1 and j′ = s2 = 2,

max0≤x≤1

dj′,j(x) ≥ dj′,j(x′) = r̄W 0(x′)− W̄ (x′) > r̄W 0(x′)− r̄W 0(x′) = 0,

thus again, by Lemma A.1, there is blockage during the first cycle.

31

Proof of Proposition 3.5. The direction that if the algorithm concludes the answer yes then it is in

fact yes follows directly from Proposition 3.4. For the other direction, suppose that the algorithm

concludes the answer no, i.e. there exists an item Wj′ in the set W and position x′ ∈ [0, 1] such that

Wj′(x′) < r̄W 0(x′). Let j be an item in W such that Wj(x

′) = W 0(x′), and consider a sequence s

in which the first two items are (Wj ,Wj′). Then

max0≤x≤1

dj′,j(x) ≥ dj′,j(x′) = r̄Wj(x′)−Wj′(x′) > r̄W 0(x′)− r̄W 0(x′) = 0,

thus, by Lemma A.1, there is blockage during the first cycle. By Definition 3.2, W cannot in fact

be a subset of some maximal universal no-blockage set because this would imply that any sequence

s is a no-blockage sequence for (W, r). This proves the other direction.

Proof of Proposition 3.6. Note that all cycles n = 1, 2, ..., J − 1 have the same cycle time of CTn =a−Wj(x∗j )

r2=

a−a r̄1+r̄

r2= ar1+r2 , where j = s

∗n. Any such cycle n starts at the steady-state hand-

off position of item j, x∗j , and ends at the steady-state hand-off position of the next item in the

sequence, x∗j′ , where j′ = s∗n+1. The fact that the items are sequenced in a decreasing order of x

∗j

indeed ensures that no blockage occurs in any cycle.

Proof of Proposition 3.7. If x0(W, r, s∗) = 0 then sequence s∗ may have blockages. Note, however,

that due to the tendency of hand-off positions of identical items towards their respective steady-

state positions when r̄ < 1, such blockages become negligible when the number of types is finite and

the number of items increases. Additionally, when a sequence s is such that the hand-off positions

xn(W, r, s) are decreasing in n, then s is a no-blockage sequence. We may apply this sufficient

condition to s∗. In a no-blockage sequence s, by Equation (A.3), the amount of work accomplished

by worker 1 at hand-off n is Wsn+1 [xn(W, r, s)] = ar̄

1+r̄ (1− (−r̄)n). Thus Wsn+1 [xn(W, r, s)] <

Wsn+1 [x∗sn+1 ] for even n, and the inequality is reversed for odd n. Applied to s

∗, for even n,

xn(W, r, s∗) < x∗s∗n+1

≤ x∗s∗n < xn−1(W, r, s∗), i.e. the hand-off position n is always strictly smaller

than that of the previous cycle, thus guaranteeing no blockage throughout the cycle. Note however

that this is not necessarily guaranteed for odd n. It follows that the condition stated in the

proposition is sufficient for s∗ to be a no-blockage sequence since hand-off n+ 1 is not greater than

hand-off n.

Proof of Proposition 3.8. This follows by Corollary B.1 since for every cycle n and all x and j = sn

and j′ = sn+1, dj′,j(x) is increasing in the work rate ratio r̄, so if the ordered pair Wj followed by

Wj′ has strong pairwise no-blockage given some r̄ then this will hold also for any smaller r̄.

32

Proof of Proposition 3.9. Suppose that s is a first-order distributional dominance sequence. Since

for each cycle n = 1, 2, ..., J − 1 and j = sn and j′ = sn+1,

max0≤x≤1

dj′,j(x) = max0≤x≤1

(r̄Wj(x)−Wj′(x)

)≤ max

0≤x≤1

(Wj(x)−Wj′(x)

)≤ 0,

the conclusion follows from Corollary B.1. That the converse holds when r̄ = 1 follows from

Definition 3.5.

Proof of Lemma 3.1. Consider any cycle n = 1, 2, ..., J − 1, and let j = sn and j′ = sn+1. As alinear combination of Wj(x) and Wj′(x), dj′,j(x) is also piecewise linear with break points in X .Since the maximum of a piecewise linear function is attained at one of the break points,

max0≤x≤1

dj′,j(x) = ma