Process Flexibility: Design, Evaluation and Applications
Mable Chou∗
Chung-Piaw Teo
†
Huan Zheng
‡
This version: August 2008
Abstract
One of the most effective ways to minimize supply/demand mismatch cost, with little
increase in operational cost, is to deploy valuable resources in a flexible and timely manner
to meet the realized demand. This notion of flexible processes has significantly changed the
operations in many manufacturing and service companies. For example, flexible production
system is now commonly used by automobile manufacturers, and work force cross-training
system is by now a common practice in many service industries. However, there is a tradeoff between the level of flexibility available in the system and the associated complexity
and operational cost. The challenge is to have the “right” level of flexibility to capture
the bulk of the benefits from a full flexibility system, while controlling for the increase in
implementation cost.
This paper reviews the latest development on the subject of process flexibility in the
past decade. In particular, we focus on the phenomenon, often observed in practice, that a
slight increase in process flexibility can reap a significant amount of improvement in system
performance. This review explores the issues in three perspectives: design, evaluation and
applications. We also discuss how the process flexibility concept has been deployed in several
manufacturing and service systems.
∗
Department of Decision Sciences, NUS Business School, National University of Singapore
Department of Decision Sciences, NUS Business School, National University of Singapore
‡
Management Science Department, Antai College of Economics and Management, Shanghai Jiao Tong Uni†
versity.
1
1
Introduction
Selling fruit juices in the canteen of the NUS Business School is a daunting task. The demands
for the different types of fruit juice are highly unpredictable, depending on the weather and
also the academic calendar of the student population. The problem is exasperated by the fact
that the supplies (i.e. fresh fruits) are perishable, and any leftovers often have to be discarded
at the end of the day. The stall owner, however, has created an ingenious product to mitigate
this problem. This product, called “surprise”, is a random mixture of fruit juices, concocted
by the owner, and tailored to the tastes and likings of the student population. You never know
what you will get when you order a “surprise” from the fruit stall, but alas, this element of
surprise has turned the product into a best-seller, and it is currently the most popular drink
in the stall. The implication for operational efficiency is also clear - the stall owner can now
cleverly re-shape the demand for the different fruits, and can deploy the available resources on
hand to meet the unpredictable demand in a more efficient manner. This is a clear win-win
design, both from a marketing and operational perspective.
Demand shaping, however, may not be feasible in all industries. In the event that customer
has specific requirements and will not switch to substitute products, some companies have
resorted to the use of a flexible product which can be customized to specific use by the customers.
Figure 1 shows two examples of flexible product - the first example shows a table that can be
customized to varying heights, depending on the needs of the users. The second example shows
the design of a flexible book shelf, where the height of the shelving can be customized by the
users depending on their needs. Note that additional features have to be built into the product
itself to allow for flexible deployment.
The main disadvantage of a flexible product is the associated increase in production cost. To
cope with the uncertain demand, the next best thing one can hope for is to exploit commonality
in parts and service requirements to make effective use of the available resources. To facilitate
rapid conversion of flexible resources and supplies to meet demands for end products, it is
necessary for the company to put a flexible product/service delivery system in place to tap into
the power of a flexible system. This is the approach adopted by several companies in dealing
with their packaging problems. For example, several companies based in Singapore have opted
to use flexible packages to economize on the shipping cost and the need to reduce wastage
in packaging. In the past, these companies used standard boxes to ship out their customers’
2
Figure 1: Examples of flexible product.
orders. However, the boxes were often less than half filled, even though the shipping charges
were computed based on the volumetric weight1 of the standard boxes used. To save on the
logistics cost, these companies have pioneered the use of flexible boxes - where the shape can be
customized based on the items packed into the boxes. Figure 2 shows an example of a flexible
box used by a Singapore company.
Figure 2: Example of a flexible package. This box can be customized to 4 different heights, by
cutting along the edges, up til the grooves as shown on the side of the box.
This strategy has also been adopted by transport authority in several countries to deal with
1
An industry jargon to convert volume into shipping weight for price calculation
3
congestion problem. In the early 70s, the New Jersey Port Authority allowed the morning
eastbound traffic to use a lane on the westbound direction of the Lincoln Tunnel, allowing each
commuter in the eastbound traffic to save up to 20 minutes every weekday morning. This little
added flexibility in the lane direction allowed the transport authority to reap an annual savings
of close to 4 million, based on an estimate of productivity value of $2.82 per hour per worker.
(Olcott, O. (1973)). This concept is still in use today in many cities, but often with the lane
direction reversal controlled by movable barrier systems in order to prevent head on collisions.
Figure 3: Eastbound buses operating on a westbound lane in the morning peak hour in Lincoln
Tunnel, New Jersey, in the early 70s. Source: Olcott, O. (1973)
The phrase “process flexibility” can be broadly defined as “the ease of changing the systems
requirements with a relatively small increase in complexity (and rework).” This is part of
the broader concept of flexibility in service and manufacturing, a rapidly developing area in
the academic and practitioner community over the last few decades. We refer the readers to
the comprehensive article by Buzacott and Mandelbaum (2008) for an historical overview of
the area, and the accompanying insights and future challenges. They presented three different
ways of thinking about flexibility: Prior flexibility, State flexibility, and Action flexibility. Prior
flexibility relates to the design of flexibility into the system by “increasing the variety of initial
actions or decisions we can make,” and state flexibility relates to the design of flexibility into
the system by increasing the ability to “cope with uncontrollable changes and uncertainty in
the environment by trying to be good under any environmental outcome.” Last but not least,
4
action flexibility relates to the ability to respond to “changes and uncertainty that are revealed
over time by taking effective recourse action.” Viewed in this framework, process flexibility
concerns the incorporation of the right level of state flexibility into the system, by accounting
for the level of action flexibility and its impact on cost and performance of the manufacturing
and service system.
Process flexibility, as a key concept to quickly respond to demand/supply uncertainties with
little cost, has already changed the operational processes in several manufacturing and service
industries. The automobile industry, for instance, has moved away from using focused plants
(where one plant produces essentially one product) to using modern flexible plants (where one
plant produces several products). The Ford Motor Company, for instance, invested $485 million
in two Canadian engine plants to renovate and retool them with a flexible system. It has also
launched a plan to equip most of its 30-odd engine and transmission plants all over the world
with flexible systems.
“...‘The initial investment is slightly higher, but long-term costs are lower in multiplies,’said
Chris Bolen, manager of Ford’s Windsor engine plant, which uses the flexible system to
machine new three-valve-per-cylinder heads for Ford’s 5.4-liter V8 engine... Ford says the
system will help it meet changes in demand. ‘If our business was hit by a significant down
sizing from V8s to V6s or V6s to (four-cylinder engines) or diesels in North America, we’ll
be able to react to that without years of turnaround,’ said Kevin Bennett, Ford director of
power train manufacturing. ’It’s essential we be able to react to the market more rapidly
than in the past.’... ”
— Mark Phelan, “Ford Speeds Changeovers in Engine Production”
Knight Ridder Tribune Business News. Washington: Nov 6, 2002.
A survey of North-American automobile industry conducted in 2004 shows that the plants
of major automobile manufacturers, such as Ford and General Motor, are more flexible than
those 20 years ago (Van Biesebroeck (2004); see also Boudette (2006)). The survey shows
that these flexible plants can produce much more types of cars to meet the rapidly changing
customer demands while their capacities do not change very much. Suh et al. (2004) contains
a case study on how flexibility can be built into the automobile business. The proposal is to
have dimensional flexibility in the floor plan of the underbody of the vehicle platform. This
can be achieved in various ways (trimming the floor plan for long wheelbase vehicles to meet
5
the requirement for short wheelbase vehicles, or to weld an extension piece to the floor plan for
short wheelbase vehicles to accommodate the long wheelbase vehicles).
The ability to respond quickly to changes in the environment is also important in troops
deployment in the military. In military tactics, the reserve force which the commander directly
controls is akin to a flexible server: it has no specific task initially in the battle plan, but can
be deployed in the most effective way, depending on how the battle evolves on the ground. Of
course, the effectiveness of the reserve force depends on the level of state and action flexibility it
can be deployed in the battlefield. Unfortunately, the flexible force deployment plan comes with
more battle preparation and movement co-ordination on the ground. The daunting task of the
commander is thus to come out with a battle plan which can be executed on the ground, and
flexible enough to adapt to changes in the environment. The ancient Chinese had apparently
mastered the art of flexibility in troops deployment. Folklore has it that the ancient “eight
elements battle formation”, a battle array formed by eight fighting units, has the ability to
change formation so quickly, for example, from attack to defense formation with fighting units
enforcing each other that the enemy “could not see the beginning from the end of the formation.”
For ease of command and control, there could only be limited ways to re-deploy the fighting
units. However, by co-ordinating their actions together, the formation is able to anticipate and
react to a wide range of possible enemy’s maneuver. Finding the structure and logic behind
the flexible deployment tactics, and thus uncovering the secrets of this ancient innovation, will
be an interesting challenge for the research community. Figure 4 depicts an artist impression
of the battle formation used in ancient warfare.
We supplement the review by Buzacott and Mandelbaum (2008) by narrowing our discussion
to the subject of process flexibility. In Section 2, we present a basic model for the process
flexibility problem, and discuss recent results obtained on the model for a variety of performance
measurements (average case, worst case etc.). We review in Section 3 the techniques that can
be used to design a sparse and yet efficient process structure, and discuss in Section 4 indices
created to measure and evaluate the performance of such a structure. We conclude the paper
with a list of recent applications exhibiting a similar theme - that a limited amount of flexibility,
properly incorporated into the system, can reap significant benefits. We do not, however, touch
on the related issue of how the capacities in the system could be configured. For this, we refer
the readers to the works in Van Mieghem (1998), Bish and Wang (2004) and Bish et al. (2005).
6
Figure 4: A depiction of battle formation in ancient Chinese history
For earlier surveys on related work on manufacturing flexibility, see Sethi and Sethi (1990) and
Shi and Daniels (2003). While we try to be comprehensive in our review, it is inevitable that
we may be unaware of other important contributions. The fault is entirely ours. Our goal in
this paper is to present an overview of some of the recent theoretical results obtained for this
class of problems, scattered over a series of papers. However, we have also included some new
results and applications that have not appeared elsewhere. For example, the expansion index,
obtained from the insight that graph connectivity is a good surrogate for the concept of process
flexibility, is described and presented in this survey for the first time. Furthermore, we applied
the theoretical results developed in earlier papers to study multi-stage supply chain and troops
deployment problems, and obtain several new insights and numerical results for these problems.
2
Models for Process Flexibility
We use a bipartite graph to represent the flexibility structure. On the left is a set A of n product
nodes while on the right is a set B of m facility/plant nodes. A link connecting product node
i to facility node j means that facility j is endowed with the capability to produce product i.
Let G ⊆ A × B denote the set of all such links; that is, the edge set of the bipartite graph.
Hence, each flexibility configuration can be uniquely represented by a bipartite graph G.
The process flexibility problem concerns the performance of flexibility at two levels. At
7
the level of action flexibility, the process flexibility problem boils down to solving the following
classical transportation problem on m supply and n demand nodes, with process structure G:
ZG∗ (D) =
max
n X
m
X
xij
i=1 j=1
s.t.
m
X
xij ≤ Di ∀i = 1, 2, . . . n;
(1)
j=1
n
X
xij ≤ Cj ∀j = 1, 2, . . . m;
(2)
i=1
xij ≥ 0 ∀ i = 1, . . . , n, j = 1, . . . , m,
(3)
xij = 0 ∀ (i, j) ∈
/ G.
(4)
The vector D = (D1 , . . . , Dn ) encodes the demand for each product, and Cj represents the
capacity/supply at plant j. Our goal is to utilize the capacities at the most efficient manner
to meet the demands, subject to the constraints imposed by the process flexibility structure G.
While we have presented the action flexibility problem as a static max flow problem, we need to
caution that the problem encountered on the ground could be more complicated than depicted
by this simple model. The information on D may not be revealed at the same time, and yet
the deployment decisions may have to be made on the spot with incomplete information. The
objective function may have to take into account the cost impact of overages and shortages, and
not mere flow maximization. The actual optimization problem encountered at the operational
level is more aptly represented by a multi-stage stochastic programming model, but we have
refrained from going into the details here in this paper for ease of exposition.
At the tactical level of state flexibility, we need to design the process structure G, so that
the system is able to respond to changes in endogenous uncertainty in the system. i.e., we want
the expected maximum flow
ED ZG∗ (D)
to be as large as possible. The process flexibility problem considered in this paper can be
succintly written as the following optimization problem:
∗
max ED ZG (D) .
G⊂A×B
Note that in essence, the process flexibility structure determines how the system can effectively allocate its capacity to handle the random demand. An alternate measurement to
8
characterize the performance of the process structure, in the absence of a probability measure
on the space of possible scenarios, is via a worst case approach. i.e., we want
∗
min ZG (D)
D∈D
to be as big as possible, so that the structure G will perform relatively well even in the worst
scenario in the set D. This changes the process flexibility problem to the following robust
optimization problem:
max
G⊂A×B
min
D∈D
ZG∗ (D)
.
For both objectives, clearly the optimal structure will be the one with full flexibility, i.e.,
the complete bipartite graph on A and B, containing all the links. However, this level of
performance comes at the expense of drastically increase operational and/or communication
costs at the operational level. For the process structure to be useful in practice, the number of
edges in the graph G should be as small as possible. The central theme of this paper is to use
a series of examples and discussions to demonstrate that in most settings, a structure with far
fewer number of edges may already perform as good as the fully flexible system. Such results
and insights are obviously important and useful in practice.
Studies on process flexibility can be traced back to 1980’s, stemming from the hot topic
“Flexible Manufacturing System” (cf. Stecke (1983), Browne et al. (1984)). The focus of
FMS is on the trade-off of investment in dedicated versus flexible capacities (cf. Fine and
Freund (1990), Van Mieghem (1998)). The challenge there is to understand the economic value
of flexible resources in the system, vis-à-vis its ability to reduce capacity investments. The
observation that a partial flexibility structure can perform nearly as well as a full flexibility
structure is pointed out explicitly by the seminal work of Jordan and Graves (1995). Their
findings were based on a study of GM production network. They calibrated the performance of
sparse partial flexibility structures with the full flexibility structure in an extensive simulation.
Surprisingly the results showed that a partial flexibility structure, if well designed, could capture
almost all the benefits in the full flexibility structure. They further proposed the use of
chaining structure (cf. Figure 5) as a good design for partial flexibility structures.
9
Product
Plant
Product
Plant
B:PartialFlexibility
A:FullFlexibility
(A chain)
Figure 5: Flexibility Structures Represented via Bipartite Graphs.
2.1
Theoretical Results: Average Performance
Note that a chain in a n by n bipartite graph has 2n (flexibility) edges, whereas a fully flexible
system has n2 edges. It is thus surprising that a simple chain structure can perform as well as
a fully flexible system in Jordan and Grave’s study using the GM data. Aksin and Karaesmen
(2004) proved that the performance of a regular k-chain (a generalization of a chain, where k
represents the degreee of each node) is increasing concave as k increase. The returns from added
flexibility into the system is thus diminishing. Chou et al. (2007a) demonstrated this effect more
succinctly by comparing the performance of the chaining structure with the fully flexible system
for an asymptotically large system. They consider the following n-plant, n-product example.
Assuming that each plant has a fixed capacity of Cj = C units for each j = 1, . . . , n, and each
product has an expected demand of Di = C units for each i = 1, . . . , n as well. When the
demand is uniformly distributed between 0 and 2C, Chou et al. (2007a) showed that
∗
ED ZChain
(D)
= 89.6%.
lim
n→∞ ED Z ∗
(D)
Full
This implies that a simple chain structure can capture close to 90% of the value of the max flow
in a fully flexible system, even when the system size is very large and the demand is uniform
over a range. The performance in the case of normal distribution is even more impressive.
Assuming that µ = 3σ, Table 1 shows the expected performance of two different structures
over the random demand as n varies. For small n (say n = 10), their simulation shows that
10
System Size n
Chaining C
Full Flexibility F
10
15
20
25
30
35
40
949.36
1434.44
1915.78
2401.94
2871.06
3352.66
3807.16
955.14
1447.00
1938.93
2441.73
2929.84
3430.70
3905.48
ED (Z ∗
(D))
Chain
(D))
Full
99.39%
99.13%
98.81%
98.37%
97.99%
97.73%
97.48%
ED (Z ∗
Table 1: Expected Sales and Chaining Efficiency for Increasing System Size
the expected sales, essentially the maximum flow, in the two systems are 949.36 and 955.14
respectively. This demonstrates that chaining already achieves most (99.39%) of the benefits of
full flexibility in this case. As the system size expands, the performance of chaining deteriorates
slightly, but still at an impressive level of 97.48% for n = 40. As n approaches infinity, it can
be shown that the limit tends to a value close to 96% (cf. Chou et al. (2007a)).
The result of Jordan and Graves (1995) is important and exciting because it is consistent
with a widely observed phenomenon: a simple and easy-to-execute structure (e.g. a chain
structure) could be as good as the most complicated one (e.g. a full flexibility structure). It
is thus reasonable to believe that a systematical and effective method to design a good sparse
flexible structure can be adopted to applications in many different fields.
Bassamboo et al. (2008) extended the theory of chaining into dynamic processing system,
through a concept called “tailored pairing.” Pairing is a configuration such that every two
classes of demands are linked by exactly one resource, and hence is different from the concept
of chaining, where there is a circular ordering of nodes and only adjacent demand nodes are
linked by a resource. They used Brownian motion approximation to show that subject to mild
conditions on the marginal cost structure as a function of the level of flexibility, tailored pairing
is optimal for symmetric system (when average demand and supply are identical and balanced)
in the heavy traffic regime. i.e., there is no need in the optimal solution to invest in server
capable of serving three or more classes of demands. The authors further postulated that the
same holds even for asymmetrical system, and supported their claims via numerical simulation.
2.2
Theoretical Results: Worst Case Performance
Chou et al. (2007b) analyzed the performance of partial flexible structure using a worst case
11
approach. They adopt the concept of graph expansion (Bassalygo and Pinsker (1973)), which
is widely used in the area of graph theory and computer science (Sarnak (2004)), to study the
process flexibility problem in this setting. Their study reveals the intimate connection between
process flexibility and graph connectivity, and shows that the graph expander structure (i.e. a
class of highly connected graphs with far smaller number of arcs than complete graph) works
extremely well as a sparse flexibility structure. This result holds in many classes of objective
functions, only requiring a mild assumption that the demand is bounded around its mean. i.e.,
demand is never more than a constant times of its mean.
Definition 1 D̃i has bounded variation of λ around its mean if D̃i ≤ λE[D̃] almost surely.
It turns out that when demand does not deviate substantially away from the mean, we can
use the action flexibility inherent in the process structure to effectively allocate capacities to
the demands. To do so, we need to understand the notion of graph connectivity associated with
every process structure.
Definition 2 A structure F is k-connected if there are at least k node disjoint paths linking
every pair of nodes in A ∪ B.
A k-chain (denoted by Ck ) is a subgraph in a n by n bipartite graph where each supply node
i is linked to demand nodes i, i + 1, . . . , i + k − 1 (modulo n). The structure Ck is clearly kconnected, with kn links. However, there are exponentially many classes of k-connected graphs
with kn edges, for k > 2, although the 2-chain is the only 2-connected graph with 2n edges.
There is a clear trade-offs in the level of connectivity with the number of edges - for higher
graph connectivity, the structure needs to have more edges.
There is a class of highly connected graph, called graph expander, which has received a lot of
attention in the literature. Basically, graph expanders are graphs where every “small” subset of
nodes are linked to a “large” neighborhood. The ratio of the size of the neighborhood and the
size of the subset measures the expansion capability of the graph. We define the neighborhood
of a subset and the concept of a graph expander formally in the following:
Definition 3 Let F be a bipartite graph with partite set A and B. For S ⊂ A, the neighborhood
of S in F is defined to be
∆
Γ(S) = {j ∈ B : (i, j) ∈ F for some i ∈ S.} .
12
Definition 4 Let F be a bipartite graph with partite set A and B. The structure F is a
(α, λ, ∆)-expander if
• for each node v in the graph, deg(v)≤ ∆ for every v ∈ A, and
• for all small subset S ⊂ A with |S| ≤ αn, we have
|Γ(S)| ≥ λ|S|.
Remark:
1. For a n × n bipartite graph which is also a (α, λ, ∆)-expander, , the number of edges is
at most ∆n.
2. A 2-chain C2 is clearly a ( n1 , 2, 2) expander, since for each subset of size 1, there are at least
2 neighbors. Furthermore, the degree is bounded by 2. It is also a ( n2 , 1.5, 2) expander,
since for every subset S of size at most 2, |Γ(S)| ≥ 1.5|S|. It is easy to check that it is
simultaneously a ( nk , (k + 1)/k, 2) expander, for all k ≤ n − 1.
3. The reason why graph expander works well in matching supply and demand is because
graph expander ensures that any suitably small group of product nodes is connected to a
relatively large number of plants. Moreover, the notion that a long chain is better than
a short chain can be cast in the same light: the expansion ratios for “small” subsets of
product nodes in long chains are higher than that in short chains.
Chou et al. (2007b) established the following main result:
Theorem 1 Consider a n × n system, where the demand D̃i has bounded variation of λ with
mean µi = µ. Assume that each plant has capacity µ. Let F be an (α, λ, ∆)-expander, with
α × λ = 1 − for some > 0. Then
"
ZF∗ (D)
≥ αλn min µ,
P
i∈A D̃i
n
!#
∗
(D)
= (1 − )ZFull
for all D.
Note that the -optimality performance holds for all demand scenario D, and is thus the
worst case performance of the expander structure, given that the demand has bounded variation of λ. This result is considerably stronger than the average case performance of the chaining
13
n
structure. Since 2-chain C2 in a n by n bipartite graph is a ( n−1
n , n−1 , 2)-expander, we have the
following immediate corollary:
Corollary 1 Suppose that (i) the demand of each product has bounded variation of 1 +
1
n−1 ,
(ii) with mean µi = µ, i = 1, . . . , n, and (iii) each of the n plants has capacity µ. Then
∗
ZC∗2 (D) = ZFull
(D)
for all D.
Remark: We notice that truncated normal distribution is often used to model product demand distribution in a lot of service and manufacturing settings. When σ = µ/3, and demand
is truncated at one standard deviation above the mean, then according to the above, 2-chain is
always as good as the fully flexible system when n = 4. However, when n increase, the worst
case performance of a 2-chain is worse off compared to the fully flexible system.
In the worst case setting, the level of flexibility needed in the structure depends on the
magnitude of demand deviation above the mean (i.e. level of uncertainty), which corresponds
to the graph expansion ratio (parameter λ) in the expander structure. For k-chain, this ratio
depends on n, and hence the performance of k-chain suffers as n increase. To find good process
structure for large n, we use a different class of graph expander structure. In particular, we use
expander where the number of edges can be much smaller than the number of edges in a fully
flexible system. The existence of such structure is well known and is by now a folklore in the
graph theory community.
Theorem 2 [Asratian et al. (1998)] For any n, λ ≥ 1, and α < 1 with αλ < 1, there exists a
(α, λ, ∆)-expander, for any
∆≥
1 + log2 λ + (λ + 1) log 2 e
+ λ + 1.
− log2 (αλ)
(5)
Note that the lower bound on the degreee ∆ is independent of n, and hence the number of
edges in this class of graph expanders is linear in n. The implication for the process flexibility
problem can be stated more succinctly as follows:
14
In the symmetrical system, for any given demand distribution with bounded variation
λ, we can find a corresponding α with
αλ ≈ 1 − , for some > 0,
such that for n sufficiently large, we can always find a process structure using at
most ∆n edges, where ∆ is given by the right hand side of (5), such that the worst
case performance of the structure is at most 1 − times of the fully flexible system.
For more general system (i.e. the number of product nodes and plant nodes might be
different and products follow different demand distributions), Chou et al. (2007b) proposed a
generalization using the concept of “Ψ-expander”, with a high Ψ (0 < Ψ ≤ 1). Suppose the
demand with mean µi is assumed to be bounded in [λi (U )µi , λi (U )µi ]. We say that the demand
has bounded variation of λi (L) and λi (U ) in this case.
Definition 5 A Ψ-expander in the process flexibility problem is a bipartite graph in A × B with



X
X
X
X
Cj ≥ min
λi (U )µi , Ψ
Cj −
λi (L)µi ,


i∈S
j∈Γ(S)
j∈B
i∈S
/
for all subset S ⊆ A.
The definition of Ψ-expander partitioned the subsets of A into two sets: (i) For small subset S
P
P
P
such that i∈S λi (U )µi + i∈S
/ λi (L)µi ≤ Ψ
j∈B Cj , we have
X
X
Cj ≥
λi (U )µi ,
i∈S
j∈Γ(S)
and hence the plants supplying to the small subsets have sufficient capacity to deal with demand
arising from the small subset. (ii) At the same time, the capacity connected to a non-small
subset is also large enough, i.e.
X
j∈Γ(S)
Cj ≥ Ψ
X
Cj −
j∈B
X
λi (L)µi ,
i∈S
/
so that at least Ψ proportion of the total capacity is utilized in the worst case. It is thus easy
to see that a structure with Ψ = 1 is as good as full flexibility.
Theorem 3 (Chou et al. (2007b)) Let F be a Ψ-expander. When D̃i has bounded variation
λi (L) and λi (U ), then for all demand realization, we can find a solution for ZF such that either
(a) all the plants are operating below their configured capacity level (because of insufficient
demand), or (b) at least Ψ proportion of the total pre-configured capacity have been utilized.
15
The above theorem suggests that a Ψ-expander has the following nice property - as long as
the demand for each product falls in the range λi (L)µi and λi (U )µi , then the process structure
guarantees a utilization rate of 100 × Ψ% in the entire system!
Example: Consider the process flexibility problem with 5 plants and 5 products. The capacity
at each plant is 100 units, whereas the demand for the 5 products are between 50 and 150,
each with mean of 100. Note that we did not specify the precise structure of the demand
distributions. A fully flexible system in this case contains 25 edges, whereas a 2-chain has only
10 edges. Note that the demand is always within 1.5 times of its mean. Hence the 2-chain has
bounded variation with λi (L) = 0.5, and λi (U ) = 1.5. It can then be shown easily that the
2-chain is a 1-expander. Thus the 2-chain structure in this case has the same performance as
the fully flexible system, for all demand realization!
2.3
2
Theoretical Results: Randomized Performance
The structural results in the worst case setting provides a glimpse to the structure of good
process structure. However, for more complicated cases (such as the non-identical and asymmetrical demand setting), such structural insights may not be readily available. Instead of
focusing on the performance of a specific structure, we can also study the average performance
of a class of process structures. This route of analysis allows us to examine the performance of
“sparse” structures in other classes of design problems. To this end, we need to generalize the
notion of process flexibility.
Consider the problem
(P ) :
X
n
Z(b, {1, . . . , n}) = max
ci xi : Ax ≤ b; xi ≥ 0, i = 1, . . . , n .
i=1
The dual problem is given by
(D) :
Z(b) = min
X
m
T
bj yj : A y ≥ c; yj ≥ 0, j = 1, . . . , m .
j=1
2
One of the authors have often played a game with students in class - the students can choose demand values
of any kind, as long as each value falls in the range of [50, 150]. The instructor will then compare the performance
of a 2-chain and a fully flexible system. Most students are amazed that a simple 2-chain always achieved the
same performance as the complicated fully flexible system.
16
(D) is a linear programming problem with m variables and n constraints. If we sample N
constraints from (D), and denote the constraints sampled by S, we obtain the problem
X
m
T
(D(S)) : Z(b, S) = min
bj yj : AS y ≥ cS ; yj ≥ 0, j = 1, . . . , m .
j=1
The dual of this problem is
X
(P (S)) : Z(b, S) = max
ci xi : AS xS ≤ b; xi ≥ 0, i ∈ S .
i∈S
Let x∗ (b) denote an optimal solution in Z(b, {1, . . . , n}). Note that since b is random, x∗ (b)
is also a random vector. We assume that problem (P ) has an optimal solution x∗ (b) with the
following property:
(Property A):
x∗i (b) ≤ λEb (x∗i (b)) almost surely for some constant λ > 0 (independent of
n), and for all i = 1, . . . , n.
The above property essentially states that for each realization of b, the optimal primal
solution x∗ (b) should not be too far above its mean value. This is similar to the bounded
variation assumptions in the process flexibility problem.
Theorem 4 Suppose Property A holds for (P ). Then there exists a set S with cardinality
N = O( λm
), such that
Eb (Z(b, S)) ≥ (1 − )Eb (Z(b)).
In the case when n >> m, S is considered a sparse set and plays the role of the sparse structure
in the process flexibility problem. This theorem identifies a natural condition under which we
can expect a suitably chosen sparse subset S (with O(m) variables) to “perform” nearly as well
as a fully flexible system with n variables, even if n >> m.
To construct the sparse set S with near optimal performance, we need to understand the
behavior of the (random) optimal solution x∗ (b). The set S in the above theorem is constructed
by repeatedly sampling from the distribution x∗ (b), up to O( λm
) times. We will see in the
later sections that for several classes of problems, the optimal solution x∗ (b) can be obtained
in closed form. For the technical details pertaining to this result, we refer the readers to Chou
et al. (2007a).
17
3
Construction Methods
The process flexibility problem can be technically modeled as a two stage stochastic programming problem, where the first stage decision concerns the selection of links subject to budget
constraints (0-1 discrete problem), whereas the second stage problem concerns finding the best
recourse action to match supply with demand, to obtain the expected second stage cost function
arising from the optimal recourse decisions. The latter is a difficult problem, since it involves
finding the expected maximum flow problem (or worst case max flow) in a general bipartite
graph. While there are numerical methods that can be used to handle this kind of problems
when the size is small, it is not widely used in practice, due to the difficulty in finding reliable distributional information on demand and accurate cost estimation. To the best of our
knowledge, this algorithmic design problem has also been largely overlooked by the research
community, in part due to the technical difficulties associated with this problem.
We describe next several heuristics that can be used to construct good process structure
quickly, exploiting the structural insights discussed in the previous section. We also review a
recent approach to address this problem, using the theory of robust optimization.
3.1
Chaining Method
The chaining concept (Jordan and Graves (1995)) is arguably the most influential strategy used
in practice to design good process structure. It is developed based on two key insights obtained
from Jordan and Graves (1995)’s landmark study:
• By adding a small amount of flexibility at the proper places in an otherwise rigid system,
we could achieve a significant improvement in its performance; and
• the links should be added to the structure to obtain long chains.
A chain refers to a path which goes through a group of distinct product and plant nodes in
consecutive order, returning to the start node at the end to form a cycle. A long chain is
preferred because it has the ability to pool the plants’ capacity and products’ demand together
and thus can deal with demand uncertainties more effectively than a short chain.
Based on the above conceptual ideas, Jordan and Graves (1995) provided three other general
guidelines to design process structure. They advise the designer to
18
• try to equalize the capacity allocated to each product node in the chain;
• try to equalize the expected demand allocated to each plant node in the chain;
• construct a long circular chain visiting as many nodes as possible.
The first two guidelines aim to help the plants to achieve higher capacity utilization and to
satisfy more demand. In the symmetric case, with each product and each plant having the
same level of mean demand and capacity, we can use the three guidelines to obtain a regular
2-chain.
The above guidelines are obtained and validated by extensive numerical simulation results.
However, these guidelines alone do not provide an implementable heuristic which can be used
to design a process structure. Jordan and Graves (1995) mentioned that they have no firm
guidelines to adding flexibility for more general cases. In fact, in their work, they use the guidelines they developed to obtain various process structures, and then used numerical simulation
to estimate the performance (e.g. average lost sales; unused capacity at each product and plant
etc.) of each structure, to determine the best process structure. This process is tedious and
time consuming. The next method tries to address this deficiency, using a heuristic guided by
the graph expansion concept.
3.2
Node Expansion Method
The previous section shows that the performance of a process structure is intimately connected
to its underlying connectivity property. We can thus design a good process structure by constructing a graph with good expansion ratio. The latter problem is recently investigated by
Ghosh and Boyd (2006). However, their approach works more for the symmetrical case (graph
expansion is after all a concept originating in graph theory), but does not address the issue
associated with asymmetrical supply and demand setting.
The notion of graph expansion is nevertheless connected to the design guidelines popularized
by Jordan and Graves (1995). The first and second guideline seek to equalize the capacity and
expected demand allocated to each product and plant respectively. This is related to the concept
of expansion ratio for all (small) subsets containing only a single node. We extend this strategy
further by fully exploiting this connection.
19
The node expansion method works by checking the expansion ratio of all singletons, although
the method can be easily extended to handle all subsets with up to k (k fixed, small) nodes.
Starting from any rigid base structure, the method augments the structure by adding links
iteratively to improve upon the node expansion ratio in a greedy manner:
In each iteration, we add a link connecting the product node with the lowest
product expansion ratio (i.e. the ratio of the total capacity of the connected plants
to the product’s expected demand), to the plant node with the lowest plant expansion
ratio (i.e. the ratio of the total expected demand of the connected product nodes to
the plant’s capacity). We skip this edge if it has already been added to the structure,
and move to the plant node with the next smallest expansion ratio. We repeat this
procedure until the number of added links reached the pre-determined limit.
Simulation studies by Chou et al. (2007b) show that this design heuristic can generate good
process structures in several applications. In fact, on the process flexibility problem encountered
in GM, the structure obtained using this direct constructive approach is already as good as the
structure developed by Jordan and Graves (1995) guided by extensive numerical simulation.
We refer the readers to Chou et al. (2007b) for details.
3.3
Sampling Method
The sampling method builds on Theorem 4. For the process flexibility problem, the optimal
solution is trivial under the full flexibility structure F. Note that in this problem, with demand
D = (D1 , D2 , . . . , Dm ) and capacity C = (C1 , . . . , Cn ), the max flow problem ZF∗ (D) has up
to nm variables, with only O(n + m) constraints. There are multiple optimal solutions to this
simple problem, and the properties of the extreme point solutions are well known. For our
purpose, however, we need a closed form solution to the problem. Fortunately, this is easy
when the structure corresponds to the full flexibility structure F.
Lemma 1 In the process flexibility problem,
x∗ij (D) =
max
D i Cj
Pm
i=1 Di ,
Pn
j=1 Cj
20
, i = 1 . . . m, j = 1, . . . , n,
is an optimal solution to ZF∗ (D) under the full flexibility structure F. Furthermore,
ZF∗ (D)
X
m
n
X
= min
Di ,
Cj .
i=1
j=1
Using this solution, we can construct (random) process structure by sampling arc (i, j) with
probability proportional to x∗ij (D). The sampling heuristic basically has two stages.
• In the first stage, the sampling probability for each link (i, j) (i = 1 . . . m, j = 1 . . . n) is
estimated by calculating the empirical average of the flow on the arc (i, j). This is used
in the case when x∗ij (D) does not have an easy closed form expression.
• In the second stage, structures can be created by selecting links using the estimated
sampling probabilities, and the structure with the best performance will be ultimately
selected.
Note that the sampling method uses numerical simulation to obtain the performance of the
structure sampled, and in a way, this is identical to the approach used by Jordan and Graves
(1995). The advantage of the sampling method is that it is systematic, and can be applied to a
wide variety of other problems, ranging from capacity pooling networks (Chou et al. (2007a)) to
transshipment network design (Lien et al. (2005)). The disadvantage, on the other hand, is that
the sampling method cannot ensure that every structure sampled will be good, and an additional
evaluation step (e.g. through simulation) is needed to identify a good structure. Furthermore,
although the method works theoretically, we do not have good qualitative understanding of the
features inherent in near optimal sparse structures.
3.4
Robust Optimization
The process flexibility problem can be recast as a two-stage stochastic maximum flow problem
as follows:
h
i
maxx EP Q(ξ̃, x)
X
s.t.
xi = K, xi ∈ {0, 1},
i
where K represent the maximum number of links in the process structure, x the design decision
variables, and ξ̃ denotes the random demand parameters in the problem, with distribution P .
21
The recourse function is given by:
∗
Q(ξ̃, x) = ZG(
x) (D)
s.t. ξ̃ = D,
e ∈ G(x) if and only if xe = 1.
When the probability measure on the scenario space is not explicitly given, we can minimize
the worst case performance of the design, over an uncertainty set U, to obtain a robust two
stage maximum flow problem:
h
i
maxx min Q(ξ̃, x)
ξ̃ ∈U
X
s.t.
xi = K, xi ∈ {0, 1},
i
To solve this robust optimization problem, the challenge is to find a nice uncertainty set U, so
that the problem
maxx tX
s.t.
xi = K, xi ∈ {0, 1},
i
∗
t ≤ ZG(
x) (ξ̃) ∀ ξ̃ ∈ U.
can be recast as a tractable convex optimization problem, even if the set U has infinitely many
scenarios. Unfortunately, when U represents the set of demand scenarios with bounded variation
around the means, the above robust formulation can be solved in polynomial time if and only
if P = N P . We refer the readers to details in Atamturk and Zhang (2007), and for some
special cases when the robust two stage network design problem can be solved in polynomial
time. The currently available approach to solve the robust process flexibility problem is via the
cutting plane approach, and is thus not computationally efficient.
4
Measurement and Evaluation
The research on the evaluation of process flexibility has so far focused on creating indices to
rank process flexibility structures, in terms of level of flexibility inherent in the structure. This
line of pursuit complements the constructive approaches, and seeks way to allow the managers
to compare different process structures quickly, without the need to evaluate its performance in
simulation, due to the lack of cost parameters and/or distributional information on the uncertain
parameters. The computation of the indices uses minimal information on the uncertain demand
parameters (mostly average values). These indices are thus usually easy-to-compute, and can
22
effectively rank the structures in terms of performance, though they cannot give an absolute
value of the performance of the different process structures.
4.1
JG Index
Jordan and Graves (1995) developed a probabilistic index to measure the performance of a
given structure. For any subset S of demand nodes, they focus on the probability that the
unsatisfied demand in that structure would exceed that for the corresponding full flexibility
structure. i.e.,
P
X
Dj −
j∈S
X
Si
X
X ≥ max 0,
Dj −
Si .
j
i∈Γ(S)
i
The largest probability among all subsets is used as a flexibility index to compare among
structures. A good flexibility structure should have a low index, since a more flexible structure
should deal with demand uncertainty more effectively, and thus the unfilled demand of the
structure should be as close to the full flexibility structure as possible. However, the JG index
is usually very hard to compute.
4.2
Structural Flexibility Index
The index developed by Iravani et al. (2005) is arguably a milestone in the development of a
measure for process flexibility. In this study, a suitably defined “structural flexibility matrix”
(SF Matrix) M was proposed to calibrate the performance of a process structure. An entry
(i, j) in M , denoted by M (i, j), represents the
maximum number of non-overlapping routes from demand node i to demand node
j,
whereas
M (i, i) = degree of arcs connected to the demand node i.
The largest eigenvalue and mean of the entries in SF matrix M are used as two alternate indices
to determine the level of flexibility in a process structure. SF indices are much easier to compute
and work better than probabilistic indices in many examples (cf. Iravani et al. (2005)), although
the computation of each entry in the matrix requires solving a maximum flow routine.
23
4.3
Expansion Index
We propose an alternate index to measure the performance of a process structure, using the
observation that graph with higher connectivity tends to be more flexible, which has been
theoretically verified in the balanced and identical case. Let N denote the number of nodes
(including both supply and demand nodes), and L the number of links in the structure. The
expansion index is defined as the second smallest eigenvalue of the Laplacian matrix
L = T T 0,
where T is a N × L matrix. For link l connecting node i ∈ A (with mean µi ) and j ∈ B (with
mean Sj ), the corresponding entry in column l of T is
Til =
p
p
µi Sj , Tjl = −
µi Sj , and Tkl = 0, ∀k 6= i, j.
The index is developed using a well-known observation in graph theory that the second
smallest eigenvalue of L, i.e. λ2 (L), is a good surrogate for measuring the connectivity of the
underlying graph (see Fiedler (1973), Ghosh and Boyd (2006)) in the case when µi = Sj = 1
for all i, j. As shown in the earlier section, a graph with good expansion ratio (high λ2 (L))
is highly connected and is more flexible in matching supply and demand. λ2 (L) can thus be
used as an alternate index to rank process structures. The index λ2 (L), as compared to the SF
index M , has the key advantage that it has been thoroughly studied in the literature.
4.4
Numerical Comparison
These indices have been tested in the following numerical experiments. The first experiment
is to rank the performance of two different graphs, i.e., the Levi graph and the regular 3-chain
(cf. Figure 6). Both graphs are regular with degree 3, and hence have equal number of edges.
The Levi graph has slightly better expansion ratio for small subsets up to order 3. In fact, it
can be shown that each subset of order 3 on one side has at least 5 neighbors, whereas it is easy
to find subsets of order 3 in the 3-chain with only 4 neighbors.
As shown in table 2, λ2 (Levi) (0.55) is higher than λ2 (Regular) (0.05), in the symmetric case
when all demand means and capacity levels are identical. This indicates that the Levi graph
could be a better process structure than the 3-chain. This is consistent with our simulation
24
A
1
A
1
B
2
B
2
C
3
C
3
D
4
D
4
E
5
E
5
F
6
F
6
G
7
G
7
H
8
H
8
I
9
I
9
J
10
J
10
K
11
K
11
L
12
L
12
M
13
M
13
N
14
N
14
O
15
O
15
P
16
P
16
Q
17
Q
17
R
18
R
18
S
19
S
19
T
20
T
20
U
21
U
21
V
22
V
22
W
23
W
23
X
24
X
24
Y
25
Y
25
Z
26
Z
26
AA
27
AA
27
B:A regular graph
A :A levigraph
Figure 6: Levi graph and a regular chain with degree 3.
experiments: the Levi graph can indeed support slightly higher amount of flow in the process
flexibility problem, compared to the 3-chain, for a variety of demand distributions.
We have also evaluated the performance of the expansion indices and SF indices, using two
representative examples studied in Iravani et al. (2005). The first example is a group of structures with random demand µ = (1.5, 1, 0.5, 0.5, 1, 1.5) and fixed capacity S = (1, 1, 1, 1, 1, 1)
(c.f.
figure 7).
The second example is a group of structures with random demand µ =
(1, 1, 1, 1, 1, 1, 1, 1) and fixed capacity S = (1, 1, 1, 1, 1, 1, 1, 1), as shown in figure 8.
We use E(ZFe (D)), the expected excess flow in structure F , as the benchmark to evaluate the
performance of structure F . E(ZFe (D)) is closely related to the max-flow objective E(ZF∗ (D)),
P
because the excess flow ZFe (D) is simply ni=1 Di − ZF∗ (D), the amount of unmet demand.
E(ZFe (D)) is a better benchmark for this experiment because it is more sensitive and does not
scale as much as the max-flow objective when problem size changes. We approximate E(ZFe (D))
by sampling 200 demand scenarios, each with demand Di uniformly distributed in (0, 2µi ).
In both cases, the ranking obtained from the expansion index is consistent with the ranking
given by (the empirical average) E(ZFe (D)). The ranking given by SF index, on the other
25
C apacity
D em and
C apacity
D em and
C apacity
D em and
C apacity
D em and
C apacity
D em and
1
1
a
1.5
1
1
a
1.5
1
1
a
1.5
1
1
a
1.5
1
1
a
1.5
1
2
b
1
1
2
b
1
1
2
b
1
1
2
b
1
1
2
b
1
1
3
c
0.5
1
3
c
0.5
1
3
c
0.5
1
3
c
0.5
1
3
c
0.5
1
4
d
0.5
1
4
d
0.5
1
4
d
0.5
1
4
d
0.5
1
4
d
0.5
1
5
e
1
1
5
e
1
1
5
e
1
1
5
e
1
1
5
e
1
1
6
f
1.5
1
6
f
1.5
1
6
f
1.5
1
6
f
1.5
1
6
f
1.5
Structure 1-1
Structure 1-2
Structure 1-3
Structure 1-4
Structure 1-5
Figure 7: SF Group 1: Structures with Demand µ = (1.5, 1, 0.5, 0.5, 1, 1.5)
C apacity
D em and
C apacity
D em and
1
1
a
1
1
1
a
C apacity
D em and
1
1
1
a
1
2
b
1
1
2
1
3
c
1
1
3
1
4
d
1
1
1
5
e
1
1
6
f
1
1
7
g
1
8
b
1
1
2
c
1
1
3
4
d
1
1
1
5
e
1
1
6
f
1
1
1
7
g
h
1
1
8
h
Structure 2-1
C apacity
D em and
1
1
1
a
1
b
1
1
2
b
1
c
1
1
3
c
1
4
d
1
1
4
d
1
1
5
e
1
1
5
e
1
1
6
f
1
1
6
f
1
1
1
7
g
1
1
7
g
1
1
1
8
h
1
1
8
h
1
Structure 2-2
Structure 2-3
Structure 2-4
C apacity
D em and
C apacity
D em and
C apacity
D em and
C apacity
D em and
C apacity
D em and
1
1
a
1
1
1
a
1
1
1
a
1
1
1
a
1
1
1
a
1
1
2
b
1
1
2
b
1
1
2
b
1
1
2
b
1
1
2
b
1
1
3
c
1
1
3
c
1
1
3
c
1
1
3
c
1
1
3
c
1
1
4
d
1
1
4
d
1
1
4
d
1
1
4
d
1
1
4
d
1
1
5
e
1
1
5
e
1
1
5
e
1
1
5
e
1
1
5
e
1
1
6
f
1
1
6
f
1
1
6
f
1
1
6
f
1
1
6
f
1
1
7
g
1
1
7
g
1
1
7
g
1
1
7
g
1
1
7
g
1
1
8
h
1
1
8
h
1
1
8
h
1
1
8
h
1
1
8
h
1
Structure 2-5
Structure 2-6
Structure 2-7
Structure 2-8
Structure 2-9
Figure 8: SF Group 2: Structures with Demand µ = (1, 1, 1, 1, 1, 1, 1, 1)
26
hand, are slightly different from (the empirical average) E(ZFe (D)). Note that lower value of
E(ZFe (D)) should ideally correspond to higher index value. Unfortunately, the SF method errs
on the performance of 1-1 and 1-2, and on 2-4 vis-à-vis the rest. The simulation results indeed
suggest that both the expansion index and SF index are easy-to-compute indices that can be
used to rank process structures rather effectively.
Table 2: Comparisons among Flexibility indices
5
Applications
The central theme in this paper is the observation that a little flexibility can go a long way in
enhancing the performance of the system. We have discussed the impact of this phenomenon
on the process flexibility problem in the earlier sections. This insight has also been observed in
numerous other settings. In the rest of this section, we review some of the key results obtained
for other related areas, and discuss some new applications.
5.1
Multi-Stage Supply Chain
Graves and Tomlin (2003) extended Jordan and Grave’s work to multi-product, multi-stage
supply chains, where each product needs to flow through several stages in the supply chain.
They proposed a supply chain flexibility measure g, where higher g indicates higher flexibility.
Unfortunately, they stop short of offering a method to design a flexible supply chain network.
27
The results established in the earlier sections can be used to establish a much stronger result
concerning the performance of sparse supply chain structure. We also give a glimpse of the
performance of the sampling method using this example as an illustration.
Consider the following supply chain design problem (see figure 9): There are n1 products,
n2 plants, and n3 suppliers. In the full flexibility scenario, each product can be produced at
any plant, using material source from any of the supplier. We assume that each unit of product
consumes a unit of material from each supplier and uses a unit capacity at a plant. We assume
further that production capacity at the plants are Ci , i = 1, . . . , n1 , and the suppliers have
limited amount of materials, at capacity Bi , i = 1, . . . , n3 . The demand for each product is
random and denoted by the random variable Di , i = 1, . . . , n1 .
Figure 9: A Supply Chain Flexibility Structure.
In the full flexibility scenario, it is easy to see that the expected sales is given by
X
n1
ED min
i=1
Di ,
n2
X
i=1
28
Ci ,
n3
X
i=1
Bi
.
The above problem can be formulated as the following set packing problem:
∗
Z (D) =
s.t.
max
n2 X
n3
n1 X
X
xijk
i=1 j=1 k=1
n
n
2
3
XX
xijk ≤ Di ∀i = 1, 2, . . . n1 ;
(6)
j=1 k=1
n1 X
n3
X
xijk ≤ Cj ∀j = 1, 2, . . . n2 ;
(7)
i=1 k=1
n1 X
n2
X
xijk ≤ Bk ∀k = 1, 2, . . . n3 ;
(8)
i=1 j=1
xijk ≥ 0 ∀ i, j, k.
(9)
For each realization of demand Di , it is easy to see that there is an optimal solution given
by
xijk (D) =
D i Cj Bk
.
Pn 1
Pn 2
Pn 1
Pn 3
Pn 2
Pn 3
max
i=1 Di ×
j=1 Cj ,
i=1 Di ×
j=1 Cj ×
k=1 Bk ,
k=1 Bk
Let S = {(i, j, k) : material from supplier k used to produce product i at plant j} denote
the supply chain configuration. We have an analogous result for this problem setting.
Theorem 5 Suppose the demand distribution satisfies
xijk (D) =
max
≤ λE
Pn 1
i=1
Di ×
D i Cj Bk
Pn 1
Pn 3
Pn 2
Pn3
j=1 Cj ,
i=1 Di ×
j=1 Cj ×
k=1 Bk ,
k=1 Bk
Pn 2
Di Cj Bk
,
Pn 1
Pn 2
Pn 1
Pn 3
Pn 2
Pn 3
max
i=1 Di ×
j=1 Cj ,
i=1 Di ×
j=1 Cj ×
k=1 Bk ,
k=1 Bk
for some λ > 0, then there exists a sparse supply chain configuration S with cardinality |S| =
O(λ(n1 + n2 + n3 )/), such that the expected demand met by the sparse supply chain system is
at least (1 − )E(Z ∗ (D)).
We can use the sampling method to obtain good sparse supply chain structure. Consider a
supply chain with 9 products, 7 plants and 5 suppliers. Demand for each product is normally
distributed. The expected demands are shown in figure 10. The standard deviation is 40%
of the expected demand. Products can be divided into 3 subgroups. Demands of products in
the same subgroup are correlated. The correlation coefficients are 0.2 pairwise in subgroup 1
29
(product 1 to 4), -0.2 pairwise in subgroup 2 (product 5 to 7), and 0.1 in subgroup 3 (product 8
and 9). There are no correlations between demands of products in different subgroups. Supplies
from the suppliers are also normally distributed. For each supplier, the standard deviation is
40% of the expected supply (shown in figure 10). Each plant is able to produce any product.
The capacity of each plant is fixed (see figure 10). In the full flexibility scenario, a plant can
produce any product using the material supplied by any supplier.
D em ands
Plants
Supplies
µ=7.6
1
µ=5.7
2
C=1.3
µ=7.4
3
C=4.7
S=8
µ=5.1
4
C=8.6
S=6
µ=4.5
5
C=13.4
S=10
µ=2.1
6
C=11.7
S=19
µ=3.6
7
C=8.3
S=12
µ=7.3
8
C=6.3
µ=8.9
9
Figure 10: A Supply Chain Network with N =12, obtained from the sampling approach.
We conduct a simulation study to test whether there exists a partial flexible supply chain
network capturing almost all the benefits of the full flexibility system. We simulate 100 scenarios
of demands for each retailer and 100 scenarios of supplies for each supplier. We use the number
of paths in the network to denote the degree of flexibility. For each degree of flexibility (N from
10 to 315) we generate 100 structures using the sampling method and return the structure with
the best empirical performance.
Figure 11 shows the effect of higher degree of flexibility on the expected sales. The figure
shows that the marginal contribution of every additional path is in general diminishing in its
returns on expected sales. When 10 paths are selected, the expected total sales is already 85%
30
Expected Statisfied D em and
50
48
46
44
42
40
Fu
19
llF
le
xi
bi
lit
y
Supply C hain Structures
18
17
16
15
14
13
12
11
10
38
Figure 11: Expected Satisfied Demand as Flexibility Increase.
of the expected sales under the full flexibility system. After the number of paths is increase to
19, the total sales is a whopping 99% of the expected sales in the full flexibility system (with a
total of 315 paths).
5.2
Military Deployment and Transshipment
Inspired by the defense in depth strategy devised by Emperor Constantine (Constantine The
Great, 274-337), Revelle and Rosing (2000) studied the following problem in troops deployment:
Each region in the empire must be protected by one or more mobile field armies (FAs) to throw
back invading enemies. It is secured if one or more FAs is stationed in the region. It is securable
if an FA can reach the region in a single step (i.e., there is a route linking the region where the
FA is stationed to it). However, an FA can be deployed from one region to another adjacent
region only when there is at least one other FA to help launch it. i.e., the FA must come from
a region with at least two FAs stationed in it. This restriction is much like the island hopping
strategy used by General MacArthur in World War II in the Pacific.
The puzzle confronting Emperor Constantine concerns the positioning of 4 FAs to protect
the 8 regions in his empire as shown in Figure 12. He chose to position two FAs at Rome,
and two at his new capital Constantinople, leaving the outskirt region Britain vulnerable to
enemies’ attack. By focusing on the troops deployment problem in the event of war in one of
the regions, Revelle and Rosing (2000) solved the above puzzle by formulating the problem into
31
Figure 12: The empire of Constantine.
an integer program. In this case, all the regions in the empire can be protected by stationing
one FA in Britain, one in Asia Minor, and two in Rome.
Note that the above deployment, however, is not securable in the event that two or more wars
happen at the same time. For instance, this deployment could not secure against joint outbreak
of wars in any two of the five regions: Gaul, Iberia, North Africa, Egypt and Constantinople.
A slightly better deployment is to station two FAs at Iberia, and another two FAs at Egypt.
This secures the regions for up to two wars, except if the wars occur at Britain and Gaul,
or Constantinople and Asia Minor. This deployment is thus more resilient against the joint
outbreak of two wars. Unfortunately, this latter deployment is politically unacceptable as no
troops are now stationed at the capital city Rome.
In general, finding the best deployment solution securing against outbreak of wars in up to
k regions (for k ≥ 2) is a challenging problem. The minimum number of FAs needed to secure
the regions will largely depend on the network structure for troops re-deployment. In general,
if the network is dense (with many links joining different regions), or has one region connecting
to many different regions, then the number of FAs needed will be low.
In this section, we consider an analogous military deployment problem. Consider a military
mission, where n strategic locations need to be defended against possible enemy’s invasion.
The army has Qi units of troops in location i. Unfortunately, the enemy’s mission could not
be predicted, and the unit of troops deployed by the enemy to attack location i is denoted by
Di . One way to strengthen the defense network is to have reinforcement troops, where units
32
in location i may be deployed to location j, if the troops can be trained to rush from i to j
within stipulated time. Of course, it will be ideal to have many reinforcement paths, as that
means the whole force can be pooled together at the right place to deal with enemy’s invasion.
However, due to the limited time in deployment, each unit in location i can only be trained to
reinforce limited number of other locations. The challenge is to design a reinforcement network
to defend against the enormous number of enemy’s possible course of action.
This problem is similar to the transshipment problem studied in the literature, although
the latter focuses mainly on the optimal inventory policy and optimal order quantity Q∗i for
each retailer (For problems with two retailers, see Tagaras and Cohen (1992); for problems
with many identical retailers, see Robinson (1990)). They all assumed complete grouping, i.e.,
a retailer could tranship its products to any other retailers. Only a few papers discuss how to
design a transshipment networks. Lien et al. (2005) studied the impacts of the transshipment
network structure. They compared the performance of different network configurations: no
transshipment, complete grouping, partial grouping, unidirectional chain and bidirectional chain
(See Figure 13). Similar to the findings in Jordan and Graves (1995), they showed that sparse
transshipment network structures can capture almost all the benefits of complete grouping.
They also indicated that the chaining structure, which is also a kind of sparse structure, would
outperform other sparse structures.
The troops deployment (and the transshipment) problem can be reduced to a variant of the
process flexibility problem, where there are n plants and n products. Each plant i has capacity
(Qi − Di )+ (the left over at retailer i), which can be used to meet demand for other products.
Each product has demand (Di −Qi )+ (unfilled demand at retailer i). Note that in this case, both
capacity and demand are random parameters in our problem, and (Qi − Di )+ × (Di − Qi )+ = 0.
From Theorem 4, and the analysis therein, the existence of a sparse support structure for
the troops deployment problem is guaranteed by the following condition:
x∗i,j (D) =
(Di − Qi )+ (Qj − Dj )+
Pn
Pn
+,
+
max
(D
−
Q
)
(Q
−
D
)
i
i
j
j
i=1
j=1
≤ λED
(Di − Qi )+ (Qj − Dj )+
Pn
Pn
+
+
max
i=1 (Di − Qi ) ,
j=1 (Qj − Dj )
almost surely for some λ > 1, and for all i, j.
33
1
1
4
2
4
2
3
3
Complete Pooling
A Bidirectional
Chain
1
1
4
2
4
2
3
3
A Unidirectional
Chain
Group Pooling
Figure 13: Different Kinds of Transshipment Network Structures
Example: When Di are i.i.d. and take values in {0, 2} with equal probability, Qi = 1 for all
i = 1, . . . , n, then (Di − Qi )+ and (Qi − Di )+ are Bernoulli variable with equal probability.
Note that
1
E((Di − Qi )+ ) = E((Qi − Di )+ ) = .
2
Furthermore,
n
X
+
(Di − Qi ) +
i=1
n
X
+
(Qj − Dj ) =
j=1
n
X
|Di − Qi | = n.
i=1
Hence
x∗i,j (D) =
(Di − Qi )+ (Qj − Dj )+
Pn
Pn
+
+
max
i=1 (Di − Qi ) ,
j=1 (Qj − Dj )
2
(Di − Qi )+ (Qj − Dj )+
n 8
+
+
≤
E (Di − Qi ) (Qj − Dj )
n
(Di − Qi )+ (Qj − Dj )+
.
≤ 8ED
Pn
Pn
+,
+
max
(D
−
Q
)
(Q
−
D
)
i
i
j
j
i=1
j=1
≤
Property A holds for this example.
34
Remarks: The sampling approach in our analysis uses the value
(Di − Qi )+ (Qj − Dj )+
ED
Pn
Pn
+
+
max
i=1 (Di − Qi ) ,
j=1 (Qj − Dj )
to obtain the variable sampling probability. This approach can gainfully employ the additional
information on the covariance structure of Di and Dj , and the total excess and unfilled demand
P
P
distribution ni=1 (Di − Qi )+ and nj=1 (Qj − Dj )+ to obtain reliable sampling probabilities.
There is a combinatorial analogue to the troops deployment problem. Suppose we distribute
2n units of troops uniformly on 2n nodes, with each location defended by exactly one unit.
Suppose also that each location will not be penetrable only if we have 2 units defending that
location. If enemy can attack up to n different locations, how would we design the reinforcement
network?
We color each node as red if enemy attacks that location, blue otherwise. The problem
can be reduced to random allocation of n red and n blue balls uniformly in the nodes of the
network. Let c(i) denote the color assigned to node i. Let e(G) denote the edge set in G. We
say that M ⊂ e(G) is a colored matching if it is a matching in G with
M = {(i, j) : c(i) 6= c(j), (i, j) ∈ e(G)}.
Let m(G) denote the cardinality of a maximum colored matching in G. Thus m(G) represents
the number of locations that can be defended in the network. Note that m(G) ≤ n for all
realizations of the color distribution, and E(m(G)) = n when e(G) = K(2n), the complete
graph on 2n nodes. Theorem 4 and Property A shows that cardinality of the edge set e(G) can
be reduced much further, while sacrificing only a little in value of E(m(G)).
Theorem 6 For all > 0, there exists n() > 0 such that for all n ≥ n(), there exists a graph
Gn with 2n nodes and O(n) edges, such that
n ≥ E(m(Gn )) ≥ (1 − )n.
Hence a sparse but yet near-to-optimal reinforcement network can be obtained with only a
small loss of locations, regardless of enemy’s course of action!
35
5.3
Sequencing with limited flexibility
Lahmar et al. (2003) considered the following sequencing problem in an automotive assembly
line: cars leaving the body shop on a moving line has to be resequenced prior to entering the
paint shop, in order to minimize the changeover cost at the paint shop. Given an initial ordering of jobs, they proposed a Dynamic Program to find the minimum cost permutation of the
sequence, so that each position is shifted not more than K1 positions to the right, and not more
than K2 positions to the left. The values K1 and K2 reflect the limited buffer space available
in the production plant, and reflect the level of flexibility within the plant. Re-sequencing is
needed to minimize, say, the changeover cost at the next station. A precise analytical measurement of the value of flexibility, however, is difficult to obtain, because the complexity of the
DP based algorithm depends on the value of K1 and K2 . Nevertheless, the numerical results
in this paper are quite convincing - the effect of flexibility diminishes rapidly, and most of the
benefits can be accrued at small values of K1 and K2 .
5.4
Call Center Staffing
Wallace and Whitt (2004) explores the use of chaining in call center staffing and skill chaining.
In a typical call center, the types of calls, and the skills involved in servicing these calls, vary
and it is not possible to train every staff to handle all calls. The authors showed that with
appropriate skill chaining, and in the scenario that the duration of service does not depend on
the call types or the agents serving it, then a simple routing policy, together with proper skill
chaining, can result in near optimal performance, even if service level constraints (eg. service
level guarantee for type k calls, or bounds on blocking probability) are taken into consideration.
This paper shows that it may be more worthwhile to pay attention to cross-training, rather
than to invest in complicated call routing software. The proper staffing level are next identified
via simulation based optimization.
5.5
Load Balancing
The concept of limited flexibility also has important application in load balancing in stochastic
network routing analysis (cf. Mitzenmacher, M. (1996)). This application follows from the
36
following interesting observation: Suppose n balls are randomly inserted into n bins, with each
bin chosen with probability 1/n. What is the expected number of balls in a bin with the
maximum load? It is not difficult to show that the bin with the maximum load should have
O(log(n)) balls with high probability. Suppose we modify the process in the following way- the
balls are inserted into the bins in sequential manner. Each ball gets to pick two bins randomly.
Depending on the load on the two bins at that time, the ball will be inserted into the bin with
the smaller load. It turns out that this simple modification reduces the peak load drastically
to O(log log n) with high probability! Having more flexibility does not help much either, since
log n
if we allow each ball to pick K bins randomly, then the peak load is reduced to O( log
log K ), for
any K ≥ 2. i.e., having more flexibility only reduces the peak load by a constant factor3 .
5.6
Other Applications
The flexibility strategy has been shown to be rather effective in various other areas such as
supply chain planning (Bish and Wang (2004)), queuing (Benjafaar (2002), Gurumurthi and
Benjaafar (2004)), revenue management (Gallego and Phillips (2004)), scheduling (Daniels and
Mazzola (1994), Daniels et al.
(1996), and Daniels et al.
(2004)), and flexible work force
scheduling (Hopp et al. (2004), Wallace and Whitt (2004), Brusco and Johns (1998)). For
instance, Hopp et al. (2004) observed similar results in their study of a work force scheduling
problem in a ConWIP (constant work-in-process) queuing system. By comparing the performances of “cherry picking” and “skill-chaining” cross-training strategies, they observed that
“skill-chaining”, which is indeed a kind of the “chaining” strategy, outperforms others. They
also showed that a chain with a low degree (the number of tasks a worker can handle) is able
to capture the bulk of the contribution from a chain with high degree.
3
This result may have ramification in appointment system design. Instead of allowing the patient to pick her
appointment day, the clinic may well benefit from balancing the load on different days by asking the patient to
come for appointment on a less congested day. Giving the patient just another choice may well be enough to
reduce the peak load by a dramatic amount!
37
6
Conclusion
We have described in this paper several settings where partial flexibility can be gainfully employed to reap the maximum benefit. We have also discussed its intimate connection with
graph connectivity problem, and have prescribed a condition under which partial flexibility will
be almost as effective as the full flexibility system. To wrap up the review, we would like to
highlight however some caveats to the adoption of partial flexibility strategy in practice.
6.1
Partially flexible structure has higher variance
Some authors have cautioned that partial flexibility has its limitation. Muriel et al. (2001)
showed that a surgery planning system (e.g. a hospital) with a limited flexibility structure
could lead to higher instances of the need for rescheduling and larger variability in resource
utilization. Bish et al. (2005) also indicated that in the make-to-order environment, partial
flexibility could introduce variability in the upstream of the supply chain, thus leading to higher
inventory cost, greater production variability and more complicated management requirement.
In another word, although partially flexible structure may attain nearly the same expected
performance as the fully flexible structure, the (random) optimal solution may exhibits more
variability compared to the optimal solution in a fully flexible system.
We illustrate this phenomenon using the simple process flexibility problem discussed earlier.
Consider Lemma 1, where the optimal flow for the full flexibility system is characterized by
x∗ij (D) =
D i Cj
,
Pm
Pn
max
i=1 Di ,
j=1 Cj
and the total max flow is merely
min
X
m
Di ,
i=1
n
X
Cj .
j=1
Consider the case when the network is balanced (n = m) with identical mean demand and
capacity at each node. In this case, the dedicated network (with one plant focusing on one
product) has total maximum flow of
n
X
min Di , Ci .
i=1
38
Although the fully flexible system has higher expected max flow, it can also have a smaller
variance, since
X
X
m
n
n
X
≤
var min
Di ,
Cj
var min Di , Ci
i=1
j=1
i=1
for many classes of demand distributions. For instance, when demand is negatively correlated,
P
Pn
so that m
i=1 Di =
j=1 Cj , then the full flexibility system has higher expected maximum flow
and lower variance than the dedicated system. Adding partial flexibility into the system will
not be able to reduce variability of the max-flow to the level attained by the full flexibility
system.
6.2
Coordination in partially flexible structure is hard
In our evaluation of the partial flexibility system so far, we have implicitly assumed that there
is a central planning agency which will dictate the optimal flow of supplies to match demands,
with complete information on demand. However, when demand signals are released in real
time and not synchronized, and when supply deployment decisions must be decided as and
when it arrives, the performance of the partial flexibility system can be far from that of the
full flexibility system. The latter does not suffer from the coordination problem, as the supplies
can be shipped to any demand destination. Hence the performance in a fully flexible system
depends only on the total demand and total supply, and is independent of the deployment
decisions in real time. This is unfortunately not the case for partial flexibility system.
We encountered this issue while working on a bread delivery problem. A group of bakeries
have agreed to donate their unsold breads at the end of each day to several old folks homes in
the area. See Figure 14 for the locations of the bakeries and the homes.
A group of volunteers, recruited under the “Food From The Heart” program, will deliver
the leftover breads each night. For ease of operations, each volunteer is in charged of one
route - from a bakery to an assigned home. The route is pre-determined and remains the
same each night (for ease of control), and does not change with the level of leftover breads
at each bakery. To minimize the mismatch between supply and demand, it will be ideal for
the volunteers to deliver the bread from one bakery to more than one home. While it is not
possible to have the flexibility to deliver the breads to all homes, it is conceivable that each
volunteer can be put in charged of two routes, and can decide which home to send the leftover
breads to based on supply information each night. The added flexibility in the operation will
39
Figure 14: Locations of Bakeries and Homes
allow the system to adjust the supply to each home appropriately to reduce the amount of
food wastages. Unfortunately, it is not easy to exploit this strategy at the operational level
because the bakeries close at different times, and the food must be handed over just before
the shops close for the night. Due to fund limitation, there is only a bare bone information
system installed to facilitate the communication between the volunteers and the manager of
the food delivery program. Currently, the volunteers can only communicate with a central
server via SMS messages. It is thus impossible to implement a centralized planning system
to coordinate the delivery operations, and to exploit fully the advantages offered by a partial
flexibility system.
This problem is also pertinent in the troops deployment problem, where the reinforcement
may have to be activated based on partial evolution of the battle on the ground. This problem
is further exacerbated by the fact that communication channels or situation reports may not
be reliable in actual combat. In this setting, coordinating the flow of goods (or forces) through
a partially flexible system is a lot more challenging.
6.3
Concluding Remarks
In summary, we have presented in this paper an overview of recent analytical results obtained for
the process flexibility problem. We show that the empirical observation that “a little flexibility
can enhance the performance of the system significantly” can be justified in a stylized model
based on maximum flow formulation in a two stage stochastic programming model. We have
40
also discussed several examples where the strategy can be put into effective use, and pointed
out some of the limitations associated with the operations of a partial flexibility system. The
challenge of coordinating the flow of goods in such system, based on incomplete information,
remains an outstanding open problem. We hope this review article can spur more research into
this furtile area, leading to useful implementation in the service and manufacturing industry.
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