1

Hierarchical Decomposition Heuristic for Scheduling:

Coordinated Reasoning for Decentralized and

Distributed Decision-Making Problems

Jeffrey D. Kelly* and Danielle Zyngier

* Industrial Algorithms LLC., 15 St. Andrews Road, Toronto, ON, M1P 4C3, Canada

jdkelly@industrialgorithms.ca

Abstract

This paper presents a new technique for decomposing and rationalizing large decision-making

problems into a common and consistent framework. We call this the hierarchical decomposition

heuristic (HDH) which focuses on obtaining "globally feasible" solutions to the overall problem, i.e.,

solutions which are feasible for all decision-making elements in a system. The HDH is primarily

intended to be applied as a standalone tool for managing a decentralized and distributed system when

only globally consistent solutions are necessary or as a lower bound to a maximization problem within a

global optimization strategy such as Lagrangean decomposition. An industrial scale scheduling example

is presented that demonstrates the abilities of the HDH as an iterative and integrated methodology in

addition to three small motivating examples. Also illustrated is the HDH's ability to support several

types of coordinated and collaborative interactions.

Keywords:

2

Decision-making, decomposition, scheduling, coordination, collaboration, hierarchical.

1. Introduction

Decomposition is a natural way of dealing with large problems. In the process industries

decomposition is found both in company organizational charts (executive-level to director-level to

management-level) and in process related decision-making elements such as production planning and

scheduling and process optimization and control systems. Other relevant instances of organizational

hierarchies can be found in Table 1.

Decision-making systems which contain at least two hierarchical levels can be decomposed into two

layers: the coordination layer and the cooperation (or collaboration) layer (Figure 1). The term

cooperation is used when the elements in the bottom layer do not have any knowledge of each other i.e.,

they are fully separable from the perspective of information, while collaboration is used when the

elements in the bottom layer exchange information.

Surprisingly, it is often the case in practice where the hierarchical systems depicted in Figure 1 do not

contain any feedback from the bottom layer to the top layer i.e., the up arrow in Figure 1. In a

management structure, when an executive sends targets1 to his/her directors (i.e., in a feedforward

manner), the directors are not necessarily expected to return a feasibility impact of the target (i.e., no

feedback). Instead, the target is usually assumed to be fixed and not subject to change. Similarly, when a

master schedule has been calculated for the entire process plant, the individual schedulers using their

collaborating spreadsheet simulator tools have very limited means of feeding or relaying information

back to the master scheduler that the master scheduling targets for the month period cannot be achieved.

The inconsistencies in the individual schedules are later apparent as infeasibilities and inefficiencies

such as product reprocessing or the inability to meet customer order due-dates on time. Therefore,

methods are needed that can coordinate and organize the different elements of the decomposed system

1 In this context a target is similar to an executive-order or a directive and later introduced as an

objective.

3

and to manage feedback between these elements. This paper presents a novel method for integrating and

interfacing these different decision-making layers with the goal of achieving global feasibility.

In the context of this paper, models may be referred to as global when they consider the full scope of

the decision-making system in terms of both the temporal and spatial dimensions of the system such as a

fully integrated oil-refinery planning its production or manufacturing months into the future. On the

other hand, local models only consider a sub-section of the decision-making system such as the gasoline

blending area or the first few time-periods of a multiple time-period planning model. In terms of the

level of detail, we classify the global models of the coordinator into decreasing order of detail as:

Rigorous: models that contain all of the known production constraints, variables and bounds

that exist in the local rigorous models of a system;

Restricted: models that are partially rigorous, i.e., they include all of the detailed constraints,

variables and bounds of sections of the entire system;

Relaxed: models that are not necessarily rigorous but contain the entire feasible region of all

cooperators with certain constraints or bounds that have either been relaxed or removed;

Reduced: models that are not rigorous and only contain a section of the feasible region of the

cooperators but usually contain, in spirit, the entire scope of the global problem.

A more detailed discussion on the effects of the different model types in the framework of the

hierarchical decomposition heuristic (HDH) is presented in Section 3. From a scheduling perspective,

master scheduling models are often relaxed, reduced or restricted global models in that they do not

contain all of the processing, operating and maintaining details of the production2 system. In the same

context, individual scheduling models are usually local rigorous models since they contain all of the

necessary production constraints in order to accurately represent the sub-system. Scheduling models

cannot usually be global rigorous models due the unreasonable computation times that this would entail.

The concepts of global versus local and reduced/relaxed/restricted versus rigorous models are of course

2 Production corresponds to the integration of process, operations and maintenance.

4

relative. That is, a local rigorous model from a scheduling perspective may be considered as global

relaxed from a process control perspective.

Another important concept is that of "globally feasible" solutions to a distributed decision-making

problem. In the context of this paper this term refers to a decision from the coordination layer which is

feasible for all cooperators in the cooperation layer. Globally feasible solutions indicate that a consensus

has been reached between the coordination and cooperation layers as well as amongst cooperators.

An important aspect of making good decisions at the coordination level is the correct estimation or

approximation of the system's capability3 i.e., roughly how much the system can produce and how fast.

If the capability is under-estimated (i.e., sand-bagged) there is opportunity loss since the system could in

fact be more productive than expected. On the other hand, if capability is over-estimated (i.e., cherry-

picked), the expectations will be too high and infeasibilities will likely be encountered at the cooperation

layer of the system. Concomitantly, in every decomposed system there also needs to be knowledge of

what constraints should be included in each of the coordination and cooperation layers and not just what

the upper and lower ranges of the constraints should be. Therefore, we introduce the notion of public,

private, protected and plot/ploy constraints which are defined as follows:

Public: constraints that are fully known to every element in the system (coordinator and all

cooperators). If all constraints in a system are public this indicates that the coordinator may be

a global rigorous model while the cooperators are all local rigorous models. In this scenario

only one iteration between coordinator and cooperators is needed in the system in order to

achieve a globally feasible solution since the coordinator's decisions will never violate any of

the cooperators' constraints;

Private: constraints that are only known to the individual elements of the system (coordinator

and/or cooperators). Private constraints are very common in decomposed systems where the

coordinator does not know about all of the detailed or pedantic requirements of each

3 Capability is defined as the combination of connectivity, capacity and compatibility information.

5

cooperator and vice-versa the coordinator can have private constraints not known to the

cooperators;

Protected: constraints that are known to the coordinator and only to a few of the cooperators.

This situation occurs when the coordinator has different levels of knowledge about each

cooperator in the system;

Plot/Ploy: situations in which one or more cooperators join forces (i.e., collusion) to fool or

misrepresent their results back to the coordinator for self-interest reasons.

The decomposition strategies considered in this paper address systems with any combination of

public, private and protected constraints. In other words, the cooperators are considered to be authentic,

able and available elements of the system in that (1) they will not deceive the coordinator for their own

benefit, (2) they are capable of finding a feasible solution to the coordinator's requests if one exists and

(3) that they will address the coordinator's request as soon as it is made.

Decomposition however comes at a price. Even though each local rigorous model in a decomposed or

divisionalized system is smaller and thus simpler than the global rigorous one, iterations between the

coordination and cooperation layers will likely be required in order to achieve a globally feasible

solution which may increase overall solution times. In addition, unless a global optimization technique

such as Lagrangean decomposition or spatial branch-and-bound search is applied there are no guarantees

that the globally optimal balance or equilibrium of the system will be found. Nevertheless, the following

restrictions on the decision-making system constitute very compelling reasons for the application of

decomposition strategies:

Secrecy/security: In any given decision-making system there may be private constraints (i.e.,

information that cannot be shared across the cooperators). This may be the case when different

companies are involved such as in a supply chain with outsourcing. Additionally the

cooperators may make their decisions using different operating systems or software so that

integration of their models is not straightforward. It is also virtually impossible to centralize a

system in which the cooperators use undocumented personal knowledge to make their

6

decisions (as is often the case when process simulators or spreadsheets are used to generate

schedules) unless significant investment is made in creating a detailed mathematical model.

Support: The ability to support and maintain a centralized or monolithic decision-making

system may be too large and/or unwieldy.

Storage: Some decision-making models contain so many variables and constraints that it is

impossible to store these models in computer memory.

Speed: There are very large and complex decision-making models that cannot be solved in

reasonable time even though they can be stored in memory. In this situation decomposition

may be an option to reduce the computational time for obtaining good feasible solutions.

The performance of any decomposition method is highly dependent on the nature of the decision-

making system in question and on how the decomposition is configured. Defining the decomposition

strategy can be a challenge in itself and of course is highly subjective. For instance, one of the first

decisions to be made when decomposing a system is the dimension of the decomposition: should the

system be decomposed in the time domain (Kelly, 2002), in the spatial equipment domain (Kelly and

Mann, 2004), in the spatial material domain (Kelly, 2004b), or in some combination of the three

dimensions? If decomposing in the time domain, should there be two sub-problems with half of the

schedule's time horizon in each one, or should there be five sub-problems with one fifth of the

schedule's time horizon in each one? Should there be any time overlap between the sub-problems? The

answers to these questions are problem-specific and therefore the application of decomposition

strategies requires a deep understanding of the underlying decision-making system.

1.1. Centralized, Coordinated, Collaborative and Competitive Reasoning

There is no question that the most effective decision-making tool for any given system is a fully

centralized strategy that uses a global rigorous model provided it satisfies the secrecy, support, storage

and speed restrictions mentioned previously. If that is not at all possible then decomposition is required.

If decomposition is needed the best decomposition strategy is a coordinated one (Figure 1) where the

7

cooperators can work in parallel. If a coordinator does not exist, the next best approach is a

collaborative strategy in which the cooperators work together obeying a certain priority or sequence in

order to achieve conformity, consistency or consensus. The worst-case decision-making scenario is a

competitive strategy in which the cooperators compete or fight against each other in order to obtain

better individual performance as opposed to good performance of the overall system (self versus mutual-

interest). This type of scenario is somewhat approximated by a collaborative framework in which the

cooperators work in parallel or simultaneously as opposed to in series or in priority as suggested above.

In this paper coordinated and collaborative decision-making strategies are discussed and are

demonstrated in the illustrative example.

Figure 2 provides a hypothetical value statement for the four strategies. If we use a "defects versus

productivity" trade-off curve, then for the same productivity (see vertical dotted line) there are varying

levels of defects, not only along the line, but also across several lines representing the centralized,

coordinated, collaborative and competitive strategies. These lines or curves represent an operating,

production or manufacturing relationship for a particular unit, plant or enterprise. Each line can also

represent a different reasoning isotherm in the sense of how defects versus productivity changes with

varying degrees of reasoning where the centralized reasoning isotherm has the lowest amount of defects

for the same productivity as expected.

Collaborative reasoning implies that there is no coordination of the cooperators in a system. As each

cooperator finds a solution to its own decision-making problem it strives to reach a consensus with the

adjacent cooperators based on its current solution. The solution of the global decision-making problem

then depends on all cooperators across the system reaching an agreement or equilibrium in a prioritized

fashion where each cooperator only has limited knowledge of the cooperator(s) directly adjacent to

itself. It is thus clear that collaborative reasoning reaches at best a myopic conformance between

connected cooperators. As previously stated, in cases where no priorities are established a priori for the

cooperators the collaborative strategy can easily become a competitive one since the cooperator which is

the fastest at generating a feasible schedule for itself gains immediate priority over the other cooperators.

8

Therefore the cooperators will compete in speed in order to simplify their decisions given that the

cooperator that has top priority is the one that is the least constrained by the remainder of the decision-

making system.

Coordinated reasoning on the other hand contains a coordination layer with a model of the global

decision-making system albeit often a simplified one by the use of a relaxed, reduced or restricted

model. As a result the conformance between the cooperators is reached based on a global view of the

system. This may entail a reduced number of iterations between the hierarchical layers for some

problems when compared to collaborative strategies, notably when the flow paths of the interconnected

resources of the cooperators are in a convergent, divergent and/or cyclic configuration as opposed to a

simple linear chain (i.e., a flow-shop or multi-product network).

Centralized systems can be viewed as a subset of coordinated systems since any coordinated strategy

will be equivalent to a centralized one when the coordination level in Figure 1 contains a global rigorous

model. Additionally, coordinated strategies are also a superset of purely collaborative systems since the

latter consist of a set of interconnected cooperators with no coordination. Collaboration can be enforced

in a coordinated structure by assigning the same model as one of the cooperators to the coordinator.

Centralized systems do not suffer from the fact that there are the arbitrary decomposition boundaries or

interfaces. This implies that in a monolithic or centralized decision-making problem it is the

optimization solver or search engine that manages all aspects of the cooperator sub-system interactions.

Examples of these interactions are the time-delays between the supply and demand of a resource

between two cooperators and the linear and potentially non-linear relationships between two or more

different resources involving multiple cooperators. In the centrally managed problem these details are

known explicitly. However in the coordinated or collaborative managed problems these are only

implicitly known and must be handled through private/protected information only.

1.2. General Structure of Decomposed Problems

As previously shown in Figure 1, decomposed or distributed problems consist of a coordination layer

and a divisionalized cooperation layer. By analyzing previous work on decomposition of large-scale

9

optimization problems, it is possible to identify two elements within the coordination layer: price and

pole coordination. Figure 3 shows the general structure of a decomposed system. In the coordination

layer there may be price and/or pole coordination where these two elements within the coordination

layer may also exchange information amongst themselves. The coordinator or super-project sends down

poles and/or prices to all cooperating sub-projects. Once that information is evaluated by the

cooperators, feedback information is sent back to the coordinator. It should be noted that the cooperators

do not communicate with each other, i.e., there is no collusion/consorting between them. This closed-

loop procedure continues until equilibrium or a stable balance point is reached between the two

decision-making layers. Most of the decomposition approaches in the literature can be represented using

the structure in Figure 3 as will be demonstrated in Section 2.

A pole refers to information that is exchanged in a decomposed problem regarding the quantity,

quality and/or logic of an element or detail of the system. The use of the word pole is taken from Jose

(1999) and has been extended to also include the logic elements of a system. The interchange of pole4

information between the decision-making layers may be denoted with what we call a protocol or parley

(Figure 4). These protocols enable communication between the layers and manage the cyclic, closed-

loop nature of decomposition information exchange. They represent and are classified into three distinct

elements as follows:

Resource protocols relate to extensive and intensive variables that may be exchanged between

sub-problems or cooperators such as flow-poles (e.g., number of barrels of oil per day in a

stream), component-poles (e.g., light straight-run naphtha fraction in a stream) and property-

poles (e.g., density or sulfur of a stream).

Regulation protocols refer to extensive and intensive variables that are not usually transported

or moved across cooperators such as resources but reflect more a state or condition such as

4 Another term for pole which is perhaps more descriptive is “peg” or pole-equilibrium-guess given that

the coordinator must essentially guess what pole values will be accepted by all of the interacting

cooperators.

10

holdup-poles (e.g., number of barrels of oil in a particular storage tank) and condition-poles

(e.g., degrees Celsius in a FCC unit).

Register protocols represent the extensive and intensive logic variables in a system and may

involve mode-poles (e.g., mode of operation of a process unit), material-poles (e.g., material-

service of a process unit) and move-poles (e.g., streams that are available for flowing material

from one process unit to the next). The former register protocols are intensive logic variables

whereas an example of an extensive logic variable would the duration or amount of time a

mode of operation is active for a particular unit.

Each protocol consists of a pole-offer to the cooperators that originates from the coordinator. The

cooperators return pole-obstacles (resource), offsets (regulation) or outages (register) to the coordinator

that indicates if the pole-offer has been accepted or not. If the cooperators accept the (feasible,

consistent) pole-offer then the pole-obstacles, -offsets or -outages are all equal to zero. The protocols

may be similarly applied to prices since these are related to adjustments of the quantity, quality and/or

logic elements of the system. From optimization theory it is known that the prices correspond to the dual

values of the poles. The adjustment of prices can be used to establish a quantity balance between supply

and demand (also known as the equilibrium price in economic theory) based on the simple economic

principle that to increase the supply of a resource its price should be increased and to increase the

demand of a resource its price should be reduced (Cheng et al. 2006).

1.3. Understanding and Managing Interactions between Cooperators

Obviously if each cooperator had no common, connected, centralized, shared, linked or global

elements then they would be completely separate and could be solved independently. Unfortunately

when there are common elements between two or more cooperators that either directly or indirectly

interact, completely separated solution approaches will only yield globally feasible solutions by chance.

A centralized decision-making strategy should have accurate and intimate knowledge of resource,

regulation and register-poles and how each is related to each other across multiple cooperators, even

11

how a resource-pole links to or affects in some deterministic and stochastic way a register-pole. As we

abstract a centralized system into a coordinated and cooperative system, details of synergistic (positively

correlated) and antagonistic (negatively correlated5) interactions are also abstracted or abbreviated. As

such, we lose information which is proxy’d or substituted by cooperator feedback along essentially three

axes: time, linear space and nonlinear space.

More specifically, the dead-time, delay or lag of how a change in a pole affects itself over time must

be properly understood given that it is known from process control theory that dead-time estimation is a

crucial component in robust and proformant controller design. Linear spatial interactions are how one

pole’s rate of change affects another pole’s rate of change which is also known as the steady-state gain

matrix defined at very low frequencies. Well known interaction analysis techniques such as the relative

gain array (RGA) can be used to better interpret the affect and pairing of one controlled variable with

another. In addition, multivariate statistical techniques such as principle component analysis (PCA) can

also be used to regress temporal and spatial dominant correlations given a set of actual plant data and to

cluster poles that seem to have some level of interdependency. Nonlinear spatial interactions define how

at different operating, production, or manufacturing-points, nonlinear effects exist which can completely

alter the linear relationship from a different operating-point. Thus, for strongly nonlinear pole

interactions, some level of nonlinear relationship should be included in the coordinator’s restricted,

relaxed or reduced model as a best practice. One such guideline for helping with this endeavor is found

in Forbes and Marlin (1994).

From an organizational and managerial perspective, understanding and managing interactions amongst

multiple diverse people across many departments and locations is a primary mandate of any

management structure. Too much antagonistic and too little synergistic interactions can lead to chaos,

low productivity and inefficiency just to name a few. Therefore a sound approach to handle this is to

first understand the interactions and then to manage them of which restructuring, removing and

5 As a point of interest note that the base of the word correlate is relate implying a relationship.

12

realigning organizational boundaries or barriers should be at the top of the list i.e., how cooperators are

divisionalized or separated across the resource, regulation and register-protocols to achieve maximum

performance.

This paper is structured as follows: first, the existing decomposition methods available in the literature

are transformed into the coordinated decomposition strategy shown in Figure 3. Second, the hierarchical

decomposition heuristic (HDH) is presented and its contributions to other decomposition strategies are

highlighted through an industrial illustrative example and three motivating examples.

2. Previous Decomposition Approaches

The need to solve increasingly larger scheduling problems has led to significant research output in the

area of decomposition. There are broadly three types of decomposition approaches classified with

respect to its drivers: pole and/or price-directed, weight-directed and error-directed. Most of the

decomposition literature contains pole- and/or price-directed approaches while process control literature

contains almost exclusively error-directed approaches. The remainder of this section provides more

details and instances of each decomposition approach.

In the same manner that optimization algorithms can be based on primal and/or dual information,

some decomposition strategies may be classified as pole and/or price-directed. The classical examples

are the Generalized Benders decomposition (Geoffrion, 1972) and Dantzig-Wolfe decomposition

(Dantzig and Wolfe, 1960) respectively. An increasingly popular price/pole method for solving large-

scale decision-making problems is Lagrangean decomposition (Wu and Ierapetritou, 2003; Karuppiah

and Grossmann, 2006). This method finds the global optimal solution of the decomposed system within

a certain tolerance and can be represented in the general decomposition structure as shown in Figure 5

(for a maximization problem).

In this method the cooperators represent temporally and/or spatially decomposed sub-problems. The

so-called linking constraints between the cooperator sub-problems are dualized in the objective function

of the coordinator with the use of Lagrange multipliers (λ) which are updated at each iteration by using

what is known as a sub-gradient optimization method. Even though this constitutes one of the most

13

efficient global optimization strategies for decomposed problems to date, there are a few drawbacks that

limit its implementation particularly with respect to solution times. Information regarding the upper-

bound (UB) of the global rigorous (maximization) decision-making problem is needed in order to

perform the update of the Lagrange multipliers. Computational times may increase significantly if the

cooperators are themselves individually difficult to solve. Additionally, there must be a solution at each

iteration which is globally feasible over all cooperators in order to calculate the lower bound which for

practical problems is not an insignificant task. It should therefore be clear that Lagrangean

decomposition benefits from improved strategies for obtaining this lower bound, i.e., strategies which

will yield a globally feasible solution such as the one presented in this work. In Karuppiah and

Grossmann (2006) solutions which were not globally feasible for all cooperators were eliminated from

the solution set and the search continued for other solutions. Our paper presents a different strategy for

obtaining globally feasible solutions that uses pole-obstacle, offset and outage information from all

cooperators in order to resolve conflict and to achieve consensus in subsequent iterations. Note that the

coordination layer in Lagrangean decomposition contains both a pole-coordinator which calculates the

new upper bound (of the maximization problem) and a price-coordinator which calculates the new

Lagrange multipliers (also known as marginal-costs or shadow-prices) for each cooperator (Figure 5).

Decomposed systems may also be managed though price/pole-directed strategies such as auctions

which are also used in multi-agent systems6. According to economic theory, if the demand for a certain

resource (pole) by a consumer (i.e., a downstream cooperator) is greater than expected, the price of that

resource must be increased in order to reduce its demand. On the other hand, a larger supply (i.e., from

an upstream cooperator) than expected for a given resource entails a reduction in its price in order to

reach price equilibrium. This is the basis of the auction-based decomposition method found in Jose

(1999). The schematic for what they call a slack resource auction as a coordinated strategy can be seen

in Figure 6.

6 Multi-agent systems require either an auctioneer or an administrator which act as the coordinator with

an individual agent, usually autonomous, representing a cooperator.

14

Jose and Ungar (2000) showed that an auction can find a set of prices corresponding to the global

optimum of the overall problem if it has separable and convex sub-problems with a single time-period.

In spite of defining the directionality of the price adjustment as a function of the pole-obstacles, they did

not explicitly state how to calculate the step-size of the pole and price increase or decrease mechanism.

Furthermore, since there is the exchange of prices, pole-offers and pole-obstacles (offsets, outages), this

strategy actually corresponds to a combined price/pole error-directed strategy (Jose and Ungar, 1998).

A similar strategy for price adjustment can be found in Cheng et al. (2006) where this price

adjustment is based on price-elasticity, i.e., the sensitivity of the prices with respect to the poles (Figure

7). In this figure, pole-observations correspond to the level of the pole-offer that each cooperator can

achieve. The method presented in Cheng et al. (2006) however suffers from one of the known

drawbacks of sensitivity analysis which is the assumption of a fixed active set or basis to keep the

elasticity information valid which can be excessively restrictive for some decision-making systems such

as scheduling problems. Cheng et al. (2006) studied process control problems which are assumed to be

linear and convex whereas production scheduling problems are inherently non-linear and non-convex.

The second class of decomposition methods is weight-directed strategies. An example is the iterative

aggregation and disaggregation strategy found in Jörnsten and Leisten (1995) where the constraints and

variables of the cooperators are aggregated in the coordinator. The coordinator then optimizes with this

new aggregated model and finds the next set of pole-offers that are sent to the cooperators (Figure 8). It

is also the responsibility of the coordinator to re-calculate the variable and/or constraint weights based

on the solutions from the cooperators. It should be noted that the constraint weights can be alternatively

applied to the pole-obstacles (offsets, outages) through the use of artificial or slack variables. Another

example of a somewhat related weight-directed decomposition strategy applied to large scheduling

problems can be found in Wilkinson (1996) who performed temporal aggregation on the constraints and

variables.

The third decomposition approach is based on error-directed strategies. In the context of our paper,

error-directed strategies imply the feedback of model parameters or biases (e.g., pole-obstacles, offsets,

15

outages and pole sensitivity information). One of the first error-directed approaches for decomposing

batch scheduling problems was proposed by Bassett et al. (1996). In their work they presented a method

for solving large-scale batch scheduling problems by decomposing the system into a planning model

(coordinator) and several scheduling sub-models (cooperators). The dimension of the decomposition

was mainly temporal and partially spatial i.e., the scheduling model in each cooperator considered a

shorter time-horizon than the coordinator. The coordinator consisted of a globally relaxed model

whereas each cooperator corresponded to a locally rigorous model. The strategy of Bassett et al. (1996)

can be described in the general coordinated decomposition structure as shown in Figure 9.

As can be seen in Figure 9, there is no price coordination in this strategy. Equilibrium is reached

solely by the interchange of pole-related information. Integer and capacity cuts are sequentially added to

the coordinator's model whenever nonzero pole-obstacles7 are generated by the cooperators. However,

this assumes that the cooperators' constraints are protected i.e., the cooperators must be willing to share

or expose these more detailed constraints with the coordinator when required.

Instead of only indicating to the coordinator that infeasibilities (obstacles, offsets, outages) have been

encountered by the cooperators, information from the cooperators can sometimes be used directly in the

coordinator model as parameters. This is the basis of another error-directed approach suggested by

Zhang and Zhu (2006) where intensive quantity (i.e., process yield) information generated by the

cooperators is used in the coordinator model. Interestingly, process yields also correspond to pole

sensitivity information since yields alter the rate at which the flow of a stream changes with respect to

the throughput of the unit-operation. In this approach yield constraints may be completely private, i.e.

only known to the cooperators and not to the coordinator given that the details of how these yields are

calculated are the responsibility of the cooperators and may be quite non-linear. It should also be noted

that similarly to the decomposition approach of Cheng et al. (2006), sensitivity information is used as

7 Bassett et al. (1996) refer to pole-obstacles as "excess" variables.

16

feedback in the communication between the decision-making layers but referring to different elements:

poles in Zhang and Zhu (2006) and prices in Cheng et al. (2006).

Similar to the previous approach of Bassett et al. (1996), the strategy presented by Zhang and Zhu

(2006) does not include any price coordination (Figure 10). Yet, the Zhang and Zhu (2006)

decomposition approach is only applied to a multi-site process optimization problem with a single time-

period which significantly decreases the complexity compared to multi-time-period scheduling

problems. Its effectiveness for multi-time-period and multi-resource scheduling problems remains

unverified especially when multiple time-periods introduces degeneracy and can increase convergence

times as shall be seen later.

3. Hierarchical Decomposition Heuristic (HDH) Algorithm

The hierarchical decomposition heuristic (HDH) achieves a coordinated or hierarchical equilibrium

between decision-making layers. The coordinator is responsible for enabling conflict resolution,

consensus reconciliation, continuous refinement and challenge research which will be explained in

more detail. An overview of the algorithm for the proposed error-directed HDH is seen in Figure 11.

The individual steps of the algorithm presented below will provide the details of the coordinated

reasoning procedure. It is assumed throughout the algorithm description that a scheduling problem is

being decomposed and it is solved with mixed-integer linear programming (MILP).

3.1. Step 1. Solve coordinator problem

Solve the relaxed/reduced/restricted problem in the coordination layer to provably-optimal if possible

or until a specified amount of computational time has elapsed. The primary assumption is that there are

common, shared or linking poles between the coordination and the cooperation layers managed through

the protocols. The initial lower and upper bounds on the quantity, quality and/or logic variables for the

pole-offers (i.e., for the resource, regulation and register-protocols) are exogenously supplied by the

modeler or user when initializing the algorithm. Note that for each resource, regulation and register-

17

protocol there will be as many poles as the number of time-periods in the decision-making or scheduling

horizon.

UB

k,ik,i

LB

k,i PPP 11 , NPi 1 (1)

The bounds or pole-outlines in constraint (1) are specified for the number of poles or equivalently the

number of protocols. The k subscript refers to the current iteration where the k-1 refers to the previous

iteration. These lower and upper pole-outlines are only managed by the coordinator and change at each

iteration if there are non-zero pole-obstacles, offsets and/or outages.

3.2. Step 2. Dispatch the solution of the coordinator problem

From the solution of the coordinator problem, quantity, quality and/or logic pole-offers Pi,k are

obtained and sent to all appropriate cooperators. Any variable in the coordinator and cooperator

problems that are not involved in the protocols are only known to themselves and are not externalized

(i.e., they are essentially private variables).

3.3. Step 3. Solve all cooperator sub-problems in parallel

Solve all local rigorous models using the pole-offers from the coordinator. In every cooperator two

pole-obstacles, offsets and outages are attached to each pole-offer which are called pole-obstacle, offset

and outage shortage and surplus ( k,sp,iP and k,sp,iP respectively). In the following equations the

expression "i sp" denotes that pole i belongs to a particular cooperator sp of which for resource-

protocols the pole-offer is sent to two cooperators (i.e., the upstream or supplier cooperator and the

downstream or demander cooperator).

k,ik,sp,ik,sp,ik,sp,i PPPP , spi,NSPsp,NPi 11 (2)

The pole-obstacle, offset and outage shortage and surplus variables i.e., k,sp,iP and

k,sp,iP must be

added to the objective function of the individual cooperators. These variables are minimized using a

suitably large weight usually an order of magnitude larger than any other term in the objective function.

Note that the shortage and surplus variables are mutually exclusive or complements to one another. In a

18

coordinated strategy the cooperators can be solved in parallel which can significantly decrease the

overall computational effort of the HDH if several CPUs are available. Additionally, the cooperators

may be solved until provably-optimal or they may be stopped after a certain amount of time has elapsed.

3.4. Step 4. Conflict Resolution

Recover and retrieve the solution of the pole-obstacles, offsets and outages of all cooperators and re-

calculate or re-adjust the pole-outlines for the pole-offers in the coordinator problem (constraint (1))

before the current iteration's coordinator problem is optimized. The pole-obstacles, offsets and outages

are the key to the HDH strategy. If all of the pole-obstacles, offsets and outages are zero then a globally

feasible solution has been found given that the coordinator and cooperators are all feasible. Else, the

HDH iterations must continue. In cases where the pole-offer Pi,k from the coordinator is sent to more

than one cooperator as is the case for a resource-protocol, the largest pole-obstacle (in absolute terms)

across the cooperators affected by that specific resource i is used.

111 k,sp,isp

k,sp,isp

LB

k,i

LB

k,i PmaxSSLPmaxSSLPP , (3a)

111 k,sp,isp

k,sp,isp

UB

k,i

UB

k,i PmaxSSUPmaxSSUPP , spi,NPi 1 (3b)

In equations (3a) and (3b) there are two lower and upper step-size adjustment parameters SSL and SSU

respectively which provide the added flexibility of defining different rates of change for the lower and

upper bounds in the coordinator problem. In the motivating examples and in the illustrative industrial

example these parameters were defaulted to a value of 1. The operator .maxsp

implies the maximum

over all sp. Alternatively, for some decomposed problems, the aggregating operator .sp

sum may be used

instead.

The rationale for the re-adjustment of the pole-outlines at every iteration is essentially taken from the

notion of capacity planning or what is also known as capacitating. The idea is for the lower level

elements (cooperators) to feedback to the higher level element (coordinator) what they are capable of

doing in terms of their achievable capacity or capability.

19

In real-life decision-making problems, short-term decisions are generally more important than long-

term decisions due to the uncertainty in decision-making systems. A long decision-making horizon

increases the chances of unforeseen events to happen and possibly change future decisions. Therefore if

deemed necessary the pole-obstacles, offsets and outages can be temporally and spatially prioritized

(weighted) in the cooperators' objective functions in order to account for uncertainty. Additionally the

stopping criterion of the HDH may incorporate an equivalent prioritization strategy. Three motivating

examples are now presented to further illustrate the details of the conflict resolution step.

Motivating Example 1

The first example of conflict resolution can be seen in Figure 12a. In this figure the original (private)

bounds for R1 and R2 in both the coordinator and the cooperators are shown. The subscripts 0, 1 and 2

refer to the resources in the coordinator, cooperator 1 and cooperator 2 respectively. The objective of the

coordinator is to maximize the usage of R1 and R2 in the system. Note that R1 and R2 may correspond

to either two different resources exchanged by the cooperators or to a single resource over two time-

periods. In addition note that there is a discrepancy in the capacity for cooperator 2 R12. Due to a

hypothetical breakdown in cooperator 2, R12 cannot be processed and has an upper bound of 0 units but

this is not known to cooperator 1 and is only known to the coordinator after the addition of feedback

yielding updates to the pole-outlines.

The results for this example can be seen in Table 2. Since the objective of the coordinator is to

maximize the usage of R1 and R2 the pole-offers in the first iteration are 10 and 5 respectively. Since

the pole-offer for R1 exceeds the upper bounds of this resource by 5 units (cooperator 1) and by 10 units

(cooperator 2) two pole-obstacles are generated for the R1 pole-offer. The R2 pole-offer is feasible for

both cooperators and therefore no pole-obstacles are generated for this pole-offer. By applying the

conflict resolution step, the bounds of R10 are adjusted by 10 units (the maximum value between 5 and

10, the two pole-obstacles) whereas the bounds of R20 remain unchanged since there were no pole-

obstacles for this resource. By maximizing R1 and R2 with the new bounds, the new pole-offers for R1

20

and R2 are 0 and 5 respectively which is a globally feasible solution, i.e., no pole-obstacles are

generated for the pole-offers.

Motivating Example 2

The second example is illustrated in Figure 12b. The only difference between motivating examples 1

and 2 is the coordinator model. In motivating example 2 the coordinator must satisfy an additional

inequality constraint involving R10 and R20 which in this case makes the coordinator problem

degenerate.

Table 3 shows the results for this example. Initially the pole-offers for R1 and R2 are 5 and 5

respectively. These pole-offers satisfy all cooperator constraints except for the R1 bound in cooperator 2

which is exceeded by 5 units. For the following iteration the bounds on R10 are shifted by 5 units and

again 5 and 5 are the new pole-offers. The same pole-obstacle exists for R1 in cooperator 2 and the

bounds on R10 are once more shifted by 5 units. In the third iteration the new pole-offers are 0 and 5

which are globally feasible. It is interesting to note that because of the introduction of degeneracy in this

problem the number of iterations required to arrive at a globally feasible solution has increased.

Motivating Example 3

In this third example (Figure 12c) a disjunction or discontinuity is introduced in cooperator 1. In this

case the upper bound of R2 can be one of two functions of R1 depending on the value of R1. The

iterations for this example can be seen in Table 4. Note that the method is able to find a globally feasible

solution in three iterations even for a degenerate problem with disjunctions. This is of particular

importance in mixed-integer linear programming (MILP) problems where the integer variables express

disjunctions in the model and there is a significant amount of degeneracy especially in the time domain.

3.5. Step 5. Consensus Reconciliation

Recover from the pole-obstacles, offsets and outages, which are plus and minus deviations from the

pole-offers as determined by the cooperator optimizations, the level of the pole-offer that each

cooperator can admit or achieve – this is also referred to as the pole-observation.

21

In order to accelerate the convergence of the HDH to a globally feasible or consistent solution, a term to

minimize the deviation of the current iteration's pole-offers from the consensus between the cooperators'

pole-observations in the previous iteration is added to the objective function of the coordinator. The

consensus between the cooperators is reached by essentially averaging adjacent cooperator pole-

observations (sp and sp') which refer to the same resource i in the coordinator (pole-opinion). This is

represented by the second, averaged term in the left-hand side of equation (4). For regulation and

register-protocols there is only one cooperator involved which is different from the resource-protocol

mentioned which has both upstream and downstream cooperators.

k,ik,i

k,'sp,ik,sp,i

k,i PPPP

P2

11, 'sp,spi,NPi 1 (4)

The pole-outliers k,iP and k,iP then must be added to the objective function of the coordinator using a

pole-outlier weight wi which is calculated as the maximum value of the pole-outlier shortage and surplus

from the previous iteration shown in equation (5).

11 k,,ik,,ii P,Pmaxw , NPi 1 (5)

At the first iteration the weights wi are set to zero which basically removes the pole-outlier minimization

for that iteration. The reason for weighting the pole-outliers by the previous iteration's maximum value

is to give more relative weight to those pole-outliers that are deviating more from the consensus or

average. If there is consensus in the previous iteration for a particular resource then the weight is zero

for the following iteration. Therefore the new objective function term to be minimized for consensus

reconciliation in the coordinator is below:

NP

i

k,ik,ii PPw1

(6)

The rationale for achieving a consensus for each resource, regulation and register-protocol is related to

the notion of harmonizing pole-opinions between one or more cooperators. Given the overriding focus

of the HDH on global feasibility, finding solutions that are consistent amongst each cooperator and the

coordinator is aided by minimizing pole-outliers between the interested parties. Both Steps 4 and 5

22

represent the tradition of solving combinatorial problems with a method called greedy-constructive

search which has the charter of finding feasible solutions at the start of the search. The notion of local-

improvement search is the guiding principle for the following two coordinated reasoning methods.

3.6. Step 6. Continuous Refinement:

Steps 1 to 5 are repeated until the pole-obstacles, offsets and outages of all cooperators are zero, i.e.,

until a coordinated or hierarchical equilibrium is reached. If a different solution is desired after reaching

the equilibrium, the incumbent solution (pole-origin) and all previous incumbent solutions can be

eliminated by using the strategy outlined in Dogan and Grossmann (2006) in their constraints (36) and

(37). The algorithm may be re-started by keeping the same pole-outlines LB

k,iP and UB

k,iP as in the current

iteration or by re-initializing the pole-outlines to their original starting values or to some other value.

This reasoning element may be considered as an improvement stage of the algorithm similar to the

notion of local-improvement search found in meta-heuristics such as simulated annealing and tabu

search.

3.7. Step 7. Challenge Research

There may be cases in which there is additional information regarding a target value for the pole-

offers as specified by some higher-level system, that is, above the coordinator layer (i.e., from the

executive to the director). For example, if the coordinator is the master scheduler and the cooperators are

the individual schedulers then the planner can provide these pole-objectives (targets). The pole-

objectives have two benefits. First, they can decrease the number of iterations given that if the plan is

achievable (thus not necessarily over-optimized or “cherry-picked”) then the pole-objectives can help

with maneuvering the coordinator to find globally feasible solutions faster. In addition, these pole-

objectives can facilitate finding better globally feasible solutions given that the planner's goal is to push

the system to regions of higher efficiency, effectiveness and economy. The use of pole-objectives is also

a similar idea to a setpoint that is sent from the plant-wide optimizer to individually distributed model

predictive controllers (MPC) (Lu, 2003).

23

When applying the challenge research step the additional variables representing the deviation of the

pole-offers from the pole-objectives are called pole-opportunities and are determined as:

k,ik,iik,i PPPPPPP , NPi 1 (7)

The additional objective function term in the coordinator is:

NP

i

k,ik,ii PPPPww1

(8)

where the pole-opportunity weights iww are determined according to the same strategy as the pole-

outlier weights iw . We liken the challenge research phase to an aiming or advancement aspect of the

reasoning which is also related to a local-improvement search.

The HDH can be represented in a block diagram in a similar fashion to model predictive controllers

(MPC) of which MPC has three forms: linear, non-linear and hybrid. Figure 13 shows the coordinator as

the MPC's feedforward engine (or economic/efficiency optimizer) that sends setpoints (pole-offers) to

the cooperators which correspond to the MPC's feedback engine (or actual plant). The measured

variables of the system are the pole-observations while the predicted variables are the setpoints. The

difference between the predicted and the measured variables corresponds to what is known as the MPC's

bias terms (pole-obstacles, offsets and outages) which are fed back to the coordinator for re-

optimization. If pole-objectives exist, they are used within the coordinator's optimization problem.

When limited feedback information such as the bias updating strategy in MPCs or linear real-time

optimizers (RTOs) is used to update model parameters there is a possibility that the closed-loop system

will not converge to the optimal operating policy in the plant even for linear systems with no structural

mismatch. Forbes and Marlin (1994) were the first to demonstrate that if the parameter values of the

left-hand side constraint coefficients deviates enough from the perfect values (from the plant) the

closed-loop system may converge to a sub-optimal corner point or active set. Due to the previous

relationship shown between the HDH and MPC the conclusions in Forbes and Marlin (1994) also

indicate that since there will usually be significant parametric and structural mismatch between the

coordinator and cooperators the system may converge to sub-optimal solutions. Mismatch primarily

24

originates due to the existence of private constraints. Zyngier (2006) developed methods that monitored,

diagnosed and enhanced the performance of such types of closed-loop optimization systems by (1)

assessing the potential effects of parameter uncertainty on the objective function (determination of a

"profit gap"), (2) detecting the main parametric contributors to the profit gap using a novel sensitivity

analysis method that did not require the assumption of a fixed active set, and (3) reducing the profit gap

by applying designed experiments to the plant. The work in Zyngier (2006) could potentially be

applicable to the HDH to gauge the difference in terms of the objective function value (profit, cost, etc.)

between using globally rigorous and relaxed or reduced coordinators. Besides, the diagnostics section of

the work would focus model improvement efforts on the most significant section of the model in terms

of objective function value.

It is also possible to align the HDH directly into the related continuous-improvement philosophies or

paradigms of the Deming Wheel, Shewhart Cycle and Kaizen. We refer to this as the plan-perform-

perfect-loop or P3-loop shown in Figure 14 which has both feedforward and feedback components

(Kelly, 2005a). In the context of the HDH, both the plan and perfect functions are included in the

coordinator while the perform function embodies the cooperators. There is feedforward from the plan

function to both the perform and the perfect functions and there is a feedback loop from the perform

function through the perfect function back to the plan function. It should be emphasized that the perfect

function’s feedback to the plan function can take several forms of which updating, re-calibrating or re-

training the decision-making model in terms of both structure and parameters inside the plan function

should always be considered inside the perfect function instead of simply resetting its bounds or

constraint right-hand-sides (i.e., capacity planning or re-capacitating). The dotted rectangle represents an

input-output relation for any manufacturing or production plant, site, enterprise or system in terms of

how orders are inputted and how objects (material, information and work products) are outputted from

the system. The P3-loop is also a useful tool in analyzing and interpreting the system dynamics in terms

25

of the many contributors to the variability8 of lead-time, cycle-time, dead-time or delay of when an order

is placed by an external or internal customer and when the objects are finally received by the customer

some time into the future. In other words, the dotted rectangle is the overall system black-box or block.

Finally, the P3-loop which is essentially an organizational or managerial archetype exists at some level

or degree in all enterprises whether in the process, discrete-parts or service industries and is also at the

heart of the HDH.

The HDH algorithm outlined in Steps 1 to 7 is a method for obtaining globally feasible solutions.

While the HDH can be used independently when any globally feasible solution is equally valuable to the

decision-making system, it can also be embedded in a global optimization framework such as

Lagrangean decomposition as a means of obtaining the globally feasible lower bound of a maximization

problem at each iteration of the overall algorithm (Wu and Ierapetritou, 2003; Karuppiah and

Grossmann, 2006).

As previously mentioned it should be highlighted that the coordinator used in the HDH may be any

combination of global or local relaxed, reduced, restricted or rigorous model. Depending on the system,

the coordinator may have detailed information about one or more of the cooperators, indicating that all

of those cooperators' constraints are essentially public. In these cases, it is not necessary to include those

cooperators explicitly as sub-problems in the HDH since the pole-offers that are made by the

coordinator will always be feasible with respect to the cooperators which only contain public

constraints. This introduces a significant amount of flexibility in handling decomposed systems since

there is more freedom to manage the trade-off between the computational complexity of solving a single

global rigorous model and the several iterations resulting from the decomposition of the system into

multiple local rigorous models.

Illustrations of the joint feasibility regions of the coordinator and cooperators are shown in Figure 15.

If using a global rigorous model as a coordinator (Figure 15a) the pole-offers that are calculated by the

8 It is well-known in queuing theory that variability causes congestion and congestion causes a reduction

in capacity or capability.

26

coordinator will always be feasible for all cooperators and thus the HDH will converge in a single

iteration. On the other hand if the coordinator is partially rigorous or restricted it contains the entire

feasible solution of a sub-section of a decision-making system (i.e., cooperator 2 in Figure 15b). In this

case explicitly including cooperator 2 as an element in the cooperation layer in the HDH is optional

since all pole-offers from the coordinator will be feasible for cooperator 2. If the coordinator is a global

reduced model it may not include part of the joint feasible solution space of the cooperation layer

(Figure 15c). When the feasible set of the coordinator model contains the feasible sets of all cooperators,

i.e., when the coordinator is a global relaxed problem, the coordination is always feasible when all

cooperators are feasible (Figure 15d).

Figure 16 illustrates how the HDH can be expressed in the general decomposition framework. It

should be noted that although we use the term "weight adjustment" this is not related to the weight-

directed method previously described. The weight adjustment method is used to determine the objective

function weights for Steps 5 and 7 iw and iww respectively.

The following section presents the application of the HDH to an industrial off-site and on-site storage

and handling system. Three different decomposition strategies are applied, namely a coordinated

strategy using a relaxed coordinator model, a coordinated strategy using a restricted or partially rigorous

model and a collaborative strategy. The results are then compared with the centralized decision-making

strategy of solving the global rigorous coordinator model. Finally, the conclusions are summarized and

directions for future research are presented.

4. Illustrative Example: Off-Site and On-Site Storage and Handling System

The layout and connectivity of off-site and on-site storage tanks, pumping and blending units at a

petrochemical process industry in Asia can be seen in Figure 17. In this figure the diamonds represent

perimeter-units through which material enters or leaves the system, triangles represent pool-units that

can store material for indefinite amounts of time, and rectangles represent pipeline-units and continuous-

process-units (blenders). This system has been modeled under the unit-operation-stock superstructure

27

(UOSS) and quantity-logic quality paradigm (QLQP) described in Kelly (2004a), (2005b) and (2006)

and using the logistics inventory modeling details shown in Zyngier and Kelly (2007). Material arrives

at the off-site terminal tanks through three supply perimeter-units (S11, S12 and S13) and is directed to

one of six pool-units (T11-T16). The material is then taken through one of two pipeline-units (P1 or

P12) to the on-site facilities. After storage in pool-units T23-T24 the material is then blended in the B21

and B22 blending process-units, stored in pool-units T25 or T26 and then sent to one of two demand

perimeter-units (D21 or D22). The objective function of the scheduling problem is to maximize profit

which in this case is only a function of D21 and D22 ($10 for every kMT of material allocated to a

demand perimeter-unit), i.e. there are no costs associated with S11, S12 S13 and S21 which is not

unreasonable when feedstock is supplied at fixed amounts according to a fixed delivery schedule. The

scheduling horizon is 8-days with 24-hour time-period durations.

This system is naturally decomposable into the off-site’s area (Figure 18) and the on-site’s area (Figure

19). Note that this illustrative example demonstrates two different strategies for handling resource-

protocols between cooperators; no regulation and register-protocols are used in this example. The first

strategy is to decompose the resource-protocol across a unit which is done on pipeline-unit P1. The

second strategy is to decompose the system across a connection between units (P12 and the pool-units)

which requires us to model an additional hypothetical process-unit (P22) which does not exist physically

in our multi-site system.

In addition to the centralized (non-decomposed) approach, three different decomposition strategies

were applied to this system: a collaborative strategy and two coordinated strategies using a relaxed and a

partially rigorous coordinator model (Figure 20). The centralized approach (Figure 20a) implies that the

coordinator is a global rigorous model and therefore the HDH is guaranteed to converge in a single

iteration although it may take an unreasonable amount time to solve. Since all pole-offers will be

feasible for the cooperators it is not necessary to include the cooperators explicitly in the algorithm. In

the collaborative strategy (Figure 20b) no information other than the resource-poles is shared across the

collaborators. This strategy can be easily inserted into the HDH structure by modeling a coordinator that

28

has perfect knowledge about the constraints at the on-sites (cooperator 2) but no knowledge about the

off-sites. Therefore the coordinator can be interpreted as a local rigorous model of the on-sites only. In

the coordinated strategy with a relaxed coordinator (Figure 20c), each cooperator is only willing to share

its connectivity information (i.e., number of units and their pumping interconnections) and the upper

bounds on the pool-unit lot-sizes but no additional operational logic details such as settling-times or

lower bounds on pool-unit lot-sizes. The coordinated strategy using a partially rigorous coordinator

model (Figure 20d) contains all of the rigorous model information for cooperator 2 and only the

connectivity information of cooperator 1 in the coordination layer. Since the rigorous model for

cooperator 2 exists within the coordinator it is not necessary to include this cooperator explicitly in the

HDH since all of the pole-offers will necessarily be feasible for cooperator 2 and no feedback is

required.

The details of the modeling can be found in Table 5 to Table 13. It should be mentioned that for this

illustrative example only conflict resolution and consensus reconciliation coordinated reasoning tactics

were applied. Continuous refinement and challenge research were not used given that there were no

available higher or upper-level objectives/targets for the resource-protocols and we felt it was necessary

to show the results without further complicating the problem with incumbent elimination quantity and

logic cuts. Hence, only the greedy-constructive elements of the HDH search are presented in the

illustrative example but not the local-improvement elements.

In Table 5 to Table 8, shut-down-when-below (SDWB) refers to the logistics constraints (involving

both quantity and logic variables) of only allowing the shut-down of an operation-mode when below a

certain threshold value. The SDWB constraint can be modeled as follows (Zyngier and Kelly, 2007):

NTtandNPLplyyXHXHXHxh tpltplplpl

SDWB

pltpp ..1 ..1 ,0,1,

maxmax

1, (9)

where t,ppxh refers to the hold-up of physical unit pp at time-period t, t,ply is the logic variable for the

set-up of logical unit pl at time-period t, max

plXH corresponds to the maximum hold-up or inventory of

logical unit pl andSUWB

plXH is the start-up-when-below lot-size of logical pool-unit pl.

29

The fill-draw delay (FDD) (Table 5 to Table 8) represents the timing between the last filling sub-

operation and the following drawing sub-operation. An FDD of zero indicates a standing-gauge pool-

unit. In the following constraints i and j indicate the specific inlet- and outlet-ports attached to the units

that are downstream and upstream of the logical pool-unit respectively. The other ipl and jpl subscripts

refer to the outlet- and inlet-port on the logical pool-unit itself.

NTttt|NT..t..tt,yy min

pl,FDDt,ipl,joutttt,jin,jpl , 101 (10)

Constraint (10) stipulates that a draw cannot occur until after the lower FDD elapses. There is also an

upper bound of FDD which indicates the maximum duration between the last filling and the following

drawing of stock out of the logical pool-unit as indicated below. For a more thorough description of

quantity, logic and logistics constraints encountered in inventory models of process industries the reader

should refer to Kelly (2006) and to a Zyngier and Kelly (2007).

NTttt|NT..t,yy max

pl,FDD

tt

ttt,jin,jplt,ipl,jout

maxpl,FDD

101

(11)

In Table 9 to Table 13 additional quantity, logic and logistics details are presented. Table 9 and Table

10 provide the semi-continuous flow logistics constraints for the outlet- and inlet-ports respectively

which model flows that can either be zero or between their lower and upper bounds within a time-

period. Table 11 shows the opening inventories, hold-ups or initial lot-size amounts in the pool-units at

the start of schedule with initial mode-operation logic setups for whether the pool-unit is in material-

operation or service A, B, C or ABC. The material balance for pool-units can be expressed as:

NT..txfxfxhxh t,joutt,jint,ppt,pp 11 (12)

where t,jinxf and t,joutxf refer to the flows entering or leaving physical unit pp at time-period t

respectively. Table 12 specifies the timing of what we call inverse-yield orders. Inverse-yields are really

recipe, intensity or proportion amounts of how much of stocks A, B and C should be mixed together in

the blending process-units; these were specified by a higher-level planning system. Table 13 shows the

supply amounts of A, B and C from the supply perimeter-units over the scheduling horizon.

30

For each of the three decomposition strategies two different demand scenarios were applied: an

aggressive scenario where all demands were fixed (Table 14) (i.e., the lower and upper bounds are

equal) and a conservative scenario where the demands had zero lower bounds (Table 15). Table 16

displays the problem statistics. These problems were generated and solved using XPRESS-MOSEL

version 1.6.3 and XPRESS-MILP version 17.10.08 with all default settings (Gueret et al., 2002). In

addition, this XPRESS-MOSEL version uses the XPRESS-MOSEL-PARALLEL which provided us with the

ability to solve problems simultaneously on different CPUs. In order to limit computation times, the

maximum time on any individual optimization problem (coordinator and cooperators) was limited to

either 30-CPU-seconds or the time to find the first integer feasible solution (whichever was the longest).

Figure 21 shows a Gantt chart with the operation modes of units throughout the scheduling horizon for

the solution from the centralized strategy. The conservative scenario allows the non-fulfillment of orders

since the lower bounds on the demands are zero thus allowing for an easier scheduling problem from a

schedule feasibility perspective. This statement is confirmed by the results in Table 17 (aggressive

scenario) and Table 18 (conservative scenario). As expected, all decomposition strategies in the

conservative scenario (Figure 22) had a significantly faster convergence time, due to requiring less HDH

iterations, than in the aggressive scenario (Figure 23). On the other hand, the overall objective function

was significantly higher in the aggressive scenario since all demands had to be fully satisfied.

In terms of the performance of the decomposition approaches the collaborative strategy found better

feasible solutions than the coordinated approaches. This is partly due to the fact that the coordinator may

under-estimate (sand-bag) the performance of the cooperators which does not occur in the collaborative

strategy. In contrast, the collaborative strategy needed more iterations than the coordinated approaches

in the more difficult aggressive scenario. It should again be highlighted that the computational

performance of the coordinated strategies is enhanced by the capability of parallelization of the

cooperator sub-problems which is not possible with a collaborative strategy given that a collaborative

approach is performed in-series or in-priority.

31

As previously mentioned, the collaborative strategy assumes that the on-sites (cooperator 2) is a

locally rigorous coordinator in the HDH and therefore the decisions made by the on-sites will have

priority over the decisions made by the off-sites (cooperator 1). This is aligned with the notion of

scheduling the bottlenecks first or focusing on the bottlenecks. The opposite collaboration strategy was

attempted where the off-sites were modeled as the coordinator and therefore its decisions had top

priority. This strategy failed to find a globally feasible solution which is easily explained by the fact that

the off-sites does not constitute a bottleneck nor is it the true driver of the decision-making in this

system. This case demonstrates that in order to successfully apply a collaborative strategy, as previously

stated, it is not only necessary to identify the correct segmentation of the decision-making system but

also the priority of its individual elements.

The use of a partially rigorous coordinator model significantly improved the speed of the HDH as

expected. In the aggressive scenario the partially rigorous coordination also provided a better globally

feasible solution than the relaxed coordinator. The centralized strategy implies the use of a global

rigorous model in the coordination layer. The first feasible solution took 10 and 4 times longer than the

collaborative strategy in the aggressive and conservative scenarios respectively. In this example the first

feasible solution of the centralized system was also the global provably-optimal one. It is interesting to

note that the global optimum was also achieved by the collaborative strategy in the aggressive approach.

This shows that while it is not guaranteed that the global optimum will be found by using the HDH, this

heuristic may in fact be able to obtain it albeit serendipitously.

5. Conclusions

The focus of this paper has been to present a heuristic which can be used to find globally feasible

solutions to usually large decentralized and distributed decision-making problems when a centralized

approach is not possible. A standardized nomenclature was established to better describe the

communication, coordination and cooperation between two hierarchical layers of a decomposed

problem. The HDH was applied to an illustrative example based on an actual industrial multi-site system

32

and was able to solve this problem faster than a centralized model of the same problem when using both

coordinated and collaborative approaches.

In addition the HDH has been contrasted with other methods structured around the notions of

price/pole-directed, weight-directed and error-directed decomposition strategies. Even though the HDH

is currently an error-directed method, future work will focus on devising weight-directed enhancements

to this heuristic using aggregation and disaggregation rules to automatically transform the global

rigorous model into both local rigorous and global relaxed/reduced/restricted models. It is expected that

such an enhancement could minimize the model-mismatch or inaccuracies introduced by the contrast

between public and private constraints. In essence, this is equivalent to using an exogenous (global

rigorous) model supplied by the modeler or scheduling analyst and programmatically generating

coordinator and cooperator endogenous models using "reduction rules".

And finally, we would like to emphasize that the HDH is only a rule-of-thumb for helping with the

diverse reasoning behind the coordination of essentially bi-level decomposition optimization problems.

For tightly bottlenecked and/or critically resource constrained problems, attention to how the

coordinator is setup and how the cooperators are separated will determine its success as is the case with

all heuristic approaches.

Nomenclature

Sets and indices

i = 1…NP number of poles

k = 1...NI number of iterations

pp= 1…NPP number of physical units

pl = 1...NPL number of logical units

sp = 1…NSP number of cooperating sub-problems

t = 1...NT number of time-periods

33

Parameters

SSL step size for lower pole-outline (bound) adjustment

SSU step size for upper pole-outline (bound) adjustment

wi weight parameters for the pole-outlier objective function terms

wwi weight parameters for the pole-opportunity objective function terms

min

plXH minimum hold-up or inventory of logical unit pl

max

plXH maximum hold-up or inventory of logical unit pl

SUWB

plXH start-up-when-below lot-size of logical pool-unit pl

min

pl,FDD minimum fill-draw delay on logical unit pl

max

pl,FDD maximum fill-draw delay on logical unit pl

Variables

Pi,k pole-offer (from coordinator) for resource, regulation and/or register i at iteration k

Pi,sp,k pole-observation for resource i at cooperating sub-problem sp at iteration k

LB

k,iP lower pole-outline (bound) of pole-offer i at iteration k

UB

k,iP upper pole-outline (bound) of pole-offer i at iteration k

iPP pole-objective i

k,iP pole-outlier shortage for resource i in the coordinator at iteration k

k,iP pole-outlier surplus for resource i in the coordinator at iteration k

k,sp,iP pole-offer shortage for resource i at cooperating sub-problem sp at iteration k

k,sp,iP pole-offer surplus for resource i at cooperating sub-problem sp at iteration k

k,iPP pole-opportunity shortage for resource i in the coordinator at iteration k

k,iPP pole-opportunity surplus for resource i in the coordinator at iteration k

t,jinxf flow that enters a physical unit at time-period t

t,joutxf flow that leaves a physical unit at time-period t

34

t,ppxh hold-up of physical unit pp at time-period t

t,jin,jouty logic variable for the movement from outlet-port jout to inlet-port jin at time-period t

t,ply logic variable for the set-up of logical unit pl at time-period t

Lagrange multiplier in Lagrangean decomposition

References

Basset, M.H.; Pekny, J.F.; Reklaitis, G.V. (1996). Decomposition techniques for the solution of

large-scale scheduling problems. AIChE J., 42, 3373.

Cheng, R.; Forbes, J.F.; Yip, W.S. (2007). Price-driven coordination method for solving plant-

wide MPC problems. J. Proc. Control, 17, 429.

Dantzig, G.B.; Wolfe, P. (1960). Decomposition principle for linear programs. Oper. Res., 8, 1.

Dogan, M.E.; Grossmann, I.E. (2006). A decomposition method for the simultaneous planning and

scheduling of single-stage continuous multiproduct plants. Ind. Eng. Chem. Res., 45, 299.

Forbes, J.F.; Marlin, T.E. (1994). Model accuracy for economic optimizing controllers: the bias

update case. Comput. Chem. Eng., 18, 497.

Geoffrion, A.M. (1972). Generalized Benders decomposition. J. Optim. Theory and Appl., 10, 237.

Gueret, C.; Prins, C.; Sevaux; Heipcke S. (revisor and translator) (2002). Applications of

Optimization with Xpress-MP, Dash Optimization, Blisworh, Northan, UK..

Jemai, Z.; Karaesmen, F. (2006). Decentralized inventory control in a two-stage capacitated supply

chain. To appear in IIE Transactions.

Jörnsten, K.; Leisten, R. (1995). Decomposition and iterative aggregation in hierarchical and

decentralized planning structures. Eur. J. Oper. Res., 86, 120.

35

Jose, R.A. (1999). Ph.D. thesis, University of Pennsylvania, PA, USA.

Jose, R.A.; Ungar, L.H. (1998). Auction-Driven Coordination for Plantwide Control. Foundations

of Computer-Aided Process Operation, Snowbird, USA.

Jose, R.A.; Ungar, L.H. (2000). Pricing interprocess streams using slack auctions. AIChE J., 46,

575.

Karuppiah, R.; Grossmann, I.E. (2006). A Lagrangean based branch-and-cut algorithm for global

optimization of nonconvex mixed-integer nonlinear programs with decomposable structures.

Submitted to J. Global Optim.

Kelly, J.D. (2002). Chronological Decomposition Heuristic for Scheduling: Divide and Conquer

Method. AIChE J., 48, 2995.

Kelly, J.D. (2004a). Production Modeling for Multimodal Operations. Chemical Engineering

Progress, February, 44.

Kelly, J.D. (2004b). Stock Decomposition Heuristic for Scheduling: A Priority Dispatch Rule

Approach. Honeywell Internal Technical Report.

Kelly, J.D.; Mann, J.L. (2004). Flowsheet Decomposition Heuristic for Scheduling: a Relax-and-

Fix Method. Comput. Chem. Eng., 28, 2193.

Kelly, J.D. (2005a). Modeling production-chain information. Chemical Engineering Progress,

February, 28.

Kelly, J.D. (2005b). The Unit-Operation-Stock Superstructure (UOSS) and the Quantity-Logic-

Quality Paradigm (QLQP) for Production Scheduling in the Process Industries. Proceedings of the

Multidisciplinary Conference on Scheduling Theory and Applications (MISTA), 1, 327.

36

Kelly, J.D. (2006). Logistics: the missing link in blend scheduling optimization. Hydrocarbon

Processing, June, 45.

Lu, J. (2003). Challenging control problems and emerging technologies in enterprise optimization.

Control Eng. Practice, 11, 847.

Wilkinson, S.J. (1996). Ph.D. thesis, Imperial College of Science, Technology and Medicine,

London, UK.

Wu, D.; Ierapetritou, M.G. (2003). Decomposition approaches for the efficient solution of short-

term scheduling problems. Comput. Chem. Eng., 27, 1261

Zhang, N.; Zhu, X.X. (2006). Novel modeling and decomposition strategy for total site

optimization. Comput. Chem. Eng., 30, 765.

Zyngier, D. (2006). Ph.D. thesis, McMaster University, Hamilton, ON, Canada.

Zyngier, D.; Kelly, J.D. (2007). Multi-Product Inventory Logistics Modeling in the Process

Industries. Accepted for publication as a chapter in the book from the DIMACS and ExxonMobil

Workshop on Computational Optimization and Logistics Challenges in the Enterprise (COLCE),

eds. K.C. Furman and I.E. Grossmann.

37

Coordination

Cooperation

Feedforward Feedback

Figure 1. Bi-level layers involved in decomposition.

38

Coordinated

Collaborative

Productivity

Defects

Centralized

Competitive

Figure 2. Defects versus productivity trade-off curves for different reasoning isotherms.

39

Pole

Coordinator

Price

Coordinator

Cooperator 2 Cooperator 3Cooperator 1

Figure 3. General structure of decomposed problems.

40

Resource-Protocols

Regulation-Protocols

Register-Protocols

Flow-Poles

Component-Poles

Property-Poles

Mode-Poles

Material-Poles

Move-Poles

Holdup-Poles

Condition-PolesProtocol

Pole-offersPole-

obstacles, offsets

and outages

Figure 4. Types of poles and protocols.

41

Update Lagrange

Multipliers ()

Pole-offers

Upper Bound

Calculation (UB)

Cooperator 2 Cooperator 3Cooperator 1

UB

Figure 5. Lagrangean decomposition.

42

Determination of

Reference Poles

Prices Pole-obstacles,

offsets, outages

Pole-offers

Cooperator 2 Cooperator 3Cooperator 1

Price Adjustment

Figure 6. Auction-based decomposition (Jose, 1999).

43

Prices Pole-observationsPrice-elasticity

(sensitivity)

No Pole

Coordination

Cooperator 2 Cooperator 3Cooperator 1

Price Adjustment

Figure 7. Auction-based decomposition with price-elasticity (Cheng et al., 2006).

44

Pole-observations

Aggregated

Model

Pole-offers

Cooperator 2 Cooperator 3Cooperator 1

No Price

Coordination

Figure 8. Iterative aggregation/disaggregation decomposition (Jörnsten and Leisten, 1995).

45

Pole-obstacles,

offsets, outages

Relaxed Global

Problem (Planning)

Cooperator 1

(Scheduling)

No Price

Coordination

Pole-offers

Cooperator 2

(Scheduling)

Cooperator 3

(Scheduling)

Figure 9. Planning to scheduling decomposition (Bassett et al., 1996).

46

Relaxed Global

Problem

Cooperator 2 Cooperator 3Cooperator 1

No Price

Coordination

Pole-sensitivity

(yields, inverse-yields)

Pole-offers

Figure 10. Yield updating decomposition (Zhang and Zhu, 2006).

47

Continuous Refinement

Yes

Stop

No

Re-adjust pole-outlines in the

coordinator based on cooperator

pole-obstacles, offsets and outages.

Are all cooperators

feasible?

No

YesExclude current solution

from coordinator keeping current

or different pole-outlines.

Solve coordinator problem.

Conflict Resolution (Equations 3a, 3b)

Consensus Reconciliation (Equations 4,5,6)pole

-off

ers

Also minimize deviation of

pole-offers

from the pole-objectives.

Challenge Research (Equations 7, 8)

Start

Is another solution

desired?

Yes

Do pole-objectives

exist?

No

Solve cooperator sub-problems.

pole-offers

pole-obstacles,

offsets and outages

Minimize deviation of coordinator

pole-offers from cooperator pole-

observations of the previous iteration.

pole-origins

Figure 11. Overview of proposed HDH algorithm.

48

0 R10 10

0 R20 5

Coordinator

Cooperator 1 Cooperator 2

0 R11 5

0 R21 10

0 R12 0

0 R22 10

(a)

0 R10 10

0 R20 5

R10 + R20 10

Coordinator

Cooperator 1 Cooperator 2

0 R11 5

0 R21 10

0 R12 0

0 R22 10

(b)

Cooperator 1 Cooperator 2

0 R11 5

For R11 5:

R21 2* R11

For R11 > 5:

R21 0.5* R11

0 R12 20

0 R22 20

0 R10 10

0 R20 15

R1c + R20 20

(c)

Figure 12. Conflict resolution – three motivating examples.

49

CooperationCooperation

Coordinator CooperatorsPole-offers Pole-observations

Pole-obstacles, offsets, outages

+-

Pole-objectives

Feedforward Engine Feedback Engine

Figure 13. Model predictive control structure of the HDH.

50

Plan Perform

Perfect

Orders Objects

Feedback

Feedforward

Feedback

System Dynamics

Figure 14. The plan-perform-perfect-loop of the HDH with orders-to-objects system dynamics.

51

CoordinatorCooperator 1

Cooperator 2

(a)

Cooperator 1

Cooperator 2Coordinator

(b)

Cooperator 1

Cooperator 2

Coordinator

(c)

Coordinator

Cooperator 1

Cooperator 2

(d)

Figure 15. Joint feasible regions of decomposed problem with different coordinator models: (a) global

rigorous model; (b) global restricted model; (c) global reduced model; (d) global relaxed model.

52

Pole-obstacles,

offsets and outages

Coordinator

Model

Cooperator 2 Cooperator 3Cooperator 1

Pole-offers

Weight

Adjustment

Figure 16. Hierarchical decomposition heuristic.

53

P1

P12

S21S11

S12

S13

D21

D22P22

B21

B22

T11

T12

T13

T14

T15

T16

T21

T22

T23

T24

T25

T26

Figure 17. Physical connectivity of the global multi-site problem (coordinator).

54

S11

S12

S13

P1

P12

T11

T12

T13

T14

T15

T16

Figure 18. Physical connectivity of local off-site sub-problem (cooperator 1).

55

S21

P1

D21

D22

B21

B22

T21

T22

T23

T24

T25

T26

P12

P22

Figure 19. Physical connectivity of local on-site sub-problem (cooperator 2).

56

Rigorous Cooperator 1 Rigorous Cooperator 2

No Cooperator 1 No Cooperator 2

(a)

No Cooperator 1 Rigorous Cooperator 2

Rigorous Cooperator 1 No Cooperator 2

(b)

Relaxed Cooperator 1 Relaxed Cooperator 2

Rigorous Cooperator 1 Rigorous Cooperator 2

(c)

Relaxed Cooperator 1 Rigorous Cooperator 2

Rigorous Cooperator 1 No Cooperator 2

(d)

Figure 20. Decomposition strategies: (a) centralized (no decomposition); (b) collaborative; (c)

coordinated – relaxed coordinator; (d) coordinated – partially rigorous coordinator.

57

B21

B22

P12 A A A

P1 B B

P22 A A A

T26

T25

T24 A

T23 A B

T22

T21 A B A B

T12 B C B C B A

T11 C A C B C B

T16 C B A B

T15 C A C B C A

T14 A A B A B A

T13

1 2 3 4 5 6 7 8

ABC

ABC

B

A

A B

ABC

ABC

B C

B

B A

CB

B

Figure 21. Gantt chart showing mode and material-operations – solution from centralized strategy.

58

0

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10 12

Iterations

Po

le-O

bst

acle

s .

Collaborative ($898.60)

Coordinated - Relaxed ($894.60)

Coordinated - Partially Rigorous ($889.60)

.

Figure 22. Total pole-obstacles in the coordinated and collaborative strategies (conservative scenario).

59

0

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10 12

Iterations

Po

le-O

bst

acle

s .

Collaborative ($935.08)

Coordinated - Relaxed ($905.08)

Coordinated - Partially Rigorous ($929.20)

.

Figure 23. Total pole-obstacles in the coordinated and collaborative strategies (aggressive scenario).

60

Table 1. Examples of organizational and managerial hierarchies.

Top Layer Middle Layer Bottom Layer

Planner Master-Scheduler Schedulers

Board of Directors Chief-Executive Executives

Chief-Executive Executive Directors

Executive Director Managers

Director Manager Workers (white-collar)

Manager Supervisor / Lead

Workers (blue-collar), i.e., Operators,

Laborers, Developers, Tradesmen,

Journeymen, etc.

Supervisor / Lead Workers (blue-collar)Equipment (Unit), Services (Utilities), Tools

(Utensils), Materials (Stocks) (UOSS)

Owner General Contractor Sub-Contractors

Administrator Doctor Nurses, Orderlies, Nursing Assistants, etc.

Sargent Corporal Private, Soldier, Troop, etc.

Manager Coach Players, Tem-Member

Principal Teacher Students

61

Table 2. Results – motivating example 1.

Iteration Resource- Pole-Outlines Pole-Offers Pole-Obstacles Pole-Obstacles

Protocol Coordinator Cooperator 1 Cooperator 2

1 R1 0 R10 10 10 -5 -10

R2 0 R20 5 5 0 0

2 R1 -10 R10 0 0 0 0

R2 0 R20 5 5 0 0

62

Table 3. Results – motivating example 2.

Iteration Resource- Pole-Outlines Pole-Offers Pole-Obstacles Pole-Obstacles

Protocol Coordinator Cooperator 1 Cooperator 2

1 R1 0 R10 10 5 0 -5

R2 0 R20 5 5 0 0

2 R1 -5 R10 5 5 0 -5

R2 0 R20 5 5 0 0

3 R1 -10 R10 0 0 0 0

R2 0 R20 5 5 0 0

63

Table 4. Results – motivating example 3.

Iteration Resource- Pole-Outlines Pole-Offers Pole-Obstacles Pole-Obstacles

Protocol Coordinator Cooperator 1 Cooperator 2

1 R1 0 R10 10 5 0 0

R2 0 R20 15 15 -5 0

2 R1 0 R10 10 10 0 0

R2 -5 R20 10 10 -5 0

3 R1 0 R10 10 10 0 0

R2 -10 R20 5 5 0 0

64

Table 5. Model parameters – cooperator 1.

Pools Lot-Size (kMT) SDWB FDD Continuous Charge-Size (kMT / h)

LB UB (kMT) (h) Processes LB UB

( min

plXH ) ( max

plXH ) ( SUWB

plXH ) ( min

pl,FDD )

T11 0 9 0 0

T12 0 4.2 0 0

T13 2 12.5 4 -

T14 2 12.7 4.05 0

T15 2 18.5 5.35 0

T16 2 18.5 6.6 0

65

Table 6. Model parameters – cooperator 2.

Pools Lot-Size (kMT) SDWB FDD Continuous Charge-Size (kMT / h)

LB UB (kMT) (h) Processes LB UB

( min

plXH ) ( max

plXH ) ( SUWB

plXH ) ( min

pl,FDD )

T21 1 6.2 - - B21 0.1 0.28

T22 1 6.2 - - B22 0.1 0.3

T23 1 6.2 1.5 - P1 0.25 0.32

T24 0 3.9 0 - P22 0.25 0.35

T25 0 0.35 - -

T26 0 0.35 - -

66

Table 7. Model parameters – relaxed coordinator.

Pools Lot-Size (kMT) SDWB FDD Continuous Charge-Size (kMT / h)

LB UB (kMT) (h) Processes LB UB

( min

plXH ) ( max

plXH ) ( SUWB

plXH ) ( min

pl,FDD )

T11 0 9 - - B21 0.1 0.28

T12 0 4.2 - - B22 0.1 0.3

T13 0 12.5 - - P1 0.25 0.32

T14 0 12.7 - - P22 0.25 0.35

T15 0 18.5 - -

T16 0 18.5 - -

T21 0 6.2 - -

T22 0 6.2 - -

T23 0 6.2 - -

T24 0 3.9 - -

T25 0 0.35 - -

T26 0 0.35 - -

67

Table 8. Model parameters – partially rigorous coordinator.

Pools Lot-Size (kMT) SDWB FDD Continuous Charge-Size (kMT / h)

LB UB (kMT) (h) Processes LB UB

( min

plXH ) ( max

plXH ) ( SUWB

plXH ) ( min

pl,FDD )

T11 0 9 - - B21 0.1 0.28

T12 0 4.2 - - B22 0.1 0.3

T13 0 12.5 - - P1 0.25 0.32

T14 0 12.7 - - P22 0.25 0.35

T15 0 18.5 - -

T16 0 18.5 - -

T21 1 6.2 - -

T22 1 6.2 - -

T23 1 6.2 1.5 -

T24 0 3.9 0 -

T25 0 0.35 - -

T26 0 0.35 - -

68

Table 9. Model parameters – semi-continuous flow constraints on outlet-ports.

Source- Mode-Operation Flows (kMT / h)

Port LB UB

S11 A,B,C 0.04 0.35

S12 A,B,C 0.02 0.45

S13 A,B,C 0 2

S21 A,B,C 0 0.25

T11 A,B,C 0 1.5

T12 A,B,C 0 1.5

T13 A,B,C 0 1.5

T14 A,B,C 0 1.5

T15 A,B,C 0 1.5

T16 A,B,C 0 1.5

T21 A,B,C 0 0.28

T22 A,B,C 0 0.28

T23 A,B,C 0 0.28

T24 A,B,C 0 0.38

T25 ABC 0 0.28

T26 ABC 0 0.3

B21 ABC 0 0.28

B22 ABC 0 0.3

P1 A,B,C 0 0.25

P12 A,B,C 0 0.35

P22 A,B,C - -

69

Table 10. Model parameters – semi-continuous flow constraints on inlet-ports.

Destination- Mode-Operation Flows (kMT / h)

Port LB UB

D21 ABC 0.06 0.28

D22 ABC 0.078 0.3

T11 A,B,C 0 2

T12 A,B,C 0 2

T13 A,B,C 0 2

T14 A,B,C 0 2

T15 A,B,C 0 2

T16 A,B,C 0 2

T21 A,B,C 0 0.28

T22 A,B,C 0 0.28

T23 A,B,C 0 0.28

T24 A,B,C 0 0.15

T25 ABC 0 0.28

T26 ABC 0 0.3

B21 A,B,C 0 0.28

B22 A,B,C 0 0.29

P1 A,B,C 0 0.35

P12 A,B,C 0 0.35

P22 A,B,C - -

70

Table 11. Initial lot-sizes for pool-units.

Pool-Units Initial Lot-Size Initial Material

(kMT)

T11 0 B

T12 0 B

T13 10.579 A (cooperators)

T14 12.675 B

T15 18.051 B

T16 2.484 B

T21 5.251 A

T22 5.52 B

T23 5.608 B

T24 0 B

T25 0 ABC

T26 0 ABC

71

Table 12. Inverse-yield orders for blender process-units.

Blender- Inlet- Start-Time End-Time Inverse-Yield

Unit Port (h) (h) LB UB

B21 A 0 24 0 0

24 48 0.95 1

48 192 0 0

B 0 24 0.95 1

24 48 0 0

48 192 0.95 1

0 192 0 0

C 0 192 0 0

B22 A 0 24 0.01781 0.01968

24 48 0.10124 0.11189

48 72 0.2804 0.30992

72 120 0 0

120 144 0.8282 0.91538

144 168 0.40206 0.44439

168 192 0 0

B 0 24 0.93219 1

24 48 0.84877 0.93811

48 72 0.6696 0.74008

72 120 0.95 1

120 144 0.1218 0.13462

144 168 0.54794 0.60561

168 192 0.95 1

C 0 192 0 0

72

Table 13. Supply profile for the scheduling horizon

Supply Start-Time End-Time Rates (kMT / h)

Perimeter-Unit (h) (h) LB UB

(Mode)

S11 (A) 48 164 0.1219 0.1219

S12 (B) 128 152 0.375 0.375

152 168 0.3125 0.3125

168 192 0.375 0.375

S13 (B) 78 96 1.47222 1.47222

126 138 0.41667 0.41667

S21 (B) 25 72 0.20426 0.20426

121 148 0.22833 0.22833

170 180 0.2055 0.2055

73

Table 14. Demand profile for the scheduling horizon – aggressive approach

Demand Start-Time End-Time Rates (kMT / h)

Perimeter-Unit (h) (h) LB UB

(Mode)

D21 0 48 0.18353 0.18353

(ABC) 48 72 0.18345 0.18345

72 96 0.18353 0.18353

96 120 0.18087 0.18087

120 144 0.18353 0.18353

144 168 0.20156 0.20156

168 192 0.18353 0.18353

D22 0 24 0.28546 0.28546

(ABC) 24 48 0.28061 0.28061

48 72 0.28372 0.28372

72 96 0.27328 0.27328

96 120 0.27465 0.27465

120 144 0.27525 0.27525

144 168 0.27639 0.27639

168 192 0.26302 0.26302

74

Table 15. Demand profile for the scheduling horizon – conservative approach

Demand Start-Time End-Time Rates (kMT / h)

Perimeter-Unit (h) (h) LB UB

(Mode)

D21 0 48 0 0.18353

(ABC) 48 72 0 0.18345

72 96 0 0.18353

96 120 0 0.18087

120 144 0 0.18353

144 168 0 0.20156

168 192 0 0.18353

D22 0 24 0 0.28546

(ABC) 24 48 0 0.28061

48 72 0 0.28372

72 96 0 0.27328

96 120 0 0.27465

120 144 0 0.27525

144 168 0 0.27639

168 192 0 0.26302

75

Table 16. Problem statistics (presolved values in brackets).

Model Constraints Continuous Variables Non-Zero Coefficients Binary Variables

Global rigorous coordinator 7234 (3010) 4450 (2187) 29780 (9484) 1170 (951)

Global partially rigorous coordinator 6809 (3134) 4450 (2406) 28056 (9996) 1170 (1045)

Global relaxed coordinator 6786 (2771) 4423 (2194) 27731 (8659) 1154 (923)

Local rigorous on-sites 3133 (1287) 2228 (1013) 12971 (4383) 607 (394)

Local rigorous off-sites 3598 (280) 1991 (266) 13314 (939) 488 (56)

76

Table 17. Results for the aggressive case.

Iterations Time Objective Function

(CPU s) ($)

1 6301.0 935.08

11 626.2 935.08

9 401.7 905.08

4 195.7 929.2

Approach

Centralized

Collaborative

Coordinated - Relaxed

Coordinated - Partially Rigorous

77

Table 18. Results for the conservative case.

Iterations Time Objective Function

(CPU s) ($)

1 1488.0 935.08

7 386.2 898.6

7 333.6 894.6

2 78.5 889.6

Approach

Centralized

Collaborative

Coordinated - Relaxed

Coordinated - Partially Rigorous