Part II:
Models
In Part I, we discussed the central role of descriptive and optimization models in supporting
supply chain decision making. In this part, we provide details about how these models are
constructed and solved. This part includes five chapters:
Chapter 3. Fundamentals of Optimization Models: Linear Programming
Chapter 4. Fundamentals of Optimization Models: Mixed-Integer Programming
Chapter 5. Unified Optimization Methodology for Operational Planning Problems
Chapter 6. Overview of Descriptive Models
Chapter 7. Supply Chain Decision Databases
In these chapters, greater attention is paid to optimization models and methods than descriptive
models and methods, although both are equally important. This emphasis is due to several
reasons. First, optimization models can provide managers with penetrating analysis of the
company’s decision options, goals, commitments, and resource constraints. Second, they can
provide a rich and robust framework for merging data, relationships, projections, and forecasts
from descriptive models.
Moreover, the fundamentals of optimization models for supply chain planning can be presented in
a compact manner. In particular, we explore linear programming (LP) models in Chapter 3 and
their extensions to mixed-integer programming (MIP) models in Chapter 4. In Chapter 5, we
extend the scope of MIP models for operational planning by presenting a unified optimization
methodology that combines model decomposition methods with heuristics. We note that the
same modeling techniques and algorithms discussed here have been applied to decision
problems arising in investment banking, engineering design, and many other areas.
By contrast, descriptive models are very broad in the methodologies they employ, which include
statistics, data mining, management accounting, deterministic and Monte Carlo simulation, and
others. It would not be possible to provide an in-depth presentation of all these methodologies in
one book. Instead we study an overview of important, illustrative descriptive models in Chapter 6,
including forecasting, simulation, and activity-based costing.
Finally, the focus on optimization models is motivated by the belief that decision makers should
possess knowledge about their form and function if they are to be used to their full advantage.
Much of this book could have been written from the perspective that the analytical engines of
optimization modeling systems are “black boxes,” which only specialists need to comprehend. We
reject such an approach because it fosters an unnecessary and sometimes dangerous lack of
understanding by managers seeking to use models to improve their decision making. If they have
little or no knowledge about how details of a model represent decision problems, managers might
be unwilling to trust its results or be too willing to blindly trust them.
Still, having committed us to study details of optimization model construction and solution, I wish
to reassure you that we are not about to embark on mathematical developments that require
graduate training in operations research. Rather, only mathematical constructions taught in high
school are used. Although this means that we are restricted to small models that do not convey
the richness of larger models for complex problems, a comprehensive knowledge of the
fundamentals can be conveyed. Moreover, without going into excessive detail, later chapters
describe how large-scale models are constructed by synthesizing smaller, simpler submodels of
the types discussed in this chapter and the next.
In Chapter 7, we examine the supply chain decision database that contains the inputs for supply
chain optimization models. The tables in this database serve as templates for populating
optimization models with data determined by descriptive models and other data input methods.
The supply chain decision database also contains optimization model outputs that are combined
with the inputs in creating managerial reports and graphical displays of supply chain decisions
determined by the models.
Chapter Five:
Unified Optimization
Methodology for Operational
Planning Problems
In the previous chapter, we examined mixed-integer programming (MIP) constructions that
extend linear programming (LP) models to more realistic representations of supply chain
problems at all levels of planning. Many large-scale MIP models have been successfully
implemented and solved. Nevertheless, such models can be challenging to optimize, especially
applications for operational planning that require considerable detail about the timing and
sequencing of decisions.
This concern arose in a project in which we implemented a modeling system to support
production scheduling at a paper mill. Major changeovers on paper machines over a planning
horizon of 30 days were a key set of decisions. For four product families manufactured on a
particular machine, the production manager wished to know the precise hour at which each
changeover from one family to another should be started. An MIP model of this problem would
have required 12 zero-one variables for each hour of the planning horizon to describe all possible
changeovers from each product family to every other product family. Because there are 720
hours in the planning horizon, the model would have required 8640 zero-one variables to capture
these decisions. However, it was unlikely that the model, or the company, would elect to make
more than five major changeovers during the planning horizon; in other words, no more than 5 of
the 8640 zero-one variables would equal 1 in an optimal solution. Needless to say, we designed
and implemented a modeling approach that was less direct but much easier to optimize.
Vehicle-routing problems invoke another class of models where brute force mixed-integer
programming needs to be tempered by more flexible methods. One example is the local delivery
problem where a company dispatches trucks on a daily basis from a central depot to make
deliveries to customers. Such problems can be modeled but not easily solved as monolithic MIP
models. For a specific class of problems, which we examine in Section 5.3, the models are made
up of submodels that describe the routing of each vehicle plus constraints stating that each
customer may be visited exactly once. The submodels are awkward to represent using mixedinteger programming, but heuristics can find good solutions very quickly. Heuristics are ad hoc
search methods customized to a specific decision problem based on rules gleaned by humans
about the problem. The simplicity and effectiveness of heuristics applied to some supply chain
problems has led to considerable applied mathematics research aimed at generalizing the
methods and providing mathematical results characterizing their performance.
The implication of these stories (and others that could be related) is as follows:
It can be necessary and desirable, especially for an operational problem, to design and
implement MIP models and solution methods that are customized to the problem. The
customized methods should aim to determine a demonstrably good, rather than an
optimal, solution to each numerical instance of the problem. Rigorous optimization
methods should be combined with problem-specific and general-purpose heuristic
methods to create schemes that rapidly compute these demonstrably good solutions.
In this chapter, we examine a general-purpose approach to such customization called the unified
optimization methodology that combines mathematical programming decomposition
methods with heuristics. Using the term methodology, which is a “body of practices, procedures,
and rules used in a discipline,” rather than the term method, which is as “an orderly arrangement
of steps to accomplish an end,” in describing this approach conveys that its implementation for a
particular problem will entail a flexible specification and integration of various algorithms and
methods appropriate to the problem.
The unified optimization methodology is presented for several reasons. First, it has and can be
successfully applied to a wide range of complex scheduling and other operational supply chain
problems. Second, because it begins by posing a complete and accurate representation of a
supply chain problem as an MIP model, we can be confident that our analysis of the problem will
be comprehensive. Moreover, the generality of mixed-integer programming permits model
extensions to capture changing problem features.
Third, the united optimization methodology exploits the complementarity between heuristic
methods and rigorous mathematical programming methods. Heuristic methods have been
oversold as all-powerful approaches for solving advanced planning and scheduling problems,
a new term that includes complex planning problems arising in operational supply chain
management. Heuristics do have an important role to play, but their effectiveness is primarily in
analyzing certain types of homogeneous submodels embedded in larger, heterogeneous models.
Moreover, heuristic methods are weak in analyzing resource allocation and other cross-cutting
(that is, systemwide) constraints that are very well analyzed by linear programming. As Section
5.2 demonstrates, decomposition methods isolate submodels that can be rapidly analyzed by
heuristics and other fast algorithms.
Fourth, the unified optimization methodology includes procedures for systematically computing
lower bounds on the cost of a minimal solution, thereby allowing computation to be terminated
with a proven, demonstrably good solution. Although there is no guarantee that a tight bound will
always be obtained within a fixed computation time, lower bounds provide the human scheduler
with critical information about whether or not, or when, to terminate computation. Fifth, the unified
optimization methodology facilitates the use of an advanced start for today’s operational planning
problem based on previous solutions of the problem and other problem-specific information.
In Section 5.1, we explore problem-specific and general-purpose heuristics and illustrate them by
analyzing submodels arising in and solutions to the local delivery problem. An overview of the
unified optimization methodology is presented in Section 5.2 where we examine its application to
a class of production scheduling problems. In Section 5.3, we return to the local delivery problem
and illustrate a numerical application of the united optimization methodology to its solution.
Similarly, in Section 5.4, we return to the production-scheduling model and illustrate a numerical
application of the unified optimization methodology to its solution. The local delivery and
production-scheduling examples are somewhat detailed, reflecting the complexity of operational
scheduling problems. The chapter concludes with final thoughts in Section 5.5 about the unified
optimization methodology. A recursive method for optimizing vehicle routes is reviewed in
Appendix 5A.