Case Article—Blending Statistics, Optimization, and Simulation in a Multiphase Case

Case Overview

In this case, a fictional company, Junko’s, is concerned about the high cost of printing and uneven utilization of presses. The problem is derived directly from Standard Register and described in Ahire et al. (2007). In that project, Standard Register needed to estimate cost and runtime profiles for print jobs with a number of attributes and then optimally assign them to printers with different characteristics and locations around the country.

The recommended solution based on a mixed integer program was an improvement over existing assignments but was not implementable because of cost and time estimate uncertainty and job size uncertainty. A heuristic solution based on optimization was implemented, which allowed for fast solutions and implementation over a rolling horizon because new customer jobs continually arrived, changing the optimal assignments. The heuristic was tested with Monte Carlo simulation to assure robustness. The case for Junko’s follows a similar but simplified path: students undertake each of these analytical steps.

The problem is ubiquitous in any job shop–style production environment with fixed setup times (and costs) and variable runtimes (and costs) that are a function of batch size. In this job shop, batch sizes are given according to the size of the order of the customer; it is not set at the discretion of the company. The core problem is a mixed integer programming problem of job assignment to press. Further, the validity of an optimal solution under the assumption of valid input parameters is tested in a sensitivity and risk analysis simulation.

Students conduct the following analyses in four parts, and each part requires one of three different analytical methods. (Part four combines two methodologies.) The case is designed to be solvable in Excel, but other tools may be used at the instructor’s discretion.

The four parts are as follows:

1. Regression: estimate relationships using regression between job size, time required, and total press cost using regression.

2. Integer optimization: formulate and solve a mixed integer optimization problem that captures the problem and then modify the formulation to evaluate two alternative scenarios.

3. Monte Carlo simulation: conduct objective function sensitivity and constraint violation risk resulting from uncertainty in the estimated parameters and job size inputs with Monte Carlo simulation using data tables (or other add-ons).

4. Reoptimization under uncertainty: evaluate the robustness of the optimal press assignments under job size and parameter uncertainty through multiple model runs with different input data with Monte Carlo simulation via a macro written in Visual Basic (or manually).

The case is customizable to the needs and interests of the instructor. Parts 1 and 2 can be combined into a single assignment. Parts 3 and 4 are risk assessment of the optimal solution and are optional.

Literature Review

To emulate a real-world project more closely, the case presented here combines multiple analytical methods rolled out in an incremental, four-part, and in-depth analysis. As implemented, a key component of the course is carefully describing the problem, method, solution, benefits, and recommendations in written reports and, occasionally, oral presentations.

The format has a number of advantages for the student: (1) intertwining of methods on the same problem; (2) conducting a multistage, incremental format; and (3) communicating the business value of analytical results. These are discussed in turn.

Intertwined Analytical Methods

It is both helpful and difficult for students to learn that analytical methods are often intertwined in practice. Gorman (2010, 2011) discusses the difficult transition from textbook methods to working with real-world problems in a capstone course with a live client, noting that students rarely identify the need for multiple methods to be employed. The case described here is designed to help students learn this in a guided fashion; it is designed to prepare students for the experience through a capstone project with a live client.

There are many cases that combine multiple methods and the benefits that such approaches contain. For example, Pachamanova et al. (2021) use multiple methods in a process improvement case in healthcare. A similar example of this approach is found in Shumsky (2009), who uses statistics, simulation, and optimization in the application of yield management to a fictitious airline. Gorman (2012) applies statistics, optimization, and simulation in the analysis of a simple game called “Pass the Pigs.” Gorman (2021a, b) applies multiple analytical techniques in the same phased fashion described here in freight rail and service vehicle inventory contexts.

Incremental, Multipass Format

Researchers express the value of using a rolled-out, or incremental, case format, expressing the problem over a number of class periods. Pachamanova (2015) discusses the naturally iterative process of applying analytical tools to a business problem and the experiential learning value of covering a problem over a number of weeks. That article suggests that such a format provides a more management consulting–type process, suggesting it is a more realistic experience for the student.

Kopcso and Pachamanova (2017) consider at length the value of a staged approach from data analysis to model building to summarizing and presenting results; these are skills, in combination, that carry great value for the student. Dobson and Tilson (2016) discuss the value of project-based learning with a three-stage format in a hospital pharmacy case, providing incremental information as the case moves forward. They posit that such a structure more closely mirrors project-based learning with a client organization, thereby providing a richer and more realistic learning experience.

These examples roll out incremental analyses on the same problem, expanding the tools used and the depth of understanding of the solution. Such a structure provides experiential and project-based learning to the student. The long-term, phased case gives students a chance to “sink their teeth” into a problem, allowing them to transition from textbook learning to problem solving they will see in practice.

Emphasis on Communicating the Business Value of Analytics

Although it is critical to be able to conduct an analysis properly and manage the analytical process, all is lost without being able to express its value. Grossman et al. (2008) examines the radical difference between presenting a model and communicating its value in a narrative fashion. A few instructors may use this case without implementing this requirement, resulting in the loss of some educational value. Students often worry about getting the answer “right” and, after spending hours working on a model, often forget its business value. As pointed out by Carraway and Clyman (2000, p. 40), “Recommendations that cannot be sold are seldom implemented. And if they cannot be implemented, then the analysis is a failure.” Grossman et al. (2008) suggests that explaining the method intuitively and selling the recommendation that results from the analysis is imperative to model success, which is a difficult skill to master. The case emphasizes this skill at each stage of the analysis.

Learning Objectives

The case is geared toward upper class undergraduates and MBA students with prior coursework in statistics, optimization, and simulation. Portions of the case might be used in a lower level class (perhaps part 1 in an introductory statistics class, part 2 in an introductory optimization class), but I do not recommend it. The case is intentionally unstructured. The objective of the case is to show students the feasibility and value of integrating these techniques in a single project, which is advanced for an introductory class.

As a real-world application that is reproducible for a student with relatively little technical training, the case shows the power of analytics. The relatively unstructured problem helps students hone their thinking and not just how to solve the problem, but how to structure it. Importantly, it improves students’ ability to express technical material in a nontechnical way.

Students benefit from the following:

1. Applying simple regression to estimate cost and runtime parameters for presses and using results to further decision making.

2. Formulating and solving an integer binary job assignment problem.

3. Conducting scenario analyses on a base model.

4. Evaluating the risk of a recommended stock under uncertainty with respect to cost and feasibility of the assignments.

5. Evaluating the robustness of a recommended job to press assignments under uncertainty.

6. Gaining experience with formal written communications of a problem and modeling results.

7. Providing insights based on optimization that allow them to search for optimization-based heuristics they might implement in the face of uncertainty.

Student Comments

This case has been administered dozens of times in upper level undergraduate optimization classes, MBA analytics case study classes, and masters in business administration classes. Generally, providing the original case to the students after the preliminary discussions is helpful. Several open-ended surveys of students have been conducted at the end of the case. Table 1 presents a small number of the comments. The comments are overwhelmingly positive and tend to fall along three broad categories.

1. Integrative/multiple analytical methods: students generally recognize the power and value of integrating techniques and, in most cases, had not done so in their academic studies.

2. Real-world/unstructured problem: students value the real-world nature of the problem. Though they struggle with the unstructured approach, they recognize that they must learn to deal with such vagaries to implement solutions in practice.

3. Connected case/course structure: students in most cases had not been exposed to a multiple-part case. They saw value in seeing a problem from many angles. For example, they actually use the regression estimates given for a purpose. They challenge the “optimum” solution, seeing that the optimum carries with it the risk of the tacit assumption of certainty.

Table 1. Excerpts of Student Comments

Supplemental Materials

I provide the student case; raw data for press operating costs, time estimation, and job assignments; and spreadsheet solutions for parts 1–4 and four exemplar solution writeups.

Conclusion

Multiphase, real-world case studies with integrated analytical tools have great value in the classroom. This work, based on Ahire et al. (2007), applies regression, integer optimization, and simulation to the job assignment problem with fixed costs. Students see the value of analytics in the published article, and students with the proper prior training can reproduce the work in a few weeks’ time. Students who have seen such integrated and in-depth cases are more likely to transition to capstone classes and adopt their textbook learning to practice because they have worked with advanced, in-depth, integrated problems. An instructor can assign any number of the four parts and, if they so choose, skip or cover other parts without assigning them.