Treffer: Teaching Students Supply Chain Risk Using a Linear Programming Model of the Fictitious Online Retailer 'Elbe'
Postsecondary Education
Weitere Informationen
The fictitious online retailer "Elbe" faces a deterministic demand for products over a geographic area and fulfills this demand using supply nodes and fulfillment centers. Experiential learning using provided solver and ILOG models (available upon request from the authors) for this supply chain allows students to experiment with disturbances in the distribution network where facilities go offline as a result of, for example, natural disasters. Students are shown how mathematical models can be used to plan the flow of product in real-world settings, as well as how these models can be used to react to supply chain risk. These models can be used in advanced undergraduate- or graduate-level classes in supply chain and operations management, leaving students with an appreciation of the models and their usefulness in real-world applications, as well as an increased understanding of supply chain risk and its impact on demand fulfillment.
As Provided
AN0142223289;jeb01apr.20;2020Mar16.03:03;v2.2.500
Teaching students supply chain risk using a linear programming model of the fictitious online retailer Elbe
The fictitious online retailer Elbe faces a deterministic demand for products over a geographic area and fulfills this demand using supply nodes and fulfillment centers. Experiential learning using provided solver and ILOG models (available upon request from the authors) for this supply chain allows students to experiment with disturbances in the distribution network where facilities go offline as a result of, for example, natural disasters. Students are shown how mathematical models can be used to plan the flow of product in real-world settings, as well as how these models can be used to react to supply chain risk. These models can be used in advanced undergraduate- or graduate-level classes in supply chain and operations management, leaving students with an appreciation of the models and their usefulness in real-world applications, as well as an increased understanding of supply chain risk and its impact on demand fulfillment.
Keywords: Experiential learning; mathematical programmingprograming; supply chain risk; transportation models
Introduction
Motivation
As supply chains expand in size and across geographical boundaries, they become more prone to disruptions. These disruptions can occur due to a variety of reasons, such as political, environmental, labor, technological changes, fluctuations in transport costs, and poor forecasting. Analyzing and calculating the risk of disruptions becomes imperative for supply chain managers. An efficient and cost effective supply chain but one with high risk of having disruptions is not necessarily a good choice. Quantifying disruption costs at the micro level can be tedious and difficult but at the macro level the calculations are straightforward—the supply chain manager calculates the increase in variable costs as goods are shipped less efficiently across the supply chain and accounts for any changes or decreases in fixed costs.
In the present article, we introduce a small supply chain model that is used to calculate variable and fixed supply chain costs as demand is fulfilled. The model is used in an MBA supply chain class or senior-level undergraduate class to introduce students to supply chain risk. Then, as a disruption is "introduced" into the supply chain at either a supply node or a fulfillment center the increase in total costs are calculated. The supply chain model is set up as mixed integer programming (MIP) model and solved in both Excel Solver and IBM ILOG CPLEX. IBM ILOG CPLEX is more efficient and will be utilized in future larger models, but Excel Solver is better at illustrating the supply chain to students and displaying how the increase in variable costs offsets any savings in fixed costs.
Literature
As early as 1980, Martin ([5]) pointed out the need for students to understand the managerial relevance of linear programming models in real-world settings and the use of mini-cases to facilitate that understanding. They specifically comment that although managers are not necessarily model builders, there must be a mutual understanding enabling the managers to have a firm grasp of the managerial relevance of the models as well as sufficient knowledge of the model to appreciate its effectiveness as well as its limitations.
McClure ([6]) remarked that many managers are hesitant to use mathematical modeling techniques that they find difficult to understand. This difficulty, in McClure's opinion, stems from the fact that the managers believe these models cannot be applied to complicated real-world problems. They opine that in many educational settings, theory and technique are emphasized more than practical industrial application, which allows the student to persist in the belief that these models cannot serve a real-world purpose.
Zahedi ([15]) examined the education of business students in management science and operations research. Zahedi suggested a real-world application as a way to ensure that students understand the power of these techniques in practice as well as motivating their use in the real world, something Zahedi referred to as semi-consulting in education. Zahedi pointed out that average business students' lack of mathematical background limited their ability to understand complicated tradeoffs and place hope in the (then) emerging field of electronic modeling.
Borsting, Cook, King, Rardin, and Tuggle ([1]) addressed the education of master of business administration (MBA) students in management science methods. In their work, they called for familiarizing the students with linear programming theory to be followed by a motivating mini-case to be solved using computer software. Students, in their opinion, should realize the usefulness of the models in real-world decision making. We concur wholeheartedly with this opinion.
Park and Nam ([7]) provided a classroom activity using linear programming to solve a dynamic lot-size problem using an MIP formulation of the Wagner-Whitin algorithm to provide an optimal solution using Excel solver. Riddle ([9]) used a simple product mix problem to familiarize students with linear programming models and Excel solver.
More recently, Yoder and Kurz ([14]) examined how linear programming is taught at a large research university in industrial engineering and management programs. Their results found a lack of comprehension of linear programming among many management students. They believed that this may be due to student perception of the importance of these models, and called for methods to be used to convey the importance of these models to business students.
Williams, Stanny, Reid, Hill, and Rosa ([13]) emphasized the need for managers to be able to communicate effectively through writing reports and memos. The authors, interestingly, found that while many students were able to formulate linear programming problems and solutions, they were less able to communicate the meaning of the results. They called for increased attention on assigning written reports where students were required to communicate and discuss these findings, something that would be necessary in their careers.
Chopra and Sodhi ([2]) provided an early glimpse into and classification of supply chain risks. They provide a discussion of various disruptions and discuss methods used to mitigate the risk of disruption. We can note that they would classify the risk modeled in the present article as internal disruption and recommend "stress testing" the supply chain and mitigating through increasing reserve capacity.
Snyder and Daskin ([11]) considered the effect of site failure in logistic networks, adding to transportation costs as demand must be fulfilled from facilities further afield. The authors develop a Lagrangian relaxation algorithm to arrive at facility locations that not only are inexpensive, but also provide for a robust reliability in the face of site failures.
Kouvelis, Chambers, and Wang ([3]) provided a review of supply chain research as well as a discussion of trends and opportunities. In their work, they stressed the importance of both supply chain education as well as the supply chain risk management. In teaching, they call for more experiential learning methods that show multifirm supply chain contexts. With respect to risk management, they called for increasing attention to disruption risk.
Manuj ([4]) examined supply chain risk in both academic and business practice. They identified several empirical concerns with the type of risk we deal with in the present article, something they classify as internal environment factor risk in the physical subchain. Interestingly, the work emphasizes the importance of training in addition to tactical mitigation methods, a perfect fit for the contribution we provide here.
Snider and Southin ([10]) used a campus cupcake experiential learning activity very early in a core operations management course, and included exposure to risk management in operations.
Most recently, and arguably most relevantly, Trent ([12]), in a discussion of trends, mentioned prominently both the need for education and talent management for supply chain professionals as well as the increase in global risk and the need for supply chain risk management.
The contribution of this work can be isolated and placed within the existing literature by commenting that, in our work, we propose a linear programming model that can be used to familiarize students with these types of models, but moreover to demonstrate their usefulness in real industrial systems, specifically in examining the risk of a distribution system in the face of disruption or the failure of facilities. Additionally, the student-friendly Excel Solver model can be a perfect segue way to introduce students to more advanced commercial solvers such as the IBM ILOG CPLEX model we provide, noting that often when a student is exposed to one commercial solver, learning others is typically much easier, in much the same way that learning a second foreign language is often easier than learning the first.
Roadmap
In the following sections, we introduce the notation, objective function, and constraints for the MIP model; validate the model by showing how it reacts appropriately to changes; introduce risk and the model used to predict the impact on the network from supply chain disruptions. The results are then discussed in the context of variable and fixed costs. Then, we provide learning goals and suggestions on using this in a classroom setting for undergraduate or graduate students. In the conclusion, we provide a synopsis of the results and future extensions to the supply chain model and the analysis of risk, as well as comments on how the model can be used for industrial sized networks.
Model
Context
The fictitious online retailer
Graph: Figure 1. Simple network showing two supply nodes, three fulfillment centers, and four demand nodes.
Notation
We introduce the following notation. Demand nodes are represented by the index
Graph
Graph
Graph
Graph
Graph
Graph
Graph
Graph
Flows of material exist between (a) supply nodes and fulfillment centers and (b) fulfillment centers and demand nodes. Variable costs per unit are denoted as
Graph
Graph
There are two types of decisions in the model, facility opening decisions and material flow decisions. The binary facility opening decisions are denoted
Graph
Graph
Graph
Graph
Objective function and constraints
With this notation introduced, we can put forth the objective function as
Graph
As can be seen in Equation 1, the first term sums facility fixed costs for supply nodes, the second sums facility fixed costs for fulfillment centers, the third calculates costs resulting from flows between supply nodes and fulfillment centers, and the last costs for flows between fulfillment centers and demand nodes.
The objective function (Equation 1) should be minimized, subject to the following constraints:
Graph
Graph
Graph
Graph
Graph
Graph
Graph
Graph
The constraints will now be briefly explained. The demand fulfillment constraint (Equation 2) ensures that the inbound flow of product into each demand node satisfies demand and must hold for each market. The supply node capacity constraint (Equation 3) ensures that capacity at the supply nodes cannot be exceeded by the outbound flow to the fulfillment centers. The supply node facility opening constraints (Equation 4) ensure that only open supply nodes can be used to ship product to the fulfillment centers. Both Equations 3 and 4 must hold for each supply node. The fulfillment center capacity constraints (Equation 5) ensure that capacity at the fulfillment centers cannot be exceeded by the flows to the markets. The fulfillment center opening constraints (Equation 6) ensure that only open fulfillment centers can ship product to markets. Similarly, both Equations 5 and 6 must hold for each fulfillment center. The fulfillment center flow constraints (Equation 7) ensures that the amount of product entering each fulfillment center from the supply nodes is equal to the amount shipped out to the demand nodes, and must hold for each fulfillment center. The nonnegativity constraints (Equation 8) prevent the flow variables from being negative, while the binary constraints (Equation 9) ensure that the facility opening decision variables are either zero or one.
As the primary purpose of this work is pedagogical, the following comments can be made with respect to the constraints. Students can be shown that constraints in Equations 4 and 6 are formulated in such a way that zero is on the right-hand side of the equation, whereas Equations 3 and 5 are formulated with a parameter (the respective capacities) on the right hand side of the equation. This is of course an arbitrary choice and yields mathematically identical constraints, a point which can be easily shown to the students.
Likewise, Equations 3 and 4 can be combined into a single constraint that serves both as a capacity constraint as well as a facility opening constraint.
Graph
The supply node capacity constraint (Equation 10) ensures that (a) the supply node can only be used if it is open and (b) that its capacity cannot be exceeded, and must hold for each supply node. In a similar way, Equations 5 and 6 can be combined into a single constraint that serves both as a capacity constraint as well as a facility opening constraint.
Graph
Analogously, the fulfillment center capacity constraint (Equation 11) ensures that (a) the fulfillment center can only be used if it is open and (b) that its capacity cannot be exceeded, and must hold for each fulfillment center. Of course, these are mathematically identical, and where Equations 3–6 are used in the ILOG formulation, Equations 10 and 11 are used in the Excel solver version of the model. As both the ILOG as well as Excel model formulations are available from the authors, students can be shown exactly how these constraints can alternatively be formulated, and that the same results will be obtained, enhancing the experiential learning goal of the exercise.
Numerical example
With the model now carefully explained, we can now parametrize the model and provide an example. The size of the example is chosen to make it accessible for students while being sophisticated enough to react appropriately to changes.
For the demand nodes, we will serve 11 markets (i.e.,
With the facility network specified, we can now turn to demands and costs. For this simple example, we start by defining the demand as 10 for all demand nodes (i.e.,
Graph
Graph
Graph
Graph
Graph
Table 1. Per unit variable costs of material flow from supply nodes to fulfillment centers.
Table 2. Per unit variable costs of material flow from fulfillment centers to demand nodes.
The last step is for the capacities for the supply nodes and fulfillment centers to be fixed, and given the specified demand for the network, the summed capacities of
Graph
Graph
Graph
Graph
Graph
Graph
PHOTO (COLOR): Figure 2. Solution for numerical example.
As can be seen in Figure 2, the optimal solution does not use the supply node Elizabeth, New Jersey, or the North-Midwest fulfillment center due to the slack capacity and relative costs of demand fulfillment. The remainder of the facilities are opened, and flows between facilities can be easily shown to students through the spreadsheet.
Validation: Showing how the model reacts to changes
We can now turn our attention to showing students how this model reacts appropriately to changes in the parameters. For this, we will utilize scenarios based on a full factorial experiment with two factors (i.e., capacities and fixed costs). We purposefully do not experiment with the proverbial "low hanging fruit" of demand, since as previously mentioned, it is an arbitrary normalization, and it is actually the demand relative to capacity that is most germane. For each factor, we will comprise two levels of high and low, yielding four scenarios, as shown in Table 3, where the top number in the cell is the fixed cost for facilities and the bottom number the capacity relative to demand. The fixed costs are the same for each type of facility and each specific facility. The capacities refer both to the supply node as well as the fulfillment centers and are expressed as percentage of summed demand.
Table 3. Illustrating the full factorial design for capacity and fixed costs.
With the full factorial study now designed, the results are obtained for the four scenarios. Now that the results are at hand we can examine them to see if the model reacts as one would expect. In educational settings, this is particularly important to convince students that these linear programming models are useful in the "real world," particularly before the crux of our analysis of risk occurs in the next chapter.
Before we start, it is helpful to recall that in the first numerical example, with costs and capacities between those presented in this section, the solution called for the supply node in Elizabeth, New Jersey, and the North-Midwest fulfillment center to be closed, with all other nodes remaining open. This was a result of the slack capacity in both the supply nodes as well as the fulfillment center network. In this case, slack capacity and nonnegligible (or substantial, if you wish) fixed costs, allowed for a cheaper solution where facilities were closed, saving fixed costs, albeit with a predictable increase in variable costs over the system. To show this to students, one can force all facility opening variables to be one, incurring fixed costs, but allowing their use. When this happens, fixed costs increase by $100, but variable costs decrease by $41.79. This is helpful to illustrate the tradeoffs present in these models to students.
In the first scenario, that with high fixed costs and high capacities, the Elizabeth, New Jersey, supply node as well as three (Northeast, North-Midwest, and Northwest) fulfillment centers are closed. Here, we can see that with more capacity and higher fixed costs, more facilities are closed compared with the previous instance. This can be rationally explained as with higher fixed costs and even more slack capacity, it behooves us to shed capacity and save fixed costs.
In the third scenario, with low fixed costs and high capacities, all supply nodes are opened, but two fulfillment centers (North-Midwest and South-Midwest) are closed. Compared with the first scenario, Elizabeth remains open since there are variable cost savings between this facility and the fulfillment centers, noting that the fixed costs are almost zero ($0.01), making any savings in variable costs to the fulfillment centers quite advantageous. This is different with the two Midwest fulfillment centers. Here, the slack capacity allows them to be closed, and even with almost zero fixed costs, they are closed. This can only be the result of these two facilities being truly redundant at this level of capacity, and the demand can be more efficiently fulfilled by the other fulfillment centers even with near zero fixed costs. Of course, this more efficient demand fulfillment comes specifically from the variable cost differences between the facilities.
In both scenarios two and four, those with low capacities, all facilities remain open. This is completely intuitive, as with no slack capacity, neither in the supply nodes, nor in the fulfillment centers, all of the facilities must remain open to fulfill the demand. Now that the model has been validated using several scenarios showing that it reacts appropriately to changes in parameters, we can now use the model to assess risk in the supply chain.
Using the model to illustrate and assess risk
Preliminary analysis
Risk in a supply chain is a complicated issue, and one that has received much attention. In this network, risk is best seen as something which will inhibit the fulfillment of demand. We will concentrate our efforts on simulating a disruption of a facility, perhaps a natural disaster, where the facility goes "offline." This will be modeled using a sudden decrease in the capacity of the facility. With this disruption, demand must be fulfilled by other facilities, which will require an increase in the variable costs. We note that for this part of the analysis, fixed costs are not relevant, as the decision maker has no ability to influence the disruption, and no ability to recoup the fixed costs.
Therefore, the risk manifests itself as an increase in the variable costs and can be quantified by a percentage penalty, which will be denoted Δ. This can be expressed, using C* as the optimal costs before disruption and C as the optimal costs after the disruption, as
Graph
With respect to capacities, it is obviously necessary to allocate additional capacity because the demand can only be fulfilled after the disruption by using slack capacity of other facilities. For this portion of the study, we will make supplier capacity 150% of demand and fulfillment center capacity 125% of demand. We will analyze the risk in two parts, first concentrating on the supplier nodes and then the fulfillment centers. For each of these, we will successively take facilities and disrupt them, placing them "offline" and taking away their capacity in the model. The optimal costs after disruption (C) will be compared with the optimal costs before disruption (C*). The solution and costs before disruption can be seen in Figure 3.
PHOTO (COLOR): Figure 3. Optimal solution and costs before disruption.
Starting with the supply nodes, we can iteratively disrupt Charleston, Elizabeth, and Long Beach. Moving to the fulfillment centers, we can disrupt the East and Northeast fulfillment centers. The results are summarized in Table 4. As can be seen, the supply node in Long Beach going offline generates the highest percentage penalty, around 30%. This is somewhat intuitive since it is the only west coast supply node, and if it is disrupted, all product must enter the system through the two east coast supply nodes. Disrupting either of the other two supply nodes results in modest percent penalties of between 5% and 8%. With the two example fulfillment centers, the East fulfillment center will increase costs by 16% if disrupted, while the Northeast fulfillment center will increase costs by 12% if disrupted.
Table 4. Results of risk analysis.
Full factorial design analysis
To make the analysis of the risk in the system more informative, we will employ a full factorial design over two experimental factors, capacity and fixed costs. Each experimental factor will have a high and a low level, resulting in four combinations. This has two purposes. First, the fixed cost and capacity levels will doubtlessly impact the network risk analysis specifically. Second, it is pedagogically advantageous to expose students to full factorial experimental design generally. For high-capacity (HC) scenarios, we will use 220 for fulfillment center capacity and 220 for supply node capacity, where in low capacity (LC) we will use 137.5 for fulfillment center capacity and 165 for the supply node capacity. For high fixed costs (HF), we use $999 for all fixed costs; for low fixed costs (LF), we use $0.01 for all of the fixed costs.
The results can be seen in Table 5, where all four scenarios are juxtaposed. Each scenario provides the base cost at the top, and costs and penalties for closing each supply node and fulfillment center. As can be seen, depending on the scenario (the levels of capacities and fixed costs) and the facility considered, the penalties can vary greatly. Students can be reminded that this is precisely the reason for the full factorial analysis being conducted and the model's importance in real-world applications given capacities and costs. Several insights can be afforded.
Table 5. Results of full factorial risk analysis.
First, and most noteworthy, we can find that some penalties for facilities are zero. This is obviously the result of the fact that some facilities were closed in the base case (before network disruption) of the scenario. In other words, some facilities were already not needed in some instances. These include supply node Elizabeth and fulfillment center North-Midwest in the LC HF scenario, supply node Elizabeth and fulfillment centers Northeast, North-Midwest, and Northwest in the HC HF scenario, and fulfillment centers North-Midwest, South-Midwest, and Northwest in the HC LF scenario. Second, we can see that in the LC LF scenario, all facilities have positive penalties, indicating that all facilities are used in the base scenario. Third, and most interesting, is that there is a pattern among the penalties, but it is not a perfectly predictable pattern. For the supply nodes, it is usually the case that they can be ranked from most to least important as Los Angeles, Charleston, and Elizabeth. This occurs in three of the four scenarios, but in the LC LF scenario, Elizabeth is more important than Charleston. This must be a result of variable cost differences and capacities, resulting in flows in the system. It can be emphasized to the students that while a general pattern has emerged (and this is certainly valuable) the specifics of an instance can result in decisions that defy this pattern. Likewise, in three of the four scenarios, the Southwest fulfillment center has the highest penalty, but in the HC LF scenario, the Southeast fulfillment center has a higher penalty. If one were to consider average penalties for fulfillment centers across the four scenarios, we would group them into the most important (i.e., Southeast and Southwest, where average penalties exceed 9%), a moderately important group (i.e., East and Northeast, with average penalties around 6%), and a less important group (i.e., North-Midwest, South-Midwest, and Northwest, with average penalties below 4%). Of course, the devil is always in the details, and we may choose to place South-Midwest in the moderately important group, for instance. In any event, students can be shown how general patterns often emerge providing some useful golden nuggets, while the specific solution of a particular instance, with given demands, costs, and capacities may well deviate from a pattern.
Class presentation and learning goals
In this section, we will provide recommendations on how this model and exercise can be implemented in a classroom setting. The learning goals and outcomes, listed from low to high on Bloom's taxonomy, would include:
To accomplish these, we would recommend that the instructor first introduce students to linear programming, unless they are already familiar with these models. This initial familiarization can be done perhaps by conducting a short lecture on the well-known "Lego my Simplex" Lego furniture linear programming experiential learning activity (e.g., Pendegraft, [8]). This simulates a simple production problem of producing tables and chairs using a finite number of Lego blocks. This activity usually does an excellent job of showing a simple problem which can be modeled using linear programming. After introducing the Lego furniture business problem, the instructor usually introduces the theory of linear programming (decision variables, parameters, objective function, and constraints). Once the students are familiar with the business problem and theory, the class is shown how Excel Solver can be used to solve the problem. This can be done either in class or through an additional online lecture that students can view outside of class. Of course, if students are already familiar with linear programming models, this step may be omitted.
Once the students are familiar with the basics of linear programming, the context should be carefully introduced (following Context section), notation (Notation section) and the mathematical model objective function and constraints (Objective function and constraints section). Once this is accomplished, the Excel Solver model can be introduced and used to solve the numerical example contained in Numerical example section. A few minor changes to the input parameters are made to illustrate how the optimal solution changes, perhaps following Validation: Showing how the model reacts to changes section. At this point, the students should be quite familiar with the model and convinced of its validity.
The students are provided the excel file and break up into groups to obtain the solutions to a series of supply chain disruptions provided by the instructors, possibly following Using the model to illustrate and assess risk section. The instructor can next lead a group discussion of the results of the supply chain disruptions focusing on "patterns" found in the solution results. The presentation ends with the instructor demonstrating the linear programming software IBM ILOG CPLEX software to model and solve the risk model. More advanced models are solved with the software. This entire classroom presentation will take around 150 min, so two 75-min sections or three 50-min sections.
The students have already learned to use Excel solver before this class so are familiar with solver. The main learning goal for students is the effect that disruptions have on the supply chain and the associated increased variable costs due to not shipping over the most cost efficient routes. Also, the students will notice how "patterns" emerge as the disruptions move around in the supply chain. Finally, solving the supply chain disruptions using solver and ILOG illustrates to students the power of linear programming and its many industrial uses.
Conclusion and outlook
In the present article, an MIP model was developed that illustrates to students how supply chain risk will cause internal disruptions in a fictitious online retailer. The purpose is twofold in that students can be further familiarized with linear programming models and their usefulness generally and the effects of supply chain risk specifically. The model minimizes the fixed and variable costs of satisfying demand at demand nodes. Product is shipped from supply nodes through fulfillment centers to the demand nodes. Fixed costs are incurred in the "opening" of the supply nodes and fulfillment centers. The variable costs are the cost per unit of shipping a unit from a supply node to a fulfillment center and a unit from a fulfillment center to a supply node. The supply chain disruptions in the present article are modeled as an involuntary loss of capacity at a supply node or a fulfillment center. The risk of disruption can be quantified by calculating the increase in variable costs, as goods are shipped less efficiently across the supply chain. The disruption costs are calculated for the involuntary closing of each of the supply nodes and fulfillment centers.
In the present study, we only considered the supply chain risk due to the involuntary closing of a supply node or a fulfillment center resulting in a complete loss of capacity for the node. This MIP model can easily be modified to consider other types of supply chain disruptions or a relaxation of assumption in the context presented here. This would include (a) shutdowns of several nodes; (b) partial shutdowns of one or more supply nodes or fulfillment centers; (c) dynamic costs or changing variable costs over time; (d) economies of scale that cause decreasing variable costs as shipment sizes increase; (e) full or partial shutdowns that cause demand to be greater than supply, and therefore leave some demand unsatisfied; (f) multiple transportation modes with different variable costs for shipments between a node and a center; and (g) capacity restrictions on shipments between a node and a center.
References
1 Borsting, J. R., Cook, T. M., King, W. R., Rardin, R. L., & Tuggle, F. D. (1988). A model for a first MBA course in management science/operations research. Interfaces, 18 (5), 72 – 80. doi: 10.1287/inte.18.5.72
2 Chopra, S., & Sodhi, M. S. (2004). Managing risk to avoid Supply-Chain breakdown. MIT Sloan Management Review, Fall, 53 – 61.
3 Kouvelis, P., Chambers, C., & Wang, H. (2009). Supply chain management research and production and operations management: Review, trends, and opportunities. Production and Operations Management, 15 (3), 449 – 469. doi: 10.1111/j.1937-5956.2006.tb00257.x
4 Manuj, I. (2013). Risk management in global sourcing: Comparing the business world and the academic world. Transportation Journal, 52 (1), 80 – 107.
5 Martin, M. J. C. (1980). Short cases for teaching management science: An extended comment. Interfaces, 10 (2), 10 – 12. doi: 10.1287/inte.10.2.10
6 McClure, R. H. (1981). Educating the future users of OR. Interfaces, 11 (5), 108 – 112. doi: 10.1287/inte.11.5.108
7 Park, J., & Nam, S. H. (2006). Teaching Lot-Sizing decisions: A spreadsheet/Mixed-Integer programming approach. Decision Sciences Journal of Innovative Education, 4 (1), 163 – 167. doi: 10.1111/j.1540-4609.2006.00110.x
8 Pendegraft, N. (1997). Lego of my simplex. ORMS Today, 24 (1), 8.
9 Riddle, E. J. (2010). An active learning exercise for introducing the formulation of linear programming models. Decision Sciences Journal of Innovative Education, 8 (2), 367 – 372. doi: 10.1111/j.1540-4609.2010.00263.x
Snider, B., & Southin, N. (2016). Operations course icebreaker: Campus club cupcakes exercise. Decision Sciences Journal of Innovative Education, 14 (3), 262 – 272. doi: 10.1111/dsji.12100
Snyder, L. V., & Daskin, M. S. (2005). Reliability models for facility location: The expected failure cost case. Transportation Science, 39 (3), 400 – 416. doi: 10.1287/trsc.1040.0107
Trent, R. J. (2018). Supply chain trends happening now. Logistics Management, 48 – 52.
Williams, J. A. S., Stanny, C. J., Reid, R. C., Hill, C. J., & Rosa, K. M. (2015). Assessment of student memo assignments in management science. Journal of Education for Business, 90 (1), 24 – 30. doi: 10.1080/08832323.2014.968517
Yoder, S. E., & Kurz, M. E. (2015). Linear programming across the curriculum. Journal of Education for Business, 90 (1), 18 – 23. doi: 10.1080/08832323.2014.968516
Zahedi, F. (1985). MS/OR Education: Meeting the new demands on MS education. Interfaces, 15 (2), 85 – 94. doi: 10.1287/inte.15.2.85
By Robert O. Neidigh and Ian M. Langella
Reported by Author; Author