Treffer: Mixed-Integer Linear Programming Optimization for the Supply Chain Game

In this article, we describe the use of linear programming in conjunction with an online simulation game in an undergraduate supply chain management class. Supply chain management is a required course for students enrolled in a Business Administration program with a concentration in management. It delivers a broad overview of the supply chain management discipline, using a simulation game by Responsive Learning Technologies to provide students with hands-on experience designing and managing a supply chain. In this teaching brief, I give an overview of the game and present the formulation of an accompanying mixed-integer linear programming model to design an optimal supply chain that meets customers, demand while maximizing company profit. Students use a spreadsheet template to develop their own implementation of the model and then apply the optimal solution during the simulation. Consequently, they gain an understanding of the role of network optimization in supply chain management. I conclude with recommendations for management of game activities using this model and a brief discussion of some of the issues to be monitored as the game is played.

As Provided

AN0153247603;q1n01oct.21;2021Oct29.06:27;v2.2.500

Mixed‐integer linear programming optimization for the Supply Chain Game 

In this article, we describe the use of linear programming in conjunction with an online simulation game in an undergraduate supply chain management class. Supply chain management is a required course for students enrolled in a Business Administration program with a concentration in management. It delivers a broad overview of the supply chain management discipline, using a simulation game by Responsive Learning Technologies to provide students with hands‐on experience designing and managing a supply chain. In this teaching brief, I give an overview of the game and present the formulation of an accompanying mixed‐integer linear programming model to design an optimal supply chain that meets customers, demand while maximizing company profit. Students use a spreadsheet template to develop their own implementation of the model and then apply the optimal solution during the simulation. Consequently, they gain an understanding of the role of network optimization in supply chain management. I conclude with recommendations for management of game activities using this model and a brief discussion of some of the issues to be monitored as the game is played.

Keywords: content areas; experiential learning; games and simulations; pedagogical approaches; supply chain management

INTRODUCTION

Simulation games that illustrate academic concepts are increasingly popular. Simultaneously, active learning and simulation‐based pedagogy are becoming more prevalent in supply chain management classrooms (Angolia & Pagliari, 2018). Simulation games have the advantage of creating a realistic model of business environment, without costly consequences of misguided business decisions. They also allow for time compression, provide quick feedback to the learner, and offer flexibility with a multitude of game choices.

The simulation game I use in the supply chain management course is offered by Responsive Learning Technologies (www.responsive.net). In this game, students are required to manage a two‐tiered vertically integrated supply chain with the objective to maximize profit. As the game progresses, teams of students must identify locations where they want to build production factories, the amount of daily manufacturing capacity to procure for each factory, and locations for distribution warehouses. They must also decide which modes of transportation to use between factories and warehouses, and which customer markets each warehouse will serve. Customer markets in the game exhibit predictable demand patterns with random components. The goal of the game assignment is to design a balanced supply chain, with total production capacity sufficient to meet total demand and a cost‐efficient distribution system.

In this teaching brief, I propose an exercise where the configuration of the supply chain is determined by formulating and solving a mixed‐integer linear program (MILP) that integrates facility location and transshipment decisions. The exercise provides an application of linear programming to a simulated game scenario that models realistic business issues in supply chain management. Network optimization models are oftentimes covered in a separate, prerequisite course on management science techniques. This exercise reduces the "silo learning" effect by helping to bridge the gap between introductory management science and supply chain management curricula.

LITERATURE REVIEW

Instructors are offered a broad array of Supply Chain Games, and Cvetić and Vasiljević (2012) developed a selection procedure specifically for logistics and supply chain management games. They use a multicriteria analysis applied to a database of various simulation games for instruction. Games were selected and evaluated based on cost, and a weighted combination of functionality, simplicity, duration, and ease of setup. The Supply Chain Game by Responsive Learning Technologies that I adopted is relatively inexpensive. At the time of writing, a single‐student license costs $30, if purchased online. The game offers rich functionality. It touches the topics of demand forecasting, inventory management, design of a production and distribution network, and selection of the most economical mode and routes of transportation. By default, the game is set up to run continuously over the course of 1 calendar week, simulating 2 years of operations. The standard game scenario is preset for instructors, but it can be modified if desired.

According to the taxonomy of simulation games proposed by Wood (2007), all games can be classified by pedagogical objectives into three categories. Insight games allow students to quickly gain an understanding of key ideas behind a certain topic. Analysis games concentrate on acquiring and applying specific skills. Capstone games provide a more comprehensive and less focused experience in comparison to insight and analysis games. Using this classification, Supply Chain Game by Responsive Learning Technologies is an analysis game. It is designed to apply and reinforce key topics discussed in a typical supply chain management course.

Simulation games also fall into several categories based on the method of implementation (Grandzol & Grandzol, 2018): spreadsheet‐based, tangible product, application software, manual, and online. Spreadsheet‐based games are probably the most prevalent, due to the popularity of spreadsheet software, and its tight integration into a typical business curriculum. For example, Shovityakool et al. (2019) present a spreadsheet‐based Flexible Supply Chain Management Game (FSCMG), where groups of students can play three modules: wholesaler, manufacturer, and supplier. FSCMG exposes students to pricing, forecasting, inventory management, and fostering collaborative relationships in a supply chain. Another spreadsheet‐based game, emphasizing cooperation among competing suppliers (known as co‐opetition), is provided by Fetter and Shockley (2014). In this game, teams of students run a simulated supply chain for both traditional and co‐opetition scenarios with varying parameters (e.g., demand variability and lead time) and compare key supply chain performance measures. DuHadway and Dreyfus (2017) discuss an Excel‐based simulation game focused on the sales and operations planning process and a supplier selection process. An Excel‐based blood supply game by Mustafee and Katsaliaki (2010) simulates broad supply chain operations. The game offers students a simulated environment, modeling the flow of materials (blood) and information from suppliers (donors) to customers (patients). An additional layer of complexity is provided by the fact that the product is perishable with limited supply.

Tangible games, where students engage in physical activities, are less common in the literature. Snider and Eliasson (2009) present a tangible product simulation, where students use Lego blocks to create inukshuks—Inuit rock structures used as landmarks to guide travelers. The game serves as a hands‐on exercise to illustrate how the JIT process can be used for mass product customization.

Application game software, although widely used in the early 2000s, has decreased in popularity simply because managing an installation of the game on lab or students' computers is an unnecessary complication. Many of these games have now been converted from manual and/or application‐based to an online format. For example, one of the most frequently used simulation games, the Beer Game, illustrates the bullwhip effect in a single‐product linear multi‐echelon supply chain (Sterman, 1989). In its classic format, the game is played manually, with students passing pieces of paper with orders written on them to each other, and manually keeping track of costs. To avoid inconveniences and errors and to streamline the game, Responsive Learning Technologies created an online version of the Beer Game (http://responsive.net/ebeer.html). A broader adaptation of the Beer Game, an online Wood Supply Game (WSG), is presented by D'Amours et al. (2017). In WSG, a common source of raw material (trees) feeds into two serial supply chains for two different products, lumber and paper, the demands for which are not correlated. Another example of an online simulation game is LINKS‐Simulation (Chapman, 2005). In this comprehensive game, students are required to manage forecasting, production, and procurement for a manufacturing company.

Rapid technological developments drive the evolution of simulation games. Liu (2017) presents a study where groups of students were exposed to either a video game‐based learning, or a traditional educational environment. Liu found that video game‐based learning has a more positive effect on the instruction of undergraduate courses in supply chain and logistics management. His paper demonstrates the value of gamification, a recent educational approach to motivate students' learning. Another example of an application game is ERPsim, developed by HEC Montreal. ERPsim is built upon the commercial SAP ERP system (Angolia & Pagliari, 2018). An advantage of this SAP‐based game is the hands‐on practice with one of the most popular ERP systems.

I have used an online simulation game provided by Responsive Learning Technologies in an undergraduate supply chain management class for several years. The focus of this article is the application of a MILP optimization model to guide strategic decisions in this game. The use of linear and integer programming for active learning in a classroom to solve business issues is documented in the literature. Drake et al. (2011) describe a teaching case where an office supplies distributor can use LP to optimize the transportation network and greatly reduce logistics costs. An example of the use of MIP in game settings is provided by Beliën et al. (2011). The authors use the Gigabike game, where students are required to create a team of ten professional bike riders who will score the highest number of points over the course of the season.

The EnergyState game scenario, used by Beliën et al. (2013), is another example and is somewhat similar to the one discussed here. The EnergyState game requires students to devise a plan to supply a country with the electrical power, using a mix of different power plants (nuclear, coal, gas, wind, and solar), in order to meet a number of objectives, including low cost, maximum power generation, and minimum CO<subs>2</subs> pollution. The optimization model described in this teaching brief deals primarily with strategic decisions in a supply chain: selecting the locations for factories and warehouses, choosing the right manufacturing capacity for production facilities, and selecting transportation modes and routes to optimize the total supply chain profit. These game decisions are of strategic importance because, if done incorrectly, they will undermine the profitability of the supply chain, regardless of the choice of inventory policies to manage production and inventory.

RESPONSIVE LEARNING TECHNOLOGIES SUPPLY CHAIN GAME

Responsive Learning Technologies offers an online Supply Chain Game ("Supply Chain Management Simulation," Responsive Learning Technologies, July 14, 2017, http://responsive.net/supply.html), which requires students to make decisions concerning the location of production facilities and warehouses, daily production capacities of the plants, transportation routes to deliver product from production plants to consumer markets, and inventory policies. The game is organized as follows. The virtual production company, Jacobs Industries, operates in four geographical areas, called Calopeia, Sorange, Entworpe, and Tyran, located on the "continent," and an island called Fardo (Figure 1).

Production factories and distribution warehouses

The company manufactures an industrial foam, which can be used as a thermal and acoustic insulator. The foam is manufactured in drums of standard size. At the beginning of the game, students have a fully operational production factory in Calopeia, as well as a distribution warehouse in the same region, as indicated in Figure 1. The company's customers are located in all five regions. Customer orders for foam drums are fulfilled by mailing drums with foam from distribution warehouse(s) to the customers. Mailing drums from the warehouses is the only option to fulfill customers' orders. Warehouses receive foam drums from the production factories. There are two possible modes of transportation for drums between the production factories to the warehouses: mail and truck. The objective of this simulation game is to design a two‐tiered supply network to fulfill customers' demand in different geographical regions, while achieving the highest cash position. It involves building production factories and warehouses in different regions of the virtual continent, in addition to the ones existing in Calopeia at the beginning of the game. Students have an option to build a production factory and a distribution warehouse in each region. If a team decides to have a factory and a warehouse in all five geographical regions, the resulting supply chain design is shown in Figure 2.

Each warehouse costs $100,000 to build, and its construction takes 60 simulated days. Each warehouse is assumed to have an unlimited storage capacity. Each production factory costs $500,000 to build, and its construction takes 90 simulated days. In order to meet demands in the customer markets, students must decide how many units of daily production capacity each factory should have, where a unit of capacity is equal to one foam drum. The game begins with the only production factory in Calopeia having an initial capacity of 70 units (i.e., 70 foam drums per day). Students can buy additional production capacity for the Calopeia factory. Each capacity unit (ability to produce one extra foam drum per day) costs $50,000, and takes 90 simulated days to "install," just like the construction of a new factory. Typically, when students decide to build a new factory in a region without one, the decisions to construct a factory and buy manufacturing capacity are done simultaneously, as having a factory without a production capacity does not make sense. If a team decides to construct a new factory with 10 units per day production capacity, it will cost $500,000+$50,000*10 = $1,000,000. Any factory's capacity can be expanded at a later point in time by purchasing additional capacity units; however, purchased capacity cannot be retired.

Logistics and transportation

Once factories and warehouses are in place, foam drums can be manufactured and sent to the warehouses from those factories by mail or truck. If mailing is chosen, units can be mailed one‐by‐one, with the cost increasing proportionally to number of units shipped. If a truck is chosen, transportation cost is charged as a flat fee, regardless of number of units shipped (up to a truck's maximum carrying capacity of 200 finished units). Students may choose to ship by truck in smaller (less than full truckload) quantities, but it will naturally increase transportation expenses per unit shipped. Therefore, if trucking is chosen as a transportation mode between factories and warehouses, there is a strong financial incentive to transport foam drums in full truckload quantities. Table 1 summarizes transportation expenses between factories and warehouses.

1 TABLETransportation costs between factories and warehouses

<table><thead><tr><th align="left">Origin and destination</th><th align="left">Cost per truckload</th><th align="left">Cost to mail one drum</th></tr></thead><tbody><tr><td>Same region</td><td>$15,000</td><td>$150</td></tr><tr><td>Different regions on continent</td><td>$20,000</td><td>$200</td></tr><tr><td>Between continent and Fardo</td><td>$45,000</td><td>$400</td></tr></tbody></table>

Transportation of finished units between factories and warehouses has a lead‐time, as shown in Table 2.

2 TABLETransportation times between factories and warehouses

<table><thead><tr><th align="left">Origin and destination</th><th align="left">Truck</th><th align="left">Mail</th></tr></thead><tbody><tr><td>Same region</td><td>7 days</td><td>1 day</td></tr><tr><td>Different regions on continent</td><td>7 days</td><td>1 day</td></tr><tr><td>Between continent and Fardo</td><td>14 days</td><td>2 days</td></tr></tbody></table>

As one can see from Tables 1 and 2, it is most economical to ship units from a factory to the warehouse located in the same geographic region. Lead times for shipping within the continent are the same, regardless of origin and destination. Both transportation expenses and lead times are highest when finished units are shipped between the continent and Fardo island in either direction.

Warehouses ship foam drums to consumers in the regional markets. Each warehouse can ship to customers located in any region (including Fardo island). The only transportation option from the warehouse to a customer is mail, and the costs are shown in Table 3.

3 TABLETransportation costs between warehouses and customers

<table><thead><tr><th align="left">Origin and destination</th><th align="left">Cost to mail one drum</th></tr></thead><tbody><tr><td>Same region</td><td>$150</td></tr><tr><td>Different regions on continent</td><td>$200</td></tr><tr><td>Between continent and Fardo</td><td>$400</td></tr></tbody></table>

As one can see by comparing Tables 1 and 3, the cost structure to mail drums to customers from warehouses is the same as to mail drums from factories to warehouses, and it is most economical to ship units to a customer from a warehouse located in the same region as a customer.

Markets' demand

In order to make informed decisions about where to build plants and warehouses, and how much capacity each plant should have, students need to estimate the total demand potential for each geographical market. Before the game begins, the instructor must choose the Supply Chain Game scenario, using the game administrator web interface, and start the game. The game engine will quickly simulate 730 days (2 years) of operations and enter the suspended mode. At this point, the instructor should enable the game for students to view. Teams have control of the game from day 730 to day 1460. Students should start by predicting the total demand in all five markets based on historical demand data. The customers in the five markets exhibit different demand patterns. Customers in the Calopeia market use foam to retrofit or repair old industrial air conditioners. The demand for foam drums in Calopeia starts from day 0 and is highly seasonal (Figure 3).

Demand in the Sorange region does not start until simulated day 640. Since students gain access to the game data when the simulator is suspended at day 730, they have 90 days' worth of demand history. Customers in the Sorange market use industrial foam in the production of hardwood floor panels to provide additional acoustic isolation. Students are advised that the demand for foam drums in Sorange will grow linearly from day 640 until day 1430. On day 1430, the demand will peak, after which it will start declining linearly, vanishing to zero on day 1460, since the foam produced by the company is displaced from the market by a new foam technology, which will become available on day 1460. The demand in the Sorange data presented to students on day 730 is shown in Figure 4, together with a simple linear trend model. In order to forecast demand in Sorange for the rest of the game, students must project the linear demand using a regression equation starting on day 640.

Customers in the Tyran market use foam for sound insulation in the production of house appliances, such as dryers and dishwashers. Demand in the Tyran market, similarly to Sorange, does not start until day 640. A month after that, on day 670, demand stabilizes to its long‐run average value, where it remains until day 1430. In the last 30 days of the game, the demand gradually decreases from the long‐run average to zero on day 1460 for the same reason as in Sorange (displacement from the market by a new product). Demand in Tyran is shown in Figure 5 along with a fitted forecast line.

On the island of Fardo, the demand for industrial foam is primarily driven by the single‐engine airplane industry, which uses the foam for insulation. As in Tyran, demand starts on day 640, and grows linearly to day 670, stabilizing to its long‐run average value after that. Figure 6 illustrates the demand in Fardo, along with its fitted forecasts.

In the last region, Entworpe, the foam is used by a single manufacturer to produce insulating quilts for insertion into walls, where both thinness and thermal insulation are important. The quilts manufacturer is ordering foam according to their fixed‐order‐size inventory policy with order size set to 250 units. Therefore, unlike other markets, Entworpe's demand quantity is stable and equal to 250. The timing of incoming orders, however, is not known. Similar to Fardo and Tyran, demand in Entworpe starts on day 640. The resulting intermittent demand pattern in Entworpe is illustrated in Figure 7, together with the fitted long‐run average.

LEVEL VERSUS CHASE STRATEGY

Before the game begins, the author recommends having a discussion with students about how to handle nonstationary demand. Demand in Calopeia varies greatly from zero to over 130 and has a clear seasonal nature. One approach to manage this demand pattern would be to purchase capacity sufficient to handle peak demand. In this case, most of the time demand in Calopeia will be below the peak, yielding low‐capacity utilization. At the same time, to avoid building excessive inventory, the team should constantly change inventory policy parameters (specifically, the reorder point [ROP]) in order to meet demand and maintain a low level of on‐hand inventory. This is a well‐known "chase" strategy. One of the disadvantages of the chase strategy is the big up‐front cost of a large capacity purchase.

An alternative approach to managing seasonal demand is the "level" strategy. Pursuant to this strategy, the company needs enough capacity to manufacture the average demand level at a steady rate. During low‐demand periods, the company will build inventory, and during high‐demand periods, it will use this inventory. Even though this strategy is associated with higher holding costs, the investment in production capacity is less than for the chase strategy. Another advantage of the level strategy is that, unlike the "chase" strategy, it does not require constant adjustments to inventory policy. In practice, since the level strategy implies steady and constant production, it does not require workforce changes through hiring and firing, resulting in better employee loyalty and morale and lower workforce training costs. This prompts a lot of companies facing seasonal demand to use the level strategy. For example, Smithfield Foods, a major U.S. pork manufacturer, starts producing and storing Christmas hams well before the end‐of‐the‐year holiday season, using existing production capacity. These meat products are stored in cold warehousing facilities and shipped to retailers and food‐service companies as demand increases toward the end of the year.

Demand in Sorange is not seasonal, but rather upward trending during most of the game. Meeting this demand using the chase strategy requires the team of students to expand the capacity of the plant continuously to keep up with the ever‐rising demand. This requirement is an obvious disadvantage of the chase strategy. Another drawback is, again, the high cost of production capacity accumulated toward the end of the game. It is much easier, using the level strategy, to procure capacity equal to the average demand, and, by setting an appropriate ROP, to make sure this capacity is constantly utilized in order to produce cumulatively the quantity required to meet demand in Sorange. In the first half of the game, the team will build the inventory since the demand in Sorange will be lower than the production capacity. This inventory will be all used up to meet demand during the second half of the game, when the Sorange demand will become higher than manufacturing capability. Inventory, therefore, will be utilized as a stored capacity.

FORMULATING THE MILP MODEL TO DESIGN AN OPTIMAL SUPPLY CHAIN

The discussion of "chase" and "level" strategic approaches to handle the demand should provide students with an understanding that a level strategy is more appropriate and less risky. Thus, to determine the optimal locations for factories and warehouses, factories' production capacities, and optimal transportation routes, students are required to formulate and solve an MILP model. The problem can be viewed as a three‐tiered supply chain network, as shown in Figure 8. In the MILP problem formulation, the geographical regions are numbered from 1 (Calopeia) to 5 (Fardo) for convenience. The optimization model is formulated in terms of total units produced and shipped to the markets over the entire game (2 simulated years, from day 730 to 1460). First, students are instructed to use the historical data (available at simulated day 730) to estimate the total number of units demanded in all five geographical markets from day 731 to 1460 (D<subs>1</subs> through D<subs>5</subs> in Figure 8). Demand in Calopeia can be estimated by adding all daily demand figures in the first simulated 730 days. Since demand in Tyran and Fardo remains stable throughout most of the simulation (with the exception of the last 30 days, when demand gradually declines to zero), it is easy to estimate the total demand in these areas. Demand in Entworpe is intermittent, as illustrated in Figure 7. To estimate the total units demanded in Entworpe, one can, again, use the average daily demand during days 640–730. Students find it most difficult to estimate the total demand in Sorange, because the daily demand will grow linearly until day 1430, after which it will drop to zero in the last 30 simulated days. To estimate total demand in Sorange, students should use a linear trend equation, projected from day 731 until day 1430, and an estimated downward linear trend in the last 30 days after that, as demonstrated in Figure 9. The resulting total demand figures D<subs>1</subs> through D<subs>5</subs> in Figure 8, therefore, are not decision variables, but estimated parameters for the MILP.

Defining the decision variables

The game begins with a warehouse present in Calopeia (region 1). Other regions do not have warehouses. We define binary decision variables for warehouses in these regions as:

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Similarly, the Calopeia region has a factory with 70 units of daily production capacity. To decide which other regions should have a factory, we define the binary decision variables:

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Each factory i is also characterized by its daily production capacity c<subs>i</subs> ≥ 0 which should be procured simultaneously with the decision to build a factory. For a factory in Calopeia (region 1), c<subs>1</subs> is the amount of additional daily manufacturing capacity, added on top of the 70 units present at the beginning of the game, for a total daily capacity of (70 + c<subs>1</subs>) units.

Arcs in Figure 8 are labeled with transshipment decision variables, followed by the cost of transporting one unit in parentheses. Solid arcs represent mailing units, and dashed arcs represent shipping by truck. Specifically,

Note that costs of mailing a unit presented in Figure 8 correspond to costs of mailing one unit in Tables 1 and 3. Costs of trucking a unit, shown in Figure 8, assume that transportation by truck is done using full truckloads. For example, the cost of trucking one unit from factory to warehouse within region 1 is ($15,000 per truck)/(200 units per truck) = $75/unit is reported next to the variable t<subs>11</subs>. Trucking to another region on the continent comes at a cost of $20,000 per truck, which translates into the cost of $100/unit for a full truckload, reported next to variables t<subs>12</subs>, t<subs>13</subs>, and t<subs>14</subs>. Sending a full truckload between a continent and the island of Fardo costs $45,000, yielding a cost of $225/unit associated with the variable t<subs>15</subs>. Note also that trucking from a factory in Fardo to any regional warehouse on a continent will also bear the cost of $225/unit. Trucking within the Fardo region yields the cost of $75/unit. Variables m<subs>21</subs>,..., m<subs>55</subs>, t<subs>21</subs>,..., t<subs>55</subs>, and s<subs>21</subs>,..., s<subs>55</subs>, and corresponding arcs, are not shown in Figure 8 to avoid clutter.

The objective function

The objective of the game is to maximize profit generated by the supply chain. Each drum is sold to the customer for $1450, and the production cost per unit (not counting the $1500 batch setup cost) is $1000. We will assume that all units supplied by the factories will be sold to customers. Since it takes 90 days to build new factories and install new production capacities, any additional daily production capacity <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12247:dsji12247-math-0003" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>&#8721;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>5</mn></msubsup><msub><mi>c</mi><mi>i</mi></msub></mrow></math> will become available only in the last (730–90) = 640 days of the game, assuming that teams will start building new factories and procure capacities for them on day 731, that is, immediately after students begin the game. Calopeia's initial 70 units of daily capacity are available during the entire game (730 days), yielding 51,100 units, if fully utilized throughout the game. Therefore, the total revenue generated by the supply chain is equal to:

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Total cost consists of cost of manufacturing units, cost of building factories and warehouses, cost of additional production capacity, cost of trucking and mailing product between the factories and warehouses, and cost of mailing product from warehouses to customers. Subtracting these costs from the revenue (1) yields the following expression for the objective function:

2 <math display="block" altimg="urn:x-wiley:15404595:media:dsji12247:dsji12247-math-0005" xmlns="http://www.w3.org/1998/Math/MathML">MaxProfit=45051,100+640&#8721;i=15ci&#8722;500,000&#8721;i=25xi&#8722;100,000&#8721;i=25yi&#8722;50,000&#8721;i=15ci&#8722;(75t11+100t12+100t13+100t14+225t15+&#8943;+225t51+225t52+225t53+225t54+75t55)&#8722;(150m11+200m12+200m13+200m14+400m15+&#8943;+400m51+400m52+400m53+400m54+150m55)&#8722;(150s11+200s12+200s13+200s14+400s15+&#8943;+400s51+400s52+400s53+400s54+150s55)</math>

Strictly speaking, the total cost should also include a production setup cost and an inventory holding cost, but these costs depend on ordering quantities and ROPs set at the warehouses. Since these parameters are not among the decision variables in the MILP formulation, we will omit setup and inventory holding costs from the objective function (2).

Constraints

The supply chain configured in the game should be balanced. All supply produced and shipped by the factories should be used to meet the demand generated by customers in the five markets. First, we must ensure that all units manufactured in the factories are shipped from the factories to the warehouses (supply constraints):

In constraints (3) and (4), the left‐hand sides are equal to total out‐bound shipments from factories 1 to 5. The right‐hand sides are equal to the total number of units produced by a corresponding factory during the game, under the assumption that all production capacities are fully utilized. Note that constraint (3) for the Calopeia region contains a fixed supply term of 51,100 units, corresponding to the total production of existing capacity of 70 units per day over the course of 730 days, as discussed earlier.

We also need a set of transshipment constraints to ensure that all units shipped from the factories to each warehouse are then shipped from that warehouse to the customers:

5 <math display="block" altimg="urn:x-wiley:15404595:media:dsji12247:dsji12247-math-0008" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munderover><mo>&#8721;</mo><mrow><mi>i</mi><mspace width="0.33em" /><mo>=</mo><mspace width="0.33em" /><mn>1</mn></mrow><mn>5</mn></munderover><mspace width="0.33em" /><mfenced separators="" open="(" close=")"><mrow><msub><mi>m</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>t</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfenced><mo>=</mo><munderover><mo>&#8721;</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>5</mn></munderover><msub><mi>s</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mspace width="0.33em" /><mo>,</mo><mspace width="0.33em" /><mi>j</mi><mspace width="0.33em" /><mo>=</mo><mspace width="0.33em" /><mn>1</mn><mo>,</mo><mspace width="0.33em" /><mi>&#8943;</mi><mo>,</mo><mspace width="0.33em" /><mn>5</mn><mo>.</mo></mrow></math>

Next, we need a set of market demand constraints. It could be that in some of the geographical areas meeting the demand does not generate profit due to high transportation costs from warehouses in other regions, or due to the insufficient demand volume to cover the expenses associated with constructing a production factory and/or a warehouse in that area. Meeting the market demand in such regions will result in a profit reduction. Therefore, demand constraints must be "≤," allowing for the possibility of not meeting the demand in any regional market where the costs of meeting demand outweigh the revenue:

6 <math display="block" altimg="urn:x-wiley:15404595:media:dsji12247:dsji12247-math-0009" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munderover><mo>&#8721;</mo><mrow><mi>i</mi><mspace width="0.33em" /><mo>=</mo><mspace width="0.33em" /><mn>1</mn></mrow><mn>5</mn></munderover><msub><mi>s</mi><mi>ij</mi></msub><mo>&#8804;</mo><msub><mi>D</mi><mi>j</mi></msub><mo>,</mo><mspace width="0.33em" /><mi>j</mi><mspace width="0.33em" /><mo>=</mo><mspace width="0.33em" /><mn>1</mn><mo>,</mo><mspace width="0.33em" /><mi>&#8943;</mi><mo>,</mo><mspace width="0.33em" /><mn>5</mn><mo>.</mo></mrow></math>

In order to formulate the rest of the constraints, an instructor must discuss with students the "big M" method of formulating constraints. The "big M" method is used to limit a set of decision variables based on the value assigned to a certain binary variable. A first set of "big M" constraints is associated with warehouses. Transshipment constraints (5) dictate that the in‐bound transportation must be equal to the out‐bound transportation for each warehouse j. At the same time, it is obvious that if the warehouse j = 2,...,5 is not built, and the corresponding binary variable y<subs>j</subs> = 0, then both in‐bound and out‐bound transportation for that warehouse must be equal to zero. To formulate this constraint, we must choose a really big value of the constant M, for example, M = 1,000,000. The corresponding constraint has the following form:

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Note that the left‐hand side of constraint (7) is equal to out‐bound transportation from warehouse j. If a warehouse j is not built, then y<subs>j</subs> = 0 and the right‐hand side of the constraint (7) is zero, which, combined with non‐negativity constraint s<subs>jk</subs> ≥ 0, implies that all out‐bound shipments from warehouse j are zeroes. At the same time, transshipment constraints (5) will guarantee that in‐bound shipments to warehouse j are zeroes as well. On the other hand, if warehouse j is built, then the right‐hand side of the constraint (7) becomes equal to M, allowing for both in‐bound and out‐bound shipments through warehouse j. Obviously, the value of M must be chosen large enough, so that if warehouse j is built, it does not impose a restriction on the amount of in‐bound and out‐bound shipments through warehouse j. Setting M to a large enough value (such as 1,000,000) serves this purpose. Note that the warehouse in region 1 (Calopeia) exists from the beginning of the game, so there is no "big M" constraint for it.

Similar to the warehouses, if a factory is not built, it cannot have any out‐bound shipments. At the same time, it is natural to impose a restriction that if factory i is not built, then its capacity c<subs>i</subs> must be equal to zero. Both of these limitations can be summarized as the following set of constraints:

8 <math display="block" altimg="urn:x-wiley:15404595:media:dsji12247:dsji12247-math-0011" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>&#8804;</mo><mi>M</mi><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo><mspace width="0.33em" /><mi>i</mi><mspace width="0.33em" /><mo>=</mo><mspace width="0.33em" /><mn>2</mn><mo>,</mo><mspace width="0.33em" /><mn>3</mn><mo>,</mo><mspace width="0.33em" /><mn>4</mn><mo>,</mo><mspace width="0.33em" /><mn>5</mn><mo>.</mo></mrow></math>

If factory i is not built, then x<subs>i</subs> = 0 and the right‐hand side of constraint (8) is zero, which, combined with non‐negativity constraint c<subs>i</subs> ≥ 0, implies that factory i capacity c<subs>i</subs> must be zero. At the same time, since c<subs>i</subs> = 0, then the right‐hand side of constraint (4) is zero as well. Since factory out‐bound shipments must be non‐negative (m<subs>ij</subs>, t<subs>ij</subs> ≥ 0), in that case constraint (4) implies that m<subs>ij</subs> = t<subs>ij</subs> = 0. Therefore, if factory i is not built, it will have no production capacity and no out‐bound shipments. If, on the other hand, factory i is built, then the right‐hand side of constraint (8) becomes equal to M, allowing the factory to produce and ship units. The value of M, again, must be selected large enough so that it does not impose a limitation on the amount of capacity that can be bought for factory i. Setting M = 1000 reasonably serves this purpose.

Expressions (2)–(8) constitute an integrated facility location and transshipment MILP model, which defines the configuration of the supply chain for the simulation game. The MILP model consists of 50 transportation variables between factories and warehouses, 25 transportation variables between warehouses and customers, 4 binary variables for the warehouse locations, 4 binary variables for the factory locations, and 5 variables for factory capacities, for a total of 88 decision variables. There are 23 constraints in the MILP model, and it can be optimized with the built‐in Excel Solver. Note that the solution for an MILP problem provides guidance for the strategic decisions in the game: where to build factories and warehouses, how much production capacity to procure, and which routes and modes of transportation to use. Facility location and production capacity decisions have high costs and cannot be reversed. If these decisions are done correctly, they set the stage for good financial results. But there is a set of other, tactical decisions, which are also important in the game. These include:

Answers to these tactical questions cannot be determined from the optimization model (2)–(8), and they require separate analysis. At the same time, these decisions are also important, because they can play a detrimental role in the financial performance of student teams. For example, setting the ROP at a regional warehouse too low can result in lost sales and subpar financial results. Some recommendations for how these decisions can be handled are discussed later in this paper.

PREGAME CLASS ACTIVITIES

To ensure that students are familiar with the game, an instructor may consider giving students a short pregame quiz, testing their knowledge of the game operations, along with its parameters and costs. A critical first step in determining an optimal supply chain design is forecasting demand in the five regional markets. As mentioned above, students find total demand in Sorange to be the most challenging to predict (Figure 9). It is worthwhile to remind students how to estimate linear trend using, for example, Excel's INTERCEPT() and SLOPE() functions.

Another issue worth addressing is students' familiarity with formulating and solving optimization problems. The majority of students (but not all) in my supply chain management class have already taken an introductory management science course where they were exposed to the most typical LP applications. Nevertheless, I have found it useful to dedicate some class time to discuss the following models with students:

At some point during class, an instructor should introduce the idea of the "big M" approach, as it is used in constraints (7) and (8) in the MILP formulation. This topic is likely going to be new for most students. To ensure a better understanding of the material, an instructor may consider posting some practice problems, and giving students a short problem on one of these topics on the pregame quiz. I found it useful to record a YouTube video explaining how to set up and solve a direct transportation LP problem with supply sources selected from a predefined set of locations using Excel. Having this video available to students eliminates many of the simple questions they may have regarding formulating and solving such problems. When teaching an undergraduate or Masters/MBA supply chain class, MILP formulations (2)–(8) should be discussed in detail in class, explaining the origin of each of the constraints. If the game is used in a PhD‐level class, the optimization model formulation may be left up to the students.

My experience indicates that it is better to provide students with a spreadsheet template already formatted for MILP (2)–(8). This way students do not have to waste time working on "cosmetic" issues on an Excel worksheet. Another advantage of providing students with a formatted template is ease of grading. An instructor may require students to submit the game report along with the Excel file containing the optimal solution. If student teams use the same Excel template, the instructor will not spend unnecessary time and effort trying to understand and trace each team's individual spreadsheet. Figure 10 shows the Excel template I use (posted as Supplementary Material). Note that this template does not contain any cost values, demand estimates, or formulas for the objective function and constraints, or Solver set‐up. Students are responsible for entering these parameters and functions in teams. In the template in Figure 10, blue cells are the placeholders for costs and total demand parameters, and yellow cells are reserved for decision variables. A green cell is designated for the objective function expression.

After the MILP formulations (2)–(8) are explained in class and an Excel template is provided, I recommend giving students at least several days, possibly a week, to solve the MILP problem and obtain the optimal supply chain configuration. At the same time, students should be warned that they must not procrastinate and leave this analysis to the last moment. My experience indicates that it may take students several hours (up to 10 for some teams) to set up Solver and to debug the spreadsheet formulas before they obtain a sensible solution. Some student teams may struggle obtaining the optimal solution for the MILP. In my experience, the most prevalent errors are:

Students may report that Excel gives them a message "Solver cannot find a feasible solution." In this case, students should be advised to check formulas for constraints one more time. When teaching an undergraduate class, the instructor should allow students to consult him/her if they spent a lot of time trying to find a solution, but still cannot make Solver work. Students can become frustrated and abandon all attempts to configure a supply chain through solving an MILP problem, which will void the value of this exercise.

An instructor should warn students that when a team buys a factory with a production capacity, the cash position will drop and overall financial standing may slide substantially, as shown in Figure 11. Students tend to view that as a bad sign and stop making required decisions until they see an improvement in the cash position. Students should be reassured that this drop in cash is expected and the initial investment will start paying for itself over the course of the game. In fact, if the team makes the right decisions, the rate of cash accumulation accelerates after these decisions come into effect, as is illustrated in Figure 11.

Some teams choose to set up Solver and perform analysis of MILP (2)–(8) in a Google Sheet document. In this case, students could be required to share this Google spreadsheet containing the optimal solution with the instructor with editing rights.

MANAGEMENT OF TACTICAL ASPECTS OF THE GAME AFTER THE OPTIMAL MILP SOLUTION IS IMPLEMENTED

As was mentioned above, solving MILP (2)–(8) provides teams with a set of strategic decisions regarding the configuration of the supply network. Students will still have to make several tactical decisions to ensure that the production meets the demand while keeping the costs low. These tactical decisions include setting:

These game parameters are easy to adjust, and, at the same time, they can have a profound effect on game results. Although making these tactical decisions is outside of the scope of this paper, I would like to provide a few pointers on them as well.

To start with a discussion of the order quantities, recall that trucking costs in the MILP formulation assume that transportation by truck between factories and warehouses is done in full truckloads. At the same time, the game manual provides all necessary information and cost parameters to calculate the economic order quantity (EOQ). For example, the production setup cost is equal to $1500. Annual unit holding cost, while not stated explicitly, can be calculated form the information provided in the game manual. Calculated EOQ will not be equal to multiples of truckloads. A well‐known fact is that in the EOQ model the total cost curve is relatively flat around the minimum point (EOQ). It implies that an actual order quantity may be different from the exact EOQ figure, and the increase in total inventory‐related cost will be small. Therefore, calculated EOQ can be rounded up or down to the nearest integer number of full truck loads. Simple calculations show that the cost increase, associated with deviation from EOQ, is outweighed by the reduction of the transportation cost due to the use of full truck loads. An instructor may find it useful to discuss the EOQ model, and "flatness" of total inventory‐related cost curve around the EOQ point, thereby providing students with a hint that EOQ should not be treated as an "ultimately correct order quantity," and deviations from EOQ are acceptable (and under certain circumstances are, indeed, required). The most advanced teams will include the corresponding cost calculations and analysis in their reports. In fact, the instructor may require that students include these calculations in the game report.

Another pitfall that may arise is setting the order quantity too high. Some teams may be under the impression that in high demand areas (such as Sorange and Calopeia), the higher the order quantity, the better. This point of view is justified by the EOQ formula, which implies that as demand is getting higher, so does the order quantity. I observed teams setting the order quantity for such regions to 1000 units and even higher. This practice may be acceptable for the factories which supply one regional warehouse exclusively. However, if a factory supplies warehouses in multiple regions, this approach may negatively impact a team's performance. Suppose, for example, a factory with a capacity of 70 units per day serves several regional warehouses, and one of them is in the region with high demand. If this warehouse has an order quantity of 1000 units, it means once the order production started, it will take (1000 units)/(70 units/day) ∼ 14 days to manufacture this batch. During this period, the plant's capacity is exclusively dedicated to this single order. Orders coming from the other warehouses will be placed in the queue, likely resulting in inventory depletion in these warehouses and lost sales (in the game, an order which is not fulfilled within one simulated day is lost). Therefore, when setting order quantities, it is important to consider if multiple warehouses compete for the same factory's production capacity. Another consideration against setting the order quantity too high is cost. In the game, when a factory starts an order, the costs associated with the order are incurred when production begins. If a regional warehouse places an order for 1000 units, it will cost $1,000,000 in direct labor and materials (each unit cost is $1000). A team may not have sufficient cash, especially at the beginning of the game. One can see in Figure 11 that after the expenses associated with the initial purchasing of factories, warehouses, and production capacity, a team may have only slightly above $1,000,000 in its virtual bank account. Under such conditions, spending all of their cash on just one order may put a team on the brink of insolvency. If that happens, no new orders can be started until sales generate enough cash in the bank. This may result in "starving" the regional warehouses and lost sales. Therefore, the instructor should advise teams not to set order quantities to very high values.

Next, consider the ROP issue. When setting the ROP, students need to be careful. The formula for the stochastic ROP

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is applicable only to the warehouses supplying the market with stochastic, but stable, demand (such as Fardo or Tyran). Markets where demand shows seasonal patterns or a linear trend should use a different approach when setting ROP. For example, if a market has a linearly growing demand (such as Sorange), the use of formula (9) will result in an ROP which is only reflective of a current average daily demand <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12247:dsji12247-math-0013" xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi>d</mi><mo>&#175;</mo></mover></math> . Since the demand demonstrates an upward trend, the ROP given by (9) will be constantly insufficient, resulting in lost sales. Further, since average daily demand changes, the ROP will need to be continually readjusted as well, which is clearly suboptimal. Another mistake that students can make is related to the value of the delivery lead time. When using formula (9), many student teams use lead times L associated with transportation as quoted in Table 2. At the same time, another part of the lead‐time is the time that it takes a factory to produce a batch, which is affected by order size and a factory's daily manufacturing capacity. The instructor may want to remind students of this before the game begins. Indeed, if a warehouse places a 200‐unit order to a factory with a daily production capacity of 70 units, it will take the factory about 3 days to make the order, and only then will it be shipped. Therefore, there is an interaction between the order quantity and the ROP. It is a good idea for students to monitor the lost sales plot. In some cases, the lost sales are an indicator of an ROP that is too low.

Finally, the game uses "priority levels" in the manufacturing process. Use of order priorities makes sense at a factory that supplies several regional warehouses. The priority level for a warehouse can take a value from 1 to 5. Upon the completion of an order, the factory checks the queue of orders placed by the regional warehouses. If orders in the queue are from several different warehouses, the order with the highest priority value will be chosen among them. If all orders have the same priority level, the factory chooses to produce an order for the same warehouse from which it received the last completed order (if this warehouse placed a new order, of course). As a result, warehouses in low‐demand markets are at risk of being undersupplied. Indeed, such warehouses place fewer orders compared to the warehouses in high demand markets. So, in the situation where all warehouses have the same priority level, when a low‐demand warehouse places an order, the factory will likely choose to produce an order from a high‐demand warehouse, because this is likely the source of the most recently completed order. To avoid this "vicious loop," orders from the low‐demand warehouses should be given higher priority than to the orders from high‐demand warehouses. Then, when the factory receives an order from a low‐demand warehouse, it will fulfill its order as soon as it is placed, which avoids the "starvation" of low‐demand warehouses. The rest of the time the factory will supply high‐demand warehouses.

SIMULATION GAME MANAGEMENT

For the simulation game, the class was divided into teams of 2–3 students per team. The game start day and time were announced in class and sent to the entire class via email 1 week in advance. It is important to emphasize in class that any decisions entered in the game while it is in the suspended mode will not be saved. Students should enter their decisions only after the game begins and the simulation is running forward.

When the game starts, students should implement the decisions obtained from solving an MILP (2)–(8) as soon as possible. Failure to do so will result in missed demand and worse performance. In my experience, some teams opted to implement only a part of an optimal decision at the beginning "just to see how things are going," and another part of it later (e.g., buy only half of the required capacity at the start). While this can be viewed as a prudent strategy, it will likely not result in the best possible financial performance.

Since the game runs continuously, the teams should be advised to keep monitoring the game at least several times per day. Doing so will allow them to spot problems early on and undertake any corrective actions. For example, some teams may see their cash position sliding down gradually over time. This is a sign that a supply chain keeps using money to run production, but at the same time, sales are low. In this case, inventory is accumulating somewhere in the supply chain. It is normal for the inventory to accumulate in high‐demand markets to be used later. For example, as discussed earlier, the Calopeia warehouse needs to accumulate seasonal inventory in order to keep selling during the busy season when demand peaks. The Sorange warehouse should accumulate inventory during the first half of the game, so that the sales continue in this market when the demand becomes large (Figure 9). At the same time, accumulation of large amounts of inventory in "stable" markets (Fardo, Entworpe, and Tyran) is not warranted. If this happens, the inventory needs to be rebalanced.

Upon the completion of the game, the teams are required to submit a report outlining their actions and reasoning/calculations behind them. Teams are also required to submit Excel files or Google spreadsheets containing the optimal MILP model solution.

The reports were graded based on an ability of the team to justify decisions made in the game, and not on the final cash position. Specifically, students were required to provide the numerical details of their analysis and to:

Part of the grade was based on team's ability to set up MILP (2)–(8) in the software, obtain the optimal solution, interpret it, and then implement it in the game. The main objective of this exercise is learning. Bearing that in mind, if a team performs poorly in the game, but provides comprehensive numerical analyses in their report, I tend to grade these teams rather favorably. At the same time, to encourage the best financial results and competition among the teams, the instructor may choose to reward top performing teams with bonus points. In my experience, top‐performing teams can get very competitive in trying to achieve the highest accumulated cash position.

STUDENTS' FEEDBACK AND DISCUSSION

Upon completion of the game, students were asked to take a survey expressing their opinion about the value of this exercise. Surveys were conducted in fall 2018 and fall 2019 for two different cohorts of students taking the undergraduate introductory supply chain management class. Responses were recorded on a 7‐point Likert scale (7 = strongly agree, 6 = agree, 5 = somewhat agree, 4 = neither agree not disagree, 3 = somewhat disagree, 2 = disagree, 1 = strongly disagree). Results of the survey are summarized in Figure 12.

Evaluations indicated that setting up an MILP and obtaining the optimal solution was challenging for some teams, as not 100% of the teams were successful. One student even commented that analyzing the Supply Chain Game was the most difficult project his team had among all classes in the School of Business. When running the simulation game in fall 2019, I made it clear in class that I expect teams to make every effort to find the MILP solution, and if any teams still struggle with MILP after spending a considerable time on it, they should consult the instructor. Most of the teams that sought help with setting up the MILP and obtaining the optimal solution were close to the correct formulation, with only a few errors in their spreadsheets. As a result, the percentage of students reporting that their team was able to solve the optimization model successfully increased substantially from 55% in fall 2018 to 83% in fall 2019. It can be easily seen from Figure 12 that, as the percentage of teams able to solve the MILP problem for the simulation game rises, the perceived value of the exercise also increases. Students' mean responses are higher across the board for 2019 cohort, when a larger percentage of teams was able to successfully find the MILP solution. During the formulation of the MILP (2)–(8) in class, I emphasized that accurate demand forecasting is critical for a good result in the game. If demand is predicted incorrectly, it results in suboptimal decisions regarding the placement of factories and warehouses, as well as production capacities needed to meet demand. This is reflected in the students' survey, as the second question received the highest response with the lowest standard deviation. At the same time, students tend not to see formulating and solving the MILP as a fun exercise, as question (5) consistently received the lowest responses.

Students were also asked to leave written comments about this exercise. Summarized below are major themes (positive and negative) in the students' reaction to the game.

Positive responses:

Negative responses:

CONCLUDING REMARKS

The Supply Chain Game by Responsive Learning Technologies is a balanced and integrated experience, which brings together the topics of design and optimization of the production and distribution network, various forecasting techniques, and inventory management. The game can be used as a practical tool which reinforces the ideas and techniques discussed in class. The MILP problem formulation used by the author in conjunction with this game is mostly appropriate for the undergraduate and Master's/MBA students. It serves to link together the concepts covered in a supply chain management class with the optimization modeling taught in a management science class. The game also allows the students to better understand the difference between strategic and tactical decisions. The MILP model assists students in making strategic decisions. At the same time, however, student teams cannot achieve the best financial performance if the tactical decisions (choice of inventory control policies at the distribution warehouses) are done incorrectly.

GRAPH: Supplementary Information