Energy Consumption and Greenhouse Gas Emissions Resulting From Tourism Travel in an Alpine Setting

Tourism—with its social, economic, and ecological dimensions—can be an important driver of sustainable development of alpine communities. Tourism is essential for local people's incomes and livelihoods, but it can also have a major impact on the local environment, landscape aesthetics, and (mainly through tourist transport) global climate change. A project currently underway is developing the Austrian mountain municipality of Alpbach into a role model for competitive and sustainable year-round alpine tourism using an integrated and spatially explicit approach that considers energy demand and supply related to housing, infrastructure, and traffic in the settlement and the skiing area. As the first outcome of the project, this article focuses on the development of the Model of Alpine Tourism and Transportation, a geographic information system–based tool for calculating, in detail, energy consumption and greenhouse gas emissions resulting from travel to a single alpine holiday destination. Analysis results show that it is crucial to incorporate both direct and indirect energy use and emissions as each contributes significantly to the climate impact of travel. The study fills a research gap in carbon impact appraisal studies of tourism transport in the context of alpine tourism at the destination level. Our findings will serve as a baseline for the development of comprehensive policies and agendas promoting the transformation toward sustainable alpine tourism.

Tourism-with its social, economic, and ecological dimensions-can be an important driver of sustainable development of alpine communities. Tourism is essential for local people's incomes and livelihoods, but it can also have a major impact on the local environment, landscape aesthetics, and (mainly through tourist transport) global climate change. A project currently underway is developing the Austrian mountain municipality of Alpbach into a role model for competitive and sustainable year-round alpine tourism using an integrated and spatially explicit approach that considers energy demand and supply related to housing, infrastructure, and traffic in the settlement and the skiing area. As the first outcome of the project, this article focuses on the development of the Model of Alpine Tourism and Transportation, a geographic information system-based tool for calculating, in detail, energy consumption and greenhouse gas emissions resulting from travel to a single alpine holiday destination. Analysis results show that it is crucial to incorporate both direct and indirect energy use and emissions as each contributes significantly to the climate impact of travel. The study fills a research gap in carbon impact appraisal studies of tourism transport in the context of alpine tourism at the destination level. Our findings will serve as a baseline for the Introduction Tourism, in particular ski tourism, significantly contributes to the national economy of Austria; 25% of all tourism spending occurs in the province of the Tyrol. In 2014, the direct added value of tourism in Tyrol was US$ 4.5 billion, representing 18% of the Tyrolean gross domestic product (Tirol Werbung 2014). Such numbers highlight the economic importance of tourism in mountainous regions like Tyrol as well as its social importance for local communities and its ecological function of sustaining attractive natural and cultural landscapes that are, in turn, the foundation of tourism. Anthropogenic climate change has the potential to significantly disturb this economically vital sector (Dawson and Scott 2013). Alpine tourism has been repeatedly identified as one of the industries most vulnerable to climate change (UNWTO et al 2008;Dawson and Scott 2013).
Tourism itself contributes significantly to global greenhouse gas (GHG) emissions, summing up to 12.5% of total global emissions, including the secondary atmospheric impacts caused by aviation, measured as ''radiative forcing'' (Scott et al 2010). Transport accounts for 72% of tourism's CO 2 emissions (UNWTO et al 2008;Peeters and Dubois 2010;Scott et al 2010;Peeters 2013), followed by accommodation (24%) and recreational activities (4%) (Peeters and Dubois 2010). Of the transport-related carbon emissions, 60% are caused by only 17% of trips, those made by plane. Car travel accounts for 36% and all other forms for 4% (Peeters and Dubois 2010).
Numerous studies have estimated tourism-related GHG emissions at the national and regional level (Høyer 2000;Becken et al 2003;Chenoweth 2009;Dwyer et al 2010;Hoque et al 2010;Perch-Nielsen et al 2010), all of them identifying transportation as the major driver of GHG emissions. Only a few attempts have been made to assess tourism-related energy consumption and GHG emissions at the destination level. Case studies have been undertaken using national emission factors and data from visitor surveys asking about distances traveled, modes of transport, and number of members in a group (G€ ossling et al 2005;Kelly and Williams 2007;Lin 2010). Kelly and Williams (2007) found that origin-to-destination transport (ODT) accounts for 80% of the energy consumption and 86% of the GHG emissions of Whistler, Canada, with over 70% of GHG emissions caused by air travel (Kelly and Williams 2007). Studies by the Mountain Riders Organization (2015) indicated that 57% of all GHG emissions of French mountain resorts are related to ODT.
While GHG emissions are a global problem that can only be solved in global agreements, progress must be made at the destination level if effective improvements are to be achieved (G€ ossling 2009). Therefore, destinationspecific baseline scenarios are needed to provide local destination planners with an in-depth view of the dimensions and relations of emissions caused by different modes of ODT. To model energy consumption and GHG emissions on a local scale, emissions factors have to be carefully defined, as these may differ substantially by region, depending on preferred modes of transport, fuel types, load factors (numbers of passengers per vehicle), and types of air travel (short and long haul) (Becken et al 2003;G€ ossling et al 2005;Becken and Patterson 2006;Peeters et al 2007;Peeters and Dubois 2010;Filimonau et al 2013).
In this article, we present a GIS (geographic information system)-based tool for calculating energy consumption and GHG emissions for tourist transport to and from a single destination, using emissions and demographic data from different sources. The Model of Alpine Tourism and Transportation (MATT) was developed as part of a study that aims to develop the mountain municipality of Alpbach, Austria (47823 0 55"N, 11856 0 40"E, Figure 1) into a role model for sustainable year-round alpine tourism. The study's objectives are to analyze Alpbach's energy demand and GHG emissions patterns and the potential of renewable energy in the region, and to initiate the development of an optimized and autonomous energy supply system, pursuing the concept of a ''carbon neutral destination'' as elaborated by G€ ossling (2009). The study contributes to Future Earth's research theme, Global Sustainable Development (Future Earth 2014), as it supports decarbonization of the tourism industry, which is crucial for many mountainous regions.
From a systems analytical approach, Alpbach as a tourist destination can be described as a system with 3 major elements: skiing and recreation, transport, and housing (including accommodation). For these sectors, the status quo of energy consumption and GHG emissions can be assessed, and pragmatic (best practice) as well as normative (self-sufficiency) scenarios of energy development can be simulated. This article focuses on the transport part of the system and shows how a carbon impact appraisal can be conducted at the destination level in a mountain area.

The Model of Alpine Tourism and Transportation (MATT)
ODT's energy consumption and GHG emissions are a function of round-trip travel distance, travel mode, load factor, and specific energy consumption and emissions factors. To develop the MATT, we relied on an electronic tourist register administrated by the local tourism association, which is part of a mandatory nationwide registration system in which data such as the accommodation's identity, the number of visitors, their places of origin, and their arrival and departure dates are recorded-hence our study does not account for same-day visitors at this stage.
We used the ArcGIS World Geocoding Service (ESRI 2016) to geocode place-of-origin data for the 2014-2015 winter season (November to April). After determining whether a geocoded data point was actually located within its respective country borders, 16,985 (about 45%) of 37,658 original entries yielded valid address points. Although the correctly geocoded share was relatively small, we consider the dataset to be representative, as the country-of-origin distribution within the 2 datasets was similar ( Figure 2).
We used QGIS, an open-source GIS, to create a density raster (heat map) of the geocoded visitor address points. Heat maps are based on kernel density estimation and allow a quick identification of hot spots and point clusters.
On this basis, 235 visitor-sending hot spots or hubs were defined, which made it possible for us to generalize and calculate journey distances ( Figure 3). In order to connect the visitor home addresses with associated hubs, ellipsoidal distances were calculated, resulting in a shapefile containing the attributes from the address layer and the number of geocoded visitors per hub as well as the spatial information. This was exported to a PostgreSQL/ PostGIS spatial database for further calculations. As only 45% of the visitor data were captured, the number of geocoded visitors associated with each hub was extrapolated to the statistical population using the equation where Visitors hub is the statistical population, Visitors geocod_hub is the number of geocoded visitors assigned to the hub, Visitors ori_co is the total number of visitors from the specific country of origin, and Visitors geocod_co is the total number of geocoded visitors from the specific country of origin.

Estimating person kilometers traveled per transportation mode
To estimate energy consumption and GHG emissions resulting from overnight visitor travel to and from Alpbach, we calculated the total amount of person kilometers traveled (PKT), based on the distances from the visitor hubs to Alpbach, for 4 modes of travel (private vehicle, bus, train, and airplane). Tourism travel to and from Alpbach is well dominated by private vehicles (75%). The high proportion of British tourists in Alpbach in winter raises aviation's share to 13%, whereas air travel constitutes only 5% of tourism travel in the province of the Tyrol as a whole. Another 7% of visitors come by bus and 5% by train (Rauch et al 2010;Manova 2014).
Based on these data, we assumed that all holiday makers from Austria and neighboring countries travel by private car, train, or bus. We used the Google Maps routing tool to calculate round-trip road distances between the hubs in these countries and Alpbach, assuming equal distances for all 3 modes of travel. For all other countries, we assumed that visitors come by plane and looked up round-trip distances from the hubs based on the ICAO carbon emissions calculator (ICAO 2015).
Calculating the overall PKT took multiple steps. We used the following equation for overland travel: where PKT overland is the total number of person kilometers traveled overland (not differing between modes and not accounting for a ''last mile'' between the railway station and the destination), Visitors hub is the number of visitors associated with each hub, and RRD hub is the round-trip road distance between hub and destination. To estimate PKT for airplane travel, we added a ''first mile'' and ''last mile'' to the flight distances (calculation based on the emissions factors for car travel). The first mile concept refers to the person's entry into the transportation network, that is, the distance from the home address to the airport. The same principle applies to the last mile. Therefore, we used the following equation: where PKT air is the total PKT by airplane, RFD hub is the round-trip flight distance from each hub assumed to be a departure airport; D fm is a 150 km radius from each home airport (fm ¼ first mile); and D lm is the last mile from either Innsbruck (60 km) or Munich (169 km) to the destination.
To calculate the total number of kilometers traveled to the destination and return, PKT overland and PKT air were  (Table 1). Kilometers traveled in each transportation mode were then calculated based on the modal split derived from recent visitor survey data, as mentioned earlier (Rauch et al 2010;Manova 2014). Energy consumption and GHG emissions were eventually calculated by multiplying PKT total by the energy and emissions factors described in the next section.

Deriving energy and emissions factors
Energy consumption for each mode of transport was calculated as direct energy (the consumption of fossil fuel and electricity during transport) and cumulative energy (which includes the energy used during the entire production process (exploration, extraction, transport, and production of fuels). Both were measured in kWh/ PKT.
Emissions were calculated as a carbon dioxide equivalent (CO 2 e) to account for the full GHG potential of emissions from burning fossil fuel. We distinguished between direct CO 2 e, which is produced at the location of energy conversion, and cumulative CO 2 e, which takes into account the entire production process (exploration, extraction, transport, and production of fuel).
Values for energy consumption and GHG emissions of private vehicles were derived from the open-source database GEMIS 4.94-Global Emissions Model for Integrated Systems (IINAS 2015). Fuel consumption and CO 2 e for diesel-and gasoline-driven medium-class passenger vehicles were calculated as average values due to the fact that the mean market share of diesel vehicles in EU-28 has oscillated between 45% and 55% since 2005 (ICCT 2015). GEMIS data do not account for car age structure but relate to specific reference years; this study used the data for 2010.
To calculate energy and emissions values per PKT, we assumed a load factor of 3 persons per car based on a local transportation study (Rauch et al 2010) and a visitor survey in winter 2015-2016 that was part of this project; the load factor for buses was derived from GEMIS 4.94 data (IINAS 2015). The values for all means of transport considered in the study are given in Table 1.
Estimates of energy and emissions values for railways were based on Germany's national fuel-production mix. Energy consumption and emissions data were taken from the Transport Emission Model, version 5.41 (Kn€ orr et al 2012), and are published online by the German Federal Environmental Agency (UBA 2014). Direct CO 2 e emissions per passenger were calculated by downscaling the cumulative CO 2 e emissions for German trains (UBA 2014) using an energy efficiency factor of 0.353 (Kn€ orr and H€ uttermann 2015)-that is, the relation of direct energy to cumulative energy (BMVI 2014). This method was considered to be appropriate since Germany's mix of electricity production is still dominated by fossil fuels (Kn€ orr and H€ uttermann 2015). To estimate mean CO 2 e and energy consumption per PKT for air travel, we used data on great-circle distances, kerosene consumption, and total direct CO 2 emissions between the various hubs and either Innsbruck or Munich. This was derived by querying the online International Civil Aviation Organization Carbon Emissions Calculator (ICAO 2015).
ICAO only accounts for direct CO 2 emissions by burning aviation fuel and quantifies neither the CO 2 emissions of preproduction processes of kerosene nor the full GHG potential of emissions in terms of radiative forcing or global warming potential. The latter is because the CO 2 e can only be calculated for GHG with a lifetime of more than 10 years (Herold 2003). As a consequence, emissions from aviation (NO x forming tropospheric ozone, methane, and water vapor forming contrails and cirrus clouds) are difficult to compare because they are short-lived and not well mixed in the atmosphere   (UNWTO et al 2008). We added a preproduction factor of 0.51 to the emissions factor of 3.16, as suggested by Myclimate (2015), and a multiplier that is supposed to account for non-CO 2 effects of aviation that affect global warming (G€ ossling and Upham 2009;Kollmuss and Crimmins 2009;Lee, Fahey et al 2009;Lee, Pitari et al 2009;Myclimate 2015). Aware that scientific knowledge about radiative forcing is still incomplete, we integrated a conservative multiplier of 2 to include the full GHG potential of air travel, as recommended in recent publications (Kollmuss and Crimmins 2009;Azar and Johansson 2012). The multiplier was only applied to CO 2 emissions directly from fuel combustion in the air and not to preproduction of kerosene. The CO 2 e (in g/pkm) for air travel was calculated for every route as follows: CO 2 e ¼ CO 2 per passenger 3:16 3ð3:16 3 2 þ 0:51Þ where 3.16 is the constant emissions factor representing the number of tons of CO 2 emitted by burning a tons of aviation fuel (ICAO 2015). As 89% of all Alpbach tourism-related flights are short haul, the CO 2 e was calculated separately for different distance classes (short, mid, and long haul). Fuel burn rates are thus weighted averages based on the aircraft typically used in the different distance classes (Myclimate 2015). Average values of CO 2 and CO 2 e were weighted by the number of visitors traveling in a given distance class, giving the short-haul distance class a higher weighting. The weighted average load factor is 131 persons per aircraft.
The amount of direct energy consumed by air travel (kWh/pkm) was calculated as follows: where fuel per passenger kilometer is the fuel burn rate divided by the average load factor for each distance class, and the specific energy of kerosene is 12 kWh/kg for aviation fuel (Air BP 2000).
To compute the cumulative energy consumption of aviation fuel, we used the 83% energy-efficiency factor given by Kn€ orr and H€ uttermann (2015): Table 2 provides an overview of the calculation of emissions factors for aviation.

Results
For aviation, direct CO 2 e (GHG) emissions per PKT were double those of vehicles, and cumulative CO 2 e emissions were 3.6 greater. The most environment-friendly transportation option in terms of direct CO 2 e emissions was the train; in terms of cumulative CO 2 e, it was the bus. Bus travel also had the lowest cumulative energy consumption score of all travel modes (Table 2).
Under current transportation patterns (same-day visitors excluded), the lion's share of cumulative energy was consumed by private vehicles, which also produced over half of total CO 2 e emissions. Air transport accounted for only a small proportion of tourism visits but a slightly larger share of energy consumption and a much larger share of CO 2 e emissions. Bus and train travel combined brought almost as many tourists to Alpbach as air travel, while using much less energy with much lower CO 2 e emissions (Table 3).
Seen in terms of a single average visitor, a visit by plane produced almost 4 times the GHG emissions of a visit by car, while a visit by bus emitted the least GHG (Table 4).

Discussion
This article's objectives were to present a new GIS-based, bottom-up approach to high-resolution analysis of energy consumption and GHG emissions patterns in transport to and from a single alpine tourism destination and to show ODT's direct and cumulative energy-use and emissions patterns. The model's approach is based on round-trip distance calculations between the place of origin of each visitor (represented by the closest hub), as given in the electronic tourist register, and the destination. Based on heat map analysis, we defined 235 hubs, which enabled us to achieve much more detailed insight into actual travel distances than would have been possible with only 1 hub defined per country of origin. The method therefore allows a much more reliable approximation of kilometers actually traveled by visitors and thus of GHG emissions than other methods. This is particularly important when the results are meant to serve as a baseline scenario for monitoring future planning strategies.
However, finding robust energy and emissions factors is challenging, as they depend heavily on a number of factors, including load factor, vehicle type, fuel type, and type of air travel (short versus long haul). Nevertheless, with some fine-tuning, the model can be adapted to other tourism destinations in similar geographical contexts, as long as detailed tourist data are available, including the visitors' hometowns and travel mode. The emissions factor used in this study for car, bus, and train transport was based on German data and thus should be reconsidered if there is a different source market. For example, the German mix of electricity production (relevant for train travel) has a low share of renewable energies.
Traveling on a bus produces more direct GHG emissions than traveling by train. However, the picture changes when the focus is on cumulative GHG emissions. In Germany's context of fossil-fuel-dominated power, travel by bus appears to be more efficient than travel by train. However, the CO 2 e factor is very different for Austrian railways, for which 92% of the power comes from renewable energies, mainly hydro power (OEBB 2013).
Aviation's contribution to ODT's overall GHG emissions is striking, as only 13% of visitors produce more than 36% of GHG emissions. This could represent an opportunity to significantly reduce emissions while affecting only a relatively small part of the tourism economy. GHG emissions from air travel are the key environmental challenge of tourism and one of the greatest challenges of climate change response in general (Becken 2007;Barr et al 2010;Scott 2011;Cohen et al 2013). However, in Alpbach, car transport dominates, accounting for three-quarters of tourist ODT and over half of total GHG emissions; train and bus travel make up only a small share of ODT and an even smaller share of GHG emissions.
Thus, to reduce GHG emissions, we recommend promoting a shift from car to train or bus transport by offering incentives-such as providing a complete chain for luggage transport (ie pickup at home and drop-off at the accommodation), climate-friendly local transportation, or reduced local travel times (for example by offering pickup service from the train station), combined with a marketing focus on short-haul rather than long-haul source markets, the promotion of longer stays (eg Peeters and Schouten 2006;Filimonau et al 2013), and on-site awareness campaigns for local stakeholders and tourists.
The latter could be challenging. Recent research has concluded that, while there is growing public awareness of   Cohen et al 2016). Therefore, soft mobility approaches to transportation, which are environment-friendly but do not involve a trade-off in comfort and flexibility, might be a key to mitigating the local climate impacts of tourism. This approach has already been recognized as a marketing instrument by the Alpine Pearls network of 25 communities across the Alpine area of Central Europe (Alpine Pearls 2016).

Conclusion
MATT demonstrates a bottom-up approach that allows a highly detailed analysis of the energy consumption and GHG patterns of ODT to a specific tourism destination, including its cumulative carbon footprint. Therefore, it fills a research gap in carbon impact appraisal studies on the destination level.
With some constraints and required adaptations, the model is transferable to other destinations in a similar geographical context. It is still limited as it does not consider same-day visitors, intracommunity transport, or employee commutes. The outcomes of existing carbon impact appraisal studies at the local scale are difficult or impossible to compare as methods for assessing GHG emissions vary substantially, and only a few studies have accounted for the indirect share of transportation's carbon footprint. The latter can make a significant difference; for instance, in our study, taking cumulative emissions and energy-use values into account reversed the comparative climate-efficiency ratings of bus and train travel. Knowing the specific gasoline/diesel market share and the national mix of power production is crucial.
MATT covers an important part of a destination's energy consumption and carbon footprint, but it is only the first module of a project aiming to transform an alpine tourism destination into a future role model for sustainable tourism. To achieve this goal, sociocultural, institutional, and political barriers have to be overcome. To this end, a detailed, holistic, and in-depth view of a tourism destination's energy consumption and GHG emissions patterns, on different spatial and temporal scales and for different sectors, is essential to communicate the need for transformative change to stakeholders, local residents, and visitors.


extfieldsofstudy:
Computer Science
Medicine

text:
A novel approach for inventory problem in the pharmaceutical supply chain

Background In pharmaceutical enterprises, keeping up with global market conditions is possible with properly selected supply chain management policies. Generally; demand-driven classical supply chain model is used in the pharmaceutical industry. In this study, a new mathematical model is developed to solve an inventory problem in the pharmaceutical supply chain. Method Unlike the studies in literature, the “shelf life and product transition times” constraints are considered, simultaneously, first time in the pharmaceutical production inventory problem. The problem is formulated as a mixed-integer linear programming (MILP) model with a hybrid time representation. The objective is to maximize total net profit. Effectiveness of the proposed model is illustrated considering a classical and a vendor managed inventory (VMI) supply chain on an experimental study. Results To show the effectiveness of the model, an experimental study is performed; which contains 2 different supply chain policy (Classical and VMI), 24 and 30 months planning horizon, 10 and 15 different cephalosporin products. Finally the mathematical model is compared to another model in literature and the results show that proposed model is superior. Conclusion This study suggest a novel approach for solving pharmaceutical inventory problem. The developed model is maximizing total net profit while determining optimal production plan under shelf life and product transition constraints in the pharmaceutical industry. And we believe that the proposed model is much more closed to real life unlike the other studies in literature.

Background
Pharmaceutical industry applies a supply chain policy that allows the continuation of a wide variety of materials with large quantities in a very fast flow. Within pharmaceutical supply chains, the product variety is a huge problem to manage within short time windows. Nevertheless, depending on the medicine drugs, amounts can be a big problem to trade with costs. In here, the requirements of small batches are particularly hard to handle. The production of pharmaceutical products has two stages as primary and secondary level. Primary production includes the production of basic molecules active components or pharmaceutical active ingredients. The secondary production also includes the processes of being formulated of these active components and the delivery to the customers. Many operations in pharmaceutical production occur in bulks called charge. Quality control also takes place with monitoring each charge. On the production line, cleaning is a matter in the case of product change (transition) and this situation is to prevent the contamination of different products. Besides, raw materials and products have a certain shelf life. All these constraints are reducing the efficiency in the pharmaceutical industry. Shelf life controls are performed for raw materials by subjecting to retest procedure at certain intervals. During retest, raw materials are kept in quarantine; they are not definitely included in the production line and if test results indicate that the raw materials can already be used, they are taken from the quarantine and transferred to production stores. In shelf life control for the products, expiration date is printed on the packaging while the product is on production line and the expiry date starts from the date of production.
Compared with supply chain of other products, pharmaceutical supply chain is very complex. The factors such as long set-up times, resource-intensive operations, short shelf life and high production of waste make the pharmaceutical supply chain different from other sectors. Pharmaceutical production is demand managed. Firms rarely deliver the product to pharmacy or patient; instead of this, they deliver products to the consumer through wholesalers (pharmaceutical warehouses).
In such a different featured sector to maintain a presence in the market despite all these constraints is possible with correctly selected supply chain management policies. To adapt to changing market conditions, sustainable supply chain policy and to compete in global market, the pharmaceutical supply chain should be carried out by mathematical models based on scientific formulas determined with correct strategies [27]. In planning, the importance of inventory management has also great importance. Considering countless complications, it is very difficult to obtain optimal schedules. However, mathematical models help to take right decisions.
For an optimal production plan in the pharmaceutical industry, cleaning and preparation times (these occur on product transition times), facility maintenance times, testing and the production of new chemicals, resource allocations, manpower utilization and inventory management must be decided and planned in an integrated way. This case requires a production planning strategy evaluating operational configurations with repeated consultations with several departments, process constraints, statistical combinations and business scenarios. The production must be planned at certain time intervals and in accordance with a hierarchical approach [11]. In planning, the importance of inventory management has also great importance. Also to compete in pharmaceutical production in the global market, it is required to develop effective inventory control policies. Companies want to meet customer demands at the highest levels and prefer product storage to avoid falling below a safety stock level. In this way, high amount of inventory cost occurs. While reducing the inventory level for minimizing costs, firms cannot meet the demands, delivery dates delays, and there are some decreases in service levels [30].
With the latest developments in information technologies, the fast and easy internet networks are used to make the information sharing easier and increase security across the supply network. Accordingly, VMI defined as "cooperation with a customer and a supplier to optimize an inventory management for least-cost on both companies" began to be used. With this model, the supplier takes responsibility for the operational management of inventory with agreed performance targets. These performance targets are continuously monitored and updated to ensure continuous improvement [12]. In order to optimize supply chain performance, the manufacturer takes the responsibility of distributor's inventory levels. Distributor also shares the demand forecast and sales data as well as inventory data. Manufacturer manages the distributor's inventory with this data. The manufacturer is responsible for determining the order quantities and time in this model [31].
In this study, in cephalosporin department of a factory making secondary pharmaceutical production, a mixed integer mathematical model is developed to obtain the best production plan while maximizing the total net profit in long term. Especially the presence of constraints related to "shelf life, product transition times that are ignored in many studies about pharmaceutical production is also added to the model and an experimental study is implemented. The proposed mathematical model is applied on two different types of supply chain (classical supply chain and vendor managed inventory) and the results of both methods are compared in terms of the total supply chain cost. In addition, the proposed model is also compared with another model from literature to illustrate effectiveness of model.

Literature review
Planning and scheduling problems have been the subject of innumerable studies in the mathematical programming literature. Various types' industrial sectors are considered with different time representations in these studies. But some studies cover various industrial sectors for planning and scheduling problem like Fleischmann and Meyr [10]. Pharmaceutical production is a batch process and a type of chemical production. Chemical productions are made in multiproduct plants. The studies for planning and scheduling in multi-product plants are Oh and Karimi [28], Alle and Pinto [1], Dogan and Grossmann [9], Mendez and Cerda [25], Liu et al. [23], Chen et al. [6]. Some studies in literature have handled production and distribution planning together (Lee and Kim [21], Bilgen and Günther [4]).
Also in this part of the study, the studies in the literature related to administrative issues, planning, scheduling and the cost optimization in the pharmaceutical supply chain are listed. The studies about optimization in pharmaceutical supply chain with mathematical methods developed by using mixed integer programming, are as follows; Papageorgiou et al. [29] stduied to optimize the problems in strategic areas such as product development, promotion strategy, capacity planning and investment strategy using a mixed integer programming in the pharmaceutical supply chain. Maravelias and Grossmann [24] discussed simultaneous optimization problem for source constrained scheduling in pharmaceutical production. They proposed MILP that maximizes expected net present value in a multi-period problem. Sundaramoorthy and Karimi [35] developed a MILP including a flexible approach increasing demand meeting ratio against changing production plans in a pharmaceutical supply chain that is the outset of contract manufacturing and new production. Levis and Papageorgiou [22] proposed a mathematical model for long-term capacity planning in a multi-site pharmaceutical industry under uncertainty. The model is an improved version of the model previously proposed by Papageorgiou et al. [29]. All problems are formulated with two-stage MILP model. Then, they developed a hierarchical algorithm for solving large-scale problems. The accuracy of their proposed method was displayed by comparing with several examples. Kim [18] applied an integrated approach to the pharmaceutical supply chain in the health sector. The aim was to reduce holding cost and to optimize inventory costs. For this, VMI was applied to reduce total supply chain cost. Amaroa and Barbosa [2] developed a planning and scheduling approach in the management of reverse flow supply chain applied in a pharmaceutical company. In their study where optimal production plans were obtained, the economical profit of the model was analyzed separately in terms of supply chain operations and customer satisfaction. Lakhdar and Papageorgiou [19] submitted a mathematical programming approach for medium-term production planning under uncertainty in biopharmaceutical manufacturing. Uncertainty in the study is related to with the fermentation concentration ratio. All problems were discussed in the two-stage multi-scenario planning problem and an algorithm was proposed for the problems in larger size. Venditti [39] developed a heuristic algorithm for production planning in the pharmaceutical industry. Baboli et al. [3] studied pharmaceutical supply chain management with two separate approaches as centralized and decentralized using mathematical programming. They reached to the conclusion that the centralized method reduced the cost much more. Sousaa et al. [34] calculated the dynamic resource allocation problem in the pharmaceutical industry with the delivery costs and the different tax rates to maximize the net profit of the company. Susarla and Karimi [37] taken into account the sequencedependent pattern change, resource using, maintenance schedules and security stocks using the mathematical model with considering integrated planning and resources in the pharmaceutical production facility. Susarla and Karimi [36] studied to optimize supply chain costs about integrated supply chain planning more than one production plant for pharmaceutical production activities. Kelle et al. [17] developed a solution with MILP for demand point in a hospital pharmaceutical supply chain organized medication requirements plan. Chen et al. [7] used a simulation-based optimization technique while increasing customer service level and reducing supply chain costs of pharmaceutical clinical trials and they planned production and distribution activities with MILP. Kelle et al. [17] developed a strategic and tactical level of decision support models in their work related to pharmaceutical supply chain and inventory solutions in a hospital. Kabra et al. [14] planned multi-stage and multi-production processes in the biopharmaceutical production as long-term by using MILP.
The studies about administrative subjects that made extended literature survey in pharmaceutical supply chain can be listed as follows; Shah [32] determined the key issues and optimization strategies for pharmaceutical supply chain in the study to determine the pharmaceutical supply chain and optimization strategies. In the study, by mentioning all pharmaceutical processes, from raw material production until delivered to the customer were explained and made some suggestions about how to increase customer service level. Besides, Shah [32] analyzed all stages one by one ensuring added value to the pharmaceutical supply chain and emphasized the important matters. Yu et al. [40] conducted a study making an evaluation for current issues and health system reform about pharmaceutical supply chain in China. Jaberidoost et al. [13] studied revealing strategic risks of supply chain management in pharmaceutical industry and they mentioned different studies about this area. Narayana et al. [26] discussed the existing studies on the pharmaceutical supply chain in their study. They classified the studies in literature according to the countries, research methods, terminology and the level of analysis. They made evaluations about the future of the studies from the administrative perspective.
In addition the studies about the implementation of VMI model in the pharmaceutical industry are limited. These are; Danese [8] discussed the project of the implementation of VMI to the entire production and distribution facilities and expressed how to manage the entire supply chain with VMI model from one side, data networks established and data systems supporting them and the performances of them in detail. This VMI model as seen in the study was a model applicable to the entire pharmaceutical industry. Shen et al. [33] tried to identify economic production amount under minimum volume constraints in the production of perishable products such as pharmaceuticals that is important for public health. They developed an approach consists of a mathematical model and VMI which was more economic in in terms of total supply chain cost. Kannan et al. [15] proposed a model revealing the benefits of VMI in pharmaceutical industry. Kannan [16] tried to identify the most appropriate supply policy to stochastic demand environment depending on the time with VMI in pharmaceutical industry. All these studies summarized as seen in Table 1.
In this study, a new mathematical model is developed to solve an inventory problem in the pharmaceutical industry. The mathematical model contains shelf life and product transition constraints together and the model is much more dealing with real life constraints unlike the studies in literature. Besides, the model contains parameters about general supply chain parameters such as costs of production, inventory holding, transition, waste product and unmet demand penalty. There are a few articles about VMI method in pharmaceutical sector. The model applied to classical supply chain method and VMI, so this study is implemented VMI method and handled a new technic in pharmaceutical supply chain.

The problem and detailed mathematical formulation
The studies in the literature discuss a simplified model of production occurs in real life. In models, they take into account only materials and machinery as resources and mathematical models try to solve the model by making critical assumptions about the case such as transit or installation times of material transfers, human resources, waste storage and treatment capacity. These assumptions prevent to be applicable of the models established in practice. However, the studies should include more comprehensive models to completely adapt to real life. To make a decision by combining entire supply chain under unique plan in real life is very difficult in dynamic market and environment conditions. Some problems in literature (Shah [32], Yu et al. [40], Jaberidoost et al. [13], Narayana et al. [26]) are studied solution suggestions by taking into account the supply, production and distribution processes. However, simple and flexible models are ensuring rapid solution and being appropriate for the real life are much needed. Although it is generally ignored in literature, pharmaceutical raw material and end product have a definite expiry date and it cannot be used after the expiry date. If there is a presence of raw material and product inventory in warehouse, they are turned into waste product and they will reflect to the supply chain as waste cost. For this reason, the raw materials and product expiry conditions must be taken into account.
In this study, a long-term planning model is developed in order to obtain the production plan that optimally fulfills a net profit objective. This criterion presents the trade between sales returns and costs issues. The main contribution of this study is to solve an inventory problem in the pharmaceutical industry by proposing a new mathematical model that contains shelf life and product transition constraints together. We believe that the proposed model is much more dealing with real life constraints unlike the studies in literature. Besides, the model contains general supply chain parameters such as costs of production, inventory holding, transition, waste product and unmet demand penalty.
A hybrid time representation is applied over a planning horizon, in which the months of the planning horizon are modeled and each month is represented by a continuous time formulation. The most effective characteristic of the problem is that, inventory amounts depend on the shelf life of the products. Also, transition conditions are handled that occur while switching from one product to another.
In the mathematical model, constraints about transitions are adopted from the models of Liu et al. [23]. However, being different from them in our model, inventory amounts changes depending on the shelf life of the product. And so; all inventory formulations are novel. The cost criterion subtracted from the total sales revenue in the objective function is novel in this model. Nomenclature of the proposed mathematical model is given at the Appendix 1. There is a little literature about VMI method in pharmaceutical sector. So this study is implemented VMI method and handled a new technic in pharmaceutical supply chain.
As close as a real life problem, a hypothetical problem is considered in cephalosporin department of a pharmaceutical factory. Our mathematical model is applied in order to obtain the best production plan while maximizing total net profit in long term. The efficiency of the model is shown on classical supply chain and VMI method. Vendor managed inventory model Danese [8], Shen et al. [33], Kannan et al. [15], Kannan [16] Objective function The total net profit is maximized to obtain from sales revenue minus supply chain costs, involving the total, production cost, product transition costs, the unmet demand costs, inventory holding cost, product transportation cost and the cost of waste products.
Product assignment constraints In each period one product is assigned as first or last product to be processed and these two equations show whether they are the first product or last product.
When the relevant product is not produced in this period, it assumes as E it = 0.

Product transition constraints
While Z ijt binary variables are representing the product transitions occurring in a period, ZF ijt variable represents the product transition between two consecutive periods.
If there is a product transition within a period, they are the variables indicating that there will be no product before the first product produced and it is not the first product, other products will give priority to this product.
Similarly, if there is a product transition within a period, there will not be any product production after the last product produced and if it is not the last product, this product will be followed by the production of other products.
If there is a product transition between two consecutive periods and if the production of a product begins in that period for the first time, there will certainly be a product transition a period before the relevant period. If a product is not the first or the last one processed, then there is not a changeover involving the product between two periods.
Travelling salesman problem formulation based subtour prevention constraints β it variable can be called as demand index or production row. The aim of writing these constraints is to determine row of the product in a period and to take the product transition cases under control.
If i th product is produced before j th product, production sequence number of i th product will be at least one more than j th product.
If that product has never been produced, demand index will be zero.
These constraints are similar to the constraints preventing the sub-tours in classical Travelling Salesman problem (TSP) [23]. In TSP problem binary variables are used to represent transition one city to another. As similar in this model to maximize net profit there should be minimum number of transition in production sequence. And Z ijt and ZF ijt variables are added to model the product transition in a period and between t-1 and t periods respectively.
This constraint enables the demand index to take at least the value of 1 and to take value up to the maximum product number.

Timing constraints
For each product produced in a period, the highest and lowest time limits are given.
The total of the production in a period and product transition times cannot exceed the existing time given for the shift.

Production constraints
The product amount produced in a period is as much as the multiplication of the production ratio and production time.

Demand constraints
The amount of the unmet demand from a product is as much as the difference between the realized demand within that period and the product amount delivered to the customer (satisfied demand) within that period.
Sales amount may be lower than or equal to demand amount that was realized; in this model the unmet demands in a period are not delivered to the customer in the next period (no backlogs).

Shelf life and inventory constraints
α i is shelf life and depends on products characteristics and it is defined as an integer multiple of t.
Inventory quantity varies depending on the product shelf life. If the relevant time is lower than the product's shelf life, that is to say, if the product's term has not been expired yet, the inventory amount of that product is as much as the difference of the sales amount within that period in the total of the stock amount transferred from the previous period and the product amount produced in that period.
But if the relevant time is longer than the product's shelf life, the products whose term is expired will be waste product.

Numerical investigation
Around the proposed mathematical model, our main research focus is addressed in numerical investigation and the following questions are answered:

Fig. 1 Factory's transportation system and warehouses
To answer these questions, an experimental study is considered based on numerical experiments and comparisons. Experimental set is summarized in Table 2.
To illustrate the applicability of our mathematical model, we consider a hypothetical pharmaceutical plant. In this case, long term production scheduling problem in the pharmaceutical secondary production is discussed.
In the cephalosporin department of this factory, injectable beta-lactam products are manufactured. In a separate facility that is completely independent from non-beta -lactam production fields, micro powder refilling is conducted in aseptic conditions and the production conditions are provided to be monitored continuously via computer aided production systems.
In our study, it is aimed to satisfy the customer demands for 15 different products produced in cephalosporin department of the factory in 24 and 30 months period to find optimal production plan maximizing the total net profit and to show in which type of supply chain (classical or vendor managed).
We hypothesized that; the pharmaceutical factory has agreements with 5 pharmaceutical warehouses. In Fig. 1 factory's transportation system and warehouses scheme is given. The factory delivers the products to pharmaceutical stores and pharmaceutical stores deliver them to the pharmacies. Before the expiry, all the products that are not delivered to the consumer return to the factory and its disposal is carried out by the factory. In this way, it is seen that the production, delivery and waste product costs (disposal cost) belonging to the products whose term is expired and that will be returned being sold, creates serious cost damages to the factory and a mathematical model is developed to minimize these costs and the model is compared by being applied in two separate types of supply chain. In this study, product name is not specified and 15 different products are named according to the active raw materials. These are shown in Table 3.
A work flow chart belonging to cephalosporin production site is seen above in the Fig. 2. As shown here, the  production is performed on a unique line. Filling machine is isolated from the production environment and in a sterile manner. The active material is loaded into the filling machine; these raw materials are again filled in pharmaceutical bottles in a sterilized way; the bottles filled are checked, labeled and boxed. The production and the expiry date determining the shelf life of the product are printed on the label and box when the product is on the line during relevant transitions.

Key model parameters Demand types
Demand forecast was made while making production plan for the next two years. Forecasted demand and sales amounts were obtained from the real old data belonging to a real pharmaceutical company. The average of sales amounts (29000) can be seen to conform to normal distribution having standard deviation (10500).
Since normal probability distribution a widely known type, in many studies related to inventory management in supply chain that demand structure is in accordance with normal distribution (Lau et al. [20]).
Equation 23 gives the distribution of forecasted demand amounts.
In current and proposed situation, the average of customer demands is 27500 and it is assumed to comply with the normal distribution, the standard deviation of which is 19000. The equation 24, gives the distribution of customer demands performed.

Production capacity
In literature there are studies in accordance with uniform distributed production capacities [5,38]. The aim of the manufacturers is always to satisfy the demand completely. Maximum production capacities are limited, but this capacity varies according to the events to occur during the processes (malfunction or periodic maintenances, or even stops). The factory works 12 h a day. Total available processing time in a month is 264 h. Production capacity is up to maximum 4500 boxes of product/hour. But this capacity vary in real life because of deteriorations or stops in production lines. When it is statistically analyzed, the capacity of cephalosporin department production line is uniformly distributed in the range of (0, 4500).

Shelf life conditions
The shelf life of each product is fixed and it is 12 months. The products whose terms are expired cannot sold within this time period, it is assumed as waste products   Table 6 The transition time (hour) from i to j

Product transition conditions
Transition cost is proportional to transition times by a factor of 10. Unit inventory holding cost data are the values such as 0.005 $/unit identified by the company for a package of product, unit sales price and rest of the other costs are given in Table 4.
Product transportation costs are varies to warehouses and given in Table 5.
The transition times from one product to another (cleaning and mold change) are variable and they are stated in Table 6 below.

Details of experiments
In this part of study, some of the experimental results are given for comparison. Firstly Experiment Number 1 and 5's details are given and compared.
In Experiment 1; factory has a classical supply chain model and produced products transfer due to the orders from pharmaceutical stores and deliver to the end user from there. The products which are expired before sold from the factory to the pharmaceutical store are sent to disposal facility. Besides, the products which is expired in the pharmaceutical store before sold to pharmacy, are sent to the producer company by the pharmaceutical stores and when these products are reached to the producer company, it is taken to "reject store" and it is delivered to the company for disposal facilities. During this returning process, all costs (transport + disposal costs) are subject to the company. A scheme of pharmaceutical supply chain processes can be seen in Fig. 3.
The above example is modeled using GAMS/CPLEX 12 for the MILP optimization. When the model is run, maximized total net profit is 42556741$ for 24 months with 10 products. The graphs including demand amounts, production amounts, stock amounts and unmet demand amounts belonging to the model are given in following Fig. 4.
Production schedule is given in Fig. 5. This figure shows how the transition times affects to the production sequence. It is hard to show all planning horizon (24 months) scheduling, so we illustrate for a part of two months planning time.
In Experiment 5, the inventory is vendor-managed. According to the agreement, pharmaceutical factory can see the inventory and sales information in the pharmaceutical stores online with the established information system. At the beginning, a production plan is created according to estimated demand amounts and with the Fig. 3 Pharmaceutical supply chain processes Fig. 4 The graph of production, demand and the unsatisfied demand for Experiment 1 information obtained from pharmaceutical store, production plans are updated and within the framework of these plans, delivery plans will be made and the product will be delivered. The amounts and time information will be communicated with the store. For each product, the level of safety stocks in the store is determined and the stocks will be controlled continuously in order not to fall under safety stock levels in the store. Safety stock levels are given in Table 7. Equation 16 in Exp. 1 turns into the equation 25 in Exp.5.
In Experiment 5, each product is sold to pharmaceutical stores according to the agreement. For this reason, in addition to the equation 18 created for sales amounts, the equation 26 above should be added. According to production plans, the products produced in the factory are directly sent to the pharmaceutical store. That is to say, sales amount is equal to the production amount. Then, the production plans are revised from month to month in the factory in accordance with the demands. Here, the aim is to prevent shortages and waste product costs and making the supply chain more profitable.
When the model is run, the maximized total net profit is 47389739 $ for 24 months with 10 products. The graphs including demand amounts, production amounts, stock amounts and unmet demand belonging to the Experiment 5 are given in Fig. 6.
The result of comparison for these two experiments is as follows in Table 8.
The mathematical model created to reveal the best production plan while minimizing supply chain costs and maximizing the total net profit. The model compared with two experiments. For the both situations, the common data and data distributions are used. Two different optimum results are obtained according to supply chain types. When these results are compared, sales revenue in Experiment 1 is less than in Experiment 5. The sales revenue difference between them is so much. Transition costs are also seen as more advantageous in Experiment 5. The unmet demand amounts are less in Experiment 5 and accordingly, its cost is lower. Production and transportation costs in Experiment 1 is less than Experiment 5. This is because the sales amounts are less than Experiment 5. Considering waste product costs, in Experiment 1, 1.72 % of the production is waste product and in Experiment 5, this rate is 0.03 %. So we can say, VMI is more advantageous in terms of waste product cost and wastage amounts. Because, the production plans advance in more controlled way in Experiment 5. A comparison of these experiments' expired product amounts can be seen in Fig. 7. When total net profit is compared, VMI is more profitable as much as 10.19 % than Classical Supply Chain. Finally, when two experiments are compared, VMI is much better in all costs except for production and transportation costs.  The model is run for all experimental set and the total net profit for all the experiments are given below in Table 9.
As seen in Table 8, VMI Method is more profitable (nearly 8,8 %) than Classical Supply Chain method.
Production planning is a crucial issue in pharmaceutical supply chain in terms of meeting customer demands just in time. Because pharmaceuticals are perishable products, the shelf life constraints have to be considered while planning, scheduling and all supply chain activities are organized.
According to our limited knowledge, some pharmaceutical companies do not take into account the shelf life constraints during their planning processes. It is assumed that all produced items will be sold after reasonable waiting time in the inventory. In reality, in some cases, there can be long waiting time of products. So their perishing day of products can be very close to shelf life because of long waiting time. Although the long waited products are still considered as "inventory products" from the planner, they must be considered "wasted" instead of inventory product.
In this study we consider shelf life constraints and product transition constraints together. In terms of the practical implications of this work in real pharmaceutical companies; the proposed model can be adapted to production planning activities in the pharmaceutical supply chain. Because; the model produces real time inventory information through the shelf life constraints and deals with "long waited (waste) and "inventory" products while the planning is done. This situation has a great opportunity to cope with wastage costs. The model balances the inventory levels, demands and lost sales. And also real time inventory information provides to reduce risks about market demands and efficient production scheduling activities. This helps to reduce all supply chain costs and the model offers a collaborative, planning and scheduling system to pharmaceutical companies while managing shelf life of products.

Comparison with another model
In this part of study, the efficiency of our model is compared with Chen et al. [6] model. To make a comparison, the Chen et al. [6] mathematical model is modified. Modified and original model are given in Appendix 2 and Appendix 3 respectively.
The objective function of proposed model consists of production cost, product delivery cost and wastage cost and it is written as follow: There is no storage capacity limit, so the (C12.) constraint is removed from the model.
In addition, backlogs are not allowed, therefore we remove the constraint (C14) from the model. But we Fig. 6 The graph of production, demand and the unmet demand amounts for Experiment 5 have to calculate unmet demand amount so a new constraint is added as given below: The stock amounts is written with considering shelf life conditions, the constraint (C13) as follow: Wastage cost is added to the model; The model is run under 24 and 30 months production period, the CPU times were 856 and 4150 respectively.
The details of comparison of these models are given in Table 10.
Models were run for 24 and 30 months under classical supply chain. Sales revenues and inventory holding costs are more profitable in our model. And the total net profits are in average %0,8 much more in our model.

Conclusion
In this study, a new mathematical model was developed to solve inventory problem and maximizing total net profit while determining optimal production plan under shelf life and product transition constraints in the pharmaceutical industry. The proposed MILP model contains "shelf life and product transition times" constraints together and we believe that the proposed model has much more real life constraints unlike the other studies in literature. Besides, the model contains parameters such as costs of production, inventory holding, transition, transport, waste product and unmet demand penalty.
As we defined at literature review section of this study; VMI method is not often used in the literature of the pharmaceutical sector. So this study implemented VMI method and handled a new technic for the pharmaceutical supply chain.
To show the effectiveness of the model, an experimental study; which contains 2 different supply chain policy (Classical and VMI), 24 and 30 months planning horizon, 10 and 15 different cephalosporin products were chosen. The results illustrated that the VMI provided much better results in terms of total supply chain costs. Especially; the waste amount that was very important in pharmaceutical sector and the cost was reduced in the VMI. The waste product amounts were 1.72 and 0.03 % of the products in experiment 1 and experiment 5 respectively. In terms of total supply chain costs, 10.19 % of an advantage was gained by the proposed VMI model. As a result, we believe that the proposed model should be adapted in the pharmaceutical industry to reduce total supply chain cost. In addition that, the proposed model is compared with a recently published study in literature. And so the results are illustrated that the proposed model is better than the other.  i,j, products c, customers t, period Sets I, J, product set C, customer set T, periods set Parameters CP i , unit production cost of i product CUD ic , unit unmet demand penalty cost of product i to customer c CC ij , product transition from product i to product j.
(change and cleaning cost on the line while passing from product i to product j) CI i , unit inventory holding cost of product i CW i , unit waste product cost of product i TC ic , unit product delivery cost of product i to customer c FD cit , forecasted demand of customer c, from i th product, at period t RD cit , the demand amount realized by customer c, from i th product, at period t M, large number PS ic , unit selling price product i to customer c α i , shelf life Ө L , the lowest production rate in a period Ө U , the highest production rate in a period TZ i,j , cleaning and mold changing time while passing from i product to j product β it , demand index rr i , production rate Variables E it = 1, 0, product i produced in t period?
F it = 1, 0, product i the first product in t period? L it = 1, 0, product i the last product in period t? Z ijt = 1, 0, product i produced before product j? ZF ijt =1,0 if there is any product transition between t-1 and t period? P it , the production amount from product i in period t S cit , sales amount from product i to customer c in period t O it , processing time for product i in period t SM it , inventory amount of product i in period t SS i , security stock amount of product i UD cit , unmet demand amount from product i to customer c in period t W it , waste product amount product i in period t NP, total net profit

Appendix 3
The proposed model by Chen et. Al.
Nomenclature Indices c, customers i,j, products k, time slots w, weeks Sets C, customers I,J, products Kw, time slots in week w W, weeks

Parameters
CBci, backlog cost of product i to customer c CIiw, inventory cost of product i in week w CTij, transition cost of product i to product j Dciw, demand of product i from customer c in week w PSci, price of product i to customer c ri, processing rate of product i Vimax, maximum storage of product i Vimin, minimum storage of product i ӨL, lower bound for processing time ӨU, upper bound for processing time τij, changeover time from product i to product j

Variables
Pro, operating profit Piw, production of product i in week w Sciw, sales of product i to customer c in week w Tkw, end time of slot k in week w Viw, volume of product i in week w Δciw, backlog of product i for customer c in week w θikw, processing time of product i in slot during week w

Binary variables
Eiw, 1 if product i is produced in week w, 0 otherwise yikw, 1 if product i is produced in time slot k during week w, 0 otherwise Zijkw, 1 if product i (slot k-1) preceds product j (slot k) in week w, 0 otherwise

Mathematical formulation
Objective function