Eindhoven University of Technology
MASTER
End-of-life products management in a closed-loop supply chain
Berzah, E.
Award date:
2018
Link to publication
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End-of-Life Products Management in a Closed-Loop Supply Chain
Company: Nike EMEA
By Emin Berzah
School of Industrial Engineering Eindhoven University of Technology
Master Thesis 1CM96
MSc Operations Management and Logistic Student Identity Number: 1036914
The project is supervised by
Dr. Boray Huang, Eindhoven University of Technology
Dr. Arun Chockalingam, Eindhoven University of Technology Dr. Maximiliano Udenio, Eindhoven University of Technology Monica Rosado, Nike, Inc.
Eindhoven, July 2018
Subject headings: closed-loop supply chain systems, Discrete Flow Model, End- of-life product management, reverse logistics, Network for Transport Measures (NTM)
i Table of Content
Table of Content ... i
Abstract...iv
Summary ... v
Development of the DFM to assess the advantages of closed-loop supply chains...vi
Generalization of the Proposed Model ...vi
Recommendation ... vii
Acknowledgements ... viii
Abbreviations ... 1
1. Introduction ... 1
1.1. Company Description ... 1
1.2. Scope of the Research... 3
1.2.1. Research Objective ... 4
1.3. Problem Definition ... 5
1.4 Research Questions... 6
1.5. Methodology ... 7
1.6. Timeline of the Project ... 9
2. Background on Sustainable Supply Chains ... 10
2.1 Background on Discrete Flow Model ... 11
2 .2 Model Explanation ... 12
2.2.1Notation of the DFM ... 12
2.3. Example DFM Financial Management... 20
2.4 Carbon Emission Quantification... 22
2.4.1 Road Transport ... 22
2.4.2 Water Transport ... 23
2.4.3 Carbon Emission Model ... 24
2.5 Reflection on the Discrete Flow Model ... 25
2.5.1. Input... 25
2.5.2 Data Collection ... 25
2.5.2 Computation of DFM ... 26
2.5.3 Relevance... 27
3. Modeling Demand ... 29
3.1. Poisson Distribution ... 29
Fitting Poisson Distribution... 29
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3.2 Normal Distribution ... 31
Fitting Normal Distribution ... 31
3.3 Negative Binomial Distribution ... 32
Fitting Negative Binomial Distribution ... 33
4. Discrete Flow Model Analysis at Nike... 35
4.1. Financial Management DFM at Nike ... 35
4.2. Transit Time Analysis ... 40
4.3 Lost Sale Analysis ... 43
4.4. Emission Analysis ... 44
4.5. Results... 46
4.5.1 Closed-Loop supply chain analysis at Nike ... 47
4.5.2. Model Implementation ... 48
4.5.3. Validation ... 49
5. Generalization of Discrete Flow Model ... 50
5.1 Industry Based Conclusions ... 50
5.2 Assumptions ... 50
6. Discussion and Managerial Insights ... 51
7. Future Research... 53
8. Reference ... 54
Appendix 1. DFM Optimization Model for End-of-Life products Management... 57
Appendix 2. Demand data for 3 years. ... 72
Appendix 3. Variable Matrix T of DFM ... 83
Appendix 4. Figures ... 84
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iv
Abstract
In this thesis, we propose a Discrete Flow Model (DFM) to assess the implication of having a closed-loop supply chain, by deployment of remanufacturing operations. Although many companies claim to have achieved green operations, assessment methods of this claim is rather subjective. Our problem is based on the fact that the Nike EMEA does not have a methodology to quantify the impact of closed-loop supply chains. We defined the sustainable supply chains in two sepearate, however, equally important fields; financial management and environmental management. We demonstrate that these two fields are associated with each other by creating a discrete flow model. Furthermore, we developed an NTM model to quantify the amount of carbon emission due to Nike EMEA operations, and showed that remanufacturing of end-of-life products are actually beneficial to companies with varying extent. NTM methodology is used since it includes a detailed structure, and it is focused on Europe. Moreover, NTM model is incorporated into our DFM to comprehend the implications of remanufacturing in terms of both financial and environmental factors. DFM is adopted to Nike supply chain, and successfully tested with a case study within Nike EMEA.
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Summary
This thesis presents a decision support system through a Discrete Flow Model to enhance the end-of-life products management. Recently, sustainability is receiving relatively more attention from companies and becoming more of a strategic competence. However, managements do not have a solid method to use in their operations (McKinsey, 2014).
Literature has insufficient information regarding the methodologies that assist in assessing the benefits and disadvantages of sustainable operations. Lack of information in this area entails our problem statement to this thesis:
How should closed-loop supply chains be managed?
This problem is solved by creating a Discrete Flow Model that represents Nike Supply Chain, thus indicating the beneficial points of including end-of-life products into the operations. We conducted this case study at Nike EMEA Headquarters.
Figure 1. Project Overview; Problems, developed model and outcome
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Development of the DFM to assess the advantages of closed-loop supply chains
As mentioned previously, Nike had the problem of not being able to quantify the impact of a closed-loop supply chain for its Operations in EMEA region. We solve this problem by introducing a DFM that incorporates both financial and environmental impacts of initiating a closed loop supply chain. Financial management is assessed through our initial DFM that calculates the cost of operations. Environmental management is assessed through NTM methodology and eventually, we integrate the NTM into our initial DFM. Discrete Flow Model is initially designed by identification of entities within the model such as supplier, manufacturing facility, re-manufacturing facility, European Logistics Center (DC), End-of- life Consolidation center and Nike Stores (See Figure 1). In order to create scientific value, we combined two separate methodologies to solve the problem at hand, which is the lack of performance visibility for end-of-life operations. We embedded the environmental emission model (NTM) into the initial DFM model, thus resulted in an integrated assessment of
Sustainable Supply Chains. Considering the complexity of Nike’s Supply Chain; this model is successful in the sense that it provides an overview of the results that would be achieved, even before implementing these decisions. Management has the confidence that this project
suffices to provide useful insights regarding sustainable supply chain decisions.
Generalization of the Proposed Model
DFM can be implemented to solve many similar problems in different industries; with small justifications on demand behaviour and parameters. We exemplified several models to capture the implications for various industries, which resulted in different optimal solutions.
Developed DFM allows for modeling different industry settings by modification of parameters. Many other industies might have different settings, however, since all these industries have highly complex, thus, similar problems in essance, by tailoring their specifics to this model, we would reveal feasible solutions.
vii Figure 2. Overview of model characterization
We believe that proposed DFM can be applicable to many supply chain settings, especially within retail industry, since some products are more prone to re-manufacturing. Applicability of this model ought to be assessed by adjusting above mentioned parameters and our
assumptions.
Recommendation
For future research, we recommend incorporating the carbon trading scheme into cost structure of DFM, so that the average cost does not only rely on operational cost, but also takes into account legislative costs. When both operational and legislative costs are considered together, they would provide a more complete picture of the cost structure of closed-loop supply chains. Moreover, including decisions about location of remanufacturing facilities would be highly relevant and useful to the literature. For instance, Nike has the manufacturing side of business mostly in Asia, which requires them to move the end-of-life products back to Asia. This operation should also cause redundant emission figures.
Searching for better locations to re-manufacturing would be an interesting subject.
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Acknowledgements
I would like to enunciate my appreciation to my academic supervisors Boray Huang, Arun Chockalingam and Maximiliano Udenio for their support and numerous consultaions. I would like to thank my company supervisor Monica Rosado, for encouraging me to comprehend much more knowledge than that could be achieved from an internship. Although it was challenging with the time restrictions between work and thesis, it was worthile to experience this term, thanks to my colleauges. I appreciate all my colleauges in Nike EMEA for creating an amazing place to work.
Finally, I wish to express that I am profoundly grateful to my family and friends for their great support.
Emin Berzah July, 2018
Abbreviations
EHQ European Headquarter. Hilversum, the Netherlands ELC European Logistics center. Laakdal, Belgium NTM Network for Transport Measures
FTW Footwear
EQP Equipment
APP Apparel
NA North America
EMEA Europe, Middle East & Africa APLA Asia-Pacific & Latin America GA Greater China
NSO Nike Stores Owned NFS Nike Factory Stores NSP Nike Stores Partner DFM Discrete Flow Model MW Manufacturing Warehouse F1 Manufacturing Facility 1 F2 Manufacturing Facility 2 nbin Negative Binomial Distribution Poi Poisson Distribution
Norm Normal Distribution
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1. Introduction
This master thesis presents a Discrete Flow Model (DFM) to assess the implications of closed-loop supply chain decisions, and to provide assistance while making supply chain decisions. Recently, executives realize the importance of sustainability, and search for ways to unlock its potential as a core competency (McKinsey, 2014). Even though the attention
received is a good indication, there are still challanges to reveal that potential. Literature treats sustainable supply chains as special cases (e.g. Sadok et al., 2015; Turki et al.,2017).
Although there are papers covering sustainable supply chains, they usually treat a specific problem instead of treating the network as a whole. Therefore, we consider a setting where problems at hand are considered as an entire system (e.g. financial&environmental), and solved by the methods from literature. Nike makes extreme efforts to increase the
sustainability awareness, aiming for closed-loop supply chains. However, they do not possess a methodology to estimate the impacts of their supply chain decisions, which creates the motivation to apply a case study at Nike. In the end, we will clarify the DFM developed for Nike, then discuss its robustness and applicability to similar cases for other industries that would benefit from closed-loop supply chains. In the following chapter we introduce Nike Inc. in section 1.1. Secondly, we mention the scope and objective of this research in section 1.2. In section 1.3 we define the problem that company faces within the scope, followed by the research questions related to that problem in section 1.4. Section 1.5 intorduces the methodology that is followed in this study to achieve an answer to the problem. Lastly, section 1.6. provides an overview on the timeline of this thesis study.
1.1. Company Description
Nike Inc. is a global sportswear company with a product portfolio comprising apparel (APP), footwear (FTW) and equipment (EQP). Company was founded in Beaverton, Oregon United States in 1972 by the partnership of Phil Knight and Bill Bowerman in order to enhance the athlete’s performance. In 1991, European Headquarters were established in Hilversum, Netherlands. In 1994 operations & logistics were centralized.
Products are diversified based on categories such as: running, basketball, Nike Sports Wear, football and Jordan. Nike is the largest supplier of the athletic products, and recently
generated the revenue of $32,376 billion in the fiscal year 2016, holding the position of 331st
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in the Fortune’s Global 500 list (Fortune Global 500, 2017). A large share of revenues (56%) is derived outside North America. Furthermore, Nike was ranked 8th in Gartner’s supply chain list in 2016-2017. Geographical allocation of business is the following: North America (NA), Europe, Middle East & Africa (EMEA), Asia-Pacific & Latin America (APLA) and Greater China (GA). Nike has no owned production facility, thus supplying products from contractors that are divided into different regions, mainly in Asia.
EMEA Headquarters contain the following functions: Design, Marketing, Product &
Merchandising, Brand, Finance, Human Resources, Innovation and Operations. Within the Operations, the department that I currently conduct my internship is “Direct to Consumer Supply Chain”. Company oversees nearly 900 stores in several continents, which entails a highly complex and dynamic supply chain.
Core of this research is regarding the impact of remanufacturing operations with end-of-life products, and their distribution flow at Nike EMEA. We will consider the
manufacturing/remanufacturing facilities, manufacturing warehouses, ELCs, stores and customers. Stores are diversified based on their product portfolio. Nike Stores Owned (NSO), Nike Factory Stores (NFS) and Nike Stores Partner (NSP). Nike owns the NSO and NFS stores, and conducts a partnership-based business with NSP, meaning that Nike does not own these stores. Nevertheless, for this study, we will simplify these to one store type.
Since we live in a resource-constrained world, Nike focuses on materials that deliver superior performance and reduce the environmental impact, as measured by Nike Materials
Sustainability Index.
Figure 1.1. Sustainability Index
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Since this study focuses on closed-loop supply chains, reuse will be the entity that we place our focus on.
1.2. Scope of the Research
This research project considers products that finish their life cycles in customer use, which are returned from customers to be remanufactured. In the next parts of this paper, we will refer to these products as end-of-life products. We assume that once the product finishes its life cycle, it can be received by Nike Stores to be used in remanufacturing, in order to have a closed- loop supply chain. Return ratio of these products will vary. Those products will be re-sold to another customer after remanufacturing. We assume that there is no difference in terms of quality between a product manufactured from raw material, and a product re-manufactured from an end-of-life product, which is a strong assumption. Cost of these different
manufacturing systems; however, can be different to assess the implications of closed-loop supply chain management. Currently, Nike can only recycle footwear products that do not carry metal part, which ought to be 70% of footwear products. Nike must decide on how to operate on these products, more specifically, “What percentage of the sold products should be deployed in the remanufacturing operations?”. In our model, we assume that not all of the end of life products are remanufactured but only the ones that are returned from the customer.
Below there is a demonstration of the scope starting from raw material to the customer. This project will formulate the problem as a discrete goods flow.
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Figure 1.2. Scope of the closed-loop supply chain network at Nike EMEA.
1.2.1. Research Objective
Objective of the study is first to analyze the current process thoroughly, then to comprehend whether it could be improved by manipulating the current operations, and finally contribute to the existing literature by creating valuable insight. Manipulating the current operations could be explained as the following: In the current setting, end of life products are incinerated or disposed in waste management facilities. Our aim is to understand whether it is more feasible to involve these products in a closed-loop supply chain. Obviously, one contribution to Nike is that company will have a method to quantify the impact of closed-loop supply chain in terms of carbon emission through NTM methodology, which ought to assist top management in decision making process. Our reference article is recent (Turki et al., 2017), however, authors only capture the financial management with some strong assumptions. Contribution to literature is that; we capture the benefits and downsides between cost figures and carbon emissions in a complex network.
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1.3. Problem Definition
Nike does not know how to quantify the impact of using end of life products, in terms of both financial and environmental aspects. In other words, there is no method of assessing the closed-loop supply chain and its implications on performance indicators; both financial and environmental.
Figure 1.3. Cause and effect diagram of Nike’s sustainable supply chain quantification problem.
This study aims to understand the implications of incorporating end of life products to create a closed-loop supply chain. Specifically, we will attempt to understand the optimal
percentage of sold products to be returned to the manufacturing operations. There are certain aspects to consider in for a retailer like Nike, such as demand behavior of the business, manufacturing failure behavior, lead time, life cycle of a product, cost-emission parameters and perhaps legislative constraints. Furthermore, it is important to understand the
transportation time in the following perspective: Obviously, transporting returned products to Asia by a plane will take less time than a maritime transportation, however, amount of carbon emission would significantly differ. Consequently, embedding a remanufacturing system to the current operations may cause additional cost and emission, which may impair supply chain performance. The problem as mentioned before, is the absence of a methodology to quantify this change in operations. We will first capture the cost structure of this problem by Discrete Flow Model (DFM). Secondly, our model will attempt to capture the environmental impact of this change by NTM methodology, which should supposedly reduce company’s footprint as well. Therefore, our model will provide insights for assessing between
environmental and financial implications of initiating a closed-loop supply chain. Obviously, there are many measurements to capture the emission by any sort of operations. We will
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quantify overall environmental impact with carbon emission values, which ought to simplify the model and direct our attention to a higher-level analysis; the number of returned products to be included in re-manufacturing operations.
1.4 Research Questions
Even though the essential idea of a closed-loop supply chain is pleasant, we need a solid basis for understanding its feasibility with a business perspective as well. For instance, the question of how many end of life products to remanufacture can be answered with our proposed methodology. First question reflects on the 2nd root cause shown in figure 1.3. regarding cost structure.
Research question 1: How should the closed-loop supply chains be operated so that they are financially beneficial?
Since Nike does not have a methodology to measure their environmental impact during supply chain operations, it is hard to make decisions. Second question reflects on this need of quantifying carbon emission, which is shown as root cause 3rd in figure 1.3.,
Research question 2: How should the closed-loop supply chains be operated so that they create environmental benefits?
Third question reflects on the root cause 4th as shown in figure 5. This question is the combination of question 1 and question 2, which eventually creates the solution of the root cause 1, for the defined problem.
Research Question 3: What is the trade-off between financial and environmental benefits while operating in a closed loop supply chain?
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1.5. Methodology
This thesis is conducted based on an empirical quantitative method to observe and test the implications of existing theory on operations research. Quantitative model-based empirical research is focused on validating the scientific models deployed by the theoretical quantitative research (Bertrand and Fransoo, 2002). The model of Mitroff et al., (1974) identifies the phases of problem solving, additionally highlights different research approaches and styles towards management science. The following four stages were implemented in this research project. We first treat the problematic areas separately, with the literature assistance. Then we create a model that conceptually captures the situation presented in the problem definition.
Scientific model is then developed to solve the problem, and to validate the proposed solution.
Figure 1.4. Research Model of Mitroff et al. (1974)
Figure 1.5. Process map of this research project
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To begin with, we identified the root causes that lead to the problem, which is the absence of a methodology to estimate the impact of a closed-loop supply chain at Nike EMEA. As already defined, research questions direct our focus to the relevant problems shown in figure 1.3. Literature review is an iterative process that takes place during the entire project. Main goal is to comprehend the relevant theories in the literature and exploit the scientific insights to shape and conduct the thesis work. We attempt to validate the relevant theory within the scope (real-life setting). Nike supply chain is highly relevant for the model being studied in this work. Initial literature study seems to have already created value to the project regarding methods to tackle the existing problem. The gap is identified in the existing literature, and we aim to contribute to scientific literature by treating this gap. Research proposal comprises the previously mentioned stages and gives an indication on the deliverables of the project.
Current state should be analyzed with empirical data to understand relevant parameters such as demand behavior. Furthermore, Nike supply chain is modeled as a closed-loop network.
After the modeling, we first assess the financial management of remanufacturing end of life products by the DFM. Moreover, we improve the model by treating the carbon emissions due to remanufacturing operations by NTM. Feasibility analysis will be conducted in terms of financial/environmental perspectives. We report on the contribution to the existing literature by explaining insights derived from this study. By the end of our research, numerical results will be shared to clarify the impact of closed-loop supply chains at Nike, and the impact of placing end of life products into the scope.
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1.6. Timeline of the Project
The project starts at 15th February 2018. Based on the required methodology and project maturity, this project will be completed in 23 weeks. Project steps are explained in Table 1.
Comprehending Supply Chain Operations at the Company Week 1
Literature Study Weeks 2-5
Data Collection/Analysis of the Problem Weeks 6-9
Gathering insight from ELC and Carriers Week 10 Modelling the Closed-Supply Chain within the Scope Weeks 11-14 Developing methods to improve goods flow Week 15-17 Implementing the Improvements in the model Week 18-20 Verification of added value from the improvements (Model) Week 21-22 Evaluation of the project/ Deliverables Week 23
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2. Background on Sustainable Supply Chains
Supply chain planning decisions include exploring optimal configurations of number, location and capacity of manufacturing and storage facilities. These strategic configurations have significant influence on the tactical and operational decisions, therefore, on the overall performance of supply chain (Rezaee et al.,2017). Substantial investments are needed to design a supply network, which entails significant decisions to be made (Li, 2013). Due to environmental concerns, legislation pressure from European countries, and possible financial benefits lead supply chain domain to place its focus on sustainability and environmental impact of remanufacturing. Consequently, focus is shifting towards sustainable supply chain management (Galve et al., 2016). Research was conducted regarding Industrial sustainability;
Despeisse et al. (2012) considered the industrial practices and environmental results. Their objective was to develop a conceptual manufacturing ecosystem model as a basis to enhance environmental performance. They proposed a model considering the waste and material flows to comprehend the interconnection within manufacturing operations. Regarding this thesis subject, in the previous decades, closed loop supply chains received considerable attention from researchers and managers (Li et al., 2014). This topic became stronger in the literature due to hazardous products, government legislations and customers’ awareness on
sustainability. Therefore, companies become more aware that closed-loop supply chain concept can likewise create increased profits and market share (Nunnen and Zuidwijk, 2004).
The reverse supply chain entails an association between forward and backward flows, which suggests a closed-loop network (Salema et al.,2007). Obviously, forward flow means raw material feeding into manufacturing, and then products reaching to customers, which means new products. On the other hand, backward flow means that end of life products come back to remanufacturing facility from customers. For the Nike case, we consider the end-of-life products returned from customers. In this research we analyze whether operating in a closed- loop is beneficial for Nike. Remanufacturing is a process that feeds in the end of life products, to produce new ones (Brito et al., 2003). Remanufacturing was available for a while to
reproduce highly valued products, currently it is also means to advertise a company’s outlook on sustainability efforts. Kenné et al. (2012) devised a production planning model with a hybrid manufacturing/remanufacturing system within a continuous stochastic demand setting.
They deployed a hedging strategy to optimize stock levels. Consecutively, these stock levels would control production rates of the system. Chung et al. (2008) evaluated an inventory
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system with a reverse flow system in supply chain, as well as a forward flow system. In their reverse material flow, returned products are remanufactured and shipped back to the retailer.
Sadok et al. (2015) analyzed a closed-loop supply chain setting with remanufacturing facilities. They estimated the optimal service levels with regards to inventory, which minimizes the total cost of inventory, lost sales, and degradation of machines. There are papers studying the lead time impact on the whole system. Teunter (2001) analyzes a composite manufacturing and remanufacturing system with disposal option, deterministic demand and lead times, in order to investigate inventory replenishment policies.
2.1 Background on Discrete Flow Model
In this chapter we provide an overview of Discrete Flow Model (DFM) that was discussed in Turki et al., (2017). In order to characterize the model, required information are: demand behavior, manufacturing failure behavior, lead time, life cycle of product, cost-emission parameters and perhaps legislative constraints. We firstly clarify the mathematical model in section 2.2. After explaining the model, we solve an example DFM problem to familiarize the subject and demonstrate the way it works in section 2.3. We argue that literature is
insufficient to provide insights regarding the environmental impact, and its trade-off with financial impact. Therefore, we dive into the entire problem that integrates both financial and environmental impact of closed-loop supply chains. In section 2.4. we introduce the carbon emission model to quantify the environmental impact. We implement environmental
assessment by deploying NTM methodology in section. We end this chapter by reflecting on the DFM in section 2.5.
12 Figure 2.1. DFM input parameters
2 .2 Model Explanation
2.2.1Notation of the DFM
t A specific point in time
t Unit time period
s(t) Inventory level at manufacturing warehouse (MW) S Manufacturing warehouse (MW) inventory capacity g(t) Number of products shipping from MW to ELC (Laakdal) r(t) Inventory level of end-of-life products returned
D Demand from Nike Store (customer) at time t y(t) Satisfied demand from ELC to Store at time t l(t) Unsatisfied demand from ELC to Store at time t w(t) Inventory level of ELC at time t
z(t) Number of end-of-life products returned at time t p Percentage of returned products to the sold products
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p* Optimal percentage of returned products to the sold products q(t) Number of products being transported from MW to ELC
Lead time between MW and ELC (this value will be the multiple of t)
Useful life cycle of a new product to the customer X European Logistics Center’s (ELC’s) capacity
V Capacity of vehicles
α(t) Production availability of Manufacturing Facility 1 (F1) at time t β(t) Production availability of Remanufacturing Facility 1 (F2) at time t U1 Maximum production rate of F1
u1(t) Production rate of F1 at time t U2 Maximum production rate of F2 u2(t) Production rate of F2 at time t MTBF1 Mean time between failures of F1 MTBF2 Mean time between failures of F2 MTTR1 Mean time to repair F1
MTTR2 Mean time to repair F2
cs Unit inventory holding cost for the Manufacturing warehouse cl Unit cost for lost sale
cr Unit cost for holding returned product’s inventory ct Unit cost for transporting a product from MW to ELC cw Unit inventory holding cost for ELC
cu1 Unit manufacturing cost of a product from raw materials at F1 cu2 Unit remanufacturing cost of a product from returned product at F2 cz Unit return cost of used product
T Total time span of the model
f(t) Cost function of the system at time t F(T) Total cost function (Objective function)
Decision variable: p*
14 Figure 2.2. Representation of the System
As demonstrated in figure 2.2. we have two manufacturing facilities. First one (F1) sources raw materials from the supplier, then manufactures the products. It is assumed that supplier has ample capacity. The second facility (F2) re-manufactures the end of life products that are coming from customers. We assume that remanufactured Nike products have the same quality level as the products manufactured from supplier’s raw material. In our model,
(re)manufactured goods consolidate in a manufacturing warehouse (MW). Consequently, goods are shipped to the European Logistics Center (ELC) in Belgium. From ELC, products are shipped to the Nike Store where they are sold to the customers. Daily customer demand are be represented by random numbers of Negative Binomial distribution (see section 3.3). If a product finishes its life cycle in customer use, and is returned, that product can be
remanufactured. End-of-Life products are consolidated in a separate center, then shipped to the re-manufacturing facility (F2). Our aim is to understand the most beneficial flow; most advantageous percentage of products that should return to manufacturing. We aim to optimize the flow by finding the returned percentage that minimizes the cost/emission. We assume 1 day as a unit time period (t=1), which is reasonable for many supply chains, as lead time is calculated with a unit of ‘a day’. Unsatisfied demand is assumed to be lost sale. Service level is highly important for Nike; therefore, we will assume that maximum production rate is always higher than demand values (U>D). After analyzing the financial management aspect,
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we extend our model into environmental impact (e.g. carbon emission). By this way, we are able to assess the result of supply chain decisions in terms of environmental concerns.
At a single point in time, t, customer demand (Nike Stores), D, occurs following a Negative Binomial distribution with parameters R (7.91391) and P (4.0812e-05), which is satisfied from the ELC in Laakdal-Belgium. Demand assumption regarding Negative Binomial distribution is validated in the section 3.3. In our model, both manufacturing facilities replenish the inventory amount of manufacturing warehouse (MW) in Asia. Manufacturing warehouse replenishes the products in ELC with vehicles, in our case via maritime
transportation with transit time . Transport vehicles have maximum total capacity of V, transporting q(t) number of products at time ‘t’ to ELC. ELC has a maximum capacity of ‘X’.
For returned end of life product’s inventory ‘r’, we have a consolidation center that is assumed to have no lead time to remanufacturing facility and to Nike stores in order to simplify the model. This consolidation center is also assumed to have no capacity constraint either. Returned products are calculated at every time point ‘t’. The time horizon subject to this study is represented by ‘T’ (1000 days).
Figure 2.3. Time horizon of model
In order to realistically design the model; at every time point ‘t’, there are certain events that may occur. These can be: the failure of manufacturing facilities F1 and F2, repair of F1 and F2, ELC being full, ELC being out of stock, losing sales, and return of end of life products.
All these events are reflected in the mathematical model.
We assume that supplier for F1 has ample capacity, therefore, F1 will never have issues regarding sourcing raw material. However, F1 may be running or down for the point in time.
α(t) =
(1)
β(t) = (2)
1 If F1 is running 0 otherwise
1 If F2 is running 0 otherwise
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Time between failures are exponentially distributed, with a mean of 30 days. On the other hand, time to repair is set to 1 day deterministically to simplify the model. Maximum production rate is set to the parameter U1 (i.e. 0 ≤ u1(t) ≤ U1). If F1 is not running, then u1(t)=0. Same approach for the F2, when F2 is running, maximum production rate of F2 is U2, (i.e. 0 ≤ u1(t) ≤ U1), and when F2 is down, u2(t)=0. In parallel with (Turki et al., 2017), we make the following assumptions:
1) We assume that time per t=1 day.
2) We assume that is the transportation time, and is a multiple of t.
3) Unlike Turki et al. (2017) deploying deterministic demand, we assume the demand ‘D’ to follow negative binomial distribution. (see section 3.3)
4) Remanufactured and manufactured products have the same quality levels, thus same prices.
5) We assume that in the first (useful life cycle of FTW) period of time, ELC has enough inventory to fulfill the demand. This periods can be seen as a warm up period for the model.
(i.e. w(0) ≥ D)
Demand ‘D’ is met with the on-hand inventory at the beginning of each period, suggesting that we use the ELC inventory level from previous period w(t-t). Thus, the
value of fulfilled demand at time t is represented in the following equation:
y(t)= ( 3)
The ELC in Belgium, which fulfills Nike Stores’ demand, gets replenished with products coming from the manufacturing warehouse. Thus, the ELC inventory level at a specific time
‘t’ is w(t) is the sum of the products in this ELC left from the previous period (t − ∆t), and the number of products that were shipped from manufacturing warehouse at time (t − τ), minus the number of products that were sold to the Nike Stores at time t:
w(t) = w(t − ∆t) − y(t) + g(t − τ) (4)
D If D ≤ w(t-t)
w(t-t) otherwise
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The ELC inventory level after the delivery and the fulfillment of the Nike stores’ demands during period τ, is represented as follows:
w(t + τ) = w(t) − τ * D + g(t) (5) Shipped units from manufacturing warehouse MW at time ‘t’ require τ units of time to be delivered to ELC. The number of units outgoing from manufacturing warehouse inventory ‘S’
at time t (i.e. g(t)) is built upon the ELC inventory level at time t + τ and its total capacity, manufacturing stock level at time w(t − ∆t), transport vehicle capacity ‘V’, and the demand from Nike stores.
g(t)= (6)
The units produced in F1 and F2 replenish the manufacturing warehouse inventory ‘S’. Total production rate from F1 and F2 is:
u(t) = u1(t) + u2(t) (7)
The manufacturing warehouse inventory level at time ‘t’ consists of the level of this inventory left from the previous period (t − ∆t), plus the units produced during the previous period, minus the units being shipped from manufacturing warehouse S at time t:
s(t) = s(t − ∆t) + u(t − ∆t) − g(t) (8)
When the demand is not satisfied from the ELC inventory, lost sales occur. Number of lost sales is defined in the following equation:
l(t) = D − y(t) (9)
We assume that the number of end-of-life products returning (z) to be remanufactured, are the percentage ‘p’ (0 < p < 1) of the satisfied demand quantity at time (t − ). As mentioned before parameter represents the useful life cycle of the product. Obviously, when (t − ) < 0, there is no returned products.
V If s(t − ∆t) ≥ V and w(t + τ) ≤ X-V+ τ * D τ * D If s(t − ∆t) ≥ τ * D and w(t + τ)=X
s(t − ∆t) If s(t − ∆t) ≥ τ * D
0 otherwise
18
z(t)= (10)
Figure 2.4. Product’s useful life cycle to the customer
As mentioned, F2 is supplied from the inventory center that consolidates the returned end-of- life products z(t). Following equation represents the number of returned products:
r(t) = r(t − ∆t) + z(t) – u2(t − ∆t) (11)
To ensure that manufactured products do not exceed the Manufacturing warehouse inventory level, we deployed a hedging point policy for production rates of F1 and F2. This policy is applied when both Manufacturing facilities are up and running (Kumar, 1986). Portioning of unit production while both facilities are in active state is represented by the following
functions:
FPU1 = (12)
FPU2 =
(13) p* y(t − ) if (t − ) ≥ 0
0 otherwise
[y/2] + 1 if y is odd
(y/2) if y is even
[y/2] if y is odd
(y/2) if y is even
19 u1(t) =
(14)
u2(t) =
(15) U1 if α(t) = 1; β(t) = 1; r(t) ≥ U2; s(t) ≤ S – U + g(t)
or α(t) = 1; β(t) = 1; r(t) < U2;s(t) < S – U + g(t) or α(t) = 1; β(t) = 0; s(t) < S – U + g(t)
or α(t) = 1; β(t) = 0; g(t) > U1; s(t) = S
or α(t) = 1; β(t) = 0; (S – s(t)) > U1; S – U + g(t) < s(t) < S g(t) – FPU2(g(t)) if α(t) = 1; β(t) = 1; r(t) ≥ FPU2(g(t)); s(t) = S
g(t) if α(t) = 1; β(t) = 0; g(t) ≤ U1; s(t) = S FPU1(g(t)) if α(t) = 1; β(t) = 1; r(t) < FPU2(g(t)); s(t) = S
S – s(t) – FPU2(S – s(t)) if α(t) = 1; β(t) = 1; r(t) ≥ FPU2(S – s(t)); S – U + g(t) < s(t) < S S – s(t) if α(t) = 1; β(t) = 0; S – s(t) ≤ U1; S – U + g(t) < s(t) < S FPU1(S – s(t)) if α(t) = 1; β(t) = 1; r(t) < FPU2(S – s(t))
0 if α(t) = 0
U2 if α(t) = 1; β(t) = 1; r(t) ≥ U2; s(t) < S – U + g(t) or α(t) = 0; β(t) = 1; r(t) ≥ U2;s(t) < S – U + g(t) or α(t) = 0; β(t) = 1; r(t) ≥ U2; g(t) > U2;s(t) = S
or α(t) = 0; β(t) = 1; r(t) ≥ U2; S – s(t)) > U2; S – U + g(t) < s(t) < S r(t) if α(t) = 1; β(t) = 1; r(t) < U2; s(t) < S – U + g(t)
or α(t) = 0; β(t) = 1; r(t) < U2;s(t) < S – U + g(t) or α(t) = 0; β(t) = 1; r(t) < U2; g(t) > r(t); s(t) = S
or α(t) = 0; β(t) = 1; r(t) < U2; S – s(t)) > r(t); S – U + g(t) < s(t) < S or α(t) = 0; β(t) = 1; r(t) < U2; S – s(t)) > r(t); S – U + g(t) < s(t) < S or α(t) = 1; β(t) = 1; r(t) < FPU2(g(t)); s(t) = S
or α(t) = 1; β(t) = 1; r(t) < FPU2(S – s(t)); S – U + g(t) < s(t) < S FPU2(g(t)) if α(t) = 1; β(t) = 1; r(t) ≥ FPU2(g(t)); s(t) = S
FPU2(S – s(t)) if α(t) = 1; β(t) = 1; r(t) ≥ FPU2(S – s(t)); S – U + g(t) < s(t) < S g(t) if α(t) = 0; β(t) = 1; r(t) ≥ g(t); g(t) ≤ U2; s(t) = S
or α(t) = 0; β(t) = 1; r(t) < U2; g(t) ≤ r(t); s(t) = S
S – s(t) if α(t) = 0; β(t) = 1; r(t) ≥ (S – s(t)); (S – s(t)) ≤ U2; S – U + g(t) < s(t) < S or α(t) = 0; β(t) = 1; r(t) < U2; (S – s(t)) ≤ r(t); S – U + g(t) < s(t) < S
0 if β(t) = 0
20
Production control for both F1 and F2 are described in the above page based on inventory levels and random events and states of the parameters. The following rules ought to assist in controlling the outcome of manufacturing facilities by incorporating returned units,
production capacity of F1, and inventory levels.
Equation 16 represents the cost function at time t, which comprises of inventor costs, lost sales costs, transportation cost, returned products costs and manufacturing/remanufacturing costs.
f(t)= cs*s(t) + cr*r(t)+cw*w(t) +ct*q(t)+cl*l(t)+cu1*u1(t)+cu2*u2(t)+cz*z(t) (16)
We define the total cost function in equation 17, where T represents the model horizon.
𝐹(𝑇) = ∑𝑡=𝑇f(t)
𝑡=0 (17)
2.3. Example DFM Financial Management
The below example ought to demonstrate the type of problem that Nike faces. Let us examine a company that aims to initiate a closed-loop supply chain by re-manufacturing its end-of-life products. That company shall introduce the re-manufacturing facility, arrange the reverse logistics operations, and define the following parameters.
Lead Time 1 day Demand D 10 units/period Life Cycle 30 days ELC Capacity X 23 units
Stock Capacity S 15 Vehicle Capacity V 15 units
MTBF1 Exp(4) MTBF2 Exp(4)
MTTR1 1 day MTTR2 1 day
U1 Max. Capacity 9 units/period U2 Max. Capacity 5 units/period
cu1 10 cu2 4
Cw 2 Cs 2
Cr 1 Cz 1
Cl 250 Cq 1
Table 2.1. Parameters of Example Model 1
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Table 2.2. DFM average costs based on percentage of sold products returning to manufacturing.
As can be seen from table 2.2, if 20% of sold products return to manufacturing, average cost tends to have relatively lower values. Although there might be slight difference at each run of model, the main conclusion is preserved, which is that average cost per period reaches nearly minimum level when number of returned products is 20% of sold products. The discrepancy between each model can be attributed to random failure behavior.
Figure 2.5. Financial Management Model with 1000 periods
0,00 500,00 1.000,00 1.500,00 2.000,00 2.500,00 3.000,00 3.500,00 4.000,00 4.500,00 5.000,00
5% 10% 15% 20% 25% 30% 35% 40% 45% 50%
Average Cost
Percentage
Example - Financial Management
Percentage 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%
Average
Cost 1,320.29 1,209.34 1,030.88 1,068.74 1,720.64 1,662.67 3,034.19 2,966.97 4,324.09 4,375.84
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2.4 Carbon Emission Quantification
In this chapter, we will define the calculation method of carbon emissions due to
transportation. In 1995, an assessment was published by Intergovernmental Panel on Climate Change (IPCC) to address the elevating greenhouse gas emissions, and their impact on the earth (IPPC, 1995). The term Greenhouse gas refers to number of cases such as carbon dioxide, methane, and CFCs (Hoen et al., 2010). However, we will only focus on carbon emission within this research. In order to achieve our targets on reducing carbon emission, measurement accuracy is highly important. Although there are several methods, this study implements NTM methodology (NTM Road; NTM Water 2008). Network for Transport Measures is a non-profit seeking organization. Another reason to deploy NTM is that because it incorporates a detailed structure, and the methodology is focused on Europe. Furthermore, NTM methodology maintains estimations for parameters, which are not known by companies (Hoen et al., 2010). Following the inclusion of carbon emission, carbon pricing scheme could be considered in the cost structure of above mentioned model; As Carbon trading is a popular mechanism that has been widely applied within nations (Zakeri et al. 2015). Considering our scope, we will analyze emission calculations for road and water transport.
2.4.1 Road Transport
Road transport provides more flexibility than other transportation methods, especially with the vehicle capacity as it can be a small or a large truck. Only significant problem related to road transport may be capacity issues. Road transport has various types, namely; rural roads, motorways, and urban roads (Road, 2008). We assume that load factor is linearly dependent on the weight of the product being carried. According to Hoen et al. (2010), amount of emissions of a vehicle can be measured by the fuel consumption, the fuel emissions, and the distance. We will analyze these three terms separately.
23 Notation:
FC Fuel Consumption (l/km)
FCempty Empty vehicle fuel consumption (l/km)
FCfull Full vehicle fuel consumption (l/km)
LF load factor
D Distance (km)
FE Fuel Emission (g/l): Gram of CO2 emitted per liter of fuel.
EMtotal Total emission due to road transportation
FC = FCempty + (FCfull – FCempty)*LF (18)
EMtotal = FC*FE*D (19)
2.4.2 Water Transport
In water transportation, vessel type has high importance for carbon emission. Depending on type of vessels, consumption figures may vary. However, due to limitations regarding data collection; we will assume one specific type to calculate emission. Water transport emission is dependent on three factors; fuel consumption (FC), the distance (Dx), and fuel emissions (FE).
For a specific vessel type, fuel consumption given in NTM Water (2008). The distance
calculations can be done through many sources such as World Port Distances even though it is not always accurate. Moreover, fuel emissions factor (1 kg of CO2 emitted while 1 liter of diesel is burnt) is necessary as well (Hoen et al., 2010). Total emission (kg) is given with the below equation:
FCvessel Fuel consumption value for a specific vessel (tonne/km) FEvessel kg of CO2 emitted when 1 l of diesel is burnt
D Distance (km)
EMtotal Total fuel emission
24
EMtotal: FCvessel * FEvessel *D (20)
2.4.3 Carbon Emission Model
St vehicle capacity
Tunits total units of raw materials shipped to Manufacturing facility, which is assumed to be the same after production.
FCempty empty vehicle fuel consumption factor (l/km)
FCfull full vehicle consumption factor (l/km)
FE_Truck emission factor for a truck (g/l)
FEvessel emission factor for a vessel (g/l)
d1 distance from supplier to manufacturing plant
d2 distance from manufacturing facility to ELC (Laakdal)
d3 distance from Nike Store to inventory center of returned products d4 distance from returned products’ inventory center to F2 e2 emissions due to producing one unit
np number of products from manufactured F1
pr percentage of products returned after useful life cycle Xm Total emission due to closed loop supply chain
Xm = (FCempty +(FCfull- FCempty)*( Tunits /St))* FE_Truck *d1+(np*e2) + (FCvessel * FEvessel *d2)
+( FCempty + (FCfull- FCempty)*(( Tunits *pr)/ St))* FE_Truck *d3
+ (FCempty +(FCfull- FCempty)*(( Tunits *pr)/ St))* FE_Truck *d4 (21)
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2.5 Reflection on the Discrete Flow Model
Implementing the DFM for our example revealed different outcomes as explained in the previous section. This section reflects on the achievements of these different methods that covers financial and environmental impacts. In order to understand these impacts, we place our focus on the following aspects:
Input - What sort of data is required for DFM?
Computation - How does the model work?
Output – What is the outcome that should assist the decision?
This section analyzes the performance of DFM model by checking above mentioned aspects.
2.5.1. Input
2.5.2 Data Collection
The project required assortment of several data regarding the demand, and operational processes. We used the following data sources for the above-mentioned requirements:
- Interviews with Nike Supply Chain, and Supply Chain Innovation professionals.
- Interviews with ELC (European Logistic Center) professionals.
- Demand data for 2015, 2016 and 2017 (Masked).
In this section we will treat the input parameters depicted in figure 2.1.
Demand Behavior: To make a comprehensive analysis, demand data is significant as the output may vary based on different demand behavior. For our study, we state that negative binomial distribution fits more appropriately as it is a retail industry trait, however, this model would provide different results for different demand distributions.
Manufacturing Failure behavior: As output is directly related to failure behavior, this parameter allows us to characterize the manufacturing setting. This makes the models more realistic as it provides flexibility. In Nike case, we assumed exponential failure behavior with a mean of 30 days, however, other settings might as well be modeled using this parameter.
26
Transit Time: Transit time is a strong determinant of inventory. Therefore, we study the behavior of the outcome as transit time increases. In such cases we incur high inventory cost as transit time increases. A benefit of this parameter allows for modeling many supply chain scenarios. For instance, Nike has the production and distribution in different continents for EMEA region, which makes lead time an interesting aspect of this study and shows the risk that lead time poses to distribution networks.
Life Cycle of Product: This input has relatively more variance as it depends on the product type, quality and customer behavior. However, changing the life cycle of product has limited impact of on the model output, as the percentage does not change, however, warm up period changes proportionally with the life cycle of product.
Cost and Emission Parameters: These parameters concern the unit cost of production, cost of lost sale, transportation, inventory cost of warehouse and ELC, return cost of unit product. We may also include a constraint on percentage, such as legislative constraints enforced by the governments.
2.5.2 Computation of DFM
A critique of both financial and environmental analysis will be provided in this section.
The weaknesses and strengths of DFM will be analyzed based on the outcome of different models.
Financial Management
As explained in the section 2.2, we have 1000 periods to run the model given the parameters, which means that for every period we exercise the 20 equations in section 2.2. Following these calculations, we end up with the total cost and total emission of these 1000 period. Then we average the cost and emissions per percentage of sold products, which provides our
essential outcome per percentage. Furthermore, cost figures do change slightly for each model run due to random behavior of demand and failure of facilities, especially due to high
variance of demand figures (see section 3.3.) Another downside of this method is that its assumptions may be unrealistic regarding the failure behavior of manufacturing facilities as we do not have sufficient data on this. The most significant drawback of this method is the highly complex equations, which causes obstacles to interpret the results, since many details
27
exist within the model. In order to understand the impact of cost structure, there are many aspects needed to explore such as lost sale behavior per percentage. Even though we decide on the percentage of sold products that should return to manufacturing, we have challenges to understand its root causes and explain it immediately. Furthermore, after calculating all the equations, output is solely the percentage of products, which does not provide insight regarding the optimal capacity of important entities such as ELC and MW.
2.5.3 Relevance
We argue that literature does not cover the integrated DFM problem, which is a combination of financial and sustainable management of end-of-life products. We first explain the gap in the literature to argue the relevance of subject. Discrete flow model is used for the design, control and the optimization of production systems. DFM is deployed as an alternative to the queuing systems to analyze discrete event systems (Turki et al, 2017). In fact, discrete flow models prove to be more realistic for discrete manufacturing systems rather than continuous flow models (Yao and Cassandras, 2012). Although there are methods to quantify the cost structure of closed-loop supply chains (see section 2.3.), this only considers financial impact.
Limited attention has been paid to the discrete flow models that incorporate the environmental impact. We argue that incorporating the environmental impact is significant. The reason is that both financial and environmental measures influence each other. Turki et al. (2017) treat a DFM problem with deterministic demand, which is modified to a variable demand in our integrated model. They consider a closed-loop supply chain that incorporates
remanufacturing, manufacturing, transportation, and selling of goods under financial management. Our thesis study will take on from the financial management and improve the model by the quantification of carbon emission. Nike is currently not able to quantify that emission due to remanufacturing operations, which makes this case relevant to the company.
Moreover, this thesis ought to add value to the existing literature by providing a framework to decide based on carbon emission whilst benchmarking with financial impact in a Retail Industry. Academia and businesses attempt to understand the profit of sustainability
(McKinsey, 2014). Whilst making decisions on sustainability, one must consider the carbon emission figures as well as the financial figures, which highlights the importance of our integrated Discrete Flow Model. Further developments would be incorporating carbon pricing to this model as well.
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29
3. Modeling Demand
Nike as a retailer, has high volatility in demand figures. This volatility may be due to promotions, weather, or dynamic pricing on retail business. We have had access to demand data of 3 full consecutive years, 2015,2016,2017 with a total of 1096 data points. In order to realistically model the Nike demand, we performed statistical analysis to model the data , and generate random values based on these distributions.
3.1. Poisson Distribution
Generally, the Poisson distribution is sufficient to model many demand data sets for many industries, using a probability function:
P{Demand=k} = e-λ λk/ k! where k=0,1,2,3, …
With a mean value of λ, which is also representing the variance. The Poisson distribution is an often assumption to model demand in continuous review systems, since it only requires a single parameter (λ), and the results are more tractable (Fisher et al., 2009). However, empirical studies (Agrawal and Smith 1996) in retail industry show that retail demands are more variable than Poisson distribution, which can be attributed to weather, promotions and competitors. Such events are not reflected in the forecast schemes, which creates the
additional variance.
Fitting Poisson Distribution
Demand data points were tested for Poisson distribution as well. Kolmogorov Smirnov test (via SPSS) was applied to assess the fitness of Poisson distribution; as can be seen from Table 5, null hypothesis is rejected (p<0.05), which means data is not a good fit for Poisson
distribution.
30
VAR00001
N 1096
Poisson Parametera,b M ean 193903.2026
M ost Extreme Differences Absolute .580
Positive .580
Negative -.403
Kolmogorov-Smirnov Z 19.199
Asymp. Sig. (2-tailed) .000c
a. Test distribution is Poisson.
b. Calculated from data.
c. The data contain a value that is too large for the Poisson distribution. A normal approximation is used.
Table 3.1.
Moreover, Demand data was fitted to a Poisson distribution via Matlab’s distribution Fitter function. As can be seen from Figure 3.1 and Figure 3.2., Poisson distribution is hardly a fit for this demand data set.
Distribution: Poisson Log Likelihood: -Inf Domain 0 ≤ y < Inf
Mean 193.903
Variance: 193.903
Table 3.2. Fitted Poisson distribution parameters
Figure 3.1. Pdf of Demand Data (Poi) Figure 3.2. Probability plot of demand data Parameter Estimate Standard error
Lambda 193.903 13.3011
31
The data set has been tested for mean/variance equality as well. However, there is a
significant difference between mean (193.903) and variance (4.992.930.763) values, showing that demand data points are not representative of a Poisson distribution.
3.2 Normal Distribution
The normal distribution may fit better due to more variation, by simply selecting a larger variance, however, empirical studies also state that the normal distribution fit poorly with low demand items. The reason for that can be explained by the negative probability values
assigned by the distribution.
Fitting Normal Distribution
The Kolmogorov-Smirnov test is first performed in SPSS software for normal distribution validity. As can be seen from table 3.3., p value is below 0.05, which suggests that the data is not a viable representation of normal distribution.
N 1096
Normal Parametersa,b M ean 193903.2026
Std. Deviation 70692.93117
M ost Extreme Differences Absolute .095
Positive .095
Negative -.059
Test Statistic .095
Asymp. Sig. (2-tailed) .000c
a. Test distribution is Normal.
b. Calculated from data.
c. Lilliefors Significance Correction.
Table 3.3. One Sample Kolmogorov-Smirnov Test for Demand Data
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Furthermore, When we fit the data to a Normal distribution in Matlab via distribution fitter function, we obtain the following parameters and graphs:
Distribution: Normal Log Likelihood: -13792.7 Domain -Inf < y < Inf
Mean 193.903
Variance: 4.997.490.000
Table 3.4. Fitted Normal distribution parameters for Demand Data
Figure 3.3. Pdf of Demand Data (Norm) Figure 3.4. Probability Plot of Demand Data In Figure 3.3. and Figure 3.4., The disrepancy between red line and demand data is apparent, which indicates that Normal distribution does not capture all the information from demand data, as suggested by Kolmogorov-Smirnov test.
3.3 Negative Binomial Distribution
Agrawal and Smith (1996) states that the negative binomial represent the store level demand data more appropriately than Poisson or Normal distributions due to capturing highly variant information in data. Negative Binomial distribution has the following probability density function:
Parameter Estimate Standard error
Mu 193.903 2135.36
Sigma 70692.9 1510.96
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Y=f (D = k | N, p) = fk (N,p) = * pN * (1 – p)k
Where 0 < p < 1, N > 0, k = 0,1,2,..
In order to assess the fitness of demand distribution, we fit the data to Poisson, Normal and Negative Binomial distributions. Moreover, we performed One sample Kolmogorov-Smirnov test to statistically validate our assumptions regarding demand behavior. Kolmogorov
Smirnov test is non-parametric, which does not require a data set to follow a normal
distribution. All nonparametric sample tests of Kolmogorov-Smirnov type can be used as a goodness-of-fit test. Furthermore, we generated probability plots to evaluate the fit of
distributions. The probability plot is a technique that assists in understanding whether or not a data set follows a given distribution (Chambers et al., 1983).
Fitting Negative Binomial Distribution
Negavite Binomial Distribution is proven to be represetative of retail demand data by the literature (Agrawal and Smith, 1996). Although we did not assess this distribution through Kolmogorov Smirnov test, demand data is fitted to a negative binomial distribution through Matlab’s distribution fitter function.
Distribution: Negative Binomial Log Likelihood: -13717.8
Domain 0 ≤ y < Inf
Mean 193.903
Variance: 4.75113e+09
Table 3.5. Fitted Negative Binomial Distribution Parameters for Demand Data
Parameter Estimate Standard error
R 7,91391 0,3312
P 4.0812e-05 1.76325e-06
N + k – 1 N – 1
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Figure 3.5. Pdf of Demand Data (nbin) Figure 3.6. Probability Plot of Demand Data As can be seen from the figure 3.5. and figure 3.6., demand data information is captured most properly with negative binomial distribution. Unlike the deterministic demand assumption of Turki et al., (2017), considering the literature suggestion (Agrawal and Smith, 1996), negative binomial will be the main distribution for our study in this model with above shared
parameters.
