An inventory model for the supply chain with partial backordering, carbon emissions, energy and imperfect process

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Instituto Tecnológico y de Estudios Superiores de Monterrey

Campus Monterrey

School of Engineering and Sciences

An inventory model for the supply chain with partial backordering, carbon emissions, energy and imperfect process

A thesis presented by

Valeria Martínez Villarreal

Submitted to the

School of Engineering and Sciences

in partial fulfillment of the requirements for the degree of Master of Science

In

Engineering Science

Monterrey Nuevo León, December 16^th, 2020

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Dedication

I dedicate my thesis work to my parents and my sister, who motivated me in the first place to continue with my master's studies, leaving behind the fears of doing it and reminding me that everything is possible with effort and dedication.

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Acknowledgements

I endlessly thank the support of my advisor, Dr. Leopoldo Eduardo Cárdenas Barrón, who was always there, working with me and helping all the time, solving all my doubts, and supporting me in everything necessary. I am very thankful for his patience, commitment, and perseverance, to help me fulfill this big step within the mastery, without stopping supporting and motivating me at any time.

I appreciate my mother, my father, and my sister, who supported me morally throughout this time to continue with my master's study and carry out my thesis.

I am very grateful to Tecnológico de Monterrey for giving me a full scholarship at its prestigious institution, as well as with CONACYT, for its maintenance support all these months.

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An inventory model for the supply chain with partial backordering, carbon emissions, energy and imperfect process

by

Valeria Martínez Villarreal Abstract

Sustainability is a highly relevant issue today. It is related to the inventory problem

because a large part of the greenhouse effect gases (GHG) come from the industry.

In the supply chain, both transport and production processes require a lot of energy that

is converted into emissions of carbon that cause global warming at the same time.

The government, therefore, has developed different policies that limit the amounts of emissions allowed to achieve a sustainable development. These regulations increase the

costs of companies, so it is convenient for them to adapt and comply with the rules.

In this research, a sustainable inventory model for the supply chain with partial backordering, lost sales, process quality and environmental costs is developed. From this inventory model, three inventory models are identified as special cases: a sustainable inventory model with process quality and environmental costs, an inventory model with process quality costs, and an inventory model with traditional inventory costs. The four inventory models address shortages with partial backordering, lost sales, energy usage, and some of them include other real topics such as environmental problems and imperfect quality. A solution algorithm is developed to determine the optimal values for the buyer’s order quantity (q), the buyer’s backordering quantity (B), the number of shipments of size q in a cycle (n), and the vendor’s production rate (P), which minimize total system costs. The optimal vendor’s production quantity (Q_v) is computed considering the buyer’s order quantity (q) and number of shipments of size (n). The optimal values for the buyer’s backordering quantity satisfied (B_q) and buyer’s lost sales quantity (B_LS) are determined taking into account the buyer’s backordering quantity (B). Numerical examples are presented and solved. Additionally, a sensitivity analysis is done with the aim to see if the parameters have a significant impact, and in turn, provide better tools and knowledge for decision-makers in companies. Finally, it is important to remark that it was found that the proposed inventory model in this research is more economical from one of Marchi et al.

(2019)’s inventory model.

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List of Figures

Figure 1. Schematic representation of the supply chain ……….. 26

Figure 2. Total cost vs. order quantity ( q ) for the sustainable inventory model with process quality and environmental ………..……….………....……….. 50 Figure 3. Total cost vs. backordering quantity (B) for the sustainable inventory model with process quality and environmental.……….……….... 50 Figure 4. Total cost vs. number of shipments (

n

) for the sustainable inventory model with process quality and environmental. ………..……….. 51 Figure 5. Total cost vs. vendor’s production rate (P) for the sustainable inventory model with process quality and environmental.………..………...… 51 Figure 6. Total cost vs. order quantity ( q ) for the sustainable inventory model with environmental costs.………..….….. 52 Figure 7. Total cost vs. backordering quantity (B) for the sustainable inventory model with environmental costs.………..……….... 52 Figure 8. Total cost vs. number of shipments (

n

) for the sustainable inventory model with environmental costs..………..………..…. 53 Figure 9. Total cost vs. vendor’s production rate (P) for the sustainable inventory model with environmental costs..……….……….... 53 Figure 10. Total cost vs. order quantity ( q ) for the inventory model with process quality cost……….………. 54 Figure 11. Total cost vs. backordering quantity (B) for the inventory model with process quality costs.………. 54 Figure 12. Total cost vs. number of shipments (

n

) for the inventory model with process quality costs………...…... 55

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Figure 13. Total cost vs. vendor’s production rate (P) for the inventory model with process quality costs. ………... 55 Figure 14. Total cost vs. order quantity (q) for the inventory model with traditional inventory costs.……….………. 56 Figure 15. Total cost vs. backordering quantity (B) for the inventory model with traditional inventory costs.……….……… 56 Figure 16. Total cost vs. number of shipments (

n

) for the inventory model with traditional inventory costs……….. 57 Figure 17. Total cost vs. vendor’s production rate (P) for the inventory model with traditional inventory costs.…..………... 57 Figure 18. Inventory level of the vendor over a cycle with a production quantity of

nq ………. 75

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List of Tables

Table 1. Equations required in the algorithm according to each inventory model... 45

Table 2. Penalties for emissions to the environment……….………. 46

Table 3. Optimal solution for Example 1 for the four inventory models……… 47

Table 4. A comparison of the proposed inventory model with Marchi et al. (2019) inventory model……… 48 Table 5. Optimal solution for Example 2 for the four inventory models……….... 49

Table 6. Effects of S , ,c and _ec c_en on total cost ( TC ), order quantity ( q ), backordering quantity (B) and number of shipments (n) for each inventory model.. 59 Table 7. Effects of ,A,h_v and h_bon total cost ( TC ), order quantity ( q ), backordering quantity (B) and number of shipments (n) for each inventory model when S = 7000 and  =0.3……….………....…... 60 Table 8. Effects of ˆ,^ˆ^, and D on total cost ( TC ), order quantity ( q ), backordering quantity (B) and number of shipments (n) for each inventory model when S = 7000 and  =0.3………..……….………... 61 Table 9. Effects of ,and r on total cost ( TC ), order quantity ( q ), backordering

quantity (B) and number of shipments (n) for each inventory model when S = 7000 and  =0.3……….……….. 62 Table 10. Effects of some parameters on total cost ( TC ) and order quantity ( q ) for each inventory model when the parameter increases………... 63 Table 11. Effects of some parameters on backorder quantity (B) and number of shipments (n) for each inventory model when the parameter increases………. 64

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n

... 85

Appendix XII. Number of shipments (

n

) ... 85

Appendix XIII. A sustainable inventory model with process quality and environmental costs ... 88

Appendix XIV. A sustainable inventory model with environmental costs ... 90

Appendix XV. An inventory model with process quality costs ... 92

Appendix XVI. An inventory model with traditional inventory costs ... 94

References ... 96

Curriculum Vitae... 103

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Chapter 1 Introduction and context of the research

1.1 Introduction

Inventories have existed since ancient times and have always been present in the history of humanity. An example of this is the people that kept large amounts of food as provisions to meet the food needs of the population for those times of crises, natural disasters, and especially droughts. Therefore, one of the main reasons why inventories exist is to avoid shortage's problems. Over time, this way of storing the goods and food necessary for survival gave rise to the existence of inventory problems to face difficult times as well as to ensure the subsistence of life and the development of daily activities.

Inventories currently serve as a strategic tool for companies, creating value for them and giving them a complementary advantage over others. For this reason, it is of great importance to building new mathematical models that meet the needs of today's industries, and that can, among other things, reduce environmental impact while minimizing total costs.

1.2 Motivation

My principal motivation for making sustainable inventory models is thinking about the current alarming situation on the planet. Companies must have options that allow them to emit fewer carbon emissions, thus minimizing pollution to the environment, taking care of their economy at the same time.

Regarding the issues of shortages and imperfect quality, they seemed to be of great relevance in conjunction with sustainability since they are scenarios that occur in real life all the time, so it is useful that there are inventory models that cover these types of situations.

With the models made in this research, companies will be able to incorporate environmental and social criteria into their policies and make better decisions based on them.

1.3 Problem statement and context

The increasing concentrations of greenhouse gas emissions in the atmosphere, which mainly result from human activities such as industry, are one of the major causes of global warming. It is one of the main challenges that our society faces because it represents a serious global threat to our future.

According to the GHG Protocol Organization (2016), greenhouse gas emissions (GHG) from the industrial and transportation sectors can be direct and indirect. The direct ones are emissions from activities carried out by companies, such as production processes,

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and the use of generators, vehicles, among other equipment (i.e. forklifts, etc.). For its part, indirect emissions come from external sources to the companies, such as electrical energy services generated by the use of electricity, heat, refrigeration, steam, among others.

Everything is polluting within the supply chain, responsible for moving inventories, from the trucks used to transport products to the forklifts within companies, among other things, since there is contamination in every system that moves products. According to Intergovernmental (2015), 45% of energy consumption for transportation comes from goods, generating 5% of global CO₂ emissions, as stated by the World Economic Forum (2019). However, as Mangan et al. (2016) affirm, for modern manufacturing industries, the transport of goods is vital, then the activity of shipping products turns out to be a significant challenge for the supply chain, Sarkar et al. (2016).

Therefore, in addition to traditional inventory costs (setup, ordering, purchasing, and holding), the supply chain incurs in the costs of CO₂ emissions, energy consumption, process quality, and transportation operations. New regulations and laws on carbon emissions try to reduce the environmental impact generated during industrial activities, and companies are responding in two ways. On the one hand, they tend to adopt more energy-efficient equipment, facilities, or vehicles. On the other hand, they can also optimize their operating decisions in production, transportation, and inventory to reduce their carbon emissions. Since the second approach can reduce more carbon emissions at less or no cost than the first, there is a clear need for optimization inventory models to help companies make better decisions in some respects.

There exists a research related to an inventory model which considers environmental, energy and imperfect process issues but it does not investigate the inclusion of a policy of allowing shortages with partial backordering and lost sales. The aim is to determine if a policy that permits shortages with partial backordering and lost sales is less costly than a policy that avoids shortages. In this direction, this research proposes a mathematical formulation that models a sustainable inventory model for the supply chain considering carbon emission, energy usage, imperfect process and shortages with partial backordering and lost sales for a single product within an infinite time horizon.

With the use of it, companies are able to make better decisions, which affect the environment to a lesser extent, and reduce their costs by optimizing the supply chain and the use of inventories.

1.4 Research questions

The following research questions, which apply to all four inventory models are formulated:

• What is the optimal buyer’s order quantity (q )?

• What is the optimal buyer’s backordering quantity (B)?

• What is the optimal number of shipments of size q in a cycle (

n

)?

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• What is the optimal buyer’s backordering quantity satisfied (B )? _q

• What is the optimal buyer’s lost sales quantity (B )? _LS

• What is the vendor's optimal production rate (P)?

• What is the vendor’s optimal production quantity (Q )? _v

• Does a policy that permits shortages with partial backordering and lost sales is less costly than a policy that avoids shortages?

1.5 Solution overview

The contributions of various authors regarding sustainable inventory models have complemented the inventory management literature in recent years. The management of working capital is a significant element to achieve the highest possible efficiency in organizations that requires finding an optimal level of inventories to minimize total costs, and in this case, being ecofriendly.

The use of different mathematical models is necessary to support decision-making and guarantee good functioning in the organization, as well as better productive results and success in the outcomes of the management.

For these reasons, the present research aims to develop different inventory models that take into account environmental issues, as well as other real-life questions such as shortages and faulty quality, allowing decision-makers to choose the option that best suits their company.

1.6 Main contributions

The main contributions of this research work are listed as follows: initially, for the preparation of chapter two, a literature review is carried out regarding the history of inventory models that considered the following topics: environment, environment with energy, energy, backordering, and quality. Later, in chapter three, a sustainable inventory model for the supply chain with partial backordering, carbon emissions, energy usage and imperfect process is developed. The mathematical formulation of the inventory model

and the algorithm are shown in the investigation. The algorithm is coded in Excel.

The numerical results obtained are validate with LINGO 17.0 software with the objective of verifying the truthfulness and effectiveness of the equations elaborated. In chapter four, a numerical example and a sensitivity analysis are performed to show the variations in the optimal values of the decision variables and show in this way how significant and sensitive each decision variable is when a parameter changes. Finally, chapter five describes the conclusions and possible future work, which is useful to give ideas in case someone wants to continue with the investigation.

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1.7 Dissertation organization

The research is organized as follows: first, in chapter two a review of the literature is carried out with emphasis on articles related to the investigation. Chapter three is divided into sections that specify the assumptions and notation used for the formulation of the sustainable inventory model and describe the development and solution procedure for the proposed inventory model. Chapter four solves a numerical example, as well as, a sensitivity analysis carried out in the latter. Finally, chapter five shows the conclusions of

the research as well as recommendations for future continuity research work.

The research also includes an Appendix section with several appendices. The Appendix I contains the acronyms used throughout this research work. The Appendix II to Appendix XII presents the detailed solution procedures for some mathematical expressions.

The Appendix XIII to Appendix XVI provides the LINGO programs for the four inventory models.

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Chapter 2 Literature review

The present chapter classifies the literature review into five categories, all of them presenting inventory models or studies thereof. The first category has studies on the subject of the environment. The second part presents a combination of papers on the environment with energy. The third part expresses articles solely on energy. The fourth part describes investigations with shortages with backordering. Finally, the fifth part contains researches related to quality. These categories are chosen because they are all used during this investigation and for the realization of the inventory models developed in chapter three. The articles on inventory models explained below were carried out by different authors during the years 2015 to 2020.

2.1 Environment

With growing environmental awareness, large companies are beginning to adopt sustainable policies and cultures. Carbon emissions are generated, among other things, during production, storage, and transport, and this has produced significant environmental problems, including climate change caused by the increase in greenhouse gases, this being a dangerous threat to the planet. For these reasons, it is essential to have inventory models that consider carbon emissions in their formulation, that support actions to mitigate climate change, and that encourage companies to modify their activities while remaining efficient but also being friendly with the environment. At the same time of seeking economic sustainability, these models help to find sustainability in business and the economy, which together is one of the main goals of today's companies.

Next, some of the articles with inventory models that take into account the environmental issue are explained in conjunction with the following topics: reverse logistics, tiered inventory model, incentives and penalties, imperfect quality, shortages and late payments, as well as environmental policies, respectively.

Reverse logistic is the same process as in traditional logistics, but in reverse, that is, it returns the product from the customer to the distributor or supplier. In this way, it generates a closer relationship among the company, the distributors, and the customers, also reducing the environmental impact and even costs. Badenes (2015) defined it as "a process of moving goods from their typical final destination, to recover value, ensuring their correct disposal, or as a simple marketing tool." Some authors included this topic in the realization of inventory models. Jayaram and Avittathur (2015) conducted a study that supports a model with emissions policies and sustainability strategies and found that reverse logistics, product recovery, as well as eco-design, are key points for strategies on ecological sustainable supply chain management. One year after discovering that reverse logistics is a reliable strategy to take care of the environment, Bazan et al. (2016) reviewed the mathematics of the inventory models with reverse logistics, paying special attention to environmental issues, and the need to model social and environmental

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problems in the inventory models of reverse logistics, to make them more sustainable and similar to reality.

The tiered inventory model is an effective way to manage the system. There are two tiered, three tiered, and multi-tiered inventory models. Hammami et al. (2015) analyzed the effect of delivery time constraints and individual and global carbon emission caps in the supply chain on total emissions in a multi-tiered inventory model. Next year, Sarkar et al. (2016) studied the effects of carbon emissions generated during transportation in a three-tier supply chain model. After that, Hariga et al. (2017) created models of economic and environmental minimization for a multi-stage cold supply chain. More recently, Huang et al. (2020) carried out an integrated inventory model with carbon emissions in a two-tier supply chain and investigated the effects that green technologies and different carbon policies can have on them; they focused on the following three carbon policies: cap-and- trade, carbon taxes, and limited total carbon emissions.

For companies to emit the least amount of carbon emissions to the environment, the government gives incentives to industries that are more environmentally friendly, and otherwise, it makes penalties, such as the carbon tax, to those that, on the contrary, pollute more than is allowed. As this is a common and important topic, several authors made inventory models taking it into account. Bazan et al. (2017) carried out a closed- loop supply chain model, considering, among other things, three important environmental issues: the energy used in manufacturing and remanufacturing, the GHG emissions from transport and production subject to penalties, and the number of times an item is remanufactured to get it back. In the same year, Wangsa (2017) carried out an economic model that considers carbon emissions from the industrial and transportation sectors and divided them into two types, direct and indirect emissions. The model respected government penalties and incentives policies for industries to reduce their emissions.

Next year, Micheli and Mantella (2018) proposed a model to inform companies about the different formats to operate optimally in the face of a heterogeneous fleet, uncertainty in demand, a carbon tax, and different policies seeking to optimize costs and reduce the impact on the environment.

In the industry, it is usual to see problems such as the following: sustainability issues, payment delays, defective quality, shortages, and arrears. For this reason, different authors analyzed and made models about these themes. Rajeev et al. (2017) carried out an analysis of the evolution of sustainability problems in supply chain management (SSCM) in industries by studying 190 articles on this topic. In the next year, Aljazzar et al. (2018) analyzed the impact of the strategy of a buyer delaying payments after receiving the items, to optimize the supply chain and reduce its carbon emissions. In their article, Tiwari et al. (2018) developed a sustainable inventory model for deteriorated articles with an imperfect quality considering the carbon emissions resulting from the storage, transport, and conservation of deteriorated articles. Taleizadeh et al. (2018) developed a

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sustainable inventory model with nonconforming items, shortages and backordering, considering carbon emissions.

On the other hand, some authors thought of decision-makers and regulatory policymakers and carried out inventory modes on this. Bouchery et al. (2017) made a model they called sustainable order quantity, which helps decision-making by providing information on regulatory policies that control carbon emissions, as well as their effectiveness and cost.

Similarly, but more recently, Wangsa et al. (2020) seek to provide the information necessary with their model, for those responsible for formulating policies to decide the quantity and frequency of delivery of the product, and the appropriate inventory level to minimize both the inventory and carbon emissions total costs.

2.2 Environment and energy

Electric energy is necessary for the development of humanity; however, it has environmental impacts as a consequence. For these reasons, it is vitally important that companies use the least amount possible. There are inventory models that help companies make this possible, some of them are explained below.

Several authors developed research on the subject of inventories, environment and energy. Some of these investigations are based on minimizing the energy used in the industry under different scenarios, on finding the optimal lot size, and finally other authors took into account the issue of vendor managed inventory (VMI) with consignment stock (CS) in set with energy and environment.

The authors Marchi, Bazan, among others, developed inventory models taking into account the energy used under different scenarios which are explained below. Bazan et al. (2015) carried out an inventory model to minimize the total cost, taking into account the energy used during the production and remanufacturing of the products, as well as the carbon emissions generated in the supply chain. Three years later, Marchi et al.

(2018) built an inventory model of the supply chain to recover waste heat from the metal industry to reduce carbon emissions, energy costs, and minimize total costs. Recently, Marchi et al. (2020a) proposed an inventory model with uncertain demand to evaluate the best decision in energy efficiency investment considering two investment options that have a different impact on the specific energy consumption curve ( SEC ): investments in the production rate and investments in organizational improvements. In another article, Marchi et al. (2020b) formulated an inventory model that considers energy consumption in a cold supply chain.

Other authors investigated the optimal batch size that helps to use less energy, and accordingly, it does not affect the environment that much. Zavanella et al. (2019) proposed a model in which they considered energy in making decisions about lot size,

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showing a more sustainable approach by taking care of environmental concerns that are affected by energy use, and at the same time, reducing total costs. In the same way, Marchi et al. (2019) proposed a model that helps to make correct decisions about the size of the batch, as well as the manufacturing time that directly affects the number of defective products that require reprocessing, thus consuming more energy. The authors considered these aspects to improve the quality of processes, products, and energy and consequently save on inventory, production, and energy costs.

Consignment stock (CS) and vendor managed inventory (VMI) are two supply chain sourcing practices among customers and suppliers. VMI favors the provider, while CS favors the client. The companies that decide it, use the one that suits them best, as long as there is sufficient trust between the supplier and the customer. For its part, in a VMI- CS agreement, the buyer's inventory remains the property of the supplier until it is withdrawn by the final demand of the client, which is useful in uncertain scenarios and with varying demands. The authors Marchi, Bazan, among others, investigated inventory models with a type of commercial practice, in conjunction with the care of environmental emissions. For example, Bazan et al. (2015) carried out two models: one with a classic policy and one with VMI-CS, considering the carbon emissions generated in transport and production, and electrical energy for production, in a system of a single manufacturer and buyer, with taxes for excess emissions, seeking to reduce energy usage which is one of the highest costs of both models. More recently, Marchi et al. (2019) introduced an inventory model for the two-tier supply chain (buyer-supplier) considering energy use, carbon emissions, and an imperfect production system, taking into account the classical inventory and the VMI-CS inventory.

2.3 Energy

Energy is necessary to have a better quality of life. It is used in many human activities, being the industry one of those that require the most amount. Nevertheless, currently, due to high energy prices, as well as policies on the efficient use of it and environmental restrictions, it is becoming increasingly important to reduce energy consumption, and therefore, the costs related to it in the industries.

Energy is used during most of the steps involved in the supply chain. For these reasons, all the authors listed below conducted researches on inventory modeling with energy issue in mind.

During the years 2016 to 2018, different authors made inventory models intending to reduce the electrical energy used in the supply chain. McBrien et al. (2016) conducted a thermal analysis on how to save energy in the supply chain of steel products through heat recovery. The next year, Xie et al. (2017) analyzed the decisions on the price of the shipbuilding and the energy efficiency of main engines (EEME) by making optimal

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decisions in the supply chain to improve sustainability. In the same year, Marchi and Zanoni (2017) presented a review of articles that talk about taking energy efficiency into account within the design of the supply chain, to improve energy performance by creating more energy-efficient processes, as well as improving competitiveness, quality, profitability, and saving costs to the company. One year after those researches, the authors Marchi et al. (2018) constructed a two-stage supply chain model, intending to give entrepreneurs the necessary information so that they can make better decisions, improving the energy performance of the entire supply chain at the lowest cost. Investing in energy efficiency leads to huge savings in energy-related cost through further improvement in energy performance.

2.4 Backordering

Diligent managers want to avoid inventory shortages as much as possible, but only to the extent that this is still feasible, and seeking a benefit more considerable than the possible costs. However, this is sometimes almost impossible because the items are exposed to shortages and pending orders, so it is necessary to assume these losses, which also, in the end, most of the time they do not turn out to have a great impact. Stock shortages can occur for different reasons such as errors in processes, carelessness in preparing and receiving items, administrative errors, loss of goods, accidents, and other special cases such as theft, own consumption, among others. Since this is a common theme in the industry, some of the researches on inventory models with shortage are described below.

The following investigations describe some inventory models with shortages and planned backorders in conjunction with carbon emissions control policies, defective items, VMI policies, spoilage rates, and other issues.

The shortages occur when a customer request to buy an item and there are not products on storage. Consequently, the demand cannot be satisfied. The shortages can be covered afterwards if and only if the customer agrees. Then these shortages become as pending orders to satisfy which are called as planned backordering. An inventory model with a shortage with planned backorders is useful when the unit value of inventory is very high. Paknejad (2015) developed three economic order quantity (EOQ) inventory models with planned backorders, taking into account performance improvement programs to evaluate their impact, determining the best program according to different scenarios, and giving guidance when deciding which one of them to apply. In the same way, Cenk Çalışkan (2021) carried out a simple solution model to find the optimal quantity, using the economic order quantity (EOQ) inventory model for damaged items with planned pending orders, considering that the articles deteriorate exponentially and in proportion to the inventory level, which reduces in the same way over time.

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On the other hand, sometimes shortages happen unplanned, so it is essential to have inventory models for these cases. The authors Zhou et al. (2016) developed an economic order quantity (EOQ) inventory model with shortages, low quality, inspection errors, and commercial credit, analyzing the effect of these four factors, both combined and independently, on inventory policies.

Planned backorders are used very frequently within the industry. It is the process of selling inventory that the company does not have a disposition at that time but knows that it is possible to do it later. Because it is something very habitual, several authors decided to make inventory models considering this issue; however, each of them also researched jointly other types of aspects as described below. Kazemi et al. (2015) derived a fuzzy (EOQ) inventory model with backorders and a long-term learning curve in the inventory planning horizon to reduce the total cost of the same and provide a more applicable model to real life. Next year, Taleizadeh et al (2016) created an EOQ inventory model with repairable imperfect products and pending orders. In 2017, Aslani et al. (2017) introduced an inventory model of the economic order quantity (EOQ) with partial delays in the orders, seeking to improve the mean and the variation of the performance of the process through two investment strategies. Kim and Sarkar (2017) carried out a stochastic inventory model on the production system to improve quality by reducing defective items to almost zero and improving the costs of pending orders by giving discounts on their prices to reduce the cost of lost sales, and in turn the total cost. In the same year, Diabat et al. (2017) proposed three inventory control models with different scenarios, including full and partial backorders, descend partially deferred payments, and upward partial advance payments, as well as impaired items. One year later, Kim et al. (2018) developed an inventory model with pending orders to calculate the number of defective items in long-term production, minimizing the costs of the supplier-buyer system. There is also very recent research on this topic. For example, Saha et al. (2020) made an inventory model considering substitute products, customer preference, and pending orders to minimize the total cost of the supply chain. Makinde and Munyai (2020) performed an economic order quantity (EOQ) inventory model that accurately estimates the penalty cost resulting from an organization's pending orders. Taleizadeh et al. (2020) developed two optimization models with partial backorders and base stock to make adequate replenishment decisions, intending to minimize the total costs of withholdings, orders, and shortages in the long term.

Since environmental issues are essential nowadays, some authors made inventory models with shortages considering the carbon emissions generated in the supply chain.

Nia et al. (2015) developed an economic order quantity (EOQ) inventory model of multiple elements and restrictions for items in short supply under a green supply chain and the VMI inventory policy of a supplier and a buyer to minimize the total costs. Three years later, Taleizadeh et al. (2018) developed four inventory models with shortages, full and partial backorders, and sale, considering emissions to the environment and economic issues. The newest research is done by Mishra et al. (2021), who built an inventory model

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with backorders and carbon emission levels in mind to achieve a green supply chain, using two types of price dependent on demands for spoilage rates and controllable carbon.

2.5 Quality

Quality is a set of characteristics of a product or service that satisfies the needs of the client adding value to them. Currently, all companies seek to have good quality in their products, to become competitive. However, there are occasions when the desired quality percentage cannot be achieved.

Due to what is mentioned in the previous paragraph, many authors decided to address this issue and make inventory models that take it into account. Some carried out inventory models with imperfect production and/or inspection, deterioration rates, VMI and consignment stock, closed-cycle supply chains (CLSC), marketing, agreements with suppliers, diffuse logistics, single-stage processes, articles growing, imperfect rates, among other issues.

Quality inspection is a routine and even mandatory in production processes, so some authors decided to include it in their inventory models, and others even took into account the errors that occurred within it. For instance, Al-Salamah (2016) developed an economic production quantity (EPQ) model for imperfect inspection and production processes to find the optimal lot size to produce to maximize the objective function. He considered the use of non-destructive and destructive sampling processes to evaluate the quality of the batches, taking into account possible classification errors. Four years later, Guha and Bose (2020) made a note to make a correction of Al-Salamah (2016)’s research article and proposed two modifications for it. The first is to correct the inventory calculations since unlike what he mentioned in the process he did not take into account the defective articles that they sent to the primary market. The second correction is in the accounting of defective items from non-destructive tests since it would change when correcting the error mentioned above. There are other investigations that include inspection, such as the one of Dey and Giri (2019) who carried out a supplier-buyer inventory model with imperfect production, defective items, and with a curve for inspection errors.

Subsequently, Khakzad and Gholamian (2020) also included the inspection process in their research. They formulated an inventory model taking into account the deterioration rate of the articles and the effect of the deteriorated articles on the other articles seeking to reduce this with inspections to minimize the total number of decayed items and decrease total costs.

An appropriate final quality is a requirement for the clients, so it is prime to identify defective products and remove them, as well as classify the deliverables products according to their status. Rezaei (2016) created an economic order quantity (EOQ)

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inventory model in which he considered defective items to separate them from the perfect ones, thereby maximizing the benefit to the buyer. Similarly, but adding recycling, Khara et al. (2020) proposed an inventory model taking into account an imperfect production system, using virgin and recycled raw materials (from manufactured products), obtaining some defective items, and with different quality levels according to their raw materials.

They considered the rate of return of the articles according to their quality and separated them conforming to this.

Some companies decide to implement a business scenario called consignment stock, which is a trade agreement based on trust, from which both the consignor and the consignee are benefited. Therefore, the authors mentioned below decided to investigate inventory models that include this aspect. The authors also included the topics explained in the previous paragraphs: the inspection processes and the quality level of the items, respectively. Khan et al. (2016) developed a model with vendor-managed-inventory (VMI) and consignment stock (CS) considering defective items and inspection for a supply chain of a single seller and buyer. Later, Taleizadeh and Moshtagh (2019) conducted a similar investigation and carried out an inventory model considering a multi-level closed-cycle supply chain (CLSC) and consignment stock, with imperfect manufacturing and remanufacturing process, as well as returns and lost sales according to the quality level of the items.

Some authors also considered external factors, taking into account marketing, the laws of supply and demand, and contemplating convenient agreements and discounts by their suppliers. Such are the cases of Zhou et al. (2015), who developed economic production quantity (EPQ) models considering items of imperfect quality in both their production and purchased products to evaluate the optimal decisions to buy or produce when the original equipment manufacturers (OEM's) offer unique discounts. In the same way, two years later, Manna et al. (2017) performed an EPQ model too, with imperfect production, defective items, and an advertising-dependent demand rate.

Some other authors decided to elaborate inventory models taking into account a fuzzy logic to make them more adaptable to the real world. Kumar and Goswami (2015) developed a production inventory model with fuzzy randomness, shortages, and items with imperfect quality. More recently, De et al. (2020) developed an economic production quantity (EPQ) model under an uncertain environment and a diffuse approach, considering processes of re-manufacturing and separation of imperfect items and partial delays to minimize costs.

In the industry, there are different types of production processes, among which are, the single-stage process. Paul et al. (2015) developed an inventory model for a single-stage imperfect production process with potential disruptions and defective products to maximize total profits after disruptions. Years later, Sarkar et al. (2020) also made an

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inventory model with an imperfect single-stage production system, adding random defective rates, remanufacturing, and multiple products, and planned backorders to make cleaner production and minimizing costs.

On the other hand, some authors focused on different types of topics. Sebatjane and Adetunji (2019) developed an economic order quantity inventory model with growing items and imperfect quality. By his side, and in the same year, Al-Salamah (2019) created an economic production quantity inventory model with an imperfect manufacturing process, defective items, and flexible rework rates to determine optimal backorders and batch sizes for processing and reworking rates.

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Chapter 3 Development of a sustainable inventory model with partial backordering

3.1 Introduction

Since the creation of the economic order quantity (EOQ) inventory model by Ford W.

Harris in 1913, as well as the lot economic production better known as economic production quantity (EPQ) inventory model developed by E.W. Taft five years later, in 1918, numerous researchers have built endless variations and extensions of these models, making them increasingly adaptable to the current reality in the industries.

For the realization of the inventory models in this chapter, the inventory model without VMI policy of the article called “Supply chain models with greenhouse gas emissions, energy usage, imperfect process under different coordination decisions” is taken as a basis. This article was published on January 25, 2019, by the authors B. Marchi, S.

Zanoni, L.E. Zavanella, and M.Y. Jaber, therefore, it is a recent investigation that can be further developed.

The inventory models proposed in this thesis contain the base model of Marchi et al.

(2019) removing the VMI policy but allowing shortages with planned backorders. Because shortages frequently occur in all companies, these inventory models are a little more complete than those without shortages, providing an advance to the current literature.

This chapter presents the development of a sustainable inventory model for the supply chain with partial backordering, lost sales, carbon emissions, energy usage and imperfect process. Three inventory models are obtained from the proposed inventory model. In total four inventory models are developed, named as: (1) a sustainable inventory model with process quality and environmental costs, (2) a sustainable inventory model with environmental costs, (3) an inventory model with process quality costs, and (4) an inventory model with traditional inventory costs.

3.2 Problem definition

This research proposes four supply chain inventory models with shortages and items of imperfect quality for a single product and with an infinite time horizon. Two of these inventory models are sustainable, which means they consider carbon emissions to the environment, while the other two contemplate inventory and quality, respectively.

The supply chain that is addressed in this research is depicted in Figure 1. The supply chain is composed of one vendor and one buyer. There is a single product that is being sold by the vendor to the buyer. The supply chain causes carbon emissions. The vendor’s production system has machine breakdowns in a random way and therefore generates defective items. The vendor manufactures a lot size of Q units within a sole run per _v production cycle. Immediately, the vendor inspects the lot Q_v in a 100% inspection

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process with the aim of identifying the defective items and then reworks these. Then, the vendor transports the lot Q to the buyer in n shipments of equal size of _v qunits. The shortages ( B ) are permitted with partial backordering (B_q) and lost sales (B ). The _LS shortages occur at the buyer side.

Pand Q _v n q, B ,B_q, andB_LS

Figure 1. Schematic representation of the supply chain

The estimated total cost of the inventory model for the supply chain is comprised of inventory, quality, and environmental costs. It is calculated as the sum of the following costs: total cost of the vendor (TC_V ), total cost of the buyer (TC_B), the costs incurred during transportation (FC+TEmC), and the penalty cost for CO₂emissions into the environment (PC).

The four inventory models seek to solve the following questions:

• What is the optimal buyer’s order quantity ( q )?

• What is the optimal buyer’s backordering quantity ( B )?

• What is the optimal number of shipments of size q in a cycle (n )?

• What is the optimal buyer’s backordering quantity satisfied (B_q)?

• What is the optimal buyer’s lost sales quantity (B )? _LS

• What is the vendor's optimal production rate (P)?

• What is the vendor’s optimal production quantity (Q )? _v

• Does a policy that permits shortages with partial backordering and lost sales is less costly than a policy that avoids shortages?

The objective function of the inventory models seeks to minimize total costs, taking into consideration the following decision variables:

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q : Buyer’s order quantity in units

B : Buyer’s backordering quantity in units

n : Number of shipments of size qin a cycle; it is an integer number P: Vendor's production rate in units per unit of time

Additionally, the following dependent decisions variables are determined with the values of q , n and B :

Bq: Buyer’s backordering quantity satisfied in units B : Buyer’s lost sales quantity in units LS

Q : Vendor’s production quantity in units v

The steps for the development of the sustainable inventory model for the supply chain with partial backordering, process quality, and environmental costs are as follows:

1. Establish the assumptions for the inventory model.

2. Define the mathematical notation.

3. Build the inventory model.

4. Optimize the inventory model.

5. Develop the algorithm for finding the optimal solution.

6. Interpret the solution.

7. Perform a sensitivity analysis.

8. Interpret the sensitivity analysis 9. Provide managerial insights

The decision-makers of different companies can choose the inventory model that best suits the budget and the company's values and beliefs. These inventory models help to managers to take the best decisions and guarantee a satisfactory function in the organization since these correspond to the optimal inventory policy that total costs are minimized. The inventory models consider real-life questions, helping to obtain a better productive outcome and a successfully management in the organization.

3.3 Assumptions

For the development of the inventory models effectively, it is necessary to have a set of assumptions that govern the environment in which it is carried out. The suppositions under which the inventory models of this research are developed are described below.

• The demand ratio is constant and known.

• The production ratio is constant and known.

• The production ratio is greater than the demand ratio.

• The planning horizon is infinite.

• The inflation rate is not considered.

• A single product is considered.

• Shortages are allowed with partial backordering at buyer side.

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• The number of shortages permitted must be less than or equal to the order quantity.

3.4 Notation

This research work uses the same symbols as in Marchi et al. (2019) in order to have a standard notation. Additionally, new symbols are incorporated. Thus, the notation is given below.

Symbol Description Parameters:

A Ordering cost ($/order)

M Corrective maintenance cost to restart the production system after a breakdown ($/breakdown)

S Setup cost ($/setup)

cec Emission tax ($/unit of weight)

,

cep i Emission penalty for surpassing emission boundi($/unit of time)

cs Inspection cost ($/unit) r Reworking cost ($/unit)

ˆ Backordering cost ($/unit/unit of time)

 Constant backordering cost ($/unit)

ˆ Good will cost ($/unit/unit of time )

 Constant good will cost ($/unit)

hb Holding cost at the buyer's side ($/unit/unit of time)

hv Holding cost at the vendor’s side ($/unit/unit of time)

cen Energy cost for the vendor ($/measure of energy)

W Total power utilized by the production system (measure of energy)

Wr Total power used by the reworking process (measure of energy)

W0, p Idle power for the production process in the ready position, related to equipment features required to support the process (measure of energy)

W0,r Idle power for the reworking process in the ready position, related to equipment features required to support the process (measure of energy)

k Constant to establish the variable part of the power utilized by the production system (measure of energy /unit)

'

k Constant to establish the variable part of the power utilized by the reworking system (measure of energy/unit)

SEC Specific energy consumption to manufacture a unit at the vendor side (measure of energy/unit)

SECr Specific energy consumption to rework a unit at the vendor side (measure of energy/unit)

et Quantity of GHG emissions from one gallon of diesel-truck fuel (unit of weight/amount of liquid)

E Greenhouse gas (CO₂) emissions from the production system (unit of weight/unit)

Er Greenhouse gas (CO₂) emissions from the reworking system (unit of weight/unit)

Eli Emissions boundi(unit of weight/unit of time)

Etr Quantity of emissions produced per period of time during transportation (unit of weight/unit of time)

D Demand rate (units/unit of time)

Pmin Minimum production rate (units/unit of time)

Pmax Maximum production rate (units/unit of time)

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3.5 A sustainable inventory model with quality and environmental cost

In this section, a sustainable inventory model is developed considering shortages with

partial backordering, carbon emissions, energy usage, and imperfect processes.

The following subsections are made to separate the equations according to their theme:

the process of quality modeling, energy specifications, and carbon emissions. It is important to remark that this research is revisiting and extending the inventory model of Marchi et al. (2019). So, the mathematical modelling is similar only for Equation (1) to Equation (17).

Pr Vendor's reworking rate (units/unit of time)

tc Truck capacity (units/mode of transport)

g Amount of gallons per truck per distance travelled (amount of liquid/mode of transport)

a Emission function parameter for the production system (unit of weight ·unit of time²/unit³)

ar Emission function parameter for the reworking system (unit of weight ·unit of time²/unit³)

b Emission function parameter for the production system (unit of weight ·unit of time/unit²)

br Emission function parameter for the reworking system (unit of weight ·unit of time/unit²)

c Emission function parameter for the production system (unit of weight/unit)

cr Emission function parameter for the reworking system (unit of weight/unit)



Percent of defective items manufactured once the production system is in the "out-of-control" state (0 1)



Non-negative parameter to estimate the quality function

 Parameter to establish the relation between the manufacturing rate and the reworking rate

 Exponential parameter for the reliability function



Non-negative parameter to estimate the quality function

 Quantity of trucks per trip

m Amount of emissions limits tp Production time (unit of time)

tr Reworking time (unit of time)

 Portion of demand that is partial backordered

(

⁰^{ }^ ¹

)

Decision variables:

q Buyer’s order quantity (units)

B Buyer’s backordering quantity (units)

n Number of shipments of sizeqin a cycle

P Vendor's production rate (units/unit of time) Dependent decision variables:

Qv Vendor’s production quantity (units), Q_v=nq

Bq Buyer’s backordering quantity satisfied (units), B_q =B BLS Buyer’s lost sales quantity (units), B_LS = −(1 )B

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3.5.1 The process of quality modeling

When a machine begins the production run of a lot for a product, it starts out producing good quality items, but after some time defective items starts to appear. The expected number of defectives in a production lot size

nq

, which was calculated by Khouja and Mehrez (1994), is given below:

( )2

( ) ( ) 2 E N f P nq

 P

= (1)

The detailed derivation of Equation (1) is presented in Appendix II.

Where ^{1 / f P}

( )

denotes the mean time to shift to an out-of-control condition and it is given by:

2

1 1

( )

f P ⁼  + P (2)

The quality in the production process is also affected by the possibility that the machine has breakdowns. The probability that a malfunction in the machine happens during the production time t_p is ^{F t}

( )

^p which represents the cumulative density function of f t( ), and

t

is a random variable that denotes the time to failure. This research considers accidental or casual failures to happen during the lifetime of the machine.

( ) 1 ( )

f t = − R t

₍₃₎

The reliability function of the system

(

^{R t}

( ) )

at a given time is given by:

( )

^t

R t = e

⁻^ ₍₄₎

where



is the machine failure rate, therefore, the probability of a machine breakdown is computed as follows:

/

0

( )

nq P p

F t dt nq

  P

=



=

(5) The detailed derivation of Equation (5) is given in Appendix III.

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3.5.2 Energy specifications

In industrial production processes, immense amounts of electricity are required.

The authors Gutowski et al. (2006) demonstrated with their research that energy consumption is a function of the speed or rate of production. In addition to this, they proved that approximately 15% of the energy used is inversely proportional to the speed with which the machine run. Additional energy is used in the starting functions of the engine, such as centrifugal energy, refrigerant, oil, among others. These operations are independent of the rate of production. The explained above is expressed mathematically below.

The equation for calculating the total power utilized by the production process is then defined as:

W =W0, p +kP

(6) where k is a constant (measure of energy/unit) that comes from the physics of the industrial production process.

Taking Equation (6) as a basis, the equation to compute the specific energy consumption ( SEC )is determined. Hence, the specific energy consumption of a product processed is calculated as follows:

(W0,_p kP t) _p

SEC nq

= + (7)

The detailed derivation of Equation (7) is shown in Appendix IV.

where:

p

t nq

= P (8)

Reworking defective items needs additional electrical energy to that already used in the process. It is given as:

0, '

r r r

W =W +k P (9)

where 'k is a constant (measure of energy/unit) that comes from the physics behind the reworking process. Here, it is considered that the reworking rate is a function of the production rate, written as:

Pr =



P (10)

Using Equation (9) and Equation (10) the speciﬁc energy consumption per unit of reworked product is obtained as follows:

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( 0. ' ) ( )

r r

r

W k P t

SEC E N



= + (11)

The detailed derivation of Equation (11) is exposed in Appendix V.

where:

( )

r

t E N



P

=

(12)

3.5.3 Carbon emissions

The quantities of GHG (greenhouse gas emission) generated during the production and rework processes by the vendor, are represented as a function of the production rate, as stated by Bogaschewsky (1995). According to Bazan et al. (2015b) these are defined respectively as:

E = aP

2

− bP c +

(13)

2

r r r r r r

E = a P − b P c +

(14)

According to Bazan et al. (2015a) the quantity of carbon dioxide (CO )₂ emissions that is caused by the movements trucks is represented as follows:

tr t

E n D ge



nq

=

(15) The detailed derivation of Equation (15) is provided in Appendix VI.

where:

c

q

= t (16)

3.6 Sustainable inventory model with process quality and environmental costs The total cost of the supply chain (TC_SC) is calculated by the sum of the following costs:

total cost of the vendor (TC ), total cost of buyer (_V TC ), total fuel cost ( FC ), total cost of _B carbon emissions ( TEmC ) and the total penalty cost ( PC ) for exceeding the limit imposed by regulations.

An inventory model for the supply chain with partial backordering, carbon emissions, energy and imperfect process

Instituto Tecnológico y de Estudios Superiores de Monterrey

School of Engineering and Sciences

An inventory model for the supply chain with partial backordering, carbon emissions, energy and imperfect process

Valeria Martínez Villarreal

An inventory model for the supply chain with partial backordering, carbon emissions, energy and imperfect process

List of Figures

n

n

n

n

List of Tables

Contents

n

n

Chapter 1 Introduction and context of the research

n

Chapter 2 Literature review

Chapter 3 Development of a sustainable inventory model with partial backordering







(

)

nq

( )

( )

t

( ) 1 ( )

f t = − R t

(

( ) )

( )

R t = e











E = aP

− bP c +

E = a P − b P c +

