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This book discusses inventory models for determining optimal ordering policies using various optimization techniques, genetic algorithms, and data mining concepts. It also provides sensitivity analyses for the models’ robustness. It presents a collection of mathematical models that deal with real industry scenarios. All mathematical model solutions are provided with the help of various optimization techniques to determine optimal ordering policy.

The book offers a range of perspectives on the implementation of optimization techniques, inflation, trade credit financing, fuzzy systems, human error, learning in production, inspection, green supply chains, closed supply chains, reworks, game theory approaches, genetic algorithms, and data mining, as well as research on big data applications for inventory management and control. Starting from deterministic inventory models, the book moves towards advanced inventory models.

The content is divided into eight major sections: inventory control and management – inventory models with trade credit financing for imperfect quality items; environmental impact on ordering policies; impact of learning on the supply chain models; EOQ models considering warehousing; optimal ordering policies with data mining and PSO techniques; supply chain models in fuzzy environments; optimal production models for multi-items and multi-retailers; and a marketing model to understand buying behaviour. Given its scope, the book offers a valuable resource for practitioners, instructors, students and researchers alike. It also offers essential insights to help retailers/managers improve business functions and make more accurate and realistic decisions.

### Chapter 1. Economic Production Quantity (EPQ) Inventory Model for a Deteriorating Item with a Two-Level Trade Credit Policy and Allowable Shortages

Abstract
This research work derives an economic production quantity (EPQ) model, and in order to make it a bit close to reality, the stockout is allowed, and this is completely backordered. In addition to this feature, it is incorporated a two-level credit scheme when both supplier and retailer are giving a delay in payment to their respective customers with the aim of enhancing the sales. The inventory model is modeled as a constrained nonlinear optimization problem, and this is resolved by the generalized reduced gradient method (GRG). Moreover, to exemplify and certify the inventory model, five instances are given and solved. Finally, a sensitivity analysis is made for studying the influence of variations of input parameters, modifying one parameter and maintaining the others at their initial input values.
Ali Akbar Shaikh, Leopoldo Eduardo Cárdenas-Barrón, Sunil Tiwari

### Chapter 2. An Economic Order Quantity (EOQ) Inventory Model for a Deteriorating Item with Interval-Valued Inventory Costs, Price-Dependent Demand, Two-Level Credit Policy, and Shortages

Abstract
In today’s competitive environment, every leading organization wishes to improve the pricing strategies in order to increase revenue, credit policy is one of the best tools of it. This research work develops an economic order quantity (EOQ) inventory model for a deteriorating item that considers interval-valued inventory costs, price dependent demand, two-level credit policy, and shortages. Due to high and uncertainty in demand, sometimes organizations have to face the situation of stock out. So, keeping this scenario in mind, this work considers the situation of partially backlogging. Here, it is developed an EOQ inventory model by considering a non-linear interval-valued constrained optimization problem. Two types of particle swarm optimization (PSO) algorithm are used to resolve it, and then the results are compared. Sensitivity analysis is done in order to investigate the impact of key parameters on decision-making. Finally, conclusions along with some managerial insights are given.
Ali Akbar Shaikh, Sunil Tiwari, Leopoldo Eduardo Cárdenas-Barrón

### Chapter 3. Inventory Control Policies for Time-Dependent Deteriorating Item with Variable Demand and Two-Level Order Linked Trade Credit

Abstract
In today’s business world to boost the demand, vendor gives a trade credit to buyer. Moreover, most of the products lose quality over time due to environmental effects. This chapter studies an inventory policy for the item which has expiry date with two levels of trade credit depending on the quantity of order. It is considered that a supplier is ready to give a mutually agreed credit period to retailer only if the order quantity purchased by retailer is more than the predetermined quantity of order. Additionally, a retailer deals a credit limit to the end consumers. Here, time- and price-sensitive demand is debated with inflation. A retailer’s main objective is to earn maximum total profit with respect to the number of replenishments throughout the finite planning horizon. Results are supported by numerical examples. Finally, a sensitivity analysis is done to develop visions for decision-makers.
Mrudul Y. Jani, Nita H. Shah, Urmila Chaudhari

### Chapter 4. Inventory Modelling of Deteriorating Item and Preservation Technology with Advance Payment Scheme Under Quadratic Demand

Abstract
This chapter comprises a single retailer and single product which deteriorates continuously. For the time-dependent deteriorating item with seasonal demand, quadratic demand is debated here which is suitable for the items whose demand with starting of the season increases initially and after end of the season, it starts to decrease. To reduce deterioration of the product, retailer needs preservation technology and due to preservation technology retailer minimizes total cost. In this chapter, the retailer has to pay a fraction of the purchase cost before the time of delivery and rest of the payment must be paid at the time of delivery. In this chapter, the optimal number of equal instalments before receiving the order quantity, replenishment time and investment of preservation technology are the decision variables that minimize the total cost. This chapter is an extension of the earlier work, as it provides the best optimal rather than the nearest minimum solution. A numerical example is delivered to demonstrate the performance of the model and to highlight certain decision-making insights.
Urmila Chaudhari, Nita H. Shah, Mrudul Y. Jani

### Chapter 5. Dynamic Pricing, Advertisement Investment and Replenishment Model for Deteriorating Items

Abstract
In practice, it is commonly observed that the quality and price of items are two important factors for customers to choose a product. The profit of a firm is greatly affected by these two factors, especially when their inventory has deteriorating items. Also, it is commonly observed that the product demand increases due to promotional efforts like advertisement through digital media, newspaper, etc. Thus, the spending on commercial promotion is a very crucial decision. This paper considers a replenishment model for perishable items with investment on promotion and retail price-dependent demand with a budget constraint. The deterioration rate is considered constant. An optimization problem is formulated in order to provide a pricing, promotional spending and replenishment policy, which maximize the total profit. Using Pontryagin’s maximum principle, the optimal advertisement investment is obtained for a given retail price and cycle time. The closed form of the inventory level is obtained by solving the respective differential equation of inventory. The model is validated by a numerical example with hypothetical parameters in result section. The results show that the model is pretty stable and the concavity is proven graphically. The sensitivity analysis is performed in discussion section. The sensitivity analysis about key inventory parameters reveals some important managerial insights. Also, the future scope is given in conclusion section, which gives a brief idea about possible extensions of this model.
Chetansinh R. Vaghela, Nita H. Shah

### Chapter 6. A Production Reliable Model for Imperfect Items with Random Machine Breakdown Under Learning and Forgetting

Abstract
This chapter considers the impact of preservation technology on an “economic production quantity” model in which the production process may not only shift from an “in-control” state to an “out-of-control” state but also may fail at any random point in time during production run time. Model is developed for multi-items with imperfect quality by considering the situation of random machine failure over infinite planning horizon. Demand rate is assumed to be multivariate. A reliable and flexible production inventory system is considered under learning and forgetting environment. We studied model in both crisp and fuzzy environment, and significant features of the model are illustrated by numerical experiments. So, numerical examples along with sensitivity analysis are given to show how the solution procedure works as well as the usages of research results.
Preeti Jawla, S. R. Singh

### Chapter 7. Inventory Policies with Development Cost for Imperfect Production and Price-Stock Reliability-Dependent Demand

Abstract
This article focuses on developing a model based on inventory dealing with the product’s sell price-stock as well as reliability-dependent demand; also it undergoes a production process which is imperfect including manufacturing of perfect as well as imperfect quality products. As such, each production firm believes in the production of perfect quality goods but because of various uncontrollable barrier factors like machinery, labor, technology, and also due to the long-run process, the production, therefore, includes imperfect quality items along with perfect quality products. The products which are perfect are ready to sell out; on the other hand, the imperfect products undergo the reworking process owing a cost to become a perfect product. By inclusion of the cost of development also by modifying the raw material quality of production system, several considerations like product’s reliability, the system’s reliability parameter, and the reworking cost can be upgraded. The aim of this article is to calculate the firm’s total profit along with the estimation of optimal values of production duration such that a manufacturer gets a maximum profit, manufacturing system’s reliability parameter, and product’s reliability. The classical optimization technique is utilized for calculating the optimal values. For the validation of developed models, numerical examples are demonstrated; then using the concept of eigenvalues of a Hessian matrix, we have proved the concave nature of the profit function of the system, and also the sensitivity analysis is done for each decision variables by fluctuating the inventory parameters for generating effective managerial insights.
Nita H. Shah, Monika K. Naik

### Chapter 8. Imperfect Quality Item Inventory Models Considering Carbon Emissions

Abstract
Research on inventory models with environmental consideration has recently become a popular research stream. The amount of energy consumption and greenhouse gas emissions is influenced by inventory decisions such as delivery quantity and delivery frequencies. This chapter focuses on a supply chain system which contains a percentage of imperfect quality items in its delivered lot; we also consider carbon emission costs under a carbon tax policy. Processing the defective items, which increases carbon emission, affects supply chain decisions. We present two economic order quantity models considering carbon emission and defective items with different shortage conditions. We then study low-carbon two-echelon supply chain inventory model considering supply chain integration and imperfect quality items. Numerical examples are provided to illustrate how these models can be applied in practice. Sensitivity analysis is performed to gain more insight on changing parameters in the numerical studies.
Hui-Ming Wee, Yosef Daryanto

### Chapter 9. Non-instantaneous Deteriorating Model for Stock-Dependent Demand with Time-Varying Holding Cost and Random Decay Start Time

Abstract
In this study, we have considered an inventory model of non-instantaneous deteriorating items with stock-dependent demand. Shortages are allowed and fully backlogged. Holding cost is not always fixed; it may depend occasionally on time. That is why we have considered holding cost as constant as well as time-dependent in the model. Also, the effect of decay start time has been considered and they are random. We categorize the model into three cases. In the Cases I and II, we consider fixed decay start time with constant and time-varying holding cost, respectively. The random decay start time has been considered in the last case. Mathematical models have been derived to determine optimal-order quantity that minimizes the total cost. Optimal solution has been illustrated with numerical examples and along with that sensitivity analysis of parameters has been carried out.
Nirmal Kumar Duari, Jobin George Varghese

### Chapter 10. Stock-Dependent Inventory Model for Imperfect Items Under Permissible Delay in Payments

Abstract
In the production process, the issue of quality is always ignored which results in defective production. These defective items can be removed from the lot through the inspection process which becomes essential for the system. Demand is considered as stock dependent. It is continuously declined to meet the customer’s demand which depends on the on-hand inventory up to the time t2. After that the inventory level declines by constant demand up to time t3. Thereafter, shortages occur and it accumulates at the rate $$\psi \left( {\tau - t} \right)\,till\,t = \tau$$ when the next batch arrives. The whole cycle repeats itself after the cycle length τ. Further, it is assumed that payment will be made to the supplier for the goods immediately after receiving the consignment. Whereas, in practice, supplier does offer a certain fixed period to the retailer for settling the account. During this period, supplier charges no interest, but beyond this period interest is being charged. On the other hand, retailer can earn interest on the revenue generated during this period. Keeping this scenario in mind, an attempt has been made to formulate an inventory policy for the retailer dealing with imperfect quality items under permissible delay in payments. Results have been analyzed with the help of a numerical example and sensitivity analysis also carried out.
Vinti Dhaka, Sarla Pareek, Mandeep Mittal

### Chapter 11. Joint Effects of Carbon Emission, Deterioration, and Multi-stage Inspection Policy in an Integrated Inventory Model

Abstract
This paper discusses an integrated inventory model between vendor and buyer for decayed type of products. The vendor produces perfect products but may arrive some defect products in the system. To control product quality, the manufacturer inspects all the products to separate the defective products. After the first-stage inspection, the defective products are reworked at a fixed cost and again inspection takes place for the reworked products in the second-stage inspection. After completion of the second-stage inspection, the defective products are disposed at some fixed cost and delivers good products to the buyer. The delivery of good products is done by single-setup multi-delivery (SSMD) policy by consideration of fixed and variable types of transportation cost. At any stage, the good quality of products may deteriorate and the constant deterioration rate is considered for vendor and buyer, separately. Carbon is emitted from every portion of the integrated system. This issue is studied in this model and finally the joint total cost of the inventory model is minimized with the help of algebraic method. To illustrate the model numerically, some numerical examples are provided along with the sensitivity analysis and the graphical representations of those examples.
Bijoy Kumar Shaw, Isha Sangal, Biswajit Sarkar

### Chapter 12. A Note on “Inventory and Shelf-Space Optimization for Fresh Produce with Expiration Date Under Freshness-and-Stock-Dependent Demand Rate”

Abstract
In a recent paper, Chen et al. in J Oper Res Soc 67(6):884–896, [4] proposed an inventory model with freshness-expiration date and stock-dependent demand, assuming nonzero ending inventory and adopting a profit maximization function. They treated the freshness index that measures the quality of produce as linear decreasing function. However, it is evident that the degradation in quality not necessarily decreases linearly for every product. Therefore, in this work, we relax this assumption and characterize the freshness index as polynomial decreasing function to strengthen the applicability of Chen et al.’s model.
Hardik N. Soni, Dipali N. Suthar

### Chapter 13. EOQ Model Under Discounted Partial Advance—Partial Trade Credit Policy with Price-Dependent Demand

Abstract
Swati Agrawal, Rajesh Gupta, Snigdha Banerjee

### Chapter 14. Effects of Pre- and Post-Deterioration Price Discounts on Selling Price in Formulation of an Ordering Policy for an Inventory System: A Study

Abstract
In present times, one of the promotional tools is offering a rebate on retail amount for raising the market needs of a product. Also, different discount rates are offered depending upon quality/originality/expediency. A non-deteriorating product maintains its quality/original conditions throughout the planning horizon. A deteriorating product may be affected by deterioration at the time of replenishment (instantaneous deterioration) or may be after some time (non-instantaneous). Retailer may offer different price discounts in each case. In this chapter, optimal ordering policies are discussed when retailer offers different price discounts to his customers, before and after deterioration starts. Moreover, the demand for a product is considered price sensitive. Pre-deterioration discount is considered to be smaller than the post-deterioration discount as per the trend. Four different situations are formulated and illustrated with support of numerical examples. Sensitivity analysis is performed to present bureaucratic insights.
Mihir Suthar, Kunal T. Shukla

### Chapter 15. Efficient Supplier Selection: A Way to Better Inventory Control

Abstract
Effective supplier evaluation during the purchasing process is important to business, as supplier selection and the success of inventory management depend on how and which suppliers are selected. Given the popularity of supplier selection with inventory control, this chapter presents an actual, complex supplier selection problem involving multiple products where conflicting inventory related attributes such as response time, delivery reliability, stock quantity, service level and the track record of the suppliers, are involved. The challenge for this case firm is to inter-twine supplier selection with inventory management so as to yield the best space utilization, lower inventory carrying cost and increase end customer satisfaction. Our main contribution is to apply fuzzy AHP and fuzzy TOPSIS to rank and choose efficient suppliers through the linguistic ratings of a set of potential suppliers. Under the environment of global competition, accurate demand fulfilment has become more significant than ever before in supply chain management. As a result, we consider stochastic demand parameters and model the problem with the help of triangular fuzzy numbers. A multi-objective mixed integer linear optimization model is formed to assign the order volume to the selected supplier(s), with a view to executing inventory control visually on a user interface.
Shuya Zhong, Sujeet Kumar Singh, Mark Goh

### Chapter 16. Supply Chain Network Optimization Through Player Selection Using Multi-objective Genetic Algorithm

Abstract
This paper is an effort to study an integrated supply chain network comprising of suppliers, manufacturers, distributors, and retailers with the key objective of minimizing the overall cost of supply chain. Players of supply chain network are selected on the basis of multi criteria. Multi-objective GA has been used to select business players under constraints. Further, the output of GA is visualized through 3D-radVis techniques with respect to location, shape, range, and distribution of non-dominated Pareto front. The paper also proposes an algorithm to analyze other integrated supply chain problems belong to this class. The model is also validated through a numerical example. This model is useful to manufacturers and distributors who involved with the industries like automobile, textile, food and electronic gadgets, etc. for the sustainable supply chain management.
Poonam Mishra, Isha Talati, Azharuddin Shaikh

### Chapter 17. Allocation of Order Amongst Available Suppliers Using Multi-objective Genetic Algorithm

Abstract
In a supply chain, procurement of items is done on the basis of individual performance, whereas the performance of supply chain can be improved by using scientific techniques. In this chapter, we discuss the manufacturer’s problem of procuring several items from the available suppliers; where, supplies from each supplier are constrained. The manufacturer needs to determine which item is to be procured from which supplier and in what quantity. The allocation of order amongst suppliers is done on the basis of multiple criteria such as unit price, quality, supply capacity, delivery time, and unit transportation cost. To demonstrate the scenario, we formulate the mathematical model, which leads to a multi-objective optimization problem. The optimization is done using multi-objective genetic algorithm, which gives a set of Pareto-optimal solutions, then we utilize 3D-RadVis technique to get the best solution. To validate the model, numerical example is presented.
Azharuddin Shaikh, Poonam Mishra, Isha Talati

### Chapter 18. Some Studies on EPQ Model of Substitutable Products Under Imprecise Environment

Abstract
In current scenario, big departmental stores used to work more efficiently with the items that can be substituted either with optimum order quantities or selling prices of the products. At the time of purchase, customer of one particular item transfers to relevant substitutable item because of difference in prices or the quantities that can be purchased in bulk. In this chapter, the inventory problem is determined in total profit maximization problem with crisp, random, fuzzy, fuzzy-random, rough, and fuzzy-rough constraints. The problem is solved through a gradient-based search technique—GRG (Generalized Reduced Gradient) method. The prices and optimal order quantities of substitutable items are obtained so that total profit for store owner is maximum.
R. L. Das, R. K. Jana

### Chapter 19. An Effective MILP Model for Food Grain Inventory Transportation in India—A Heuristic Approach

Abstract
In this work, we investigate a real-life inventory transportation problem faced by the Food Corporation of India (FCI). FCI is the central agency responsible for procurement, storage, and transportation of food grains over a large geographical area of India. Due to lopsided procurement and consumption of major food grains (i.e., rice & wheat) transportation of food grains across the warehouses becomes inevitable. FCI faces a significant challenge to find the optimal amount of food grains to be stored at each warehouse and transported among the warehouses to meet the demand during each period. In this study, we formulate an MILP model to determine the optimal inventory transportation decisions related to food grain transportation in India and demonstrate it via a case study. Commercial optimization packages can be used to solve the problem of this class. However, as we see, they fail to provide a solution for large size problem instances. Therefore, we propose a heuristic-based solution approach to solve the problem. It is seen that under a practical time limit, the proposed heuristic performs significantly well in terms of accuracy as compared to commercial optimizations packages. The nature of the study is generic in nature and can be also applied to various similar real-life problem scenarios.
Sayan Chakraborty, Kaushik Bhattacharjee, S. P. Sarmah

### Chapter 20. Fuzzy Based Inventory Model with Credit Financing Under Learning Process

Abstract
Mahesh Kumar Jayaswal, Isha Sangal, Mandeep Mittal

### Chapter 21. A Fuzzy Two-Echelon Supply Chain Model for Deteriorating Items with Time Varying Holding Cost Involving Lead Time as a Decision Variable

Abstract
In this paper, we have developed a two-stage supply chain production-inventory model for deteriorating product with time-dependent demand under fuzzy environment. Here we describe an EOQ model with changeable lead time and time-dependent holding cost. This situation is very common in the market, once an enterprise has some key technology or product that others have not, as a supplier, it can decide the prices and lead time of the technology or product to the buyers or retailers according to its need. Then the retailer determines his optimal order strategy, i.e., decides on the quantity of products to order from the suppliers. Under this circumstance, the problem that lead time, as a controllable variable of the supplier, and how it affects the cost to the supplier, retailer and whole supply chain is very important to the supplier and retailer because double-win benefits is a base of existence for the supply chain. In reality it is seen that we cannot define all parameters precisely due to imprecision or uncertainty in the environment. So we have defined the inventory parameters, such as set up cost, stock-out cost, and deterioration cost as triangular fuzzy numbers. The signed distance method and graded mean integration method have been used for defuzzification. To illustrate the proposed model a numerical example and sensitivity analysis with respect to different associated parameters has been presented.
Srabani Shee, Tripti Chakrabarti

### Chapter 22. Transportation-Inventory Model for Electronic Markets Under Time Varying Demand, Retailer’s Incentives and Product Exchange Scheme

Abstract
Exchange offers are popular in many businesses to attract new customers. Availability of novel varieties, stiff competition, and regulatory restriction of discarding old products enhances to facilitate such offers in the market. Although, the reduction of out-of-pocket expenses to the customers helps to increase the sale of products; it also elevates the decision problem for managing the inventories of the exchanged products for the retailers. In view of this, the present article addresses inventory decision modeling in a system where a customer can buy products from the electronic-market retailer either by paying the full price or getting some price-discount for exchanging old products. Retailer bears transportation costs for the new and exchanged products. Four models are formulated and numerical examples are presented. Sensitivity analysis is also performed to understand the effect of various parameters in the models.

### Chapter 23. Electronic Components’ Supply Chain Management of Electronic Industrial Development for Warehouse and Its Impact on the Environment Using Particle Swarm Optimization Algorithm

Abstract
The electronic component model inventory is a method of balancing investments to achieve the service-level goal. Here you can see the electronic components warehouse and the distribution centers for the electronic components inventory. Environmental heritage policy, electronic components, and electronic component warehouses are very important issues, as the supply chain of electronic components for chemicals is directly linked to people’s lives. Variable electronic component order quantity, economical electronic component order quantity, electronic component time order quantity, electronic component removal order quantity, and electronic environment component order quantity inventory rules are usually used in the modern trend for inventory management of parts of electronics. However, effectively managing the inventory of a substance in electronic components is a difficult problem due to its properties. We have proposed the PSO inventory policy for resellers of electronic component warehouses in the electronic supply chain. We also model a model designed to measure the effectiveness of administrative strategies. The proposed algorithm for artificial bee colony (PSO) determines the optimal product at the time of the order using the current stock. The simulation results show that the ABC algorithm for artificial bee colonies is an effective way to manage electronic component warehouses and electronic component supply chains.
Ajay Singh Yadav, Anupam Swami, Navin Ahlawat, Dhowmya Bhatt, Geethanjali Kher

### Chapter 24. Interpretive Structural Modeling to Understand Factors Influencing Buying Behavior of Air Freshener

Abstract
In order to gain “loyalty”, a firm has to maintain “quality” that defines acceptability of its offerings. In the era of globalization, all industries face cutting-edge competition and it becomes difficult to survive for less qualitative products. This paper is an attempt to study attributes preferred by air freshener buyers. A multi-scaling technique, interpretive structural modeling (ISM), has been applied to understand and find the contextual relationship amongst the various attributes under study. Furthermore, MICMAC classification has been done to determine the autonomous, dependent, linkage and independent nature of the factors. Out of 14 attributes, only 5 comes out to be independent which are the deriving attributes for the rest of attributes. The findings can also help company policymakers with understanding the interrelated attributes associated with trustworthiness in the context of air freshener industry and implement them in effective strategic planning. This study has been carried out in one of the metropolitan cities of India and results obtained are significant.
Deepti Aggrawal, Jagvinder Singh, Anurag Kumar, Adarsh Anand

### Chapter 25. Decision-Making with Temporal Association Rule Mining and Clustering in Supply Chains

Abstract
Timely identification of recently rising patterns is required in business process. Data mining methods are most appropriate for the characterization, valuable examples extraction, and predications which are essential for business support and decision-making. Some research studies have also expanded the use of this idea in inventory management. However, not very many research analyzes have considered the utilization of the data mining approach for supply chain inventory management. In this chapter, two unique cases for supply chain inventory management dependent on cross-selling effect are presented. First, the cross-selling effect in different clusters is characterized as a basis for deciding the significance of items. Second, the cross-selling in different time periods is considered as a criterion for ranking inventory items. An example is devised to approve the outcomes. It is illustrated that by using this modified approach, the ranking of items may get affected resulting in higher profit.
Reshu Agarwal