Economic growing quantity

Journal Pre-proof Economic Growing Quantity,

Abolfazl Gharaei, Eman Almehdawe PII:

S0925-5273(19)30338-X

DOI:

https://doi.org/10.1016/j.ijpe.2019.107517

Reference:

PROECO 107517

To appear in:

International Journal of Production Economics

Received Date:

01 February 2019

Accepted Date:

07 October 2019

Please cite this article as: Abolfazl Gharaei, Eman Almehdawe, Economic Growing Quantity,, International Journal of Production Economics (2019), https://doi.org/10.1016/j.ijpe.2019.107517

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Journal Pre-proof

𝐔𝐆𝐅[𝐭𝟏, 𝐭𝟐]

The optimal status of the EGQ inventory system, when G(t)≠ V(t)

Journal Pre-proof Economic Growing Quantity

Abstract The essence of some items is continuously growing and gaining weight over time. These are known as growing items, which their weights and values increase over time. This paper provides a new generation of inventory models, entitled Economic Growing Quantity (EGQ), which is designed by focusing on growing items of agricultural industries such as fisheries, poultry, and livestock. Our EGQ inventory model considers the probability density functions of survival and mortality for a growing item. It also considers the growth functions of live and dead grown items. Accordingly, the total cost for EGQ model consists of the costs of purchasing process, the disposal cost of dead items, the holding cost, the feeding cost, and the setup costs for both the live and the dead items. Our goal is to determine the optimal economic growth/slaughter cycle and the economic growing quantity in order to minimize the total costs. Subsequently, we optimize the weight of slaughtering, the Utility of Growth Functions (UGF), and other important performance metrics of the system. The applicability and validity of our model is demonstrated based on numerical examples of the poultry industry. The results of sensitivity analyses recommend some strategic implications and insights for the managers of growing items. Keywords: Growing Items; Economic Order Quantity; Agriculture; Inventory Management; Survival Functions; Mortality Functions. 1. Introduction Inventory management is an important operation to decrease the total costs. Harris developed the Economic Order Quantity (EOQ) inventory model to determine the optimum quantity and replenishment cycle (Harris, 1913; Sivashankari & Panayappan, 2014; Rezaei, 2014). For many years, the provided EOQ model by Harris (1913) was expanded and optimized (Schwarz, 1972). Accordingly, Taft (1918) developed the Harris model for production systems, where the products are gradually produced at a known rate. His efforts led to the advent of a new generation of inventory models, named Economic Production Quantity (EPQ) models. The EOQ and the EPQ models were developed in the sequence of inventory models to include the backorder levels (Hadley & Whitin, 1963). Contemporary to the paper of Hadley et al (1963), an inventory model was 1

Journal Pre-proof presented for decayed items (Ghare & Schrader, 1963). Their paper was a spark to design the inventory models of deteriorative items. In this regards, variable deterioration rates were considered by (Covert & Philip, 1973). Afterwards, researchers' efforts focused on the inventory models of perishable items (for deep review, see (Goyal & Giri, 2001; Bakker, Riezebos, & Teunter, 2012)). Many of the research in this area have recently focused on expanding the amelioration inventory models (Zhang, Li, Tian, & Feng, 2016; Hwang, 1999; Sana, 2010; Mondal, Bhunia, & Maiti, 2003). In contrast to the deteriorating items, the economic value or utility of the amelioration items increases over time by ameliorating activation such as wine, whose economic value or utility increases over time (Zhang, Li, Tian, & Feng, 2016; Mahata & De, 2016). During more than a century, more efforts have been made to develop and optimize the inventory models in different systems. This is while the nature of some products and items is constantly growing and gaining weight over time. These are known as the growing items, which value and size increase over time (Rezaei, 2014; Zhang, Li, Tian, & Feng, 2016; Nobil, Sedigh, & Cárdenas-Barrón, 2018; Sebatjane & Adetunji, 2018). The products from certain industries, such as aquatics and fisheries, poultry, livestock, and cereal industries are among the growing items or products (Rezaei, 2014; Nobil, Sedigh, & Cárdenas-Barrón, 2018; Sebatjane & Adetunji, 2018). The mentioned models in the literature cannot be applicable for such industries, which their products grow and gain weight over time. To this end, we develop a more realistic inventory management model for growing items, entitled "Economic Growing Quantity (EGQ)" model as a new generation of inventory models by building on previous models, especially the provided model by (Rezaei, 2014). One of the key differences between the EGQ model and previous works of growing items is considering the mortalities. A traditional way of explaining mortality patterns is by the probability of survival (Ravindranathan, 1994). Contrary to Rezaei's model (2014) and its subsequent developments, we consider the probability density functions of survival and mortality in EGQ model in order to define the expected mortality and survival rates and the historical trends of survival and mortality (Weon, 2004) for growing items. On the other hand, not only we consider the growth function, the feeding cost, and the holding cost for live items, but we also consider them for mortalities, based on the growth function of dead grown items in order to get closer to the reality and nature of growing items. Taking into account the probability density 2

Journal Pre-proof function of mortality, imposes the disposal cost of mortalities or dead grown items to the inventory system. Accordingly, disposal cost of dead items as an important term (Tabler, Berry, Xin, & Barton, 2002) is considered in our model, which was not included in previous models of growing items. Moreover, ignoring the lead times for ordered items can lead to inappropriate ordering decisions and increased costs (Noblesse, Boute, Lambrecht, & Van Houdt, 2014). The lead time reduction is especially beneficial in the case of high demand products (Glock, 2012), such as inventory systems of growing items. Unlike the (Rezaei, 2014; Zhang, Li, Tian, & Feng, 2016; Nobil, Sedigh, & Cárdenas-Barrón, 2018; Sebatjane & Adetunji, 2018), we consider the lead time as an essential factor to achieve the economic Reorder Point (ROP) based on weight of live or dead items per growth cycle. Furthermore, we calculate the average inventory per growth cycle of growing items. Because, inventory turnover acts as an excellent metric for evaluating how long products remain in inventory (Reynolds, 1999). In addition, in most industries with growing items, a setup cost is required (Nobil, Sedigh, & Cárdenas-Barrón, 2018), along with the ordering cost of purchasing process, which depends on the weight of the new born items. In this regards, we also consider the ordering cost of the new born items in purchasing process. Finally, we provide a new concept for growing items, entitled "Utility of Growth Functions (UGF) to evaluate the quality of growth functions. To our knowledge, there is a gap in the literature in the field of growing items. 2. Literature Review The main difference between inventory models of the ameliorating, deteriorating, perishable, and regular items with growing items is the item growth or the item weighing over time, constantly. In other words, the weight of the inventory level of growing items increases over time period (Rezaei, 2014; Nobil, Sedigh, & Cárdenas-Barrón, 2018). Moreover, the value and size of items increase over time (Nobil, Sedigh, & Cárdenas-Barrón, 2018), along with the fact that their quantity may also increase due to reproduction during the growth cycle. Because of the fundamental differences among growing items, ameliorating items, deteriorating items, regular items, and perishable items, we only focus on inventory models for growing items in the literature review. However, we compare the most popular inventory models of the ameliorating items, deteriorating items, perishable items, and regular items with our EGQ model. 3

Journal Pre-proof The closest to our work is the EOQ model for growing items by Rezaei (2014). His paper's goals were to 0

obtain the economic order quantity per growth cycle, to optimize the growth cycle length and the total profit. In his model, the probabilities of survival and mortality during the growth cycle were not investigated. Furthermore, the ordering costs of the new born items in purchasing process and the disposal costs of dead grown items were ignored. Later, (Zhang, Li, Tian, & Feng, 2016) considered the effects of a carbon tax on the total cost to find the optimal growth cycle and order quantity. They modeled a situation in which a retailer ordered the new born chickens from a supplier and slaughtered them based on their weights. In the following, (Nobil, Sedigh, & Cárdenas-Barrón, 2018) developed an EOQ model of poultries when shortages are permitted. The goal of their study was to optimize the shortage and the cycle length in order to minimize the total inventory costs. Recently, (Sebatjane & Adetunji, 2018) designed a model in which items were screened in order to distinguish the good quality items from the poor quality items. The objective was to maximize the expected total profit. Table (1) compares our model with other related inventory models. Please Insert Table (1) Based on Table (1), the contribution of our model is such that it is applicable to general industries, which produce the growing products. Our model is not confined and exclusive to a specific growing product or industry. In addition, the feeding costs, the holding costs, the operational costs, and the growth patterns are considered not only for the live items, but also for the dead grown items until their death. As a result, the costs of dead grown items are calculated until their death. Moreover, we consider both the ordering costs of the new born items and the disposal costs of dead items. We also provide a new concept for EGQ inventory systems, entitled Utility of Growth Functions (UGF) in order to evaluate the utility and quality of growth functions of live and dead grown items. In addition to formulating the economic growing quantity per growth cycle, we formulate the economic growth/slaughter cycle, economic ROPs per growth cycle, average inventory, and the weight of slaughtering as performance metrics of our model. The rest of the paper is organized as follows: The problem along with its assumptions is defined in Section 3. Mathematical formulation of the EGQ model is provided in Section 4. Two Cases of model optimizations are provided in Section 5. A real numerical example is solved in Section 6 followed by some sensitivity analyses. Finally, the managerial implications and conclusions are provided in Sections 7 and 8, respectively. 4

Journal Pre-proof 3. Problem Definition and Assumptions 3.1. Problem Definition In this paper, a new generation of inventory models named, EGQ model will be designed for growing items. In this model, the newborn growing items with an approximated initial weight (w0) are ordered in order to satisfy the demand at the end of the growth cycle. The growable items feed and grow up based on their growth pattern per cycle. During the growth period, the dead grown items should be disposed from the inventory system. The total weight of live grown items should meet the demand at the end of cycle. The demand is known, deterministic, and constant per growth cycle. For the next cycle, the newborn items should be replaced, instantly. So, before the start of the next growth cycle, the newborn items should be ordered based on lead-time and ROP in order to fulfill the demand of the next cycle. Accordingly, the aim of EGQ model is to minimize the total inventory costs for growing items in order to optimize the growth cycle, the quantity of products, the ROPs, the average inventory, the weight of slaughtering, and finally the utility of growth functions. In this regards, the EGQ model considers the growth functions, feeding costs, holding costs, and the operational costs both for live items and for dead grown items until their death. Also, in order to formulate and optimize the ROPs per growth cycle, the EGQ model considers the lead times of ordered items. Most importantly, the probability density functions of mortality and survival, along with the ordering cost of the newborn items and disposal costs are considered. 3.2. Assumptions The models of Rezaei (2014) and Sebatjane & Adetunji (2018) were configured based on some limiting assumptions, which focus on the poultry industry, while we provide a general inventory model of growing items by considering the dead grown items in addition to live items. Below is a summary of our model assumptions: 1.

The live grown items are slaughtered at the end of the growth cycle, instantaneously.

2.

All the purchased new born items at the beginning of each cycle are healthy and growable.

3.

Mortalities only occur over the growth cycle.

4.

The dead products are not recoverable and recyclable; they should be disposed from the system.

5.

There are not surplus production and shortage per cycle. 5

Journal Pre-proof 6.

The products do not have any reproduction or proliferation during the growth cycle.

7.

The profitable byproducts of items are not considered.

8.

Feeding costs, holding costs, and set up costs are associated not only with the live items, but also with the dead grown items, until their death.

9.

The lead time for new items is known and deterministic.

10. The products do not deteriorate, perish or ameliorate during the growth cycle. In addition to the mentioned assumptions, we assume that the growth and mortality functions don't change from one cycle to another. Moreover, discount for purchased items is not considered. Finally, based on conducted works by (Guchhait, Maiti, & Maiti, 2013; Matsuyama, 1995; Matsuyama, 2001), the setup cost can be assumed to depend on both production quantity and time. 4. Mathematical Formulation 4.1. Notations The following notations, including the parameters and the decision variables are used for mathematical formulation of the EGQ model: 𝑠𝑡(𝑡): The probability density function of survival 𝑚𝑡(𝑡): The probability density function of mortality 𝑚: Disposal cost of dead items per weight unit per time unit (€/gram/week) 𝑉(𝑡): The growth function of dead grown items 𝐺(𝑡): The growth function of live items 𝐿: Order lead time (Week) 𝐷: Demand rate per growth cycle (Gram) 𝑝: The costs of purchasing process per weight unit (€/gram) 𝑓: Feeding cost per weight unit per time unit (€/gram/week) 𝐹𝑙: The total feeding cost for live items (€) 𝐹𝑑: The total feeding cost for dead grown items (€) ℎ: Holding cost per weight unit per time unit (€/gram/week) 6

Journal Pre-proof 𝐻𝑙: The total holding cost for live items (€) 𝐻𝑑: The total holding cost for dead grown items (€) 𝑘: The total variable costs of setup per weight unit per time unit (€/gram/week) 𝑘𝑙: The total variable costs of setup for live items (€) 𝑘𝑑: The total variable costs of setup for dead grown items (€) 𝑤0: The approximated initial weight of new born items (Gram) 𝑊𝑇: The total weight of live items at the end of the growth cycle (Gram) 𝑦: Number of ordered items 𝑇: The growth/slaughter cycle (Week) 𝑊𝑆∗ : The optimum weight of slaughtering (Gram) 𝑈𝐺𝐹: The Utility of Growth Functions (Percent) 𝑊 ∗ : The initial economic weight for ordering per growth cycle (Gram) 𝐼 ∗ : The optimum average inventory (Gram) 𝑅𝑂𝑃𝑙∗ : The economic reorder point based on the weight of live items (Gram) 𝑅𝑂𝑃𝑑∗ : The economic reorder point based on the weight of dead items (Gram) Next, we explain the details of the cost structure for the EGQ model. 4.2. The Total Cost Figure (1) illustrates the behavior of the EGQ model for a growing item over time in order to provide a background for the terms of the total cost function. Please Insert Figure (1) The costs of EGQ inventory system per cycle consist of the total costs of purchasing process, the total holding, feeding, and setup or operational costs for both the live items and the dead grown items until their death, and finally the total disposal costs of dead items. Accordingly, the total cost per cycle (TC) is: 𝑇𝐶= 𝑃+ 𝐻+ 𝐹 + 𝐾 + 𝑀

(1)

This subsection demonstrates the terms' formulations of total cost per cycle in details as follows: 7

Journal Pre-proof The Total Costs of Purchasing Process (P): The EGQ inventory system starts a new growth cycle with ordering and purchasing the new born items. This cost can be computed as follow: 𝑃= 𝑝𝑦𝑤0

(2)

The Total Holding Costs (H): The total holding cost not only considers the live items, but also considers the dead grown items until their death. According to Figure (1), the total holding cost for live items is: 𝐻𝑙= ℎ𝑦 ∫

𝑇 𝑇 (𝑡) 𝑑𝑡 ∫ 𝐺(𝑡) 𝑑𝑡 𝑠 𝑡 0 0 .

(3)

.

This is basically the multiplication of the number of survived items by weight of a live item based on the growth function of live items. The total holding cost of dead grown items is similar, with the difference that the number of ordered items is multiplied by the mortality probability function and the weight of a dead grown item based on the growth function of dead gown items as follows: 𝐻𝑑= ℎ𝑦 ∫

𝑇 𝑇 (𝑡) 𝑑𝑡 ∫ 𝑉(𝑡) 𝑑𝑡 𝑚 𝑡 0 0 .

(4)

.

Accordingly, the total holding cost per growth cycle is the sum of the total holding costs for both live and dead grown items as shown in Equation (5):

(∫𝑇0𝑠𝑡(𝑡) 𝑑𝑡 ∫𝑇0𝐺(𝑡) 𝑑𝑡 + ∫𝑇0𝑚𝑡(𝑡) 𝑑𝑡 ∫𝑇0𝑉(𝑡) 𝑑𝑡)

𝐻= 𝐻𝑙+𝐻𝑑= ℎ𝑦

.

.

.

.

(5)

The Total Feeding Cost (F): The total feeding cost is formulated for both live and dead grown items per growth cycle and can be obtained by Equation (6).

(∫𝑇0𝑠𝑡(𝑡) 𝑑𝑡 ∫𝑇0𝐺(𝑡) 𝑑𝑡 + ∫𝑇0𝑚𝑡(𝑡) 𝑑𝑡 ∫𝑇0𝑉(𝑡) 𝑑𝑡)

𝐹= 𝐹𝑙+𝐹𝑑= 𝑓𝑦

.

.

.

.

(6)

The Total Setup and Operational Costs (K): Based on (Matsuyama, 2001), (Guchhait, Maiti, & Maiti, 2013), and (Matsuyama, 1995), we assume that the setup cost varies with the lapse of time and depends on order quantity or production rate. The setup costs in the EGQ model like the personnel cost of operational tasks over growth cycle, as well as the consumable materials are considered as time dependent costs, which should be spent based on the weight of items. In the classical inventory theories, setup cost at 𝑡, 𝑘(𝑡) = 𝑘 is often assumed. In this same sense our model deals with a more general model than the classical inventory

8

Journal Pre-proof theories (Matsuyama, 1995). Accordingly, the total setup and operational costs for live and dead grown items per growth cycle can be obtained by Equation (7):

(∫𝑇0𝑠𝑡(𝑡) 𝑑𝑡 ∫𝑇0𝐺(𝑡) 𝑑𝑡 + ∫𝑇0𝑚𝑡(𝑡) 𝑑𝑡 ∫𝑇0𝑉(𝑡) 𝑑𝑡)

𝐾= 𝑘𝑙+𝑘𝑑= 𝑘𝑦

.

.

.

(7)

.

The Total Disposal Cost (M): Considering the mortality probability function imposes the disposal cost of mortalities or dead items to the EGQ inventory system. Accordingly, the total disposal cost per growth cycle is formulated as Equation (8): 𝑀= 𝑚𝑦 ∫

𝑇 𝑇 (𝑡) 𝑑𝑡 ∫ 𝑉(𝑡) 𝑑𝑡 𝑚 𝑡 0 0 .

(8)

.

The Total Cost of the EGQ Inventory System (TC): Based on Equations (1) to (8), the total cost of the EGQ inventory model can be simplified as follow:

(

𝑇

𝑇

𝑇

)

𝑇

(9)

𝑇𝐶=𝑦 𝑤0𝑝 + (ℎ + 𝑓 + 𝑘 + 𝑚)(∫0 𝑚𝑡(𝑡)𝑑𝑡 ∫0 𝑉(𝑡)𝑑𝑡) + (ℎ + 𝑓 + 𝑘)(∫0 𝑠𝑡(𝑡)𝑑𝑡 ∫0 𝐺(𝑡)𝑑𝑡

Accordingly, the total costs of the EGQ inventory system per unit time (TCU) can be calculated as follow: 𝑦

(

𝑇

𝑇

𝑇

𝑇

)

𝑇𝐶𝑈= 𝑇 𝑤0𝑝 + (ℎ + 𝑓 + 𝑘 + 𝑚)(∫0 𝑚𝑡(𝑡)𝑑𝑡 ∫0 𝑉(𝑡)𝑑𝑡) + (ℎ + 𝑓 + 𝑘)(∫0 𝑠𝑡(𝑡)𝑑𝑡 ∫0 𝐺(𝑡)𝑑𝑡

(10)

In order to optimize Equation (10), we first need to determine the probability density functions of survival and mortality, and the growth functions of live and dead grown items. In this regards, we consider the Polynomial models, which have been fit to growth data for male broiler chickens (Tzeng & Becker, 1981), California turkeys (Ersoy, Mendeş, & Keskin, 2007) and the brown frog (Tong, Du, Hu, Cui, & Wang, 2018). Furthermore, the growth patterns of the specific types of broiler chickens using polynomial functions were used by (Tompic, Dobsa, Legen, Tompic, & Medic, 2011). They mentioned that the Polynomial models are much easier to handle than nonlinear growth models, because they are linear from a statistical point of view and can be solved easily using linear regression. Finally, they stated that a linear model (firstorder polynomial) is especially attractive, because of its easy interpretability. The polynomial growth model 𝑟

provided by (Hadeler, 1974) is 𝑊 = 𝑑0 + ∑𝑖 = 1𝑑𝑖𝑡𝑖. Where: 𝑟 defines the first to the rth order of fit, 9

Journal Pre-proof 𝑑0 defines intercept, and 𝑑𝑖 defines the regression coefficient. We use the first order polynomial model (𝑟=1) as growth functions of live and dead grown items as follow: (11)

𝐺(𝑡) = 𝑉(𝑡) = 𝑑0 + 𝑑1𝑡

Moreover, we consider the uniform distribution as the probability density functions of survival and mortality, where Seal (1954) assumed a uniform distribution of animal deaths over the interval (0, T) (Kimball, 1960; Seal, 1954). Accordingly, we use the below uniform distributions where the sum of the probabilities of survival and mortality per cycle is equal to one: ∼

𝑠𝑡(𝑡)

Unif

(a,

b)

—›

𝑚𝑡(𝑡)

𝑎 , 𝑏―𝑎―1

Unif

∼

(

𝑏 ) 𝑏―𝑎―1

(12) In the probability distributions of (12), ( a ) and ( b ) define the parameters of the uniform probability function. The order quantity and the reorder point are main decision variables of inventory systems. In the next section, we optimize the aforementioned decision variables, along with other variables in two Cases, where the growth function of live items is/isn't equal to the growth function of dead grown items in Case (1)/Case (2). 5. Model Optimization 5.1. Case 1: 𝑮(𝒕) = 𝑽(𝒕) = 𝒅𝟎 + 𝒅𝟏𝒕 5.1.1. The Economic Growth/Slaughter Cycle (T) Based on assumed probability and growth functions in previous section, Equation (10) can be reformulated as follow:

(

𝑇𝐶𝑈= 𝑦

𝑤0𝑝 𝑇

(

+ (ℎ + 𝑓 + 𝑘 + 𝑚)

)( )∫ (𝑑0 + 𝑑1𝑡)𝑑𝑡 + (

𝑏―𝑎―1 𝑏―𝑎

1 𝑇

𝑇 0

)( )∫ (𝑑0 + 𝑑1𝑡) 𝑑𝑡)

ℎ+𝑓+𝑘 𝑏―𝑎

1 𝑇

𝑇 0

(13)

The simpler form of Equation (13) is:

(

𝑇𝐶𝑈= 𝑦

𝑤0𝑝 𝑇

1

(

+ 𝑏 ― 𝑎 𝑑0 +

)((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘)))

𝑑1𝑇 2

(14)

The total cost function in Equation (14) is a convex function (See the Appendix A for a proof). Accordingly, it can be optimized by differentiation in order to find the economic growth/slaughter cycle. Therefore, we obtain the optimum value of T (T*) in a closed-form as follows: 10

Journal Pre-proof ∂𝑇𝐶𝑈 ∂𝑇

(

=𝑦

―𝑤0𝑝 2

𝑇

+

( )( )((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘))) = 0 𝑑1 2

1 𝑏―𝑎

(15)

Where, y≠0; and

(

―𝑤0𝑝 𝑇2

+

𝑑1

)

( )( ) 2

1 ((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘)) = 0 𝑏―𝑎

(16)

By some simplifications, the optimum cycle length (T*) for Case (1) can be obtained using Equation (17): 𝑇∗ =

2𝑤0𝑝(𝑏 ― 𝑎)

(17)

𝑑1((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘))

Based on Equation (17), the optimum growth cycle for the EGQ model is directly affected by the initial weight of new born items and the costs of purchasing. Moreover, mentioned variable is inversely affected by the holding cost, feeding cost, operational cost, and disposal cost of dead items. 5.1.2. The Economic Growing Quantity (EGQ) Regardless of the shortage, the demand should be equal to the total weight of live items at the end of the growth cycle. In other words, the final weight of all live items should be equal to the demand, as follows: 𝐷 𝑇∗

∫

= 𝑊𝑇 = 𝑦

𝑇∗

∫

𝐺(𝑡) 𝑑𝑡 ― 𝑦

0

(

0

𝑇∗

)∫

𝑏―𝑎―1 1― 𝑏―𝑎

0

𝑇∗

𝑚𝑡(𝑡) 𝑑𝑡∫

𝑇∗ 0

(

𝑇∗

)∫

𝐺(𝑡) 𝑑𝑡 =

0

𝑇∗

( )∫

𝑦 𝐺(𝑡)𝑑𝑡 = 𝑏―𝑎

∫

𝑉(𝑡) 𝑑𝑡 = 𝑦

0

𝑏―𝑎―1 𝐺(𝑡) 𝑑𝑡 ― 𝑦 𝑏―𝑎

𝐺(𝑡)𝑑𝑡

(18)

0

By replacing the growth function of Equation (11) with Equation (18), we have: 𝑦 𝐷 = 𝑊𝑇 = 𝑏―𝑎

𝑇∗

∫

0

𝑑1𝑇 ∗ 𝑦 ∗ 𝑑0 + 𝑑1𝑡 𝑑𝑡 = 𝑑𝑇 + 𝑏―𝑎 0 2

(

2

)

(19)

After some simplifications, the Economic Growing Quantity (EGQ) for Case (1) can be obtained by:

𝐸𝐺𝑄 = 𝑦 ∗ = 𝑄𝐴𝑏𝑒𝑚 The phrase "Abem" is taken from the first two letters of the authors' names, Abolfazl & Eman 2𝐷(𝑏 ― 𝑎) = (20) 2 2𝑑0𝑇 ∗ + 𝑑1𝑇 ∗ Based on Equation (20), the optimum number of ordered items directly depends on demand rate and reversely on parameters of growth functions. The recent finding confirms the obtained result by (Rezaei, 11

Journal Pre-proof 2014) regarding the economic order quantity for growing items. However, based on our model, EGQ is inversely affected by the optimum growth cycle, when the growth functions of live and dead grown items are same. For some growing items such as cereals, which cannot be ordered based on quantity, we can place an order in terms of initial economic weight. Accordingly, the initial optimum weight per growth cycle can be ordered upon Equation (21). 2𝑤0𝐷(𝑏 ― 𝑎)

∗

𝑊 ∗ = 𝑤0𝑦 =

2𝑑0𝑇 ∗ + 𝑑1𝑇 ∗

(21)

2

5.1.3. The Optimum Total Cost By substituting 𝑦 ∗ and 𝑇 ∗ into Equation (14), the optimum total cost of Case (1) can be calculated as: 𝑇𝐶𝑈 ∗ = 𝑦 ∗

(

𝑤0𝑝 𝑇∗

1

(

+ 𝑏 ― 𝑎 𝑑0 +

𝑑1𝑇 ∗ 2

)((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘)))

(22)

5.1.4. The Reorder Points (ROPs) In this subsection, the ROP for the EGQ inventory system is formulated on the basis of the weight of live and dead items. The ROP Based on the Weight of Live Items: In accordance with Figure (1), we have: 𝐷 ― 𝑅𝑂𝑃𝑙= 𝑦

(∫𝑇𝑇 ― 𝐿𝑠𝑡(𝑡)𝑑𝑡 ∫𝑇𝑇 ― 𝐿𝐺(𝑡)𝑑𝑡) .

.

(23) As a result, the optimal status of 𝑅𝑂𝑃𝑙 can be obtained by Equation (24): 𝑅𝑂𝑃𝑙∗ = 𝐷 ― 𝑦 ∗

𝑇∗ 𝑇∗ ∫𝑇 ∗ ― 𝐿𝑠𝑡(𝑡)𝑑𝑡 ∫𝑇 ∗ ― 𝐿𝐺(𝑡)𝑑𝑡

(

.

.

)

(24)

Moreover, we can directly obtain the optimal status of 𝑅𝑂𝑃𝑙 based on Figure (1) through the Equation (25). 𝑅𝑂𝑃𝑙∗ = 𝑦 ∗

(∫𝑇0

∗

―𝐿

𝑇∗ ― 𝐿 𝐺(𝑡)𝑑𝑡

𝑠𝑡(𝑡)𝑑𝑡 ∫0 .

.

)

(25)

By substituting the 𝑦 ∗ and 𝑇 ∗ in Equation (25), the 𝑅𝑂𝑃𝑙∗ is obtained as:

12

Journal Pre-proof 2

2𝑑0(𝑇 ∗ ― 𝐿) + 𝑑1(𝑇 ∗ ― 𝐿)

( )( 𝑦∗ b―a

𝑅𝑂𝑃𝑙∗ =

2

)

(26) The ROP Based on the Weight of Dead Items: We also formulate the ROP based on the weight of dead items. In this regards, if the access to the weight of live items is not possible, the ROP can be obtained based on the weight of dead items, which may be measurable. According to the Figure (1), the optimal status of 𝑅𝑂𝑃𝑑 can be formulated by Equation (27):

(∫𝑇0

𝑅𝑂𝑃𝑑∗ = 𝑦 ∗

∗

𝑏―𝑎―1 𝑇∗ ― 𝐿 (𝑡)𝑑𝑡 𝑉(𝑡)𝑑𝑡 = 𝑦 ∗ 𝑚𝑡 ∫0 𝑏―𝑎

―𝐿

.

.

)

(

2

2𝑑0(𝑇 ∗ ― 𝐿) + 𝑑1(𝑇 ∗ ― 𝐿)

)(

2

)

(27)

5.1.5. The Average Inventory Considering the average inventory as a median value for inventory level of the EGQ system in order to estimate the inventory turnover per growth cycle is inevitable. Accordingly, based on Equation (28), we can calculate the average inventory (𝐼) as:

𝐼=

𝑇

𝑇

(∫0𝑠𝑡(𝑡)𝑑𝑡 ∫0𝐺(𝑡)𝑑𝑡)

(

𝑦𝑤0 + 𝑦

.

)

.

2

(28)

By considering (𝑦 ∗ ) and (𝑇 ∗ ), the optimal status of average inventory can be calculated by Equation (29): ∗

𝐼 =𝑦

∗

2𝑤0(𝑏 ― 𝑎) + 2𝑑0𝑇 ∗ + 𝑑1(𝑇 ∗ )

(

4(𝑏 ― 𝑎)

2

)

(29)

5.1.6. The Optimum Weight of Slaughtering The optimum weight of slaughtering can be obtained by dividing the demand to the number of live items. Accordingly, we have: 𝑊𝑆∗ =

𝐷 𝑇∗

𝑦 ∗ ∫0 𝑠𝑡(𝑡) 𝑑𝑡

=

𝐷

( )

1 𝑦∗ b―a

=

𝐷(𝑏 ― 𝑎) 𝑦∗

(30)

5.1.7. Utility of Growth Functions (UGF), a new concept for EGQ systems The UGF is a novel index defined for growing items in order to evaluate the utility and quality of growth functions of live and dead grown items at a specific time interval or throughout the growth cycle. The UGF 13

Journal Pre-proof can be utilized by the inventory managers of EGQ systems as an efficient quality control (QC) tool for evaluation and monitoring the utility and quality of growth functions in different time frames of growth cycle. In order to more clarify the concept of UGF, we describe the mathematical framework of this concept. The UGF[t1, t2] is the result of dividing the enclosed area between the growth functions of live and dead grown items over the total enclosed area due to both demand line and time interval [t1, t2]. In other words, UGF[t1, t2] is the difference between the weight of live and dead grown items divided by total enclosed weight due to demand line and interval [t1, t2], as illustrated in Graphical Abstract. This index is expressed as a

[

percentage, a value such that

―1 𝑡2 ― 𝑡1

+1

]

≤ UGF[t1, t2] ≤ 𝑡2 ― 𝑡1 , where the closer values to

+1 𝑡2 ― 𝑡1

indicate the

strongest quality and high utility of growth functions. To clarify the concept of the UGF index, consider Equation (31) as follow: 𝑡2

UGF[t1, t2] =

𝑡2

𝑡2

𝑡2

𝑦∫𝑡1𝑠𝑡(𝑡)𝑑𝑡 ∫𝑡1𝐺(𝑡)𝑑𝑡 ― 𝑦∫𝑡1𝑚𝑡(𝑡)𝑑𝑡 ∫𝑡1𝑉(𝑡)𝑑𝑡

(31)

(𝑡2 ― 𝑡1)𝐷

Accordingly, the UGF[0, T] for Case (1) by considering (T*) and (y*), the growth functions, and the probability density functions of survival and mortality can be obtained by Equation (32). 𝑇∗

UGF[0, T] =

𝑇∗

𝑇∗

𝑇∗

𝑦 ∗ ∫0 𝑠𝑡(𝑡)𝑑𝑡 ∫0 𝐺(𝑡)𝑑𝑡 ― 𝑦 ∗ ∫0 𝑚𝑡(𝑡)𝑑𝑡 ∫0 𝑉(𝑡)𝑑𝑡 𝑇∗𝐷

𝑦 ∗ (2𝑑0𝑇 ∗ + 𝑑1𝑇 ∗ )(2 ― 𝑏 + 𝑎) 2

=

2(𝑏 ― 𝑎)𝑇 ∗ 𝐷

(32)

Based on Equation (32), if the inventory managers of EGQ systems intend to achieve the desired status of UGF, they can increase the ordered items and shorten the slaughter cycle length, simultaneously. 5.2. Case 2: (𝑮(𝒕) = 𝒅𝟎 + 𝒅𝟏𝒕) ≠ (𝑽(𝒕) = 𝒛𝟎 + 𝒛𝟏𝒕) In Case (1), we assumed the same growth function for live and dead items. The growing items may have different growth functions of live and dead items. We therefore consider the more general scenario in which the growth functions of live and dead items are different. 5.2.1. The Economic Growth/Slaughter Cycle Similar to Case (1) and based on assumed probability and growth functions in the previous section, Equation (10) can be reformulated for Case (2) as follow:

(

𝑇𝐶𝑈= 𝑦

𝑤0𝑝 𝑇

(

+ (ℎ + 𝑓 + 𝑘 + 𝑚)

𝑧1𝑇

)(𝑧0 + ) + (

𝑏―𝑎―1 𝑏―𝑎

2

)(𝑑0 + ))

ℎ+𝑓+𝑘 𝑏―𝑎

𝑑1𝑇 2

(33) 14

Journal Pre-proof The total cost function in Equation (33) is a convex function (See Appendix B for a proof). Similar to Case (1), we can obtain (𝑇 ∗ ) in a closed-form status using the derivation of the Equation (33) with respect to T. 𝑇∗ =

2𝑤0𝑝(𝑏 ― 𝑎)

(34)

(𝑧1(ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + 𝑑1(ℎ + 𝑓 + 𝑘))

5.2.2. The Economic Growing Quantity (EGQ) The final weight of live items should equal to demand as follows:

(∫

𝑇∗

𝐷 = 𝑊𝑇 = 𝑦

0

(

)

𝑇∗

)∫

𝑏―𝑎―1 𝐺(𝑡)𝑑𝑡 ― 𝑏―𝑎

𝑉(𝑡) 𝑑𝑡

0

(35)

By replacing the growth functions of live and dead grown items with Equation (35) and after some simplifications, the (EGQ) for Case (2) can be calculated by Equation (36) as follow:

𝐸𝐺𝑄 = 𝑦 ∗ = 𝑄𝐴𝑏𝑒𝑚 =

2𝐷(𝑏 ― 𝑎) (𝑏 ― 𝑎)(2𝑑0𝑇 ∗ + 𝑑1𝑇 ∗

) ― (𝑏 ― 𝑎 ― 1)(2𝑧0𝑇 ∗ + 𝑧1𝑇 ∗ 2)

2

(36)

As stated earlier, for some growing items, which cannot be ordered based on quantity, we can place an order based on the initial economic weight. Accordingly, the initial optimum weight per growth cycle can be ordered based on Equation (37). 2𝑤0𝐷(𝑏 ― 𝑎)

∗

𝑊 ∗ = 𝑤0𝑦 =

(𝑏 ― 𝑎)(2𝑑0𝑇 ∗ + 𝑑1𝑇 ∗

) ― (𝑏 ― 𝑎 ― 1)(2𝑧0𝑇 ∗ + 𝑧1𝑇 ∗ 2)

2

(37)

5.2.3. The Optimum Total Cost By substituting (T*) and (y*) in Equation (33), the optimum total cost for Case (2) can be obtained as follow: ∗

𝑇𝐶𝑈 = y

∗

(

𝑤0𝑝 𝑇∗

(

+ (ℎ + 𝑓 + 𝑘 + 𝑚)

𝑧1𝑇 ∗

)(𝑧0 + ) + (

b―a―1 b―a

2

)(𝑑0 + ))

ℎ+𝑓+𝑘 b―a

𝑑1𝑇 ∗ 2

(38)

5.2.4. The Reorder Points (ROPs) The ROP for EGQ inventory system can be formulated, not only based on the weight of live items, but also based on the weight of dead items. The 𝑅𝑂𝑃𝑙∗ and 𝑅𝑂𝑃𝑑∗ for Case (2) can be obtained by: 𝑅𝑂𝑃𝑙∗ =

2

2𝑑0(𝑇 ∗ ― 𝐿) + 𝑑1(𝑇 ∗ ― 𝐿)

( )( 𝑦∗ b―a

2

)

(39)

15

Journal Pre-proof

(

𝑅𝑂𝑃𝑑∗ = 𝑦 ∗

)(

b―a―1 b―a

)= (

𝑇∗ ― 𝐿 𝑧0 + 𝑧1𝑡 𝑑𝑡 ∫0 .

2

2𝑧0(𝑇 ∗ ― 𝐿) + 𝑧1(𝑇 ∗ ― 𝐿)

)(

𝑏―𝑎―1 𝑦∗ 𝑏―𝑎

2

)

(40) 5.2.5. The Average Inventory Similar to Case (1), the optimal status of average inventory is: ∗

𝐼 =𝑦

∗

2𝑤0(𝑏 ― 𝑎) + (𝑏 ― 𝑎)(2𝑑0𝑇 ∗ + 𝑑1𝑇 ∗

(

) ― (𝑏 ― 𝑎 ― 1)(2𝑧0𝑇 ∗ + 𝑧1𝑇 ∗ 2)

2

4(𝑏 ― 𝑎)

)

(41)

5.2.6. The Optimum Weight of Slaughtering The optimum weight of slaughtering can be obtained by dividing the demand to the number of live items as shown in Equation (42). 𝑊𝑆∗ =

𝐷 𝑇∗

𝑦 ∗ ∫0 𝑠𝑡(𝑡) 𝑑𝑡

=

𝐷

( )

1 𝑦∗ b―a

=

𝐷(𝑏 ― 𝑎) 𝑦∗

(42)

5.2.7. Utility of Growth Functions (UGF) The UGF[0, T] for Case (2) can be calculated by Equation (43), which engages the optimum values of (T) and (y), the different growth functions, and the probability functions of survival and mortality. 𝑦∗

((2𝑑0𝑇 ∗ + 𝑑1𝑇 ∗ 2) ― (𝑏 ― 𝑎 ― 1)(2𝑧0𝑇 ∗ + 𝑧1𝑇 ∗ 2))

UGF[0, T] =

2(𝑏 ― 𝑎)𝑇 ∗ 𝐷

(43)

6. Numerical Analyses In this section, we solve some numerical examples to show the application of our model and the validity of our solutions. Then, we perform some sensitivity analyses to better understand the behavior of EGQ inventory system and to provide the strategic implications and insights of our model. 6.1. Numerical Examples Rezaei (2014) analyzed an EOQ inventory model of male broiler chickens as a specific type of growing items. We applied the same values of his example for the parameters of mentioned item. However, we added some new values for new defined parameters of our model. Moreover, we changed some values of his example because of the different nature of our defined parameters. Finally, we utilized the Polynomial growth model with the first order of fit of male broiler chickens (Tompic, Dobsa, Legen, Tompic, & Medic, 16

Journal Pre-proof 2011). Accordingly, our numerical example considers a poultry company, which orders and buys the male broiler 45-gram chickens. The chickens are delivered one week after ordering process. Then, the company feeds/grows the chickens in order to fatten them until they reach a targeted weight for slaughtering at the end of growth cycle, when the market demand should be satisfied, completely. The chicken mortalities occur during the fattening process, while the company should dispose the carcasses from the farm. The company is looking for the optimal values of ordered chicks, cycle length, and slaughtering weight. Moreover, the necessity of ROP is inevitable for poultry company, because it reflects the inventory level of company that triggers the placement of an order of chicks for next cycle (Chen, 1998). In this regards, an EGQ inventory system of mentioned company is modeled and optimized with the following parameters: 𝑤0= 45 (gram); ℎ= 3.83562E-05 (€/gram/week); 𝑝= 0.04 (€/gram); 𝑓= 0.000698082 (€/gram/week); 𝐷= 1,917,808 (gram); 𝑚= 0.00005 (€/gram/week); 𝑑0= 357.067; 𝑑1= 124.671; 𝑧0= 150.612; 𝑧1= 119.277; 𝑘= 0.000055 (€/gram/week); 𝑎= 9.9; 𝑏= 11; 𝐿= 1 (week). By considering the Equations (17), (20), (21), (22), (26), (27), (29), (30), and (32), the optimum values for Case (1) are as follows: 𝑇𝐶𝑈 ∗ =422 (€); 𝑇 ∗ = 6.0230 (Weeks); 𝐸𝐺𝑄(𝑄𝐴𝑏𝑒𝑚) = 478 (Chickens); 𝑊 ∗ = 21,517 (Gram); 𝑅𝑂𝑃𝑙∗ = 1,463,292 (Gram); 𝑅𝑂𝑃𝑑∗ = 146,329 (Gram); 𝐼 ∗ = 969,662 (Gram); 𝑊𝑆= 4,412 (Gram); 𝑈𝐺𝐹= + 0.149

The results of optimization mean that we can meet the demand by placing an order of 478 chickens, which are to be fed for 6.0230 weeks as optimal slaughter time, when the weight of chickens reaches to 4,412 gram. In other words, we should place an order of 21,517 grams of 45-gram chickens per cycle and sell/slaughter them when their weights reach to 4,412 grams. Moreover, we should replace the new order of the next cycle when the weight of live items reaches to 1,463,292 grams or when the weight of dead items reaches to 146,329 grams. Accordingly, the optimum total cost of the EGQ system is 422 €, and the optimal level of average inventory is 969,662 grams. The proximity of UGF value to +1/(𝑇) indicates the high quality and proper utility of growth functions. We can also calculate the optimum values for Case (2) as follows:

17

Journal Pre-proof 𝑇𝐶𝑈 ∗ =400 (Euro); 𝑇 ∗ = 6.0356 (Weeks); 𝐸𝐺𝑄(𝑄𝐴𝑏𝑒𝑚) = 463 (Chickens); 𝑊 ∗ = 20,817 (Gram); 𝑅𝑂𝑃𝑙∗ = 1,420,885 (Gram); 𝑅𝑂𝑃𝑑∗ = 95,492 (Gram); 𝐼 ∗ = 969,312 (Gram); 𝑊𝑆= 4,560 (Gram); 𝑈𝐺𝐹= + 0.150

The related plots to 𝑇𝐶𝑈 for Cases (1) and (2) based on the decision variables (𝑇) and (𝑦) are illustrated in Figure (2). Please Insert Figure (2) Our poultry example differs from other models in similar problems (Rezaei, 2014; Zhang, Li, Tian, & Feng, 2016; Nobil, Sedigh, & Cárdenas-Barrón, 2018; Sebatjane & Adetunji, 2018) because of six streams of novelties: considering the mortality of poultries by defining the probability density functions of survival/mortality of poultries, considering the disposal cost of poultry carcasses by defining the growth function for dead grown poultries, considering the lead time to achieve the economic ROP as a vital variable in case of high demand products such as poultry industries, evaluation of the quality of growth functions to improve/modify/adjust the growth patterns of poultries for next cycles, considering the costs for both live and dead poultries in order to adapt to the real nature of poultry models, and finally considering the average inventory to obtain the inventory turnover of poultry company. 6.2. Sensitivity Analyses 6.2.1. Sensitivity Analysis of Parameters Based on Table (2), the change rates in holding cost ℎ, disposal cost 𝑚, and setup cost 𝑘 of the EGQ model have very insignificant effects on optimal status of decision variables and dependent variables for both Cases (1) and (2). In this regards, the change effects of the initial weight 𝑤0, feeding cost 𝑓, and costs of purchasing process 𝑝 on optimum values of variables, including 𝑇, 𝑦, 𝑇𝐶𝑈, 𝑅𝑂𝑃𝑙, 𝑅𝑂𝑃𝑑, 𝑊𝑠, 𝑊, and 𝑈𝐺𝐹 are examined with incremental steps of one tenth of an interval [0.5, 1.5]. Please Insert Table (2) The following observations are extracted based on Table (2): 

The decision variables and dependent variables of the EGQ model are most sensitive to initial weight of items, feeding cost, and purchasing process costs for both Cases (1) and (2). Moreover, the change patterns of initial weight 𝑤0 and costs of purchasing process 𝑝 are same. As the change rates of 18

Journal Pre-proof parameters 𝑤0, 𝑝 decrease, the change rates of 𝑇𝐶𝑈, 𝑦, and 𝑈𝐺𝐹 increase and vice versa. On the other hand, when the change rates of parameters 𝑤0, 𝑝 increase, the change rates of 𝑊𝑠, 𝑇, 𝑅𝑂𝑃𝑙, and 𝑅𝑂𝑃𝑑 increase and vice versa. Figure (3) shows the change rates of the most important variables due to change rates of parameters 𝑤0, 𝑝 for Case (1), which has the same behavior and shape as Case (2) with very insignificant differences. Please Insert Figure (3) 

Large decrease rates of feeding cost result in large increase rates of 𝑊𝑠 and 𝑇 variables and vice versa. Moreover, as the change rate of parameter 𝑓 increases, the change rates of 𝑇𝐶𝑈, 𝑦, and 𝑈𝐺𝐹 increase and vice versa. Same as parameters 𝑤0 and 𝑝, the intensity of change rates of variables in the decremental state of parameter 𝑓 is greater than its incremental state. Figure (4) shows the change rates of the most important variables due to change rates of parameter 𝑓 for Case (2), which has the same behavior and shape as the Case (1). Please Insert Figure (4)

6.2.2. Simultaneous Sensitivity Analysis of Decision Variables Based on the obtained results of simultaneous sensitivity analyses of decision variables, the effects of the simultaneous changes of the both variables 𝑇, and 𝑦 on dependent variables of the EGQ model are much more significant and impressive than the parameter effects. Accordingly, this subsection of the sensitivity analyses focuses on investigating the simultaneous changes of the decision variables 𝑇, 𝑦 on main dependent variables, including 𝑊, 𝑈𝐺𝐹, 𝑅𝑂𝑃𝑙, 𝑅𝑂𝑃𝑑, 𝑊𝑠, and 𝑇𝐶𝑈 for both Cases (1) and (2). The following observations are extracted based on related drawn surfaces to each one of the simultaneous sensitivity analyses: 

Based on Figure (5), when the change rates of variable 𝑦 increase and the change rates of variable 𝑇 decrease simultaneously, the change rates of 𝑊 and 𝑈𝐺𝐹 increase and vice versa. Please Insert Figure (5)



Based on Figure (6), when the change rates of variable 𝑦 increase and the change rates of variable 𝑇 increase simultaneously, the change rates of 𝑅𝑂𝑃𝑙 and 𝑅𝑂𝑃𝑑 increase and vice versa. 19

Journal Pre-proof Please Insert Figure (6) 

Based on Figure (7), when the change rates of variable 𝑦 decrease and the change rates of variable 𝑇 increase simultaneously, the change rates of 𝑊𝑠 increase and vice versa.



Based on Figure (7), when the change rates of variable 𝑦 is decremental and the change rates of variable 𝑇 is incremental simultaneously, the change rates of 𝑇𝐶𝑈 will be decremental and vice versa. Please Insert Figure (7)

The behavior of Case (1) is same as Case (2) with very insignificant differences.

7. Managerial Implications and Insights In this section, the managerial implications and insights are provided based on the results of sensitivity analyses. Our goal is to provide some practical recommendations for decision makers of EGQ systems and agriculture industries. The following recommendations are valid for both Cases, where the growth function of live items is/isn't equal to the growth function of dead grown items: 

When the decision makers are faced with low utility of growth functions for a growing item, they are encouraged to simultaneously increase the ordered items and shorten the slaughter cycle length. Moreover, they can order items with a lesser initial weight at the beginning of each growth cycle in order to enhance the utility of growth functions.



If the EGQ managers have to satisfy a limitation on the minimum weight of slaughtering, which is usually determined by buyers, it is recommended to simultaneously decrease the number of ordered items and lengthen the slaughter cycle. Furthermore, it is suggested to order higher initial weight of items in order to increase the weight of slaughtering.



When the lead time decreases, it is suggested to both place a smaller quantity of newborn items with higher initial weights and lengthen the growth cycle length simultaneously, along with increasing the ROPs and the weight of slaughtering. However, mentioned policies may decrease the utility, because of the increase in the weight of slaughtering. 20

Journal Pre-proof 

When the feeding cost decreases, it is recommended to slaughter the grown items at higher weights and order a smaller quantity of items along with increasing the ROPs to higher weights in order to decrease the total cost of the EGQ system.



If the costs of purchasing process increase, it is recommended to place a smaller quantity of newborn items and prolong the growth cycle length in order to decrease the total cost of system.



If the decision maker's objective is to satisfy a limitation on minimum initial weight of new born items, which is usually determined by suppliers, they should decrease the ordered items and prolong the growth cycle, simultaneously.

8. Conclusion and Future Research In this paper, the Economic Growing Quantity (EGQ) model, as a new generation of inventory models is designed, formulated and optimized. The EGQ inventory model can be utilized as a practical model by agricultural industries with growing items. The optimal decision variables were derived by minimizing the total inventory costs. Moreover, the dependent variables of the model were derived including 1) the ROP, 2) the weight of slaughtering, 3) the UGF, and, 4) the average inventory. Considering both the probability density functions of survival and mortality and the growth functions of live and dead grown items is one aspect of our model, which makes it suitable for different types of agricultural industries with growing items. The results of the sensitivity analyses demonstrated that the decision variables and dependent variables are most sensitive to three parameters, including the initial weight, the feeding cost, and the costs of purchasing process. As a future research direction, we are interested to work on EGQ models with reproducible and profitable byproducts under shortage conditions. Considering shortages and byproducts is expected to enrich the proposed EGQ model and make it more applicable to a wider range of industries. Acknowledgments

21

Journal Pre-proof The authors are thankful to anonymous reviewers for meticulously reviewing the manuscript and helpful suggestions, which helped to improve the paper. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Reference Bakker, M., Riezebos, J., & Teunter, R. H. (2012). Review of inventory systems with deterioration since 2001. European Journal of Operational Research, 221(2), 275–284. doi:10.1016/j.ejor.2012.03.004 Chen, F. (1998). Echelon Reorder Points, Installation Reorder Points, and the Value of Centralized Demand Information. Management Science, 44(12-part-2), S221–S234. doi:10.1287/mnsc.44.12.s221 Covert, R. P., & Philip, G. C. (1973). An EOQ model for items with Weibull distribution deterioration. AIIE Transactions, 5(4), 323-326. doi:10.1080/05695557308974918 Ersoy, I. E., Mendeş, M., & Keskin, S. (2007). Estimation of parameters of linear and nonlinear growth curve models at early growth stage in California Turkeys. Arch. Geflügelk, 71(4), 175-180. Ghare, P. M., & Schrader, G. F. (1963). A Model for an Exponential Decaying Inventory. Journal of Industrial Engineering, 14, 238-243. Glock, C. H. (2012). Lead time reduction strategies in a single-vendor–single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. International Journal of Production Economics, 136(1), 37–44. doi:10.1016/j.ijpe.2011 Goyal, S. K., & Giri, B. C. (2001). Recent trends in modeling of deteriorating inventory. European Journal of Operational Research, 134(1), 1–16. doi:10.1016/s0377-2217(00)00248-4 Guchhait, P., Maiti, M. K., & Maiti, M. (2013). Production-inventory models for a damageable item with variable demands and inventory costs in an imperfect production process. International Journal of Production Economics, 144(1), 180-188. doi:10.1016/j.ijpe.2013.02.002 Hadeler, K. P. (1974). Mathematik Fur Biologen (Heidelberger Taschenbucher). Springer Berlin Heidelberg Publication. Hadley, G., & Whitin, T. M. (1963). Analysis of inventory systems. Englewood Cliffs, N. J. : PrenticeHall. Harris, F. W. (1913). How many parts to make at once Factory. The Magazine of Management, 102, 135136. Hwang, H. S. (1999). Inventory models for both deteriorating and ameliorating items. Computers & Industrial Engineering, 37(1-2), 257–260. doi:10.1016/s0360-8352(99)00068-6 22

Journal Pre-proof Kimball, A. W. (1960). Estimation of Mortality Intensities in Animal Experiments. Biometrics, 16(4), 505. doi:10.2307/2527758 Mahata, G. C., & De, S. K. (2016). An EOQ inventory system of ameliorating items for price dependent demand rate under retailer partial trade credit policy. OPSEARCH, 53(4), 889–916. doi:10.1007/s12597-016-0252-y Matsuyama, K. (1995). Inventory policy with time-dependent setup cost. International journal of production economics, 42(2), 149-160. doi:10.1016/0925-5273(95)00196-4 Matsuyama, K. (2001). The EOQ-Models modified by introducing discount of purchase price or increase of setup cost. International Journal of Production Economics, 73(1), 83-99. doi:10.1016/S09255273(00)00181-X Mondal, B., Bhunia, A. K., & Maiti, M. (2003). An inventory system of ameliorating items for price dependent demand rate. Computers & Industrial Engineering, 45(3), 443-456. doi:10.1016/s03608352(03)00030-5 Nobil, A. H., Sedigh, A. H., & Cárdenas-Barrón, L. E. (2018). A Generalized Economic Order Quantity Inventory Model with Shortage: Case Study of a Poultry Farmer. Arabian Journal for Science and Engineering. doi:10.1007/s13369-018-3322-z Noblesse, A. M., Boute, R. N., Lambrecht, M. R., & Van Houdt, B. (2014). Lot sizing and lead time decisions in production/inventory systems. International Journal of Production Economics, 155, 351–360. doi:10.1016/j.ijpe.2014.04.027 Ravindranathan, N. (1994). On a Statistical Model of Mortality in Chicken. Biometrical Journal, 36(3), 353–361. doi:10.1002/bimj.4710360313 Reynolds, D. (1999). Inventory-Turnover Analysis. Cornell Hotel and Restaurant Administration Quarterly, 40(2), 54–58. doi:10.1177/001088049904000217 Rezaei, J. (2014). Economic order quantity for growing items. International Journal of Production Economics, 155, 109–113. doi:10.1016/j.ijpe.2013.11.026 Sana, S. S. (2010). Demand influenced by enterprises’ initiatives — A multi-item EOQ model of deteriorating and ameliorating items. Mathematical and Computer Modelling, 52(1-2), 284–302. doi:10.1016/j.mcm.2010.02.045 Schwarz, L. B. (1972). Economic Order Quantities for Products with Finite Demand Horizons. AIIE Transactions, 4(3), 234–237. doi:10.1080/05695557208974855 Seal, H. L. (1954). The estimation of mortality and other decremental probabilities. Scandinavian Actuarial Journal, 1954(2), 137–162. doi:10.1080/03461238.1954.10414206 Sebatjane, M., & Adetunji, O. (2018). Economic order quantity model for growing items with imperfect quality. Operations Research Perspectives. doi:10.1016/j.orp.2018.11.004

23

Journal Pre-proof Sivashankari, C. K., & Panayappan, S. (2014). Production inventory model for two-level production with deteriorative items and shortages. The International Journal of Advanced Manufacturing Technology, 76(9-12), 2003–2014. doi:10.1007/s00170-014-6259-8 Tabler, G. T., Berry, I. L., Xin, H., & Barton, T. L. (2002). Spatial Distribution of Death Losses in Broiler Flocks. The Journal of Applied Poultry Research, 11(4), 388–396. doi:10.1093/japr/11.4.388 Taft, E. W. (1918). The most economical production lot. Iron Age, 101(18), 1410-1412. Tompic, T., Dobsa, J., Legen, S., Tompic, N., & Medic, H. (2011). Modeling the growth pattern of inseason and off-season Ross 308 broiler breeder flocks. Poultry Science, 90(12), 2879–2887. doi:10.3382/ps.2010-01301 Tong, Q., Du, X. P., Hu, Z. F., Cui, L. Y., & Wang, H. B. (2018). Modelling the growth of the brown frog (Rana dybowskii). PeerJ, 6, e4587. Tzeng, R. Y., & Becker, W. A. (1981). Growth Patterns of Body and Abdominal Fat Weights in Male Broiler Chickens. Poultry Science, 60(6), 1101–1106. doi:10.3382/ps.0601101 Weon, B. M. (2004). Complementarity between survival and mortality. arXiv preprint q-bio/0403017. Zhang, Y., Li, L., Tian, X., & Feng, C. (2016). Inventory management research for growing items with carbon-constrained. 2016 35th Chinese Control Conference (CCC). doi:10.1109/chicc.2016.7554880

Appendix (A): Convexity Proof of TC Function for Case (1): The Hessian matrix for Case (1) is as follow:

𝐻𝑒𝑠𝑠𝑖𝑎𝑛 =

( )( ( )

|

2𝑦𝑤0𝑝

―𝑤0𝑝

3

𝑇 2𝑤0𝑝

2

𝑇

+

)

𝑑1

( )( )

1 ((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘)) 2 𝑏―𝑎 0

𝑇3

|

Eq. (A.1)

Since the (T) as growth cycle and (y) as order quantity cannot have negative values, so we will have: 2𝑦𝑤0𝑝 𝑇3

0

Eq. (A.2)

Then, we should calculate the Hessian determinant as follow: |𝐻𝑒𝑠𝑠𝑖𝑎𝑛| =

(

2(𝑤0𝑝)2 5

𝑇

―

( )( )( ) 2𝑤0𝑝 𝑑1 3

𝑇

)

1 ((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘)) 2 𝑏―𝑎

Eq. (A.3)

The Hessian matrix will be positive semi-definite and also the TC function will be convex, if we have:

24

Journal Pre-proof

(

2(𝑤0𝑝)2 𝑇5

―

1 ( ( 𝑇 )( 2 ) 𝑏 ― 𝑎)((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘))) ≥ 0 2𝑤0𝑝 𝑑1 3

Eq. (A.4)

After some simplifications, we have: 𝑇2 ≤

2𝑤0𝑝(𝑏 ― 𝑎) 𝑑1((ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + (ℎ + 𝑓 + 𝑘))

= 𝑇∗2

Eq. (A.5)

The right hand side of the above Equation is exactly equal to T ∗ 2 in Case (1). Accordingly the TC function of Case (1) is convex.

Appendix (B): Convexity Proof of TC Function for Case (2): The Hessian matrix for Case (2) is as follow: 𝐻𝑒𝑠𝑠𝑖𝑎𝑛 =

( )( ( )

|

2𝑦𝑤0𝑝

―𝑤0𝑝

3

𝑇 2𝑤0𝑝)

2

𝑇

+

(

)

)

1 (𝑧 (ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + 𝑑1(ℎ + 𝑓 + 𝑘)) 2(𝑏 ― 𝑎) 1 0

𝑇3

|

Eq. (B.1)

Since the (T) as growth cycle and (y) as order quantity cannot have negative values, so we will have: 2𝑦𝑤0𝑝

0

𝑇3

Eq. (B.2)

Then, we should calculate the Hessian determinant as follow: |𝐻𝑒𝑠𝑠𝑖𝑎𝑛| =

(

2(𝑤0𝑝)2 5

𝑇

―

( )( 2𝑤0𝑝 3

𝑇

)

)

1 (𝑧 (ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + 𝑑1(ℎ + 𝑓 + 𝑘)) 2(𝑏 ― 𝑎) 1

Eq. (B.3)

The Hessian matrix will be positive semi-definite and also the TC function will be convex, if we have:

(

2(𝑤0𝑝)2 𝑇5

―

( )( 2𝑤0𝑝 𝑇3

)

)

1 (𝑧 (ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + 𝑑1(ℎ + 𝑓 + 𝑘)) ≥ 0 2(𝑏 ― 𝑎) 1

Eq. (B.4)

After some simplifications, we will have:

(

2(𝑤0𝑝)2 ≥ 𝑇2(2𝑤0𝑝)

)

1 (𝑧 (ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + 𝑑1(ℎ + 𝑓 + 𝑘)) 2(𝑏 ― 𝑎) 1

Eq. (B.5)

Finally, we will have: 𝑇2 ≤

2𝑤0𝑝(𝑏 ― 𝑎)

(𝑧1(ℎ + 𝑓 + 𝑘 + 𝑚)(𝑏 ― 𝑎 ― 1) + 𝑑1(ℎ + 𝑓 + 𝑘))

= 𝑇∗2

Eq. (B.6) 25

Journal Pre-proof The right hand side of the above Equation is exactly equal to T ∗ 2 in Case (2). Accordingly the TC function of Case (2) is convex.

26

Journal Pre-proof

Figure (1): Behavior of the EGQ inventory model over time, when G(t)≠ V(t) (Without shortage)

Figure (2): The related plots to 𝑇𝐶𝑈 for Cases (1) and (2) according to 𝑦 and 𝑇

Figure (3): Change rates of variables and unknowns due to change rates of Parameters 𝑝, 𝑤0

Journal Pre-proof

Figure (4): Change rates of variables and unknowns due to change rates of Parameter 𝑓

𝑇 ∗ and 𝑦 ∗ due to change rates of 𝑈𝐺𝐹 and 𝑊Figure (5): Change rates of

Figure (6): Change rates of 𝑅𝑂𝑃𝑙 and 𝑅𝑂𝑃𝑑 due to change rates of 𝑦 ∗ and 𝑇 ∗

Journal Pre-proof

Figure (7): Change rates of 𝑊𝑆 and 𝑇𝐶𝑈 due to change rates of 𝑦 ∗ and 𝑇 ∗

Journal Pre-proof 

Economic Growing Quantity for Growing Items



Utility of Growth Functions of Live and Dead Grown Items



Probability Density Functions of Survival and Mortality



Reorder Points Based on Weight of Live and Dead Items



Economic Growth Cycle and Economic Weight of Slaughtering

Journal Pre-proof

Table (1). Comparison among EGQ model and relevant inventory models

Growing

*

*

*

*

Mortality / Live and dead items Order lead time The main assumptions

*

*

*

*

*

*

*

*

* *

* *

*

*

*

Disposal of dead items Gradual sell

*

*

Instantaneous sell

* *

*

* * *

*

*

*

Gradual replenishment Instantaneous replenishment The economic quantity

* *

* *

*

* * *

The economic growth cycle Decision variables/ Dependent variables

*

*

*

*

*

*

*

*

* *

* *

* *

* *

ROPs based on weight of live and dead items The average inventory

*

*

Utility of Growth Functions (UGF)

* *

Shortage quantity Purchasing and production costs

*

*

*

*

* *

*

*

*

*

*

*

*

* *

* *

* *

* *

Feeding costs for dead grown items Holding costs of live items

*

*

*

*

*

*

*

*

*

Holding costs of dead grown items

* *

Disposal cost Setup costs

*

Heuristic Solution methodology

* * * * * * *

*

*

*

* * * * * * *

*

Exact Differentiation/ Closed-Form

* * * *

*

Feeding costs for live items The main terms of objective function

Our Model

Sebatjane et al. (2018)

*

* *

Nobil et al. (2018)

Ameliorating

* *

Zhang et al. (2016)

*

Rezaei (2014)

*

Sana (2010)

Deterioration/ Perishable

Mondal et al. (2003)

*

Hwang (1999)

*

Covert et al. (1973)

Product type

*

Ghare et al. (1963)

Regular

Hadley et al. (1963)

Comparison sub-criteria

Taft (1918)

Comparison criteria

Harris (1913)

The conducted works

*

*

*

*

Journal Pre-proof

Table (2). Sensitivity Analyses: Change rates of variables and unknowns due to change rates of Parameters Parameter

𝑤0

𝑓

𝑝

ℎ

𝑚

𝑘

𝑎, 𝑏

Change Rates 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

𝑇

𝑦

𝑇𝐶𝑈

𝑅𝑂𝑃𝑙

𝑅𝑂𝑃𝑑

𝑤𝑠

𝑊

𝑈𝐺𝐹

case 1

case 2

case 1

case 2

case 1

case 2

case 1

case 2

case 1

case 2

case 1

case 2

case 1

case 2

case 1

case 2

0.7071 0.7746 0.8367 0.8944 0.9487 1.0000 1.0488 1.0954 1.1402 1.1832 1.2247 1.3345 1.2411 1.1649 1.1012 1.0470 1.0000 0.9588 0.9224 0.8898 0.8604 0.8338 0.7071 0.7746 0.8367 0.8944 0.9487 1.0000 1.0488 1.0954 1.1402 1.1832 1.2247 1.0123 1.0098 1.0073 1.0049 1.0024 1.0000 0.9976 0.9952 0.9928 0.9905 0.9882 1.0014 1.0011 1.0009 1.0006 1.0003 1.0000 0.9997 0.9994 0.9991 0.9989 0.9986 1.0177 1.0141 1.0105 1.0070 1.0035 1.0000 0.9966 0.9932 0.9898 0.9865 0.9832 1.0298 1.0196 1.0125 1.0072 1.0032 1.0000 0.9974 0.9953 0.9935 0.9919 0.9906

0.7071 0.7746 0.8367 0.8944 0.9487 1.0000 1.0488 1.0954 1.1402 1.1832 1.2247 1.3346 1.2412 1.1650 1.1013 1.0470 1.0000 0.9588 0.9224 0.8898 0.8604 0.8337 0.7071 0.7746 0.8367 0.8944 0.9487 1.0000 1.0488 1.0954 1.1402 1.1832 1.2247 1.0123 1.0098 1.0073 1.0049 1.0024 1.0000 0.9976 0.9952 0.9928 0.9905 0.9882 1.0014 1.0011 1.0008 1.0005 1.0003 1.0000 0.9997 0.9995 0.9992 0.9989 0.9986 1.0177 1.0141 1.0105 1.0070 1.0035 1.0000 0.9966 0.9932 0.9898 0.9865 0.9832 1.0079 1.0052 1.0034 1.0020 1.0009 1.0000 0.9993 0.9987 0.9982 0.9978 0.9974

1.6640 1.4596 1.3044 1.1820 1.0826 1.0000 0.9302 0.8703 0.8183 0.7726 0.7322 0.6397 0.7171 0.7915 0.8633 0.9327 1.0000 1.0654 1.1291 1.1912 1.2518 1.3111 1.6640 1.4596 1.3044 1.1820 1.0826 1.0000 0.9302 0.8703 0.8183 0.7726 0.7322 0.9817 0.9854 0.9890 0.9927 0.9964 1.0000 1.0036 1.0073 1.0109 1.0145 1.0182 0.9978 0.9983 0.9987 0.9991 0.9996 1.0000 1.0004 1.0009 1.0013 1.0017 1.0022 0.9737 0.9790 0.9843 0.9895 0.9948 1.0000 1.0052 1.0104 1.0156 1.0208 1.0260 0.4782 0.5826 0.6870 0.7913 0.8957 1.0000 1.1043 1.2086 1.3129 1.4172 1.5215

1.6569 1.4550 1.3015 1.1804 1.0819 1.0000 0.9307 0.8713 0.8196 0.7742 0.7340 0.6418 0.7190 0.7930 0.8643 0.9332 1.0000 1.0649 1.1280 1.1895 1.2495 1.3082 1.6569 1.4550 1.3015 1.1804 1.0819 1.0000 0.9307 0.8713 0.8196 0.7742 0.7340 0.9819 0.9855 0.9891 0.9928 0.9964 1.0000 1.0036 1.0072 1.0108 1.0144 1.0180 0.9979 0.9984 0.9988 0.9992 0.9996 1.0000 1.0004 1.0008 1.0012 1.0016 1.0021 0.9739 0.9792 0.9844 0.9896 0.9948 1.0000 1.0052 1.0103 1.0155 1.0206 1.0258 0.5894 0.6838 0.7714 0.8529 0.9289 1.0000 1.0666 1.1291 1.1879 1.2433 1.2956

1.3337 1.2366 1.1600 1.0974 1.0449 1.0000 0.9610 0.9266 0.8960 0.8685 0.8437 0.4406 0.5416 0.6485 0.7607 0.8780 1.0000 1.1265 1.2573 1.3922 1.5310 1.6735 1.3337 1.2366 1.1600 1.0974 1.0449 1.0000 0.9610 0.9266 0.8960 0.8685 0.8437 0.9660 0.9728 0.9796 0.9864 0.9932 1.0000 1.0068 1.0137 1.0206 1.0274 1.0343 0.9960 0.9968 0.9976 0.9984 0.9992 1.0000 1.0008 1.0016 1.0024 1.0032 1.0040 0.9514 0.9610 0.9707 0.9805 0.9902 1.0000 1.0098 1.0196 1.0295 1.0394 1.0493 0.4600 0.5679 0.6758 0.7839 0.8919 1.0000 1.1081 1.2162 1.3243 1.4325 1.5406

1.3222 1.2288 1.1549 1.0944 1.0436 1.0000 0.9621 0.9286 0.8988 0.8720 0.8478 0.4432 0.5441 0.6506 0.7623 0.8789 1.0000 1.1255 1.2551 1.3886 1.5258 1.6667 1.3222 1.2288 1.1549 1.0944 1.0436 1.0000 0.9621 0.9286 0.8988 0.8720 0.8478 0.9663 0.9730 0.9797 0.9865 0.9932 1.0000 1.0068 1.0136 1.0204 1.0272 1.0340 0.9966 0.9973 0.9980 0.9986 0.9993 1.0000 1.0007 1.0014 1.0020 1.0027 1.0034 0.9517 0.9613 0.9710 0.9806 0.9903 1.0000 1.0097 1.0195 1.0293 1.0391 1.0489 0.6828 0.7560 0.8238 0.8867 0.9453 1.0000 1.0512 1.0993 1.1444 1.1870 1.2271

0.9025 0.9306 0.9530 0.9713 0.9867 1.0000 1.0116 1.0218 1.0309 1.0391 1.0465 1.0642 1.0493 1.0356 1.0230 1.0111 1.0000 0.9895 0.9795 0.9699 0.9608 0.9520 0.9025 0.9306 0.9530 0.9713 0.9867 1.0000 1.0116 1.0218 1.0309 1.0391 1.0465 1.0030 1.0024 1.0018 1.0012 1.0006 1.0000 0.9994 0.9988 0.9982 0.9976 0.9970 1.0004 1.0003 1.0002 1.0001 1.0001 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 1.0043 1.0034 1.0026 1.0017 1.0009 1.0000 0.9991 0.9983 0.9975 0.9966 0.9958 1.0072 1.0048 1.0030 1.0018 1.0008 1.0000 0.9994 0.9988 0.9984 0.9980 0.9977

0.8986 0.9277 0.9509 0.9700 0.9861 1.0000 1.0122 1.0229 1.0326 1.0412 1.0491 1.0680 1.0521 1.0376 1.0242 1.0117 1.0000 0.9889 0.9785 0.9685 0.9590 0.9498 0.8986 0.9277 0.9509 0.9700 0.9861 1.0000 1.0122 1.0229 1.0326 1.0412 1.0491 1.0031 1.0025 1.0019 1.0013 1.0006 1.0000 0.9994 0.9988 0.9981 0.9975 0.9969 1.0004 1.0003 1.0002 1.0001 1.0001 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 1.0045 1.0036 1.0027 1.0018 1.0009 1.0000 0.9991 0.9982 0.9973 0.9964 0.9956 1.1952 1.1502 1.1086 1.0698 1.0337 1.0000 0.9684 0.9387 0.9109 0.8846 0.8598

0.9025 0.9306 0.9530 0.9713 0.9867 1.0000 1.0116 1.0218 1.0309 1.0391 1.0465 1.0642 1.0493 1.0356 1.0230 1.0111 1.0000 0.9895 0.9795 0.9699 0.9608 0.9520 0.9025 0.9306 0.9530 0.9713 0.9867 1.0000 1.0116 1.0218 1.0309 1.0391 1.0465 1.0030 1.0024 1.0018 1.0012 1.0006 1.0000 0.9994 0.9988 0.9982 0.9976 0.9970 1.0004 1.0003 1.0002 1.0001 1.0001 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 1.0043 1.0034 1.0026 1.0017 1.0009 1.0000 0.9991 0.9983 0.9975 0.9966 0.9958 4.5323 3.4162 2.3070 1.2021 0.1001 1.0000 2.0987 3.1963 4.2930 5.3892 6.4848

0.8238 0.8708 0.9103 0.9441 0.9737 1.0000 1.0236 1.0449 1.0644 1.0823 1.0989 1.1395 1.1052 1.0748 1.0475 1.0227 1.0000 0.9790 0.9596 0.9415 0.9245 0.9085 0.8238 0.8708 0.9103 0.9441 0.9737 1.0000 1.0236 1.0449 1.0644 1.0823 1.0989 1.0061 1.0048 1.0036 1.0024 1.0012 1.0000 0.9988 0.9976 0.9964 0.9953 0.9941 1.0007 1.0005 1.0004 1.0003 1.0001 1.0000 0.9999 0.9997 0.9996 0.9995 0.9993 1.0087 1.0070 1.0052 1.0035 1.0017 1.0000 0.9983 0.9966 0.9949 0.9932 0.9916 5.3882 3.9156 2.5517 1.2844 0.1034 1.0000 2.0333 3.0031 3.9150 4.7742 5.5851

0.6010 0.6851 0.7666 0.8460 0.9237 1.0000 1.0750 1.1490 1.2221 1.2943 1.3658 1.5633 1.3945 1.2634 1.1584 1.0722 1.0000 0.9386 0.8857 0.8395 0.7988 0.7627 0.6010 0.6851 0.7666 0.8460 0.9237 1.0000 1.0750 1.1490 1.2221 1.2943 1.3658 1.0186 1.0148 1.0111 1.0074 1.0037 1.0000 0.9964 0.9928 0.9892 0.9857 0.9822 1.0022 1.0017 1.0013 1.0009 1.0004 1.0000 0.9996 0.9991 0.9987 0.9983 0.9978 1.0270 1.0214 1.0160 1.0106 1.0053 1.0000 0.9948 0.9897 0.9846 0.9796 0.9747

0.6035 0.6873 0.7683 0.8472 0.9243 1.0000 1.0744 1.1477 1.2201 1.2916 1.3624 1.5580 1.3909 1.2611 1.1570 1.0716 1.0000 0.9391 0.8865 0.8407 0.8003 0.7644 0.6035 0.6873 0.7683 0.8472 0.9243 1.0000 1.0744 1.1477 1.2201 1.2916 1.3624 1.0185 1.0147 1.0110 1.0073 1.0036 1.0000 0.9964 0.9928 0.9893 0.9858 0.9823 1.0021 1.0016 1.0012 1.0008 1.0004 1.0000 0.9996 0.9992 0.9988 0.9984 0.9979 1.0268 1.0213 1.0159 1.0105 1.0052 1.0000 0.9948 0.9898 0.9847 0.9798 0.9749

0.8320 0.8758 0.9131 0.9456 0.9743 1.0000 1.0232 1.0444 1.0637 1.0816 1.0982 0.6397 0.7171 0.7915 0.8633 0.9327 1.0000 1.0654 1.1291 1.1912 1.2518 1.3111 0.8320 0.8758 0.9131 0.9456 0.9743 1.0000 1.0232 1.0444 1.0637 1.0816 1.0982 0.9817 0.9854 0.9890 0.9927 0.9964 1.0000 1.0036 1.0073 1.0109 1.0145 1.0182 0.9978 0.9983 0.9987 0.9991 0.9996 1.0000 1.0004 1.0009 1.0013 1.0017 1.0022 0.9737 0.9790 0.9843 0.9895 0.9948 1.0000 1.0052 1.0104 1.0156 1.0208 1.0260

0.8285 0.8730 0.9111 0.9443 0.9737 1.0000 1.0238 1.0455 1.0655 1.0839 1.1010 0.6418 0.7190 0.7930 0.8643 0.9332 1.0000 1.0649 1.1280 1.1895 1.2495 1.3082 0.8285 0.8730 0.9111 0.9443 0.9737 1.0000 1.0238 1.0455 1.0655 1.0839 1.1010 0.9819 0.9855 0.9891 0.9928 0.9964 1.0000 1.0036 1.0072 1.0108 1.0144 1.0180 0.9979 0.9984 0.9988 0.9992 0.9996 1.0000 1.0004 1.0008 1.0012 1.0016 1.0021 0.9739 0.9792 0.9844 0.9896 0.9948 1.0000 1.0052 1.0103 1.0155 1.0206 1.0258

1.4142 1.2910 1.1952 1.1180 1.0541 1.0000 0.9535 0.9129 0.8771 0.8452 0.8165 0.7493 0.8057 0.8584 0.9081 0.9551 1.0000 1.0429 1.0842 1.1239 1.1622 1.1994 1.4142 1.2910 1.1952 1.1180 1.0541 1.0000 0.9535 0.9129 0.8771 0.8452 0.8165 0.9879 0.9903 0.9927 0.9952 0.9976 1.0000 1.0024 1.0048 1.0072 1.0096 1.0120 0.9986 0.9989 0.9991 0.9994 0.9997 1.0000 1.0003 1.0006 1.0009 1.0011 1.0014 0.9826 0.9861 0.9896 0.9931 0.9965 1.0000 1.0034 1.0069 1.0103 1.0137 1.0171

1.4149 1.2915 1.1955 1.1182 1.0542 1.0000 0.9534 0.9128 0.8769 0.8450 0.8163 0.7490 0.8054 0.8582 0.9079 0.9551 1.0000 1.0430 1.0843 1.1241 1.1625 1.1997 1.4149 1.2915 1.1955 1.1182 1.0542 1.0000 0.9534 0.9128 0.8769 0.8450 0.8163 0.9879 0.9903 0.9927 0.9952 0.9976 1.0000 1.0024 1.0048 1.0072 1.0096 1.0120 0.9986 0.9989 0.9992 0.9995 0.9997 1.0000 1.0003 1.0005 1.0008 1.0011 1.0014 0.9825 0.9861 0.9896 0.9931 0.9965 1.0000 1.0035 1.0069 1.0103 1.0137 1.0172

1.0456 1.0298 1.0189 1.0109 1.0048 1.0000 0.9961 0.9929 0.9902 0.9878 0.9859

0.8484 0.8774 0.9074 0.9379 0.9688 1.0000 1.0313 1.0628 1.0944 1.1261 1.1578

0.4782 0.5826 0.6870 0.7913 0.8957 1.0000 1.1043 1.2086 1.3129 1.4172 1.5215

0.5894 0.6838 0.7714 0.8529 0.9289 1.0000 1.0666 1.1291 1.1879 1.2433 1.2956

1.5644 1.4603 1.3498 1.2355 1.1187 1.0000 0.8801 0.7591 0.6375 0.5153 0.3926

1.6713 1.5193 1.3767 1.2430 1.1176 1.0000 0.8895 0.7855 0.6876 0.5951 0.5077