APPLICATION OF ECONOMIC ORDER QUANTITY MODEL FOR THE PURCHASE AND GROWTH OF ORGANIC POULTRY WITH INCREMENTAL QUANTITY DISCOUNTS

ABSTRACT

The purpose of the study is to develop a lot sizing model for growing chickens if the supplier of the products offers incremental quantity discounts. A mathematical model was derived to determine the Optimal Inventory Policy which minimizes the total inventory cost in both the owned and rented facilities. A solution procedure for solving the model is developed and illustrated through a numerical example. Sensitivity Analysis was performed to demonstrate the response of the Order Quantity and total costs to some key parameters. Incremental quantity discounts in reduced purchasing costs. However, ordering very large quantities has downsides as well. The biggest downsides includes the increase holding cost, the risk of running  out  of  storage  capacity  and  item  deterioration  since  the  cycle  time increases if larger quantities are purchased. Owing to the importance of growing items in the food chains, the model presented in this thesis can be used by procurement and inventory managers when making purchasing decisions.

CHAPTER ONE

1.0  INTRODUCTION

1.1 Background to the Study

Economic Order Quantity (EOQ) is the quantity that sets the holding cost and the ordering  cost at equilibrium. It is the quantity of stock that must be ordered in order to minimizes total variable cost.

Growth model methodology has been widely used in the modeling of organisms such a plants  and  animals  growth.  Growth  study  in  many  branches  of  science  have demonstrated that more complex nonlinear functions are justified and required, if the range of the independent variable encompasses juvenile, adolescent, matured and senescent stages of growth (Khamis & Sakomura 2005). The growth potential has been described by the Gompertz function in several studies (Sakomura et al., 2005). The Richards’   models   have   been   shown   to   give   good   descriptions   of   growth (Goonewardene, 2003). The Richards’ function, first introduced in 1959 (Amanulla et al., 2007). The Richard’s model is a more generalized, four parameter function with a variable  inflection  point  that  provides  a  more  complete  description  of  the  growth process.

Inventory  management  is  concerned  with  ensuring  the  right  quantity  of  goods  is available    at the right time that is, when customer demand the goods. The two major decisions in inventory management are the quantity and the timing of the orders. These decisions were first addressed by Harris (1913). Harris proposed a model popularly referred to as the Economic Order Quantity (EOQ), which seeks to balance the fixed cost of ordering items against the variable cost of keeping stock, thereby determining the best quantity to order per procurement cycle. While the basic EOQ model has found some practical applications, it makes a number of assumptions which do not reflect most real-life inventory systems. In order to model more realistic systems, various researchers have revised the classic EOQ model by relaxing the model assumptions in some ways (Holmbom and Segerstedt, 2014). In an attempt to create a new variant of the EOQ model.

In poultry breeding, selection of strain for meat production is based on weight at the age given (Mignom-Gasteau and Beamond, 2016). They suggested that, the improvement of weight at a certain age would alter heavily the entire growth curve and after induce side effect on to fattening stage, the reproduction, the movement troubles or also sexual dimorphism, thereby necessitating consideration of the totality of growth curve.

Many mathematical functions like Richards model (Knizetova et al., 1991). Janoschek model (Gille and Salomon, 1994), logistic model (Grossman and Bohren, 1985), Gompertz model (Barbato, 1991; N’dri et al., 2006), were used for describing growth of poultry.  Indeed,  the mathematical model permits to recap the information in some parameters and strategic points (Knizetova et al, 1997), to describe the range of weights according to age.

Thus it is possible to compare animals at the same physiological stage where the growth speed is maximal, which is not possible to measure through the traditional body weight study (Mignom-Gasteau and Beamond, 2000). Moreover, the non-linear investigation of the growth process has some advantages in not only mathematical explaining growth, but also estimating the relationship between feed requirements and body weight, and plays a crucial role in animal husbandry (Sengul and Kiraz, 2005).

Many  broiler  growth  data  analyses  have  been  conducted  using  the  well-known Gompertz     growth function, which describes a single sigmoidal growth phase (Wang and Zuidhof, 2004). In recent years, there are many studies that have been performed with respect to growth analyses in slow-growing broilers. (Santos et al., 2005 ),used the Gompertz model to analyze growth in two slow-growing broiler lines housed in two different systems. (Dourado et al., 2009), also used the Gompertz model to examine growth of slow- growth broilers reared in the free range system. Indeed, (N’dri et al., 2006), made estimates of genetics parameters for Gompertz model parameters in slow- growing broilers reared in the label range system. It is cleared that Gompertz, Logistic and Richards models were used in the analyses of the growth of living organisms.

Modern poultry breeding would be an interesting solution for mitigating the problem  of animal protein supply in every town having increase demography. Chicken production in developing countries  serve as important source of animal protein and source of income especially for women (Zaman et al., 2004). Many mathematical functions like Richards  model  and  others  were  used  for  describing  growth  of  poultry.  However, despite the growing demand for poultry products, poultry farmers worldwide face numerous problems. In Nigeria and Ghana, for example, poultry farmers have suffered setbacks in poultry production due to rising costs of farm inputs and some other challenges that have hampered the production and growth process.

This work proposes an inventory system where the items being ordered grows during the course of the inventory replenishment cycle and the supplier offers incremental quantity discount.

1.2 Statement of the Research Problem

In recent times, the poultry industry in Nigeria has been experiencing a steep decline in output.  The decline has  been  attributed  to  soaring cost  of production  driven  some farmers out of the business and prospective investors increasingly unwilling to invest in the industry.

Poultry farming occupies a vital place in the economy of Nigeria. As human population increases, the poultry continues to grow to meet the demand for meat and eggs. The significant of poultry production lies in the quality of products that are provided to humans. Some of the factors that are responsible for successful poultry keeping are selection of proper breeds and site, economic housing, feeding and management policy. Our study focuses on inventory management when the unit purchasing cost decreases with the order quantity Q, in other words, a discount is given by a seller if the buyer purchases large number of units. Our objective is to determine the optimal ordering policy for the buyer when dealing with such items. We will discuss two types of quantity discount contracts; all units’ discounts and incremental quantity discounts.

Quantity discounts are usually offered by suppliers as a means of encouraging buyers to purchase larger volumes. Most inventory models which consider incremental quantity discounts assume that demand is deterministic.

1.3 Aim and Objectives

1.3.1 Aim

The aim of this work is to apply Economic Order Quantity model in order to   determine the  optimal number of live newborn items (chicks) at the beginning of the growth cycle as well as to minimize total inventory costs.

1.3.2 Objectives

The objectives are to:

I.  Apply a lot  sizing model  for growing items  if  the supplier of the items  offers incremental quantity discounts.

ii. Determine the optimal inventory policy which minimizes the total inventory cost. iii. Develop solution procedures for solving the model through numerical example.

iv. Economic order quantity with quantity discount will be modeled to achieve optimal level of inventory. This implies cost saving in inventory control and achievement of maximum profit, carrying cost will be reduced to the lowest possible value.

1.4 Significance of the Study

Most price discount models are very useful in food chain. This is because a number of food items like livestock and fish products are greatly influenced by time. It may be necessary to consume the food items within a limited time period, usually the shelf life. This is motivated by the inherent nature of most food items. In addition, most food items are functional products, and for each product categories, profit is usually driven by  sales  volume  rather  than  margins.  This  also  means  that,  most  food  items  are, therefore, produced in volumes in order to take advantage of economy of scale to drive down the unit cost as result of the fairly large over head costs.       This is enough motivation for suppliers within this chain to provide quantity discounts in so many instances so that the food items are moved away from them to the next level of the supply chain as quickly as possible in order to avoid losses due to spoilage and deterioration. It has, however, been observed that there seems to have been no study that has consider  the implication of marginal discount on the lot sizing policy of growing items. This is probably because growing items models in inventory management is relatively young area and researchers are just beginning to study it. Also, it is important to  focus  on  incremental  discount,  because  all  quantity discounts  are  more  straight forward with standard algorithm, and hence, more commonly studied than the marginal discount pricing models.

This study seeks to fill this gap as such lot sizing model may be important for the procurement  manager  in  charge  of  decisions  in  the  supply  chain  of  fresh  items especially, due to the fact that quantity discount is not uncommon in this area.

1.5 Scope and Limitation of the Study

The study covers purchasing and growth of items, such as chickens with incremental quantity discount, the weight of items is determined, the consumption period is used to determine the cycle time (slaughtered age) and the objective function, which is the total cost per unit cycle.

The study is limited to purchasing and growth of items with incremental quantity discounts and the possible price breaks of ordered items.

1.6 Organization of the Study

The study consists of five chapters, with chapter one being the introduction, chapter two  deals with the literature review, that is, the chapter reviews the literature of the past researches that had been conducted by various scholars, chapter three is the methodology(formulation of mathematical model). A numerical example is presented in chapter four illustrating the proposed solution procedure and to provide managerial insights through a sensitivity analysis on the major input parameters, while chapter five contains summary of the findings, conclusions and recommendations.

1.7 Definition of terms

i. Feasibility: is an assessment of the practicality of a proposed plan or method.

ii.  Replenishment  cycle:  A  term  used  in  inventory management  that  describes  the process by which stocks are resupplied from some central location.

iii. Holding cost: is the cost of maintaining inventory in stock. It includes the  interest on capital and cost of storage, maintenance and handling.

iv. Setup cost: Represents the fixed charge incurred when an order (no matter  the size) is  placed.

v. Inventory: Is commonly thought of as the finished goods a company  accumulates before selling them to end users, it can also describe as the raw materials used to produce the finished goods.

vi.  Constraints:  Limitation  or  restriction  (  something  that  imposes  a  limit  or restriction)

vii. Price breaks: It is a reduction in price, especially for bulk purchase.

viii. Optimal quantity: An efficient quantity of items when its marginal    benefit equals its   marginal cost.

ix. Algorithm: An algorithm is a step by step method of solving problems.

x. Discounts: A deduction from the usual cost of items

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