CURRICULUM VITAE/RESUME

P.O BOX 21884,                                                             
UNIVERSITY OF IBADAN,                                         
IBADAN.
Phone Number: 07031078201, 08158328924.
E-mail: olanreleoladeji@yahoo.com

 

OLANRELE OLADEJI OLUNIYI

Objective: To act locally and think globally, utilizing available resources to achieve global
economic scales and scope in a competitive environment.

Sex : Male
Marital Status : Single
Date of Birth : 18/11/1980
Nationality : Nigerian
State of Origin : Oyo State
Languages : English & Yoruba.
Home Town : Eruwa
Home Address : 21/IFL, Beside Adabi’s Mosque, Awotan Area, Apete Ibadan.
Blog Address : http://www.olanreleoladejio.blogspot.com

Education:2006 - 2010 : University of Ibadan, Ibadan.
2001 – 2005 : The Polytechnic Ibadan, Ibadan.
2000 : Sidney Computers, Ibadan.
2000 : Renascent High School, Agugu, Ibadan.
1987 – 1992 : I.M.G School III, Mokola, Ibadan.

Qualifications:2010 : Bachelor of Science Degree in Industrial & Production Engineering.
Second Class Honour (Upper Division).

2005 : National Diploma in Electrical/Electronics Engineering.
Upper Credit Division.

2000 : Certificate in Desktop Publishing.

2000 : Secondary School Leaving Certificate.

2001 : Primary School Leaving Certificate.
 
Professional Membership/Certification: Associate Membership Institute of strategic Management of Nigerian (ISMN).
Graduate Membership Nigerian Society of Engineers (NSE).
Graduate Membership Institute of Productivity and Quality Management (IPQM).
Health, Safety and Environment (HSE) Level 2 Certificate.

ProfessionalExperience/Skills:
March 2011 – March 2012 :Delta State Post Primary Education Board.
Job Description : Teaching (NYSC).

March – August 2009 : European Soaps and Detergents Limited (ESDL).
No. 13E, LSPDC Industrial Estate, Amuwo-Odofin, Lagos.
Job Description : Line Operation/Supervision, Supervision of Saponification Plant, Updating
of daily saponification plant production and Updating of daily stock report using Microsoft Excel.

June – August 2008 : TAT Enginerring
Jericho Idi – Ishin Quarters, Ibadan.
Job Description : Engineering Trainee.

Skills : Proficient in the use of Microsoft office tools, Internet, General computer usage, A team player,Attentive to details, and Deadline oriented.
 
Positions of responsibility:
SERVICOM Community Development Leader (CDS).2011/2012: National Youth Service Corps, Isoko North, Delta state.

Public Relations Officer(PRO).2008/2009 Session : Industrial Engineering Students Association (IESA), University of Ibadan.

Member Editorial Board.2004/2005 Session : Association of Electrical/Electronics Engineering Students, The Polytechnic Ibadan.

Member Electoral Committee.2004/2005 Session : Association of Electrical/Electronics Engineering Students, The Polytechnic Ibadan.

Award:Dean’s Honour Award for High Academic Standing (Performance) - 2008/2009 Academic Session

Hobbies:Researching, Meeting People & Travelling.

Project and Report:

•  Application of dynamic programming to production planning, in an animal feed mill.

•  The dynamics of soap making, a practical perspective.

•  The making of a CGPA CALCULATOR, using Microsoft Excel.

Referees:Engineer. H.I. Anyanwu (FCPQM)
Registrar/Chief Executive,
Institute of Productivity and Quality Management,
Suite 62, New Apapa Plaza,
Off Koffo AbayomiAvenue, Lagos.
08037257700, 08023216199
instituteofproductivity@yahoo.com
hyanyanwu@yahoo.com

Dr. C.O Anyaeche
Department of Industrial & Production Engineering,
Faculty of Engineering,
University of Ibadan,
Ibadan.
08033356609.
Osita.anyaeche@mail.ui.edu.ng

Mr. A.D Adeyeye
Department of Industrial & Production Engineering,
Faculty of Technology, University of Ibadan.
08029324164.
ademola.adeyeye@mail.ui.edu.ng.
adadeyeye@yahoo.com.

APPLICATIONS OF INDUSTRIAL ENGINEERING IN A MANUFACTURING SYSTEM.

AN ARTICLE WRITTEN TO BE PUBLISHED IN INDUSTRIAL ENGINEERING STUDENT ASSOCIATION (IESA) MAGAZINE.

APPLICATIONS OF INDUSTRIAL ENGINEERING IN A MANUFACTURING SYSTEM.

Various industrial engineering processes are been employed mostly in the developed world and it has since brought about a great appreciation in their systems. Corporations like Walt Disney Wonderland, Pfizer, General Motor and General Electric have extensively implement various industrial engineering tools which synchronized  development, improvement, implementation and evaluation of integrated systems of people, money, knowledge, information, equipment, energy, materials, analysis and synthesis, as well as the mathematical, physical and social sciences together with the principles and methods of engineering design to specify, predict, and evaluate the results to be obtained from such systems or processes.

Manufacturing is the use of machines, tools and labour to produce goods for use or sale. The term may refer to a range of human activity, from handicraft to high technological application, but is most commonly applied to industrial production, in which raw materials are transformed into finished goods on a large scale. Such finished goods may be used for manufacturing other, more complex products, such as aircraft, household appliances or automobiles, or sold to wholesalers, who in turn sell them to retailers, who then sell them to end users – the "consumers". The manufacturing sector is closely connected with engineering and industrial design. Examples of major manufacturers are General Motors Corporation, General Electric, and Pfizer, Volkswagen Group, Siemens, Michelin, Toyota, Samsung, and Bridgestone.

Industrial engineering is not physically demanding, but frequently takes the Engineer out of the office into production and manufacturing areas. Today, this often means travelling across the country or around the world to the manufacturing site. Industrial Engineers spend much of their time asking questions. They may talk with production workers, as well as technical or administrative staff. It is not unusual for these Engineers to be involved in several projects at once. Therefore, they must be flexible enough to drop one project and pick up another at a moment’s notice.

Much of an Industrial Engineer’s output is used by management for making decisions. As a result, these workers must be accurate; their recommendations may affect the size of their firm’s profits, its labour relations, as well as its productions costs. Because of this, stress may be considerable at times. Industrial Engineers usually work a 40-hour workweek. However, long or irregular hours may be necessary to meet deadlines or when working on special projects.

Industrial Engineers, who may find themselves in manufacturing industries, will be responsible for the following tasks:

ü Operations:

·        Review schedules or forecasts, specifications, and customer requirements to understand what activities, and in what order, things should be done.

·        Develop methods, labour utilization standards, and cost analysis systems for efficient staff and facility operation.

·        Monitor workflow schedules according to established best practices to come up with improved cycle time.

·        Study operations sequence, material flow, functional statements, organization charts, and project information to determine systems (labour, tools, computers) design and workplace layout.

·        Apply statistical methods to determine processes, staff requirements, and production standards.

·        Project system deliveries based on marketing forecasts, supply chain design, storage and handling facilities, and maintenance requirements.

ü Logistics and Distribution (Supply Chain Management):

·        Design methods of transporting goods from one location to another. This could mean locating, designing, and building of warehouses for large national merchandisers so their stores can be stocked on a timely basis. It could mean designing the system of trucks, rail and air to supply parts for assembly or repair (as in the auto industry).

·        Design systems for handling materials from differing transportation modes and redistributing them in a minimum amount of time; for example, long haul trucks, local trucks, air cargo delivery, and containers.

·        Design systems for automated replenishment of stock; such as, scanning a bar coded product in a store triggers a system that orders new stock to be delivered back to that same store.

·        Design systems for the transport of people in a municipal setting, such as rail, bus, and train.

·        Design public facilities, such as parking garages, public transportation stations or centers, for the efficient flow and safety of people.

ü Facilities Planning:

·        Draft and design layout of equipment, materials, and workspace to illustrate maximum efficiency, using drafting tools and computer simulation.

·        Plan and establish sequence of operations to fabricate or assemble parts or products, or service customers, and to promote efficient utilization of resources.

ü Quality Control:

·        Coordinate quality control objectives and activities to resolve production problems, increase product reliability, and minimize cost with partners around the world.

·        Analyze statistical data and product specifications to establish quality and reliability objectives of finished product.

·        Formulate sampling procedures and forms for recording, evaluating, and reporting quality and reliability data.

·        Implement methods for disposition of defective material or parts, and assesses cost and responsibility.

·        Estimate production cost and effect of product design changes for management review, action, and control.

·        Record or oversee recording of information to ensure currency of engineering drawings and documentation of production problems.

·        Direct workers engaged in product measurement, inspection, and testing activities to ensure quality control and reliability.

ü Material management:

Industrial engineer’s are involved in planning and building design for the movement of materials, or with logistics that deal with the tangible components of a supply chain. Specifically, this covers the acquisition of spare parts and replacements, quality control of purchasing and ordering such parts, and the standards involved in ordering, shipping, and warehousing the said parts.

The goal of materials management is to provide an unbroken chain of components for production to manufacture goods on time for the customer base. The materials department is charged with releasing materials to a supply base, ensuring that the materials are delivered on time to the company using the correct carrier. Materials is generally measured by accomplishing on time delivery to the customer, on time delivery from the supply base, attaining a freight budget, inventory shrink management, and inventory accuracy. The materials department is also charged with the responsibility of managing new launches.

In some companies materials management is also charged with the procurement of materials by establishing and managing a supply base. In other companies the procurement and management of the supply base is the responsibility of industrial engineer. The industrial engineer is then responsible for the purchased price variances from the supply base. In large companies with multitudes of customer changes to the final product over the course of a year, industrial engineer is responsible for all new acquisition launches and customer changes. He ensures that the launch materials are procured for production and then transfers the responsibility to the plant materials management.

There are various applications of industrial engineering tools that are essential for everyday planning in the manufacturing system e.g. Lean manufacturing, Agile manufacturing, Flexible manufacturing e.t.c.

-         Lean manufacturing:

Lean manufacturing implementation depends largely on industrial engineer and it is focused on getting the right things to the right place at the right time in the right quantity to achieve perfect work flow, while minimizing waste and being flexible and able to change. These concepts of flexibility and change are principally required to allow production levelling, the flexibility and ability to change are within bounds and not open-ended, and therefore often not expensive capability requirements. More importantly, all of these concepts have to be understood, appreciated, and embraced by the industrial engineer who build the products and therefore own the processes that deliver the value. The cultural and managerial aspects of Lean are possibly more important than the actual tools or methodologies of production itself. There are many examples of Lean tool implementation without sustained benefit, and these are often blamed on weak understanding of Lean throughout the whole organization.

Lean aims to make the work simple enough to understand, do and manage. To achieve these three goals at once there is a belief held by some that Toyota's mentoring process,(loosely called Senpai and Kohai, which is Japanese for senior and junior), is one of the best ways to foster Lean Thinking up and down the organizational structure. This is the process undertaken by Toyota as it helps its suppliers improve their own production. The closest equivalent to Toyota's mentoring process is the concept of "Lean Sensei," which encourages companies, organizations, and teams to seek outside, third-party experts, who can provide unbiased advice and coaching and this leads to the concept of project outsourcing.

-         Agile manufacturing:

It is the duty of an industrial engineer to create the process, tools, and training to enable his employer respond quickly to customer needs and market changes while still controlling costs and quality, and this concept is known as agile manufacturing.

An enabling factor in becoming an agile manufacturer has been the development of manufacturing support technology that allows the marketers, the designers and the production personnel to share a common database of parts and products, to share data on production capacities and problems particularly where small initial problems may have larger downstream effects. It is a general proposition of manufacturing that the cost of correcting quality issues increases as the problem moves downstream, so that it is cheaper to correct quality problems at the earliest possible point in the process.

Agile manufacturing is seen as the next step after LEAN in the evolution of production methodology. In manufacturing, when companies have to decide what to be, they have to look at the Customer Order Cycle (the time the customers are willing to wait) and the leadtime for getting supplies. If the supplier has a short lead time, lean production is possible. If the Customer Order Cycle is short, agile production is beneficial. It is therefore the duty of industrial engineer to find out which one is appropriate for his company and apply accordingly.

-         Flexible manufacturing system (FMS):

A manufacturing system in which there is some amount of flexibility that allows the system to react in the case of changes, whether predicted or unpredicted. This flexibility is generally considered to fall into two categories.

v The first category is machine flexibility and it covers the system's ability to be changed to produce new product types, and ability to change the order of operations executed on a part.

v The second category is called routing flexibility, which consists of the ability to use multiple machines to perform the same operation on a part, as well as the system's ability to absorb large-scale changes, such as in volume, capacity, or capability.

Most flexible manufacturing systems consist of three main systems. The work machines which are often automated CNC machines are connected by a material handling system to optimize parts flow and the central control computer which controls material movements and machine flow. The main advantages of a flexible manufacturing system are its high flexibility in managing manufacturing resources like time and effort in order to manufacture a new product. The best application of a flexible is found in the production of small sets of products like those from a mass production.

OLANRELE OLADEJI. O

Bsc (INDUSTRIAL AND PRODUCTION ENGINEERING)

UNIVERSITY OF IBADAN, IBADAN. (2010).

ND (ELECTRICAL AND ELECTRONICS ENGINEERING)

THE POLYTECHNIC IBADAN (2005).

olanreleoladeji@yahoo.com.

Application of dyamic programming to production planning in an animal feedmill

APPLICATION OF DYNAMIC PROGRAMMING MODEL TO PRODUCTION PLANNING, IN AN ANIMAL FEEDMILL.

BY

OLANRELE, OLADEJI OLUNIYI

MATRIC NO: 134315

SUBMITTED TO THE DEPARTMENT OF INDUSTRIAL AND PRODUCTION ENGINEERING, FACULTY OF TECHNOLOGY, UNIVERSITY OF IBADAN.

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF BACHELOR OF SCIENCE IN INDUSTRIAL AND PRODUCTION ENGINEERING.

OCTOBER 2010.

CERTIFICATION

This is to certify that OLANRELE, OLADEJI OLUNIYI with Matriculation Number 134315 carried out this project in the Department of Industrial and Production Engineering, Faculty of Technology, University of Ibadan, Nigeria.

...................................................                                          .................................................

  DATE                                                       Supervisor

                                                          Mr. A.D Adeyeye

                                                          B.Sc (IFE)., M.Sc (Ibadan).

                                                          Department of Industrial and

                                                                        Production Engineering,

                                                                        Faculty of Technology,

University of Ibadan.

...................................................                                        .................................................

  DATE                                                 Head of Department

                                                          Dr. Festus .A. Oyawale

                                                          BSIE., M.Sc.,(Kalamazoo.,

Ph.D. (Benin), MNSE.

                                                          Department of Industrial and

                                                                        Production Engineering,

                                                                        Faculty of Technology,

University of Ibadan.

DEDICATION

Dedicated to my hero;

 Late Madam Oyepeju Ajala.

ACKNOWLEDGEMENT

I appreciate God Almighty, the repository of all knowledge, for His faithfulness, favour, grace, protection, love and divine guidance, throughout the period of my study.

Lastly, I doff my cap for my amiable and always accessible Project Supervisor

Mr. A.D Adeyeye, for his readiness and determination to impact in me all I need for the successful completion of this project, despite his busy schedule; I pray that heaven will make provisions for your children, even on foreign soil (AMEN).

My parents Mr Joseph Olanrele, Mrs Taiwo Olanrele and Mrs Kehinde Thomas for their moral, financial and parental supports. Inclusive, are my brothers Femi & Dokun Olanrele, my cousins Bisi Olanrele, Kolawole & Damilola Abubakar, and Tinuke Oyelakin. Not forgetting my uncle, Pastor Bode Olanrele (UK) and my aunties Mrs Joke Alake and Mrs Deyo Ojedapo.

I would like to show my profound gratitude to the personnel and supervisors in the company where I collected the data used for this study. My gratitude goes to Dr Adigun and Mr Lekan Alese for their assistance in solving the formulated problem.

My appreciations go to all members of staff in the Department of Industrial and production Engineering, Faculty of Technology, University of Ibadan.

All my colleagues and friends, I cannot afford to forget Adeolubodun Muyiwa, Temiloluwa .O. Bamgbose, Olasebikan Abbey, Allu Ayobami, Ogunmola Sarah, Akinyele Gbenga, Otokiti Bayo, Sola, Seyi, Isaac, Busari Taiwo my roomate, Amokeodo Oluwatobbiloba and all members of the Baptist Student Fellowship, University of Ibadan.

TABLE OF CONTENTS

Page

Title page             .........................................................................i

Certification                   .........................................................................ii

Dedication            .........................................................................iii

Acknowledgement                   ...............................................................iv-v

Table of contents  .........................................................................vi-viii

List of tables        .........................................................................ix

List of figures       .........................................................................x

Abstract      ....................................................................................xi

CHAPTER ONE.

1.0     Introduction                   ...............................................................1-2

1.1     Justification of the study                   ...........................................2-3

1.2     Objective of the study             ...........................................3

1.3     Outline of succeeding chapters          ...........................................3-4

CHAPTER TWO.

2.0     Literature review  ................................................................5-9

2.1     Dynamic programming  ......................................................9-10

2.2     Aggregate production planning         ............................................10-11

2.3     Inventory    ...........................................................................11-12

2.4     Inventory transactions            ............................................12-15

2.5     Inventory carrying cost in summary ..................................15

2.6     Wagner-Whitin algorithm        ............................................15-16

2.7     Forecasting ...........................................................................16-19

2.8     Microsoft Excel Spreadsheet   ............................................19-20

CHAPTER THREE.

3.0     Problem formulation and solution methodology  .............21

3.1     Background information                   ............................................21-23

3.2     Model assumption                   ......................................................23-24

3.3     Model notation     ................................................................24

3.4     Model construction        ......................................................25

3.4.1  Objectives of the model ......................................................25

3.4.2  Constraint of the problem       ............................................25-26

3.4.3  Mathematical model of the problem ..................................26

3.5     Solution procedure                  ......................................................27-28

CHAPTER FOUR.

4.0     Model application                    .......................................................29

4.1     Process description        .......................................................29-30

4.2     Data collection and analysis             ..................................31-32

4.3     Model application                    ......................................................32-33

4.4     Stage by stage solution procedure              .......................34-37

4.5     Discussion of results      ......................................................38-39

CHAPTER FIVE.

5.0     Conclusion and recommendation               .......................40

5.1     Conclusion ..........................................................................40

5.2     Recommendations                   ......................................................41

REFERENCES            .................................................................42-45

APPENDIX         ...........................................................................46-47
LIST OF TABLES

Table                            Title                                                             Page

I                  Demand for layers and growers feed ..... ..............................30

II                 Table of cost        ...................................................................31

III                Quarterly production and inventory plan that will        ................38

LIST OF FIGURES

Figure                                     Title                                                  Page

1                           Stage diagram for the constraint                 ...........................26

2                           Diagram of an animal feed production line .................29

ABSTRACT

The problem of this study is that of determining the quantity of products to produce and inventory level to carry from one period to the other, with the objective of minimizing the total costs of production and the annual inventory, while at the same time meeting the customer’s demand. A mathematical model was formulated for a multi-product problem using Dynamic Programming approach. The model was solved using the solution procedure proposed by Wagner and Whitin. The results show that the minimum total cost will be achieved with production in periods 1, 2, and 4. While demand for period 3 are satisfied with inventory from period 2. The total cost of this plan is N225, 704, 210.00, which is N6, 141, 765.00 less than the existing plan.

CHAPTER ONE

1.0    INTRODUCTION.

It is a known fact that the most measurable goal of a profit oriented organizations is to make profits and achieve a sustainable growth for the future amid such uncertainty and stiff competition.

It is now not just a matter of producing and selling, but making decisions on setting production schedules and meeting it, satisfying customer’s demands, boosting the company’s image, maintaining the quality of products and keeping inventory cost as low as possible.

In making decision on production planning, the production quantity for each period as well as that of each product is stated. Some key reasons for developing this kind of plan include:

v To supply the finished products as specified in the sales programme on or before the due date.

v To ensure a balance or economic utilization of labour, plant and capital in the most efficient way possible.

In specifying these quantities therefore, consideration must be given to such natural limitations as storage and plant capacities, labour as well as material availability e.t.c.

Production planning presents a little problem, if one is dealing with a constant demand throughout the planning periods, in which case, only what can be sold for these periods will be produced in order to avoid carrying inventory of finished products.

In reality however, demand fluctuates from period to period, exceeding regular capacity and even overtime sometimes, while at the other time, it may be far below the regular capacity. In an attempt to adjust to these fluctuations in demand, a good and well planned production schedule will be appropriate such that demand will be met promptly. This of course may sometimes lead to building inventories.

It must also be emphasized here that in making production decisions, not only is it planned to meet customer’s demands, effort is also being made to put total production cost to the bearest minimum as well as reducing waste in other to maximize profit.

1.1    JUSTIFICATION OF THE STUDY.

The importance of using management science techniques in production planning cannot be over emphasized. It is clearly evident that developed countries of the world which engage in management science technique have fewer problems of production management and yield higher profits than developing countries. Despite the use of these techniques and the millions spent on overstocked inventories to improve deliveries, industry still continues to deliver only fifty to seventy percent of promised orders.

A good production plan reduces waste e.g. labour, time and material. And it also synchronizes well the material inventory system with the production expectation in meeting the overall corporate objectives.

1.2    OBJECTIVE OF THE STUDY.

Objectives of this study are:

I.                   To determine the product quantities to produce per period so as to meet the demand for each product.

II.                To determine the inventory level to be carried out during the periods in the planning horizon.

These could be achieved through studying the market records to determine the quantities and rate of demand, the delivery dates and therefore plan for an economic production schedule. And through the collection and analysis of all necessary data gotten from appropriate personnel.

1.3    OUTLINE OF SUCCEEDING CHAPTERS

The remaining part of this project is divided into four parts. Chapter two consists of review of works in areas of application of management science technique, dynamic programming, aggregate production planning, inventory, Wagner and Whitin Algorithm, and forecasting.

Chapter three outlined the theoretical framework and solution methodology for carrying out this study: general introduction, mathematical formulation (Model development), while the process description, analysis, interpretation and inference were in chapter four.

Chapter five contains the conclusion and recommendation as well as the references.

CHAPTER TWO

2.0    LITERATURE REVIEW.

The aggregate production planning, as observed by Zoller (1971), is concerned with a minimum cost adaptation of a production process to demand fluctuations by means of determining and controlling overall production rate, workforce size and inventory levels that begin with a forecast of expected demand for the product groups over a planning horizon, say a year.

Monks (1982), defined operation management as that activity whereby resources, flowing within a defined system are combined and transformed in a controllable manner to add value in accordance with policies communicated by management.

Operation research model is an abstract representation of some real world process, system or subsystem (Dilworth, 1992). They are used in all aspect of life. We think and speak with models rather than with actual tangible objects. Accountants do not collect actual naira in different containers to keep track of various accounts. Instead, they write numbers in amounts in the account. Words and numbers are symbols that stand for something else, which is they are abstractions of reality, or models. In our memory we carry a vast array of useful models. When we construct a paragraph to describe something, we are producing a model of the aspects or features that we feel are pertinent and wish to convey.

Everret and Ronald (1999), state it that operations management is the management of the conversion process which converts land, labour, capital and management inputs into desired outputs of goods and services.

According to Teslang (2000), Production management can be define as the process of decision making related to production processes so that the resulting goods or services are produced according to specification in the amounts and by the schedule demanded and at minimum cost. He later stated the objectives of production management to include:

i.)                Helps to achieve uninterrupted flow of material through the production line in order to meet customers varied demand with respect to quality and committed delivery schedule.

ii.)              To achieve the production objectives with respect to quality, quantity, cost and timelines of delivery.

iii.)           For effective utilization of the firms resources.

iv.)           To help the company to supply a good quality product to the customer on the continuous basis at competitive rates.

After a mathematical model is formulated for a given problem, the next phase in an OR study is to develop a procedure (usually a computer-based procedure) for deriving solutions to the problem from this model (Hillier and Lieberman, 2001). Sometimes, in fact, it is a relatively simple step, in which one of the standard algorithms (systematic solution procedures) of OR is applied on a computer by using one of a number of readily available software packages.

Operations research methods are supposed to develop and analyze mathematical models of systems that incorporate factors, such as chance and risk, to predict and compare the outcomes of alternative decisions or strategies (Timothy, 2004). In operations research there is modelling of complex systems, analysis of system models using mathematical and statistical techniques, and application of the techniques to engineering problem paradigms. The resulting data help decision makers determine policy, allocations, and the best courses of action in the control of complex systems.

Planning and scheduling are decision making processes that are used on a regular basis in many manufacturing and service industries, these forms of decision making play an important role in procurement and production, in transportation and distribution, and in information processing and communication (Michael, 2005). The planning and scheduling functions in a company rely on mathematical techniques and heuristic methods to allocate limited resources to the activities that have to be done. This allocation of resources has to be done in such a way that the company optimizes its objectives and achieves its goals.

Taha (2007), states it that the peculiarity of most OR techniques is that solutions are not generally obtained in formula like closed forms. Instead they are determined by algorithm. An algorithm provides fixed computational rules that are applied repetitively to the problem, with each repetition called iteration, moving the solution closer to the optimum. Because the computations associated with iteration are typically tedious and voluminous, it is imperative that these algorithms be executed on the computer. Among the mathematical programming techniques as stated by Taha are;

i.                   Linear Programming Technique: This is useful for solving many complex problems which otherwise are more difficult to solve.

These problems include;

-         Production scheduling

-         Product mix decisions

-         Capital budgeting

-         Plant location

-         Resource allocation and optimal utilization of resources e.t.c

ii.                 Dynamic Programming Technique: Designed to solve problems that can be partitioned into time-dependent stages. These problems include;

-         Time dependent production scheduling

-         Time dependent inventory control

-         Determining reject allowances

-         Distributing scientists to research team e.t.c

iii.              Goal Programming Techniques: This technique is essentially used when the decision criteria is more than one i.e multi objective function problems.

Others include;

iv.              Non-Linear Programming Technique.

v.                 Integer Programming Technique.

2.1    DYNAMIC PROGRAMMING.

A general dynamic programming model can be easily formulated for a single dimension process from the principle of optimality. The programming situation involves a certain quantity of economic resources (space, finance, people, and equipment) which can be allocated to a number of different activities (Michael, 2005), The objective is to allocate the available resources amongst the various activities so as to maximize the total return. In dynamic programming, optimization is viewed as a stage wise process. When analytic expressions are not available or are too complicated to manipulate, search algorithms have the merit of using more realistic representations of the costs than the optimization method does.

Dynamic programming is handy in solving a problem with multi-stage problem simultaneously, (as in this case production quantity and inventory quantity to be carried). A particular situation in which there is appreciable variation in average monthly demand and availability of raw materials among the different periods under consideration. A general Dynamic Programming Algorithm; is applicable in a situation in which there is absence of shortage, the inventory model is based on minimizing the sum of production and holding cost for all i periods and it is assumed that the holding cost for period i is based on end of period inventory (Taha, 2007). 

2.2    AGGREGATE PRODUCTION PLANNING.

Since demand has consistently remained the essence of continued production, it is highly imperative to establish demand forecasting which will definitely serve as a basis for planning production operations adequately. Production planning and scheduling as defined by Adams and Ebert (1986) is concerned with the process of setting production goals for future operations in an organisation by focusing adequate attention on the volume and timing of outputs, the utilization of operations capacity and therefore balancing these outputs with capacity at desired levels for feasibility and competitive effectiveness. The end result of setting these production goals is a programme or schedule, which specifies the quantity of each product to produce in each time period. Regular, overtime or subcontracting hours required the levels of inventory and backlog that will be held and subsequent workforce size that will be employed in order to meet demand and satisfy the sales programme must be known.

2.3    INVENTORY.

An inventory is the quantity of commodity that a business must maintain to ensure smooth operation, with the goal of minimizing the total cost of inventory (Lucey, 2002). Inadequate inventory can lead to undue costs, production delays and inefficiencies including lost orders or even loss of customers. More than adequate inventories results into excessive inventory holding cost. The control of production activities rest firmly upon the control of inventories, quality and cost. Lucey classified the different categories of inventory into:

(1) Raw material inventory: These are input materials employed in the manufactured of end products.

(2) Work-in-process inventory: Concerns partly finished goods and materials held between manufacturing stages.

(3) Finished goods inventory: Completed products ready for sale or distribution.

Weiss (1989) defines inventory as a stock of goods that is maintained by a business in anticipation of some future demand. According to Monks (1982), it is an idle resource that possesses economic value. Inventory levels may increase or decrease and are related to expected demand levels from either products or supplies.

2.4    INVENTORY TRANSACTIONS.

Companies use inventory transactions to move items within and among their facilities (Edwards, 2003). The Inventory Management system defines inventory transactions as:

ü Issues

ü  Adjustments

ü  Transfers

Issues

Issues are typically used to remove inventory from a location. An issue can be used in each of the following situations:

• Damaged goods: Product can be damaged; you can issue this product to a loss location or account.

• Marketing demonstration: A sales representative may require an inventory item for demonstration purposes during the sales cycle. To maintain accountability, this item can be issued to the sales representative.

• Internal use: Some businesses need to remove product from inventory for internal use. For example, an oil company might use product for its delivery fleet. You can use an issue to move an item from inventory to internal disposition.

Adjustments

Adjustments are used to reconcile discrepancies between physical inventory counts and on hand system quantities. You can use an adjustment the following situations:

• Shrinkage: Items occasionally disappear from inventory through theft or loss. Adjustments can be used to document these losses.

• Unrecorded gain: Sometimes a missing item reappears. Adjustments can be used to document the gain in inventory.

• Initial balances: When creating records for a new warehouse, adjustments can be used to record initial inventory levels.

Transfers

A transfer documents the movement of an item from place to place. You can use a transfer in the following situations:

• Movement from location to location. When it is necessary to move an inventory item between locations in a warehouse or on the shop floor, you can make a transfer to document this type of movement.

• Movement from vehicle to location. Product movements from vehicles to locations in a warehouse are common. You can use a transfer to track this type of movement.

• Movement from plant to plant. Inventory movements among facilities must be recorded to accurately maintain inventory records. You can use a transfer for this type of movement.

Monitoring feedback and control is the final and most crucial aspect of production planning and control activities. At this stage, from the system’s inbuilt feedback arrangement, it is basically determined if the system in implementation maintains standard output, quality and cost, so as to help achieve the firm’s objective of minimizing both production cost and resource wastages and finally maximizing production output and profit.

Typically, holding costs are estimated to cost approximately 15-35% of the material's actual value per year (Charles, 2005). The primary factors that drive this up include additional rent needed, great insurance premiums to protect inventory, opportunity costs, and the cost of capital to finance inventory.

According to REM Associates Management Consultant (1995), over 65% of most companies do not compute inventory carrying costs, they use rough estimates. The standard “rule of thumb” for inventory carrying cost is 25% of inventory value on hand. The cost of capital is the leading factor in determining the percentage of carrying cost.

2.5    INVENTORY CARRYING COSTS IN SUMMARY.

Cost of Money                                  6% - 12%

Taxes                                                          2% - 6%

Insurance                                          1% - 3%

Warehouse Expenses                        2% - 5%

Physical Handling                                       2% - 5%

Clerical & Inventory Control            3% - 6%

Obsolescence                                     6% - 12%

Deterioration & Pilferage                            3% - 6%

 Total                                                25% - 55%

Source: REM Associates Management Consultant 1995.

2.6    WAGNER-WHITIN ALGORITHM.

A forward algorithm for a solution to the following dynamic version of the economic lot size model is given: allowing the possibility of demands for a single item, inventory holding charges, and set up costs to vary over N periods, we desire a minimum total cost inventory management scheme which satisfies known demand in every period. Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N period.

As Wagner-Whitin algorithm (1958), presented a precise method for determining the optimal lot size for a product with dynamic demand for single-stage production without consideration of capacity constraints. The Wagner-Whitin problem is also called “Single-Level Lot Sizing Incapacitated problem”.

As in the classical lot-size formula is an infinite production rate and a steady consumption over the period assumed. The Wagner and Whitin method produces the optimum even if fixed costs vary from period to period, condition is that the current fixed costs are used in the periods. An important implication of the Wagner-Whitin- algorithm is the Zero Inventory Property, a production then takes place only if the camp is empty. Then the optimum of the complete needs of a term is either fully covered from stock or from production of that period. A situation that is where the demand is partly in the production and partial satisfaction of the camp, which means that in a period of storage and set-up costs, cannot be minimal costs because the setup costs can be saved by the Pre-Placement of production. This general finding has been used in the heuristic method for determining the lot size as a basis.

2.7    FORECASTING.

Forecasting can be defined as attempting to predict the future by using qualitative or quantitative means. In an informal way, forecasting is an internal part of all human activity, but from the business point of view increasing attention is being given to formal forecasting systems involve very advanced statistical techniques (Lucey, 2002);

-         Quantitative techniques are techniques of varying levels of statistical complexity which are based on analysing past data of the item to be forecast e.g. Sales figure, store issues, cost incurred e.t.c.

-         Qualitative techniques are techniques which are used when data are scarce e.g. the first introduction of a new product.

Virtually every form of decision making and planning activity in business involves forecasting. Typical applications include:

v Production planning.

v Demand forecasts.

v Inventory control.

v Advertising planning

v Investment cash flows.

v Corporate planning.

v Cost projections.

v Budgeting.

Most organizations use forecasts as input to comprehensive planning processes such as financial planning, budgeting, sales planning, and finished goods inventory planning that are charged with accomplishing particular goals. This implies that the forecast needs not only to be accepted by external parties, but also to guide efforts of the organization. Thus, an important measure of forecast effectiveness is how much they support these planning needs. The fit between forecasting and planning is an under-studied relationship in the literature, but at a minimum level, the forecast process needs to match the planning process in terms of the frequency and speed in which the forecast is produced. The forecasting horizon and accuracy of the forecast should be such that it allows the elaboration and execution of plans to take advantage of the forecast (Mentzer and Bienstock, 1998).

A planning approach such as Quick Response (Hammond, 1990) requires as input a sense of the uncertainty surrounding the forecasts in order to manage production. Thus, the forecasting process complementing such a planning process should have a means of providing a relative measure of uncertainty (Fisher and Raman, 1996).

Nevertheless, forecasting is not an exact science. In an organizational setting, the forecasting process requires information from multiple sources (e.g., intelligence about competitors, marketing plans, channel inventory positions, etc.) and in a variety of formats, not always amenable to integration and manipulation (Makridakis et al., 1998). The multiplicity of data sources and formats creates two major challenges for a forecasting process. First, since not all information can be accurately reflected in a statistical algorithm, judgment calls are a regular part of forecasting processes (Armstrong, 2001). The judgmental criteria to make, adjust, and evaluate forecasts can result in individual and functional limitations and biases that potentially compromise the quality of the forecasts. Second, since the vast majority of the information providers and the makers of those judgment calls are also the users of the forecast, there are strong political forces at work explicitly attempting to bias the outcome of the process.

Thus the forecasting process, in addition to fitting with the organization planning requirements, needs to explicitly manage the biases (whether individual or functional) that might affect the outcome of the process. We recognize two potential sources of biases in the organization intentional and unintentional that incorporates the judgmental, informational, and political dynamics that affect forecasting performance. In the following subsections, we provide analytical context from relevant literature to articulate frameworks and expectations that will help the reader to assimilate the case details in these two dimensions.

2.8    MICROSOFT EXCEL SPREADSHEET.

Spreadsheets are considered a very popular form of computer modelling for many applications and have become an indispensable business tool. Spreadsheets allow the user to combine data, mathematical formulas, text and graphics together in a single report or worksheet. Some spreadsheet packages like “Microsoft Excel” contain or allow ad-in modules with optimization features. “Solver” is the optimization tool of Excel (Anderson et al, 2003).

CHAPTER THREE

3.0    PROBLEM FORMULATION AND SOLUTION METHODOLOGY.

3.1    BACKGROUND INFORMATION.

One of the problem often encountered in production planning in industries with large product demand, is production planning requirement. The problem is that of determining the quantity to be produced and the inventory quantity to be carried, such that the demand of each period will be met at minimum total production cost. The above problem has characteristics of dynamic programming problem as stated by Hillier and Lieberman (2001), these characteristics are:

Ø The problem can be divided into stages, with a policy decision required at each stage.

Ø Each stage has a number of states associated with the beginning of that stage.

Ø The effect of the policy decision at each stage is to transform the current state to a state associated with the beginning of the next stage.

Ø The solution procedure is designed to find an optimal policy for the overall problem.

Ø  Given the current state, an optimal policy for the remaining stages is independent of the policy decisions adopted in previous stages.

For the following reasons a general dynamic programming algorithm and specifically Wagner-Whitin dynamic programming inventory model solution procedure will be adopted in this study. A major advantage of the dynamic programming technique is that it provides the solution, with much less effort than exhaustive enumeration, which is required to solve the same problem by other means. Dynamic programming is characterized by three types of equations (Michael, 2005), namely;

(i)                Initial conditions

(ii)             A recursive relation and

(iii)           An optimal value function.

Wagner and Whitin (1958) used dynamic economic lot size model as a guide to formulating a model that will handle both the production rate and inventory levels simultaneously. The decision variables are the production rate and inventory levels. Because the number of combinations, in general, can be as large as the product of the number of possible values of the respective variables, the number of required calculations tends to “blow up” rapidly when additional state variables are introduced (Hillier and Lieberman, 2001), this phenomenon is known as “curse of dimensionality”. In this study, only one constraint, the conservation of material constraint is important. The requirement that no demand must be back ordered or lost leads to the non negativity constraint on the inventory.

Dynamic economic lot size model developed by Wagner and Whitin, (1958) is a special case of the general dynamic programming models, which reduce the volume of computations. One of the basic assumptions of this dynamic lot-sizing model is that the aggregate demand of a given period must be satisfied in that period by either the new production or from entering inventory.

The first step of the study involves the formulation of the problem, which involves a description of the objectives which must be reflected as an accurate representation of the overall system and the second step involves the construction of the model which depends on the first step. After formulating the model of the problem as stated above, the method of dynamic programming algorithm will be employed to solve the model to obtain the quarterly production rate (number to produce).

And lastly the result of the model is translated into the detailed operating instructions, issued in an understandable form to the individual who will administer and operate the recommended system.

3.2    MODEL ASSUMPTION.

The following assumptions are set to construct the mathematical model of the production planning problem.

1.     The average periodic demand varies appreciably among the different quarters.

2.     Raw materials are available, but there is periodic change in their prices.

3.     The model will handle both the production rate and inventory level simultaneously.

4.     Multiple products are produced.

5.     Only conservative of material constrained will be considered.

6.     Single objective i.e minimizing total cost.

7.     The model is deterministic.

8.     Shortages are not allowed.

9.     Unit production cost vary from period to period

10.                         Unit holding cost is unchanged for all period.

3.3    MODEL NOTATIONS.

G     -       Total cost (Objective function)

z_ij     -       Quantity of product j produced in period i (Kg).

c_ij       -        Cost of producing one unit of product j for period i (N/Kg).

h_ij      -        Unit cost of storage of product j for period i (N/KG).

x_ij       -        Inventory of product j at the start of period i (KG).

x_{j, i+1}   -        Inventory of product j at the end of period i (KG).

d_ij      -        Demand of product j at period i (KG).

K_ij      -        Setup cost in period i for all j.

3.4    MODEL CONSTRUCTION.

3.4.1 OBJECTIVES OF THE MODEL.

A production plan is required which will state the quantities of each product to be produced per period i so as to meet the demand for the period at a minimal total cost. The cost function is made up of two components (production and inventory costs). The Production cost for product j in period i is given by;

Production cost for product j =                                                                                         (1)

The first term is the cost of producing z_ij units of product j in period i, while the second term is the setup cost. The cost of carrying x units of product j from period i to period i + 1 is given by;

Inventory cost of product j =                                                            (2)

The cost function for product j for period i is given by;

                                                                                        (3)

The total cost for all the products over the planning horizon is given by;

                                                                             (4)

3.4.2 CONSTRAINT OF THE PROBLEM.

The only constraint of the problem is the material balance constraint. From fig.1, the sum of inventory brought into period i, and production at period i must be equal to the demand of period i plus inventory carried from period i to period i + 1. That is, sum of materials entering period i must be equal to the sum of materials leaving period i.

                                                                                                                                                   (5)

:.                                                                                       (6)

FIGURE 1: STAGE DIAGRAM FOR THE CONSTRAINT

3.4.3 MATHEMATICAL MODEL OF THE PROBLEM.

The planning problem may be stated as:

Minimize (G) =

                                                                               (7)

for i = 1, 2,..........., I.

      j = 1, 2,..........., J.

       

3.5    SOLUTION PROCEDURE

Wagner and Whitin solution procedure was used and under the given conditions it can be proved that:

1.     Given the initial inventory x_i = 0, then at any period i of the I periods model, it is optimal to have a positive production quantity z_i* or positive entering inventory x_i* but not both; that is z_i*x_i* = 0.

2.     The amount produced z_i at any period i is optimal only if it is zero or if it satisfies the exact demand of one or more succeeding periods. (z_i = 0, d_i, d_i + d_i+1, d_i + d_i+1 + d_i+2 e.t.c). These succeeding periods are such that if the demand in period i + m (< I) is satisfied by z_i* then the demands of period i, i+1, i+2,....i+m-1, must also be satisfied.

The solution procedure begins by finding the optimal policy for the first stage.

The optimal policy for the first stage prescribes the optimal policy decision for each of the possible states at that stage. There is a recursive relationship that identifies the optimal policy for period i, given the optimal policy for period i+1 is available. This recursive relationship is;

G_i(x_{j,i + 1}) = Minimize ((c_ijz_ij + k_ij) + h_ijx_{j,i + 1})                                                 (8)

Therefore, finding the optimal policy decision at period i require finding the minimizing value of x_i and the corresponding minimum cost is achieved by using this value of x_i and then following the optimal policy when you start at period i  + 1. The precise form of the recursive relationship differs somewhat among dynamic programming problems. However, notation analogous to that introduced in the section will continue to be used here, as summarized on page 25.

The recursive relationship keeps recurring as we move from period to period. When the current period i is increased by 1, the new function is derived by using the G_i+1(x_{j,i + 1}) function that was just derived during the preceding iteration, and then this process keeps repeating, until it finds the optimal policy starting at the final period. This optimal policy immediately yields an optimal solution for the entire problem.

CHAPTER FOUR

4.0    MODEL APPLICATION.

4.1    PROCESS DESCRIPTION.

The case under consideration in this project is an animal feed producing company. The company produces animal feeds like;

·        Chicken feed

·        Fish feed

·        Pig feed

·        Cow feed

Chicken feed which consist of layers feed and growers feed are with the highest demand rate and price per unit of its raw materials are not stable, it’s on these on which this study is based. The company operates a flow line work-system, consisting of scale, grinder, conveyor and mixer. The materials are weighed on the scale, and the maize is poured into the grinder, which has a sieve through which the grinded maize passes into the conveyor. The conveyor transport the grinded maize into the hood of the mixer, where other materials are been added and are properly mixed together through the auger. The mixing takes up to twenty minutes (20minutes). After which it is bagged in 25kg or 50kg depending on the demand of each.

 

Grinder

Mixer

Conveyor

 

FIGURE 2: DIAGRAM OF AN ANIMAL FEED PRODUCTION LINE.

4.2    DATA COLLECTION AND ANALYSIS.

Data for the period of the year under consideration are collected and analyzed for easy model application. The demand for three consecutive months is summed up and grouped in one quarter and this is repeated for the rest of the months. Aggregate demand for four (4) quarters of the year is therefore got and the setup cost corresponding to each quarter is shown in Table II below.

TABLE I:  DEMAND FOR LAYERS AND GROWERS FEED.

QUARTERS	LAYERS FEED DEMAND (KG)	GROWERS FEED DEMAND (KG)
PERIOD 1	1,112,750	325,500
PERIOD 2	1,478,500	371,750
PERIOD 3	1,400,255	357,000
PERIOD 4	1,004,000	327,000

Likewise, data on unit production cost, inventory holding cost and setup cost were collected as obtained from the company under study. And the production cost is made up of labour cost, machine cost and cost of raw materials.

TABLE II: TABLE OF COST.

PERIODS	LAYERS FEED		GROWERS FEED		SETUP COST kij (N)
	Unit Production Cost (N).	Inventory Holding Cost (N).	Unit Production Cost (N).	Inventory Holding Cost (N).
1	31.00	9	28.00	9	375,000.00
2	33.00	9	29.00	9	311,500.00
3	45.00	9	42.00	9	263,500.00
4	37.00	9	33.00	9	417,500.00

4.3    MODEL APPLICATION.

The model applied for this case is a production and inventory model as stated in equation (7), and it can be expressed as;

Minimize (G) = ((c₁₁z₁₁ + k₁₁) + h₁₁x₂₁) + ((c₂₁z₂₁ + k₂₁) + h₁₂x₃₁) + ((c₃₁z₃₁ + k₃₁)

+ h₁₃x₄₁) + ((c₄₁z₄₁ + k₄₁) + h₁₄x₅₁) + ((c₁₂z₁₂ + k₁₂) + h₂₁x₂₂) + ((c₂₂z₂₂ + k₂₂)

+ h₂₂x₃₂) + ((c₃₂z₃₂ + k₃₂) + h₂₃x₄₂) + ((c₄₂z₄₂ + k₄₂) + h₂₄x₅₂)                           (8)

Putting the values of k_ij, c_ij and h_ij from Table I. The model can be expressed as in equation (9) below:

Minimize (G) = ((31z₁₁ + 375000) + 9x₂₁) + ((33z₂₁ + 311500) + 9x₃₁)

 + ((45z₃₁ + 263500) + 9x₄₁) + ((37z₄₁ + 417500) + 9x₅₁)

+ ((28z₁₂ + 375000) + 9x₂₂) + ((29z₂₂ + 311500) + 9x₃₂)

+ ((42z₃₂ + 263500) + 9x₄₂) + ((33z₄₂ + 417500) + 9x₅₂)                                  

S.t;                                                                                                               

x₂₁ = x₁₁ + z₁₁ – d₁₁

x₃₁ = x₂₁ + z₂₁ – d₂₁                                                                                                                                                    

x₄₁ = x₃₁ + z₃₁ – d₃₁                                                                                                                                                (9)                  

x₅₁ = x₄₁ + z₄₁ – d₄₁

x₂₂ = x₂₁ + z₂₁ – d₂₁

x₃₂ = x₂₂ + z₂₂ – d₂₂

x₄₂ = x₃₂ + z₃₂ – d₃₂

x₅₂ = x₄₂ + z₄₂ – d₄₂

Non negativity constraint:

z₁₁, z₂₁, z_31, z₄₁,z₁₂, z₂₂, z₃₂, z₄₂, x₂₁, x₃₁, x₄₁, x₅₁, x₂₂, x₃₂, x₄₂, x₅₂  ≥ 0

4.4    STAGE BY STAGE SOLUTION PROCEDURE

The models were solved according to the procedure proposed by Wagner and Whitin (1958), and under the given conditions it can be proved that:

1.     Given the initial inventory x_i = 0, then at any period i of the I periods model, it is optimal to have a positive production quantity z_i* or positive entering inventory x_i* but not both; that is z_i*x_i* = 0.

2.     The amount produced z_i at any period i is optimal only if it is zero or if it satisfies the exact demand of one or more succeeding periods. (z_i = 0, d_i, d_i + d_i+1, d_i + d_i+1 + d_i+2 e.t.c). These succeeding periods are such that if the demand in period i + m (< I) is satisfied by z_i* then the demands of period i, i+1, i+2,....i+m-1, must also be satisfied.

The software used in solving this problem is Microsoft Excel Solver and the stage by stage solution procedure as presented in appendix 1 and 2 is stated as follow:

PRODUCT 1 (LAYERS FEED)

Stage 1: Beginning at period 1 we already know that quantity of product produced in period 1 (z₁₁*) must satisfies the demand (d₁₁) for the period, also there is possibilities of producing more than the demand (d₁₁) for period 1, which requires carrying the excess as inventory to period 2, period 3 and up to period 4. But the production option with the minimum cost is taken as the production policy for period 1.

G₁(x₂₁) = Min ((c₁₁z₁₁ + k₁₁) + h₁₁x₂₁)                                                                         (10)

= (((31 × 1112750) + 375,000) + (9 × 0)) = 34870250                                              (11)

= (((31 × 2591250) + 375,000) + (9 × 1,478,500)) = 94010250                                (12)

= (((31 × 3595505) + 375,000) + (9 × 2,482,755)) = 150020450                              (13)

= (((31 × 5071505) + 375,000) + (9 × 3,958,755)) = 190180450                              (14)

Production policy: produce z₁₁* = 1,112,750kg of layers feed.

Stage 2: For the problem consisting of just the last three stages (i = 3), the recursive relationship reduces to;

G₂(x₃₁) = ((c₂₁z₂₁ + k₂₂) + h₂₁x₃₁) + f₁(x₃₁ + d₂₁ + z₂₁)                                                             (15)

= (((33× 1478500) + 311,500) + (9 × 0)) + 35092800 = 83972250                                      (16)

= (((33 × 2482755) + 311,500) + (9 × 1,004,255)) + 35092800 = 142782960                      (17)

= (((33 × 3958755) + 311,500) + (9 × 2,482,755)) + 35092800 = 184950960                   (18)

Production policy: produce z₂₁* = 2,482,755kg of layers feed.

Stage 3: The third stage (i = 3) and fourth stage problems (i = 4) are solved in a similar fashion. Thus, for i = 3,

G₃(x₄₁) = ((c₃₁z₃₁ + k₃₁) + h₃₁x₄₁) + f₂(x₄₁ + d₃₁ + z₃₁)                                                             (19)

= (((45 × 1004255) + 263,500) + (9 × 0)) + 81533500 = 142782960                                    (20)

= (((45 × 2480255) + 263,500) + (9 × 1,476,000)) + 81533500 = 193986960                      (21)

Production policy: no production i.e z₃₁* = 0.

Stage 4: For the last stage i = 4

G₄(x₅₁) = ((c₄₁z₄₁ + k₄₁) + h₄x₅₁) + f₃(x₅₁ + d₄₁ + z₄₁)                                                              (22)

= (((37 × 1476000) + 417,500) + (9 × 0) + 113129756 = 180348460                                   (23)

Production policy: produce z₄₁* = 1,476,000kg of layers feed.

Production policy for Product 1 (Layers feed):

z₄₁* = 1,476,000kg, z₃₁* = 0, z₂₁* = 2,482,755kg, z₁₁* = 1,112,750kg.

x₂₁ = 0, x₃₁ = 1,004,255kg, x₄₁ = 0, x₅₁ = 0.

PRODUCT 2 (GROWERS FEED).

Stage 1: Beginning at period 1 we already know that quantity of product produced in period 1 (z₁₂*) must satisfies the demand (d₁₂) for the period, also there is possibilities of producing more than the demand (d₁₂) for period 1, which requires carrying the excess as inventory to period 2, period 3 and up to period 4. But the production option with the minimum cost is taken as the production policy for period 1.

G₁(x₂₂) = Min ((c₁₂z₁₂ + k₁₂) + h₁₂x₂₂)                                                                         (24)

= (((28 × 325500) + 375,000) + (9 × 0)) = 9489000                                                  (25)

= (((28 × 697250) + 375,000) + (9 × 371,750)) = 23243750                                    (26)

= (((28 × 1054250) + 375,000) + (9 × 728,750)) = 36452750                                  (27)

= (((28 × 1381250) + 375,000) + (9 × 1,055,750)) = 48551750                                (28)

Production policy: produce z₁₂* = 325,500kg of growers feed.

Stage 2: For the problem consisting of just the last three stages (i = 3), the recursive relationship reduces to;

G₂(x₃₂)= ((c₂₂z₂₂ + k₂₂) + h₂₂x₃₂) + f₁(x₃₁ + d₂₁ + z₂₁)                                                              (29)

= (((29 × 371750) + 311,500) + (9 × 0)) + 9489000 = 20581250                                          (30)

= (((29 × 728750) + 311,500) + (9 × 357,000)) + 9489000 = 34147250                               (31)

= (((29 × 1055750) + 311,500) + (9 × 684,000)) + 9489000 = 46573250                             (32)

Production policy: produce z₂₂* = 728,750kg of growers feed.

Stage 3: The third stage (i = 3) and fourth stage problems (i = 4) are solved in a similar fashion. Thus, for i = 3

G₃(x₄₃) = ((c₃₂z₃₃ + k₃₃) + h₃₃x₄₃) + f₂(x₄₃ + d₃₃ + z₃₃)                                                             (33)

= (((42 × 357000) + 263,500) + (9 × 0 )) + 20209500  = 34147250                                      (34)

= (((42 × 684000) + 263,500) + (9 × 327,000)) + 20209500 = 49516250                             (35)

Production policy: no production i.e z₃₂* = 0.

Stage 4: For the last stage i = 4

G₄(x₅₂) = ((c₄₂z₄₂ + k₄₂) + h₄x₅₂) + f₃(x₅₂ + d₄₂ + z₄₂)                                                              (36)

 = (((28 × 327000) + 417,500) + (9 × 0) + 30469000 = 4535575                              (37)

Production policy: produce z₄₂* = 327,000kg of growers feed.

Production policy for Product 2 (Growers feed):

z₄₂* = 327,000kg, z₃₁* = 0, z₂₁* = 728,750kg, z₁₁* = 325,500kg.

x₂₁ = 0, x₃₁ = 357,000kg, x₄₁ = 0, x₅₁ = 0.

4.5    DISCUSSION OF RESULTS.

The model was solved and the results showing quantity of each product to produce and inventory to be carried from one period to the other is presented in Table III. From the result it is required that only 1,112,750Kilograms of Layers feed and 325,500Kilograms of Growers feed should be produced to meet the demand for period 1 leaving no inventory. For the 2nd quarter 2,482,755Kilograms of Layers feed should be produced to meet the demand of 1,478,500Kilograms, leaving 1,004,255Kilograms of Layers feed in inventory. While 728,750Kilograms of Growers feed should be produced to meet the demand of 371,750Kilograms, leaving 357,000Kilograms of Growers feed in inventory.

For the 3rd quarter no production should take place, and the inventory of 1,004,255Kilograms of layers feed and 357,000Kilograms of growers feed, carried over from period 2 will satisfy the demands in period 3. The reason for this is because prices of the major raw materials go up most time around this period and this is justifiable because the storage cost incurred here is not to be compared with the cost of producing in this period. For the 4th quarter 1,476,000Kilograms of layers feed should be produced to exactly meet the demand leaving no inventory. While 327,000Kilograms of growers feed should be produced to exactly meet the demand leaving no inventory.

TABLE III:  QUARTERLY PRODUCTION AND INVENTORY PLAN.

PERIOD	LAYERS FEED		GROWERS FEED
	PRODUCTION QTY (KG)	INVENTORY LEVEL (KG)	PRODUCTION QTY (KG)	INVENTORY LEVEL (KG)
1	1,112,750	-	325,500	-
2	1,478,500	1,004,255	371,750	357,000
3	-	-	-	-
4	1,476,000	-	327,000	-

CHAPTER FIVE

5.0    CONCLUSION AND RECCOMENDATION.

5.1    CONCLUSIONS.

Production planning problem as it relates to an animal feed producing company was observed and tackled, using a dynamic programming approach to make production and inventory level decisions, the objective is to minimize the total cost of production and the annual inventory cost, at the same time meets the customer’s demand.

Wagner and whitin inventory model was used, stipulating the minimum quantities of the product to produce per quarter and the corresponding inventory levels such that total production cost is minimized over the planning period. The model considered the four quarters planning horizon from June 2009 to May 2010 and the corresponding production forecast data was used.

The results show that a total minimum cost will be achieved with production in period 1, 2, and 4. While demand for period 3 are satisfied with inventory from period 2. The total cost of this plan is N225, 704,210.00 for the two products, which is N6, 141, 765.00 less than the existing plan.

5.2    RECOMMENDATIONS.

From the above study, the following recommendations can be drawn:

       i.            Operations research or management science techniques as used in this study are very useful in providing mathematically feasible solution to the problem of production planning. However, the management must still play a major role of reconciling the scientific solution with the environmental conditions and other intangible effects to arrive at wise decisions.

     ii.            It is advisable that this type of model is adopted when dealing with making decisions on production and inventory levels for varying period. It helps, without exhaustive enumeration to determine the minimum quantities of product to produce to meet demand at the same time not incurring excessive storage cost by way of inventory in an attempt to meet all demand.

  iii.            This project as an attempt to improve on the use of Wagner and Whitin model to solve more than one product problem can be improved upon in the nearest future to solve more than two products problem through the proper exploitation of the Microsoft excel solver or other applicable software.

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