Clonal integration in Panicum trugidum Article Example | Topics and Well Written Essays - 1250 words

Clonal integration in Panicum trugidum - Article Example Many invasive plants are clonal, however little is known about clonal integration. The discussion expounds more on clonal integration on Panicum turgidum, a drought and salt tolerant plant that is used for erosion control, thatching, fodder, and production of flour (Al-Khateeb 2006). Further, it focuses on the establishment of raments either randomly and genetically. The discussion also outlines the major benefits of clonal integration in plants. Panicum turgidum is plant that is very common in Arabia, Senegal, Pakistan and in most parts across the Sahara desert (Al-Khateeb 2006). The plant is widely referred as Tuman, Taman or Thaman in Arabia and Egypt (Al-Khateeb 2006). In Sahara Arabics, it is referred to as Markouba or Merkba. Other names that commonly refers to Panicum turgidum is Guinchi and Du-ghasi especially in Somalia (Al-Khateeb 2006). Most plants do not tolerate in saline areas, however, Panicum Turgidum is a salt resistance plant (xerohalophyte). Photosynthesis in saline plants is generally lower compared to non-saline environment. This is attributed to the limited uptake of carbon dioxide, reduced stomata size, and chlorophyll content; this leads to reduction in plant growth (Hartnett 1993). Competition among the plant also reduces the quantum yield of photosynthesis. It also limits the growth of leaves, stolon length and ramets. Connected raments of clonal Panicum turgidum plants share carbohydrates, water, and nutrients through clonal integration. Studies have shown that clonal integration in Panicum turgidum facilitates establishments of newly produced ramets. Clonal integration also improves chances of survival and reproduction of adult ramets in Panicum turgidum. Importantly, it also helps genets to occupy more open space. The discussed positive effects of clonal integration in Panicum turgidum help the plants to have competitive edge over plants

Strategic analysis paper Research Example | Topics and Well Written Essays - 1250 words

Strategic analysis - Research Paper Example In 1928, the General Mills acquired other companies due to its fast growth. The companies that were involved in the merger included the Wichita Mill and Elevator Company. Between the years 1929 and 2004, the General Mills was involved in several other mergers (Brucato 2012). The main aim of the mergers was to enable the company to produce high quality products that would be acceptable in the market. For example, in the year 1965, the General Mills bought Rainbow Crafts, which was the manufacturer of Play-DOH. This buying of the company benefited General Mills since it played a big role in reducing the costs of production in the company and also the revenue that the company got from it transactions increased drastically. Merging of companies has a great advantage to the companies involved in the merger (Doz and Hamel 1998). The company sells different food products in several brands in the world market. These include brands such as Betty Crocker, Yoplait, Colombo, Trix and Lucky Charm s to mention a few. Since the year 2004, the General Mills Company has produced products of high quality but their main market has been the wealthy that are conscious with their health. In 2012, the company was ranked among America’s largest corporations. The company was also ranked as the third-largest consumer products company in the United States of America. The purpose of studying the General Mills was to understand in great depth its current financial situation and also its current position among the words’ food producing companies so that a strategy and action can be taken to improve the quality of the products it produces. This would also help to trigger its rise in the ranking of food producing companies worldwide. During the study, the possible big competitors to the company were also examined. The study looks at the strengths of the company, its weaknesses, opportunities and threats that it might be facing since its foundation. The threats that the General Mi lls experiences in other words are also the problems that the company faces. The study therefore helped to come up with the solutions to the problems. General Mills, Inc just like several other food producing companies has strengths and weaknesses that must first look at and considered so that the company can be successful and be at par with other food producing companies in the United States and in the world (Brucato 2012). There are also opportunities that have come up in the company recently as a result of its success and production of quality products. Despite all these strengths and opportunities, there exist several threats that are a great problem in the development of the company (Dess, Gregory, Lumpkin and Taylor 2005). The threats have negative impacts such as causing losses in the company. Among the strengths of the company, one is that the company has ready availability of resources that is important in the production of its products. This helps it to be consistent in it s production thus having a constant and a wide market for its goods. The other strength of the company is that it has good and fast transportation system for its finished products. General Mills has wide market worldwide and within the US. Therefore it has several aircrafts that are efficient in the exportation of the perishable food products to other countries. The other strength of the company is the availability of a ready market for its products. The company is ranked as the third largest food

The Mathematical Mystery Behind Sudoku Mathematics Essay

The Mathematical Mystery Behind Sudoku Mathematics Essay Puzzle games can be very enjoyable and is popular amongst kids as well as adults. Many of you may know the game Sudoku; where by the goal of the game is to fill in the remaining empty cells with each number from 1-9 appearing no more than once from each column, each row and each of the nine sub-grids. Sudoku is a type of logic-based numerical puzzle game that has a unique solution once completed. The most common form of a Sudoku is constructed as a 99 grid with nine 33 sub-grids and is primarily partially completed. Sudoku has become appealing among puzzle enthusiasts and involves complex thinking and practice. Available daily in newspapers, mobiles and many more, this addictive and brain-teasing puzzle game has become one of the most popular games to play since the time of the Rubiks cube. This dissertation discusses the mathematical side involved in Sudoku. There is no mathematics in actually solving a Sudoku but more of how it is used from a creators side. The 99 grid will be considered in the majority of the report; however a glimpse into other size grids will be discussed briefly also known as variants. Mathematicians have been questioning How many unique solutions are there in a Sudoku? Essentially meaning what are the possible ways of filling in an empty Sudoku grid so that each row, column and sub-grid contains the numbers 1 through 9. Your first thought of an answer may be a couple of thousands, but as you understand the concepts behind a Sudoku, you begin to grasp a whole new aspect. Combinatorics and permutation group theory are largely interwoven with analysing Sudoku. For that reason, I aim to explore these theories and understand how it applies to the methods of enumerating Sudoku grids. In particular I will be looking at Felgenhauer and Jarviss approach to enumerating all possible Sudoku grids where they employ several mathematical concepts. Furthermore I will uncover the importance of Latin squares and its use of constructing Sudokus. There are many constraints in regards to when are similar solutions considered different such as solutions of similar structure, symmetry etc. Preserving symmetries are known as relabeling symbols, band permutations, reflection, transposition and rotation. Burnsides Lemma theorem is one of their techniques in computing the number of essentially different solutions. Many difficult problems are of the type called nondeterministic-polynomial known as an NP-complete problem. This will direct me onto the debate on whether Sudoku is an NP-complete problem. Sudokus can take many forms and shapes. These are called Sudoku variants and consist of rectangular regions, Sudokus with a large region having no clues (numbers), an empty row, column or sub-grid and many more! Here I will research the logic behind irregular Sudokus as well as examining any occurring patterns or whether it has occurred by chance. 1.2 Latin squares and Sudoku Sudoku is also a special case of Latin squares. The Swiss mathematician, Leonhard Euler made many fundamental discoveries during 1782 including Latin squares. A Latin square is an N x N matrix where by a set of N characters are arranged such that each row and column contains one of each character. This is also in the case of a Sudoku, when complete, with an additional constraint that the nine sub-grids must hold the numbers 1-9. A reduction can be made to any Latin square by permuting the rows and columns. This arrangement is an aspect of combinatorics and is most commonly referred to as enumeration. Enumerative combinatorics is a classic area of Combinatorics and involves counting the number of infinite class of finite sets. Counting combinations and counting permutations are two of the most common forms. The number of valid Latin squares is known to be approximately 5.525 x 10 ²Ãƒ ¢Ã‚ Ã‚ ·. Write about Colbourns proof 1.3 Combinatorics and Permutation group theory Combinations and permutations have slightly different meaning. Combinations are the number of different ways of selecting n objects from a set but the order of events is not important. From a set of 3 objects, lets call these 1, 2 and 3. If for example I was asked to pick the number of ways of selecting 2 objects out of the 3, there would be three combinations 12, 23 and 13. 12 = 21 since the order of each pair is not important. A permutation on the other hand does consider the position. Therefore if I was to use the above example, there would be six permutations. A simpler way to calculate a larger set would be to use formula 1: Formula 1. = = Where is the combination formula, is the permutation formula, n is the total number of objects and r is the number to be arranged Both methods are one way of computing the number of possible Sudoku solutions and this will be looked at later in the report. Chapter 2 Enumerating possible Sudoku solutions 2.1 Distinct Sudoku solutions There are many approaches to enumerating possible Sudoku solutions. To enumerate every possible Sudoku solution, a Sudoku differs from another if they are not identical. Thus all solutions will be consider unless they are like for like. Felgenhauer and Jarvis was the first to enumerate the Sudoku grid solutions directly in 2005. There approach was to analyze the permutations of the top row used in valid solutions. Their knowledge of the complexity in computing the number of Latin squares has made them aware of how they should go about getting an answer with fewer computations. Hence by using relabeling this could shorten the number of counts. To make it easier, each sub-grid is given an abbreviation seen in figure 3. B1 B2 B3 B4 B5 B6 B7 B8 B9 Figure 1. Abbreviated sub-grid with top band (Felgenhauer and Jarvis, 2006) Firstly they consider every solution to filling in blocks B2, B3, given that B1 is in standard form. To work out every possible way of arranging B1 on its own would essentially be computing the number of permutations of 9 symbols. There are 9! of filling in B1. The main operation they use is called relabeling. 1 2 3 4 5 6 7 8 9 Figure 2. B1 in standard form (Felgenhauer and Jarvis, 2006) Felgenhauer and Jarvis have found that B2 and B3 is the same as the transpose of B2 and B3. Therefore the number of ways of arranging B1, B2 and B3 and B1, B2 and B3 to a complete grid is equally the same. This means that computing one set of possibilities will cut down the number of solutions. Inevitably, there are few pairs of B2 and B3 that needs to be worked out and as well as using reduction the number of possibilities for the top band of a Sudoku grid is 9! x 2612736 = 948109639680. The next section involves brute force computation. As running through all 2612736 possibilities would be exceedingly tedious for B2 and B3, Felgenhauer and Jarvis attempts to identify configurations of the numbers in these blocks which give the same number of ways of completing to a full grid. This in return, will cut down the number possibilities. Permuting B2 and B3 in every way such that the result gives a unique solution will preserve the number of complete grids. This is the same for B5 and B6, and B8 and B9. However this changes B1 from its standard form, so an additional relabeling of B1 needs to be performed. Another approach to reducing the number of possibilities is to permute the columns in each block and permute the rows of any block. Reducing the number of possible ways by permuting. Lexicographical reduction Permutation reduction Column reduction As a result of these methods, Felgenhauer and Jarvis have found that there are approximately 6670903752021072936960 à ¢Ã¢â‚¬ °Ã‹â€  6.671 x 10 ²Ã‚ ¹ Sudoku solutions. In light of this result, there are fewer solutions than Latin squares due to the fact that there is that extra restriction of 9 sub-grids. That being said, there will be no shortage of Sudoku puzzles any time soon. Verification of this result has been confirmed by several other mathematicians Ed Russell to be more precise. 2.2 Essentially different Sudoku grids Whether symmetrical Sudoku grids are considered as two separate solutions is another method of enumerating the possible solutions. In this case, the only solutions are ones that are essentially different. Lets say two Sudoku grids are equivalent if one is a transformation of the other by applying any number of symmetries. If however, no such chain of symmetries can occur between two grids, it is essentially different. Two Sudoku grids are the same provided the first grid can be converted to the second by applying some sort of symmetry. For instance, take figure 3 4 below; the set of 3s in the first grid can be interchanged by the placements of the set of 1s, effectively producing the second grid. 1 7 2 8 6 4 9 3 5 4 9 3 5 1 7 2 8 6 6 5 8 2 9 3 1 7 4 2 4 7 3 5 9 6 1 8 8 6 5 1 7 2 3 4 9 3 1 9 4 8 6 5 2 7 5 8 6 7 2 1 4 9 3 9 2 4 6 3 8 7 5 1 7 3 1 9 4 5 8 6 2 Figure 3. Valid Sudoku grid 3 7 2 8 6 4 9 1 5 4 9 1 5 3 7 2 8 6 6 5 8 2 9 1 3 7 4 2 4 7 1 5 9 6 3 8 8 6 5 3 7 2 1 4 9 1 3 9 4 8 6 5 2 7 5 8 6 7 2 3 4 9 1 9 2 4 6 1 8 7 5 3 7 1 3 9 4 5 8 6 2 Figure 4. Another valid Sudoku grid from Figure 1 As well as this, a solution is said to be the same as another if any two columns or rows are swapped. The first column and second column in figure 3 can be exchanged to give figure 5. The two solutions are said to be symmetrical because the transformation still produces a valid Sudoku grid. 7 3 2 8 6 4 9 1 5 9 4 1 5 3 7 2 8 6 5 6 8 2 9 1 3 7 4 4 2 7 1 5 9 6 3 8 6 8 5 3 7 2 1 4 9 3 1 9 4 8 6 5 2 7 8 5 6 7 2 3 4 9 1 2 9 4 6 1 8 7 5 3 1 7 3 9 4 5 8 6 2 Figure 5. First and second column swapped from Figure 1. Another form of symmetries includes rotational grids. A rotation of Figure 3 by 90 degrees generates a new valid Sudoku grid shown in Figure 6. 7 9 5 3 8 2 6 4 1 3 2 8 1 6 4 5 9 7 1 4 6 9 5 7 8 3 2 9 6 7 4 1 3 2 5 8 4 3 2 8 7 5 9 1 6 5 8 1 6 2 9 3 7 4 8 7 4 5 3 6 1 2 9 6 5 9 2 4 1 7 8 3 2 1 3 7 9 8 4 6 5 Figure 6. Rotational of 90 degrees from figure 1 These operations performed above maintain the property of it being valid and this is known as symmetries of a grid. When an object is subject to these operations, certain properties are preserved. An example would be if one performs symmetry on to a Sudoku grid and repeats this operation once more, the final transformation is itself symmetric. In addition a symmetrical object can be transformed back to its original state by another form of symmetry. Performing several symmetries on a Sudoku grid can also be achieved by grouping its neighbouring pair. So the first symmetry can be paired with the second or the second can be paired with the third and so on. The resulting transformation is nevertheless the same either way. From these properties, it is inevitable to say that the set of symmetries form a group of any Sudoku grid. A group is a set G if it satisfies the following properties: CLOSURE If f and g are elements of G, then f ·g is also an element of G. ASSOCIATIVITY If f, g, and h are elements of G, then f ·(g ·h)=(f ·g) ·h must satisfy. IDENTITY ELEMENT There is an element e in G such that g ·e=e ·g=g for all g in G. INVERSE For any element g of G, there is another element d of G such that g ·d=d ·g=e, where e is the identity element. (The element d = g-1.) The symmetry group is thus generated by the transformations of: re-labelling the nine digits, permuting the three stacks (3 vertical blocks of a Sudoku), permuting the three bands (3 horizontal blocks of a Sudoku), permuting the three columns within a stack, permuting the three rows within a band, and any reflection or rotation. Any two transformations can be merged to shape other elements and together they comprise of the symmetry group G. Given that any element of G can be mapped so that it takes one grid to another, we can say that the set of valid Sudoku grids has a finite number of elements. Thus G has finitely many symmetries. The association between symmetrical Sudoku grids are in fact an equivalence relation and satisfies the following three properties: for grids A, B and C in set G Reflexivity A = A Symmetry If A = B then B = A Transitivity If A = B and B = C then A = C Let A be any valid Sudoku grid, we must consider all the grids that are equivalent to a valid Sudoku grid A. To do this, we firstly have to group together grids that are essentially the same so that we can partition the set of grids. This will break the set of Sudoku grids into subsets, with groups that contain no relating elements within each other. The term subset can be called equivalence classes. This can also be referred to as X/G. In any equivalence class, there are elements that are equivalent to each other by symmetry. The total number of elements in X/G is equal to the number of essential Sudoku grids. To enumerate all essential Sudoku grids, we shall look at all the symmetries neglecting the re-labelling of the nine digits for the time being. The number of distinct symmetries founded by Russell and Jarvis (2006) is said to contain 3359232 (pg 4). In this finite group H, we need to find the average number of grids fixed by an element of H, up to re-labelling. Next we need to verify the number of fixed points of all elements in H. Russell and Jarvis have found that there are 275 classes of symmetries using a software package called GAP. It is interesting to note that some of the elements in H contain equivalent fixed grids. In other words, it is now easier to work out as each of the classes contains one symmetry. However a number of symmetries in H have no fixed points. Subsequently, it is not necessary to calculate the number of fixed grids for those that have no fixed points. That being said, there are only 27 out of 275 classes that contain fixed points, meaning fewer computation s. Rotman. J. J (1995) demonstrate that if X is a finite G-set and |X/G| is the number of G-orbits of X, then Formula 2 holds where, for gцG, X is the number of xцX fixed by g (pg 58-61). Using this notion, we have established that the number of valid Sudoku grids is of a finite set and X/G is the number of essentially different Sudoku grids, so we can obtain the number of essentially different Sudoku grids by using the Burnside Lemma Theorem. Formula 2. Burnside Lemma Theorem (Rotman, 1995) Burnside Lemma Theorem is a useful tool when dealing with symmetry with a set of countable objects. When used to enumerate the essentially different Sudoku grid, the set of equivalent grids form an orbit of the symmetric group. The number orbits are essentially the number of different grid solutions. This may sound slightly (ALOT) trickier to compute, nonetheless Russell and Jarvis have shown that the number of essentially different Sudoku grids is 5,472,730,538 with the implementation of Burnsides Lemma Theorem. Chapter 3 Nondeterministic polynomials 3.1 NP-complete and Sudoku Sudokus may relate to a variety of problems, in particularly, whether Sudoku is an NP-complete problem. It is known that NP-complete problems are one of the most complicated cases in NP, also referred to as nondeterministic-polynomial. Its rival, P problems relates to NP as both being in the same complexity class. Mathematicians have yet to solve whether NP-complete problems can be solved in polynomial time or more commonly whether P = NP. Consequently being one of the greatest unsolved mathematical problems. The majority of computer scientists believe that P à ¢Ã¢â‚¬ °Ã‚   NP, as a result would mean that NP-complete problems are significantly trickier to compute than to verify. Unfortunately, nobody has yet found an efficient algorithm, not even with the use of computers available today. A problem is said to be NP-complete when its solution can be proved in polynomial time. And if that problem can be solved in polynomial time, all problems in NP can be solved too. An interesting characteristic of NP-complete problems is that the time frame to solve the problem increases rapidly as the size of the problem gets larger. If that is the case and Sudokus are NP-complete, solving a Sudoku of higher order (say 17 ² x 17 ²) will become increasingly challenging algorithmically then the standard 3 ² x 3 ² version were talking trillions of years. It has been shown that Sudoku does belong to the category of NPC problems by Takayuki Yato of the Univeristy of Tokyo (2003). An exchange for the notation ASP-completeness (shorthand for Another solution problem), led the proof of NP-completeness of ASP. Their proof uses reduction in order to obtain the required polynomial-time ASP from the problem of Latin squares by Colbourn (1984) who has verified, the NP-completeness of ASP of Latin square completion Another accountable source by Provan states that, It is known that solving general-sized Sudoku puzzles is NP-hard, even for square grids with blocks consisting of the sets of rows and columns (Latin Squares) or for p2 x p2 grids with blocks consisting of rows, columns, and the p2 partitioned p x p subsquares. Mathematical programmes such as the 0-1 linear programming and the knapsack problems are also cases of NP-complete problems. A full list of other problems that are NP-complete can be found in Garey and Johnson (1979). Chapter 4 Sudoku Variants 4.1 Variation The classic form of a 99 Sudoku are polyominoes. There are other variations of Sudokus that can be applied to the rules of Sudoku. There are puzzles of the size 66 with 23 regions or a 1212 grid of 43 regions. More so, there are other fascinating Sudoku variants such as Greater than Sudoku. Chapter 5 Personal Critical Review The progress I have made during the duration of this project, have been fairly slow but surely getting there. Having said this on many occasions, I have still not conquered my time management skills! The project started very slow which meant I was behind schedule. Nevertheless my organisational skills have kept me on balance. The GANT chart has been of great help in doing so. What has kept me going throughout this project in particular would be self discipline and motivation. This project has proven that I am capable of working to my own initiative, but also well within a group; my time during the group project. Furthermore, my time on this project has definitely promoted a better mentality of my future ambitions. I have learnt that it is crucial to read a lot, as well as reading as broadly as I can. This in turn have aided in the running of my project. With other coursework deadlines, I made that a priority and had no time to meet with my supervisor. I understand that meeting with my supervisor is equally important because a supervisor is there to encourage and to advice on any difficult obstacles I may encounter. An area of interest to proof whether NP-complete problems can be solved in polynomial time, was left open as future work. This could be the next step of extending this report that little bit further. Chapter 6 Conclusion A challenging problem for further research is to proof whether NP-complete problems can be solved in polynomial time. This has yet to be solved and anyone who has a formal proof will be rewarded $1 million dollars by The Clay Mathematics Institute.

A College Student’s Approach to Courtly Love Essay -- Relationships Li

A College Student’s Approach to Courtly Love The term "courtly love" is a highly ambiguous one. As it applies to works of literature, it spans over hundreds of years and over a half dozen countries. Hence finding its specific literary and allegorical definition and impact on literature is difficult. It is important to understand the roots of courtly love. To do so means that one gains a greater understanding of the most foundational element of any society- the relationship between men and women. If a student of literature holds only a vague understanding of courtly love, then he or she holds only a vague understanding of medieval culture. In turn when this student moves on to various other periods of British literature, they will have a nearly impossible time determining in what ways the dynamics of romantic relationships and marriage have changed. In this paper I will work to find a concise yet comprehensive definition for courtly love that may be useful to students of literature. Additionally I will explore the impact of courtly love on the literature in which it makes appearances. Finally I will examine the contemporary understanding of this term and how it is relevant to contemporary times. Gaston Paris first coined the term "amour courteious" in 1883. In this, it is clear that the French have had a tremendous impact on the spread of this phenomenon in literature (as with the French troubadours). Courtly love certainly functions on two levels. We must distinguish these two uses of the word- to describe experience and to denote a genre. In the first it is a set of codes that regulates the interaction between two lovers. There are set rules that were often unspoken at the time. In this sense it is not so much a term ... ...e if we have no starting point? Women searching for liberation from social customs and restrictions cannot find it without empathizing with those who came before them. Inner conflicts between human nature and social well being that men experience must be dealt with in some way. If he knows the attempts proposed before him, he can save time and effort in reserving from investing in these. For any reader of any time period of the courtly love tradition, these questions and should remain in mind. The answers may give us a new direction to move in as we reread the classics and write our own. Works Cited Ford, Boris. Medieval Literature. Part Two: The Eurpean Inheritance. New York, 1983. O’Donogue, Bernard. The Courtly Love Tradition. New Jersey, 1982. Stevens, J.E. Medieval Romance. London, 1973. Zumthor, Paul. Speaking of the Middle Ages. Lincoln, 1986.

Once More, America, Before I Go Essay

The explication of poetry demands close reading of a single short poem or several stanzas of a longer work. Its goal is to unearth the hidden meaning/s of the poem by using the poetic techniques and elements employed by the author. Some of these techniques and elements include â€Å"diction, stanza and line structure, meter, rhythm and imagery (â€Å"Poetry Explication,† n. d. ). Walt Whitman’s poem, â€Å"Once More, America, Before I Go,† benefits from the use of explication due to its abstract nature, as it lacks concrete and specific imagery. To offset this problem, an in-depth look at the way Whitman uses rhythm and language will help to expound on the theme of the American democracy, of which he was an outspoken supporter. For Whitman, rhythm and language are intertwined, as the rhythm of the poem is inevitably linked with the type of language used. The work begins with the lines from which the poem takes its title: â€Å"One song, America, before I go / I’d sing, o’er all the rest, with trumpet sound, / For thee—the Future (Whitman, 1872). † This first stanza is notable: it establishes and introduces the readers to Whitman’s radical departure from traditional poetics. Note that the stanza seems like one continuous line, as if it were written in prose. Yet, this prose unit is broken in erratic intervals to form lines and not one continuous sentence. Whitman’s experimentation encapsulates perfectly his view of the democratic American society. This society, he believed, was the best form of society because it allowed for the individual’s self-expression and self-formation. Written as if spoken from his deathbed, as signaled by the first line, he tells American that it is the â€Å"Future. † The first letter of future is capitalized, which indicates it to be a proper noun. As such, future was become synonymous with future, and, at the same time, it implies the American democracy is the future, the mold for everybody to follow. Words such as these pepper the work, as can be seen in succeeding stanzas. In the second stanza, he elaborates on the other things he would do for America before dying: â€Å"I’d sow a seed for thee of endless Nationality; / I’d fashion thy Ensemble, including Body and Soul; / I’d show, away ahead, thy real Union, and how it may be accomplish’d (Whitman, 1872). † Nationality, ensemble, body and soul, and union all have their first letters capitalized. Again, Whitman’s unique use of language here gives the poem a deeper meaning. By using the same technique he used with the word â€Å"future† in the preceding stanza, he again turns these abstract concepts into concrete proper nouns. Furthermore, through such technique, he emphasizes the America will inevitably be the paradigm of all these because of democracy. In the second line, three words are capitalized: ensemble, body, and soul. All of these points to Whitman’s desire to form the perfect citizenry of America. In order to do this, he had to start with perfecting the individual person, a goal that can easily be reached because of democracy. The third stanza is different from the rest of the poem, being set off in parenthesis. It indicates a plan he will only start, but not accomplish, unlike those tasks he mentioned initially: â€Å"(The paths to the House I seek to make, / But leave to those to come, the House itself. ) (Whitman, 1872)† Here, Whitman is broaching on the continuation of time from the past to the future, and the fact that the experiment in democracy will see its final form in the future. He will only blaze the trails, but the final form will be for the future. The poem ends with an assertion of his belief. However, he says that simply believing will not bring results – they must also prepare: â€Å"Belief I sing—and Preparation (Whitman, 1872)† Both must act together to fulfill the goal not only for the present but also for the future: â€Å"Life and Nature are not great with reference to the Present only, / But greater still from what is yet to come, / Out of that formula for Thee I sing (Whitman, 1872). † He believes that the present is already good, as emphasized by the words life, nature, and present having their first letters capitalized. However, he believes that with the coming of future comes the fulfillment of the promise afforded by democracy.

Demand Forecasting and Production Planning

ScienceAsia 27 (2001) : 271-278 Demand Forecasting and Production Planning for Highly Seasonal Demand Situations: Case Study of a Pressure Container Factory Pisal Yenradeea,*, Anulark Pinnoib and Amnaj Charoenthavornyingb a Industrial Engineering Program, Sirindhorn International Institute of Technology, Thammasat University, Patumtani 12121, Thailand. b Industrial Systems Engineering Program, School of Advanced Technologies, Asian Institute of Technology, P. O. Box 4, Klong Luang, Patumtani 12120, Thailand. * Corresponding author, E-mail: [email  protected] tu. ac. th Received 24 May 2001 Accepted 27 Jul 2001 ABSTRACT This paper addresses demand forecasting and production planning for a pressure container factory in Thailand, where the demand patterns of individual product groups are highly seasonal. Three forecasting models, namely, Winter’s, decomposition, and Auto-Regressive Integrated Moving Average (ARIMA), are applied to forecast the product demands. The results are compared with those obtained by subjective and intuitive judgements (which is the current practice). It is found that the decomposition and ARIMA models provide lower forecast errors in all product groups. As a result, the safety stock calculated based on the errors of these two models is considerably less than that of the current practice. The forecasted demand and safety stock are subsequently used as inputs to determine the production plan that minimizes the total overtime and inventory holding costs based on a fixed workforce level and an available overtime. The production planning problem is formulated as a linear programming model whose decision variables include production quantities, inventory levels, and overtime requirements. The results reveal that the total costs could be reduced by 13. % when appropriate forecasting models are applied in place of the current practice. KEYWORDS: demand forecasting, highly seasonal demand, ARIMA method, production planning, linear programming, pressure container factory. INTRODUCTION Most manufacturing companies in developing countries determine product demand forecasts and production plans using subjective and intuitive judgments. This may be one factor that leads to production inefficiency. An accuracy of the demand forecast significantly affects safety stock and inventory levels, inventory holding costs, and customer service levels. When the demand is highly seasonal, it is unlikely that an accurate forecast can be obtained without the use of an appropriate forecasting model. The demand forecast is one among several critical inputs of a production planning process. When the forecast is inaccurate, the obtained production plan will be unreliable, and may result in over- or understock problems. To avoid them, a suitable amount of safety stock must be provided, which requires additional investment in inventory and results in an increased inventory holding costs. In order to solve the above-mentioned problems, systematic demand forecasting and production planning methods are proposed in this paper. A case study of a pressure container factory in Thailand is presented to demonstrate how the methods can be developed and implemented. This study illustrates that an improvement of demand forecasts and a reduction of total production costs can be achieved when the systematic demand forecasting and production planning methods are applied. The demand forecasting and production planning methods are proposed in the next section. The background of the case study, including, products, production process, and the forecasting and production planning procedures being used in the factory, are briefly described in Section 3. The detailed analyses of the forecasting methods and the production planning method are explained in Section 4 and Section 5, respectively. Finally, the discussion and conclusion are presented in Section 6. 272 ScienceAsia 27 (2001) P ROPOSED D EMAND F ORECASTING PRODUCTION PLANNING METHODS AND The proposed demand forecasting and production planning methods are depicted in a step-by-step fashion in Fig. . Most factories produce a variety of products that can be categorized into product groups or families. Individual products in the same product group generally have some common characteristics. For example, they may have the same demand pattern and a relatively stable product mix. As a result, it is possible to forecast the aggregate demand of the product group first, and then disaggregate it in to the demand of individual products. Since the forecast of the aggregate demand is more accurate than that of the individual demand1, it is initially determined in Step 1. Then the demands of individual products are determined in Step 2 by multiplying the aggregate demand with the corresponding product mix that is normally known and quite constant. Since the demand forecasts are always subject to forecast errors, safety stocks are provided to avoid stock-out problems. Based on the standard deviation of the forecast errors and the required service level, the safety stocks for individual products are determined in Step 3. Production planning decisions are so complicated and important that they should not be subjectively and intuitively made. Consequently, an appropriate production planning model should be formulated to determine the optimal decisions. With this model, its parameters, eg, demand forecasts, safety stocks, holding cost, overtime cost, machine capacity, inventory capacity, and available regular time and overtime, are entered or updated (Step 4). In step 5, the optimal decisions regarding the production quantities, inventory levels, and regular production time and overtime for each product in each production stage are obtained by solving the production planning model. Step 6 indicates that only the optimal production plan of the current month will be implemented. After one month has elapsed, the demand forecasts and the production plan will be revised (by repeating Steps 1 to 5) according to a rolling horizon concept. BACKGROUND OF THE CASE STUDY The pressure container factory manufactures 15 products, ranging from 1. 25 to 50 kg of the capacity of pressurized gas. The products are divided into eight product groups, namely, Group 1 to Group 8. The first six groups have only two components, â€Å"head† and â€Å"bottom†, while the last two groups have three components, â€Å"head†, â€Å"bottom†, and â€Å"body†. The production process can be divided into five stages as shown in Fig. 2. Stage 3 is only required to produce the products having three components (ie, those in Groups 7 and 8). Stage 4, the circumference welding, is found to be a bottleneck stage due to its long processing time. Presently monthly demand forecasts are subjectively determined by the Marketing Department based on past sales and expected future market conditions. No systematic method is used in forecasting. Using these forecasts and other constraints, such as availability of raw materials, equipment, and production capacity, the monthly production plan for a three-month period is intuitively determined without considering any cost factor. This results in inaccurate demand forecasts and, subsequently, an inefficient production plan. Stage 1 Blanking 1) Forecast the monthly demands of each product group throughout the planning horizon of 12 months 2) Determine the demand for each individual product 3) Determine the safety stock for each individual product Stage 2 Forming of bottom and head Stage 3 Forming of body 4) Update the parameters in the production planning model Stage 4 Circumference welding 5) Run the planning model to obtain the optimal planning dicisions ) Roll the plan by repeating Steps 1 to 5 after one month has elapsed Stage 5 Finishing Fig 1. Proposed forecasting and planning steps. Fig 2. The production process to manufacture a pressure container. ScienceAsia 27 (2001) 273 FORECASTING METHODS Steps 1, 2, and 3 of the proposed forecasting and planning process are discussed in detail in this section. Firstly, the aggregate demand forecasts of eight product groups throughout the planning horizon of 12 mont hs will be determined. Secondly, the demand forecasts of the product groups will be disaggregated into those of individual product. Thirdly, the safety stocks of individual product will be calculated based on the forecast error. Aggregate Demand Forecasts of Product Groups The typical demand pattern of each product group is seasonal. As an example, Fig. 3 shows the demand pattern of Product Group 3. Thus, three forecasting models that are suitable for making seasonal demand forecasts are considered. They are Winter’s, decomposition and Auto-Regressive Integrated Moving Average (ARIMA) models. 2-5 Because of their simplicity, the Winter’s and decomposition models are initially used to forecast the aggregate demand of each product group. If the Winter’s and decomposition models are inadequate (ie, the forecast errors are not random), the ARIMA model which is more complicated and perhaps more efficient will be applied. The Winter’s model has three smoothing parameters that significantly affect the accuracy of the forecasts. These parameters are varied at many levels using a computer program to determine a set of parameters that give the least forecast errors. There are two types of the decomposition model, namely, multiplicative and additive types. The former is selected since the demand pattern shows that the trend and seasonal components are dependent. The forecast errors of the Winter’s and decomposition models are presented in Table 1. Based on the calculated mean square error (MSE) and the mean absolute percentage error (MAPE), it is seen that the decomposition model has lower Original Series (x 1000) 16 forecast errors in all product groups than the Winter’s model. Thus, it is reasonable to conclude that the decomposition model provides better demand forecasts than the other. One way to check whether the forecasting model is adequate is to evaluate the randomness of the forecast errors. The auto-correlation coefficient functions (ACFs) of the errors from the decomposition model for several time lags at the significant level of 0. 05 of each product group are determined. The ACFs of Groups 1 and 3 are presented as examples in Fig. 4 and 5, respectively. The ACFs of Groups 4, 5, 6, 7, and 8 are similar to those of Group 1 in Table 1. Forecast errors of the Winter’s and decomposition models. MSE Products MAPE (%) Winter’s Decomposition Winter’s Decomposition 9,879,330 4,363,290 2,227,592 4,507,990 10,039,690 574,108 636,755 883,811 36. 14 48. 94 24. 25 30. 08 18. 80 53. 86 61. 99 46. 52 26. 97 31. 86 15. 97 23. 4 13. 14 34. 80 34. 45 28. 76 Group 1 16,855,149 Group 2 8,485,892 Group 3 5,433,666 Group 4 6,035,466 Group 5 23,030,657 Group 6 1,690,763 Group 7 2,034,917 Group 8 1,884,353 Estimated Autocorrelations 1 0. 5 coefficient 0 -0. 5 -1 0 4 8 lag 12 16 20 Fig 4. ACFs of the residuals from the decomposition model for Group 1. Estimated Autocorrel ations 1 0. 5 16 demand 3 coefficient 0 8 -0. 5 4 -1 0 0 10 20 30 time index 40 50 60 0 4 8 lag 12 16 20 Fig 3. Actual demand of Group 3. Fig 5. ACFs of the residuals from the decomposition model for Group 3. 274 ScienceAsia 27 (2001) Fig 4, while those of Groups 2 and 3 are similar. It can be seen from Fig. 4 that the ACFs of all lags are within the upper and lower limits, meaning that the errors are random. However, the ACF of lag 1 in Fig. 5 exceeds the upper limit. This indicates that auto-correlations do exist in the errors and that the errors are not random. From the ACFs, we can conclude that the decomposition model is adequate for forecasting the demands of Groups 1, 4, 5, 6, 7, and 8, but inadequate for forecasting those of Groups 2 and 3. Therefore, the ARIMA model is applied to Groups 2 and 3. From the original time series of the demand of Group 3 (in Fig. 3), and the ACFs of its original series (in Fig. ), it can be interpreted that the original series has a trend, and a high value of ACF of lag 12 indicates the existence of seasonality. 2 Hence, a non-seasonal first-difference to remove the trend and a seasonal first-difference to remove the strong seasonal spikes in the ACFs are tested. Fig. 7 shows the ACFs of the ARIMA (p,1,q)(P,1,Q) 12 model afte r applying the first difference. The nonseasonal plot indicates that there is an exponential decay and one significant ACF of lag 2. Thus, the AR(1) and MA(1) process denoted by ARIMA (1,1,1)(0,1,0)12 is identified. The ACFs of the residuals after applying this ARIMA model shown in Fig. reveals that there is a high value of ACF of lag 12. Therefore, the AR(1) and MA(1) process for the seasonal part or ARIMA (1,1,1)(1,1,1)12 can be identified. The ACFs of the residuals generated from this model are shown in Fig. 9. Since all ACFs are within the two significant limits, the ARIMA (1,1,1)(1,1,1)12 model is adequate. Using the Statgraphic program, the model coefficients can be determined. The demand forecast for Group 3 is presented in Eq. 1. Ft = 1. 197 X t ? 1 ? 0. 197 X t ? 2 + 0. 54408 X t ? 12 ? 0. 65126 X t ? 13 + 0. 10718 X t ? 14 + 0. 45592 X t ? 24 ? 0. 54574 X t ? 25 + 0. 08982 X t ? 26 ? 1. 6699et ? 1 ? 0. 7154et ? 12 + 0. 76332et ? 13 + 29. 34781 (1) where Ft is the demand fo recast for period t Xt is the actual demand for period t et is the forecast error for period t Similarly, the forecasting model for Group 2 is ARIMA (3,0,0)(3,0,0). 12 The demand forecast of Group 2 is presented in Eq. 2. Estimated Autocorrelations for Original Series 1 Estimated Residual ACF 1 0. 5 0. 5 coefficient coefficient 0 0 -0. 5 -0. 5 -1 0 5 10 lag 15 20 25 -1 0 5 10 lag 15 20 25 Fig 6. ACFs of the actual demand for Group 3. Fig 8. ACFs of the residuals of ARIMA (1,1,1)(0,1,0)12 model for Group 3. Estimated Residual ACF 1 Estimated Autocorrelations for 1 Nonseasonal Differences 1 Seasonal Differences 1 0. 5 0. 5 coefficient coefficient 0 0 -0. 5 -0. 5 -1 0 5 10 lag 15 20 25 -1 0 5 10 lag 15 20 25 Fig 7. ACFs after first differencing for Group 3. Fig 9. ACFs of the residuals of ARIMA (1,1,1)(1,1,1)12 model for Group 3. ScienceAsia 27 (2001) 275 Ft = 0. 36951X t? 1 + 0. 30695X t? 2 – 0. 18213X t? 3 + 0. 20132 X t? 12 ? 0. 07439 X t? 13 ? 0. 06180 X 14 + 0. 03667 X t? 15 ? 0. 03325X t? 24 + 0. 01228 X t? 25 + 0. 01021X t? 26 ? 0. 00606 X t? 27 + 0. 68660 X t? 36 ? 0. 25371X t? 37 ? 0. 21075X t? 38 + 0. 12505X t? 39 + 354. 4515 2) The forecast errors of the decomposition and ARIMA models for Groups 2 and 3 are presented in Table 2. It reveals that the ARIMA model has lower Table 2. Forecast errors of the decomposition and ARIMA models. MSE Products Group 2 Group 3 Decomposition ARIMA 4,363,290 2,227,592 3,112,974 1,235,788 MAPE (%) Decomposition ARIMA 31. 86 15. 97 29. 05 13. 18 MSE and MAPE than t he decomposition model. Therefore, the ARIMA model should be used to forecast the aggregate demands of Groups 2 and 3. For other product groups, however, the decomposition model should be used because it is more simple yet still adequate. The comparison of the demand forecast errors obtained from the forecasting models and those from the current practice of the marketing department (as presented in Table 3) indicates that the errors of the forecasting models are substantially lower than those of the current practice. Demand Forecasts of Individual Products The demand forecast of product i for period t, dit, is obtained by multiplying the aggregate demand forecast of the product group (obtained from the previous steps) by the corresponding product mix (as presented in Table 4). Table 3. Forecast errors of the current practice, decomposition, and ARIMA models. MSE Product Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Current practice Decomposition 16,672,342 4,394,693 4,988,962 4,754,572 19,787,102 795,621 849,420 1,060,301 9,879,330 4,507,990 10,039,690 574,108 636,755 883,811 ARIMA 3,112,974 1,235,788 MAPE (%) Current practice Decomposition 30. 58 34. 68 23. 50 25. 73 17. 54 42. 70 38. 36 37. 93 26. 97 23. 24 13. 14 34. 80 34. 45 28. 76 ARIMA 29. 05 13. 18 – Table 4. Product mix. Product group Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0. 17 0. 20 0. 26 0. 23 0. 14 1. 0 0. 53 0. 47 0. 65 0. 35 1. 0 1. 0 1. 0. 3 0. 7 2 3 4 5 6 7 8 276 ScienceAsia 27 (2001) Calculation of Safety Stock The safety stocks of finished products must be provided to protect against stock-out problems due to inaccurate demand forecasts. Based on the forecast errors obtained from the demand forecasting models, the amount of the safety stock is calculated using the following formula. 12 SSit = sf * ? j * ? ij (3) PRODUCTION PLANNING METHO D The production planning model is developed by initially defining decision variables and parameters, and then mathematically formulating the production planning model. Step 4 of the method requires that the model parameters be estimated and entered into the model. The model is solved for the optimal solution (Step 5). Step 6 recommends that the model parameters are updated, and the model is solved again after one planning period has passed. The production planning problem of the factory under consideration belongs to the class of multistage, multi-item, capacitated production planning model. The models in this class have been discussed extensively in. 6-11 They differ in assumptions, objectives, constraints, and solution methods. Our production planning model is a modification of the multi-stage, multi-product model discussed in Johnson and Montgomery. 6 Its objective is to minimize the total overtime and inventory holding costs. Costs of laying off and rehiring are not considered because laying off and rehiring are not allowed according to the labor union regulation. Since the production cost is time-invariant and all demands must be satisfied, the regular time production cost is thus not included in the objective function. Relevant parameters and decision variables are defined as follows: Parameters : hik = Holding cost per unit of product i at stage k (baht/unit/period) co = Cost per man-hour of overtime labor (baht/man-hour) dit = Demand forecast of product i for period t (units) aik = Processing time for one unit of product i at stage k (hours/unit) (rm)kt = Total available regular time excluding preventive maintenance and festival days at stage k for period t (man-hours) (om)kt = Total available overtime excluding preventive maintenance and festival days at stage k for period t (man-hours) W = Warehouse capacity (units) SSit = Safety stock of product i for period t (units) Iik0 = Initial inventory of product i at stage k (units) N = Total number of products (15 products) T = Total number of periods in the planning horizon (12 periods) K = Total number of stages (5 stages) where SSit = Required safety stock level of product i for period t sf = Safety factor = 1. 64 for a required service level of 95 % of the standard normal distribution ? j = Standard deviation of forecast errors of Group j. ?ij = Product mix of Product i in Group j. Since the errors of the recommended demand forecasting models are lower than those of the current practice, it is clear that SSit based on the use of the models must be lower than that determined from the current practice (assuming that the service levels of both cases are the same). Table 5 presents the required safety stocks of the current practice and the recommended forecasting models at 95 % service level. Table 5. Required safety stock of current practice and of recommended forecasting models. Safety stock (units) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Current practice 1,138 1,339 1,741 1,540 937 3,438 1,941 1,722 2,324 1,252 7,295 1,463 1,511 507 1,182 Recommended forecasting models 887 1,043 1,356 1,200 730 2,905 979 868 2,274 1,224 5,258 1,245 1,323 460 1,072 ScienceAsia 27 (2001) 277 Decision variables: Xikt = Quantity of product i to be produced at stage k in period t (units) Iikt = Inventory of product i at stage k at the end of period t (units) Rkt = Regular time used at stage k during period t (man-hours) Okt = Overtime used at stage k during period t (man-hours) LP model: Minimize Z = ? ? ? hik Iikt + ? ? co Okt , (4) i =1 k =1 t =1 k =1 t =1 N K T K T Eq. 7 represents the material balance constraint in Stage 3, which produces the body of threecomponent products, for Products 13, 14, and 15. Constraint (13) must be included since the finished products are very bulky and require significant warehouse space that is quite limited. Work-inprocess inventory does not require significant storage space because it can be stacked. The non-negativity constraint (16) ensures that shortages of work-inprocess inventory do not occur. Input Parameters The initial inventory of product i at stage k, Iik0, is collected from real data of work-in-process or finished good inventories on the factory floor at the beginning of the planning horizon. The inventory holding cost of product i at stage k, hik, is estimated by assuming that the annual inventory holding cost is 25% of the cost per unit of the product at the respective production stage. Since the cost per unit is constant over the planning horizon, the annual inventory holding cost is time-invariant. The factory has enough space in the warehouse to store not more than 40,000 units of finished products. The total available regular time, (rm)kt, is estimated based on the fact that the factory is normally operated 16 hours a day and six days a week, and the total available overtime, (om)kt, is calculated by assuming that the overtime could not be more than six hours a day. The overtime cost, co, is assumed to be constant throughout the planning horizon, and is estimated to be 60 Baht per man-hour. After all related parameters have been estimated and entered into the planning model, the optimal values of all decision variables are calculated using the LINGO software. The computation time takes less than one minute on a Pentium PC. Results of the Production Planning Models with Different Levels of Safety Stock In this section, two production planning models with different safety stock levels (as shown in Table 5) are solved to determine the total cost savings when the recommended forecasting models are applied in place of the current practice. The inventory holding, overtime, and total costs of both models are presented in Table 6. Based on the optimal total cost of the current practice (4,078,746 Baht per year) and the optimal total cost of the recommended forecasting models (3,541,772 Baht per year), the total cost saving is 536,974 Baht per year, or 13. 2 %. It can be also seen Subject to – Finished product requirement constraints I i 5,t? 1 + X i 5t ? I i 5t = dit – ? i, t ; k = 5, (5) Material balance between stages constraints ? i, t ; k = 4, (6) (7) ? i, t ; k = 2, (8) ? i, t ; k = 1, (9) I i 4 ,t? 1 + X i 4 t ? I i 4 t = X i 5t I i 3,t? 1 + X i 3t ? I i 3t = X i 4 t ?t ; i = 13, 14, 15; k = 3, I i 2,t? 1 + X i 2t ? I i 2t = X i 4 t I i1,t? 1 + X i1t ? I i1t = X i 2t Capacity constraints ? aik X ikt ? Rkt + Okt i= 1 N ?k , t , (10) – Available regular and overtime constraints. Rkt ? (rm) kt Okt ? ( om) kt ?k , t , ? k , t , (11) (12) – Inventory capacity of finished product constraints. ? I ikt ? W i= 1 N ?t ; k = 5, (13) – Safety stock of finished product constraints. I ikt ? SS it ?i, t ; k = 5, (14) – Non-negativity conditions X ikt ? 0 I ikt ? 0 ?i, k , t , ? i, t ; k = 1, 2, 3, 4 (15) (16) 278 ScienceAsia 27 (2001) Table 6. Comparison of the optimal costs of production planning models. Optimal costs (Baht/year) Model based on the current practice Inventory holding cost Overtime cost Total cost 2,117,051 1,961,695 4,078,746 Model based on recommended forecasting models 1,775,552 1,766,220 3,541,772 REFERENCES 1. Nahmias S (1993) Production and Operations Analysis, 2nd ed, Irwin, New York. 2. Vandaele W (1983) Applied Time Series and Box-Jenkins Models, Academic Press, New York. 3. Winters PR (1960) Forecasting Sales by Exponentially Weighted Moving Average. Management Science 6(4), 324-42. 4. Box GE and Jenkins GM (1970) Time Series Analysis, Forecasting, and Control, Holden-Day, San Francisco. 5. Makridakis S Wheelwright SC and McGee VE (1983) Forecasting Methods and Applications, 2nd ed, John Wiley & Sons, New York. 6. Johnson LA and Montgomery DC (1974) Operations Research in Production Planning, Scheduling, and Inventory Control, John Wiley & Sons, New York. 7. Bullington P McClain J and Thomas J (1983) Mathematical Programming Approaches to Capacity Constrained MRP Systems: Review, Formulation, and Problem Reduction. Management Science 29(10). 8. Gabbay H (1979) Multi-Stage Production Planning. Management Science 25(11), 1138-48. 9. Zahorik A Thomas J and Trigeiro W (1984) Network Programming Models for Production Scheduling in MultiStage, Multi-Item Capacitated Systems. Management Science 30(3), 308-25. 10. Lanzanuer V (1970) Production and Employment Scheduling in Multi-Stage Production Systems. Naval Research Logistics Quarterly 17(2), 193-8. 11. Schwarz LB (ed) (1981) Multi-level Production and Inventory Control Systems: Theory and Practice, North-Holland, New York. 12. Tersine RJ (1994) Principles of Inventory and Materials Management, 4th ed, Prentice Hall, New Jersey. that the optimal inventory holding cost and overtime cost in the production planning model based on the recommended forecasting models are almost equal which indicates that the model can efficiently achieve a tradeoff between both costs. Normally, the optimal decisions in the first planning period will be implemented. After the first period has passed, the new forecasts will be determined, and the model parameters will be updated. The updated model is solved again to determine the optimal decisions in the current period. This is called a rolling horizon concept. However, the details and results of this step are not shown in this paper. DISCUSSION AND CONCLUSION The ARIMA model provides more reliable demand forecasts but it is more complicated to apply than the decomposition model. Therefore the ARIMA model should be used only when the decomposition model is inadequate. When compared against those of the current practice of the company, the errors of our selected models are considerably lower. This situation can lead to substantial reductions in safety stocks. Consequently, the lower safety stocks result in decreased inventory holding and overtime costs. The results of the production planning model are of great value to the company since the model can determine the optimal overtime work, production quantities, and inventory levels that yield the optimal total overtime and holding costs. The production planning method is more suitable than the existing one that does not consider any cost factors. Moreover, it has been proven that an application of appropriate forecasting techniques can reduce total inventory holding and overtime costs significantly. In conclusion, this paper demonstrates that an improvement in demand forecasting and production planning can be achieved by replacing subjective and intuitive judgments by the systematic methods.