The Gnumeric Spreadsheet for Solution of Widespread Non-Homogeneous Model of Renewal Machines
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Transfer inovácií 31/2015 2015 THE GNUMERIC SPREADSHEET FOR SOLUTION OF WIDESPREAD NON-HOMOGENEOUS MODEL OF RENEWAL MACHINES Ing. Alena Pešková use Windows. Users can find the instructions for Technical University of Košice work with this spreadsheet on the Gnumeric page Faculty of Mechanical Engineering (http://projects.gnome.org) Thus it creates favorable Department of Production Systems conditions for a broad base of users. The advantage Němcovej 32, 042 00 Košice, Slovakia is that it supports export and import from/to e-mail: [email protected] Microsoft Excel. It is part of office suite GNOME Office (OpenOffice.org). The benefit of this office Ing. Hana Sokolová suite is modest system requirements. The objective Technical University of Košice of Gnumeric is to attract the general public. The Faculty of Mechanical Engineering spreadsheet can handle large tables. It is able to Department of Process and import and export data to several different formats, Environmental Engineering for example CSV, Microsoft Excel, HTML, LaTeX, Park Komenského 5, 042 00 Košice, Slovakia Lotus 1-2-3, OpenDocument, etc. Format of Gnumeric is .gnm format or .gnumeric. Abstract MATTERS OF RENEWAL The objective of the article is to show a Renewal theory explores the optimal solution of discrete model of renewal machines time of appropriate substitution of objects and using Gnumeric spreadsheet. We will demonstrate scheduled plan of repairs. Solving examples searching of solution on an example of non- regarding the renewal of machines is based on homogeneous extended discrete model of renewal process modeling. Renewal of machines gives us machines. We determine what is the supposed the answer to the question what is the expected time of appropriate objects exchange via modelling number of objects that need to be replaced due to process using discrete models. wear or failure. In paper [3] we deal with the basic knowledge of homogenous and non-homogeneous Key words: Gnumeric, renewal of machines, models and procedure of solving renewal discrete discrete models of renewal, non-homogeneous models - basic homogenous and non-homogeneous models model. The models were solved in Excel. INTRODUCTION NON-HOMOGENOUS WIDESPREAD Renewal theory gives us the answer to DISCRETE MODEL OF RENEWAL the question of determining the expected number of Non-homogeneous model has diverse objects [1] Renewal discrete models observe the initial structure. Hence we have machines that can process of exchanging of these objects. Calculation be not only new but also have age t too (one year, with Gnumeric represents an available free two years, etc.) at the beginning of the operation solution. Although nowadays Excel is more start. We extend by another elements for this type preferred. Many users using this software does not of discrete model in engineering company. know about the availability of open spreadsheet However, we must distinguish between the Gnumeric and its benefits. We can also mention elements that are new E at the beginning and the another open software for spreadsheet besides O(t) elements that we include in the operation b : Gnumeric i.e. OpenOffice.org Calc, Calligra (t) Tables, KCells, LibreOffice Calc, NeoOffice, Siag. b(t) = EO(t)+∆(t) ∆(t) - accession of elements Open software Gnumeric We apply the calculation procedure in Gnumeric and its source code belongs to Gnumeric on the following example [4] : the open, freely available spreadsheet that is 1000 machines of the same type is used of which available for free, licensed under the terms of the 500 is new, 300 is one year old and 200 two-years GNU General Public License [2] The first version old is in operation. In following years these statuses of Gnumeric was released in December 2001. The of the machines are planned: 1st year - 1050, 2nd founder of Gnumeric was Miguel de Icaza. The year - 1150, 3rd year -1300, 4th year - 1500, 5th latest version designed for users is Gnumeric year - 1800. Probabilities of failure are: p(0)=0,2; 1.12.22 released in April 2015. It is mainly used by p(1)=0,3; p(2)=0. The maximum lifetime of objects the users who work under the operating system is 3 years. We observed the machines for the 5- Linux, but it is easily accessible even for those who 115 Transfer inovácií 31/2015 2015 years. Determine how many new machines we will Tab.5 Non-homogenous dicrete model with include in each year to extended renewal. expanded features Solution of the example: We are listing values needed for calculation from task of example for better clarification Tab.1 We calculate specific probabilities for individual years according to the We will explain in detail the solving formulas for the probabilities of survival Tab.2 procedure, where we are dealing with two separate These data are assumptions for creating a table of phases Tab.3, Tab.4. We have calculated the age structure. probabilities of survival according to Tab.2. We will create a table of age structure for non- Tab.1 Input data homogeneous discrete model Tab.3 Horizontal row tells us about monitored period, the vertical row about life cycle of the object (age). We present survival probability p(T>t) for clarity, clearness and the accurate calculation next to the table. Objects have different ages at the beginning because it is a non-homogeneous model. We have 500 new machines, 300 one year machines and 200 two years machines in our case. We have 1000 machines of the same type with different age structure together in operation. Sum of columns C44: H44 is the same, i.e. 1000. First line gives us the number of renewed objects that have age=0. We won’t calculate according to formulas for renewal for simplicity than in our previous article [3] calculation =C41*A42 applies for cells D42. We get value 400 when we multiply the number of Tab.2 Calculation for probabilities of survival objects (500 of new machines - machines with age 0) with probability of survival p(T>1)=0,8. We will calculate an conditional probability (p(T>2)/ p(T>1))*E1 for cell D43. Relationship = (D33/D32)*C42 applies. Therefore the probability of failure of At first, we are solving two stand alone object in the second year provided that it survived phases of the model, each separately. We calculate the first year multiplied by the number one year a classic example of non-homogeneous discrete objects - 300. We get the formula = D44- (D42 + model Tab.3 in the first phase, where we will use D43) in cell D41, where we calculate the number of the conditional probability. We deal with advanced renewed elements with age zero. We are counting features Δ(t) in the second phase Tab.4. the next column similarly. We will begin with cell D41*E42=A42 then E43. We can calculate the cell Tab.3 Non-homogenous dicrete model (1.phase) E41 since we know the sum of the column. We continue similarly in next columns. We calculated a table for an easy renewal for non-homogeneous discrete model by this way. We move on to the second table for calculation of expanded renewal. At first, we need to calculate additions of extended Tab.4 Extension of elements (2.phase) elements Δ(1)=1050-1000=50, analogous Δ(2)=150; Δ(3)=300; Δ(4)=500; Δ(5)=800 We write them to the last row in the table. We have to calculate number of new included elements in the first line. The value 50 will be in the cell D49 because we included so many elements to renewal Later, merge of both phases occurs Tab. in the first period. We will continue counting in the 5. We must be careful in the extended model second and next periods similarly to a simple because the final sum is the same, but has a renewal. We will calculate a table in this way. In growing character. 1st year - 1050, 2nd year - 1200, the next table single renewal and renewal phase get 3rd year - 1300, 4th year - 1500, 5th year - 1800. merged and we get nonhomogeneous expanded 116 Transfer inovácií 31/2015 2015 discrete model. After the union of both phases we References calculate according to the formula: [1] Valenčík, Š.: Metodika obnovy strojov. Košice. b (t) = Eo (t) + Δ (t) EVaOL Strojnícka fakulta TU Košice, Košice b(1)=50+413; b(2)=420+110; b(3)=458+187; 2011, 330 s., ISBN 978-80-533-0679-7. b(4)=424+295; b(5)=432+470 the number of [2] http://www.gnumeric.org (cit. 20.6.2015) renewed elements in the first row. We get from the [3] Pešková, A.: Using Excel to Solve Discrete example, that we will include into to the extended Models for Machine Renewal in Education renewal in the first period 50 elements, in the Process In: ATP journal plus. Č. 1 (2013), s. 57- second 110 elements, in the third 187 elements, in 59. - ISSN 1336-5010 the fourth 295 elements and the last 470 elements. [4] http://fsi.uniza.sk/ktvi/leitner/2_predmety/OA/C CONCLUSION vicenia/14_TO.pdf (cit. 20.6.2015) Gnumeric was proved to be a good choice for solving examples of renewal discrete This paper was supported by Project models mainly due to the availability of this VEGA n. 1/0124/15 Research and development software. We can use Gnumeric to solve models by of advanced methods for virtual prototyping of using Markov chains and optimal renewal strategies production machines. (optimal life cycle of object, the optimum cycle of renewal machine in terms of cost of its fail, optimal renewal cycle of the machine in terms of its access) as a spreadsheet to determine age structure of objects.