### Application of Fuzzy Number Max-Plus Algebra to Closed Serial Queuing Network with Fuzzy Activity Time

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

dipresentasikan pada: The 6-th SEAMS-GMU 2011 International Conference on Mathematics and Its Applications. July 12-15, 2011.

Abstract. The activity times in a queuing network are seldom precisely known, and then could be represented into the fuzzy number, that is called fuzzy activity times. This paper aims to determine the dynamical model of a closed serial queuing network with fuzzy activity time and its periodic properties using max-plus algebra approach. The finding shows that the dynamics of the network can be modeled as a recursive system of fuzzy number max-plus linear equations. The periodic properties of the network can be obtained from the fuzzy number max-plus eigenvalue and eigenvector of matrix in the system. In the network, for a given level of risk, it can be determined the earliest of early departure time of a customer, so that the customer's departure interval time will be in the smallest interval where the lower bound and upper bound are periodic.

Key words and Phrases: max-plus algebra, queuing network, fuzzy activity times, periodic.

### Pemodelan Jaringan Antrian Seri Tertutup Waktu Aktifitas Interval dengan Menggunakan Aljabar Max-Plus Interval

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

ABSTRAK

Artikel ini membahas tentang pemodelan dinamika dan sifat periodik jaringan antrian seri tertutup waktu interval dengan menggunakan aljabar max-plus interval. Hasil pembahasan menunjukkan bahwa dinamika jaringan antrian seri tertutup waktu interval dapat dimodelkan ke dalam suatu persamaan matriks atas aljabar max-plus interval. Dapat ditentukan pula saat keberangkatan awal tercepat pelanggan agar saat keberangkatan pelanggan selanjutnya berada dalam interval terkecil yang batas bawah dan batas atasnya periodik.

Kata Kunci : aljabar max-plus, interval, nilai eigen, jaringan antrian, periodik.

### A Max-Plus Algebra Approach to Critical Path Analysis in The Project Network With Fuzzy Activity Times

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

dipresentasikan pada: IndoMS International Conference on Mathematics and Its Applications (IICMA 2009). FMIPA UGM, Yogyakarta 11-12 Oktober 2009.

Abstract. The activity times in a project network are seldom precisely known, and then could be represented into the fuzzy number, that is called fuzzy activity times. This paper aims to determine the fuzzy earliest starting times, fuzzy total duration time and fuzzy critical path using max-plus algebra approach. The finding shows that the project network with fuzzy activity times can be represented as a matrix over fuzzy number max-plus algebra. The project networks dynamics can be represented as a system of fuzzy number max-plus linear equations. From the solutions of the system we can determine the fuzzy earliest starting times and fuzzy total duration time. The fuzzy critical path with a certain degree of critically can be determine through an interval critical path determination. We know that an alpha-cut of the fuzzy activity times are an interval activity times. The interval critical path can be determine through a crisp critical path determination. Meanwhile, the crisp critical path can be determine via a computation using max-plus algebra approach.

Key words and Phrases : max-plus algebra, project network, fuzzy activity times, fuzzy critical path.

### ALJABAR MAX-PLUS BILANGAN KABUR

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

dipresentasikan pada: Seminar Hasil Penelitian Mahasiswa S3 dalam Dies ke-54 FMIPA UGM. 10 Oktober 2009.

Extended Abstract
Aljabar max-plus telah digunakan untuk memodelkan dan menganalisis jaringan untuk waktu aktifitas deterministik. Waktu aktifitas dalam jaringan kadang tidak dapat diketahui dengan pasti dan dapat dimodelkan dalam suatu bilangan kabur (fuzzy number). Makalah ini akan membahas suatu aljabar dengan elemen-elemennya berupa bilangan kabur dengan operasi maksimum dan penjumlahan yang didefinisikan di dalamnya. Aljabar ini diharapkan dapat memberikan landasan analisa jaringan dengan waktu aktifitas kabur melalui pendekatan aljabar max-plus .
Aljabar max-plus bilangan kabur merupakan perluasan aljabar max-plus melalui aljabar max-plus interval dan menggunakan Teorema Dekomposisi dalam himpunan kabur. Aljabar max-plus interval sendiri merupakan perluasan aljabar max-plus di mana elemen-elemennya berupa interval-interval real dengan operasi maksimum dan penjumlahan interval di dalamnya. Aljabar max-plus interval merupakan semiring idempoten komutatif.
Dapat ditunjukkan operasi maximum dan penjumlahan yang didefinisikan melalui potongan-alfa bersifat tertutup dalam himpunan semua bilangan kabur, di mana potongan-potongan-alfa tersebut merupakan interval tersarang. Selanjutnya himpunan semua bilangan kabur yang dilengkapi dengan operasi maximum dan penjumlahan tersebut merupakan semiring idempoten komutatif. Hal ini sebagai konsekuensi dari aljabar max-plus interval yang merupakan semiring idempoten komutatif, di mana sifat-sifat pada interval juga berlaku pada potongan-alfa.

Kata-kata kunci: semiring , idempoten, aljabar max-plus, bilangan kabur.

### Artikel: Determining the Latest Completion Times in Project Networks with Fuzzy Activity Times Using Fuzzy Number Max-Plus Algebra

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

dipresentasikan pada: International Conference on Natural and Material Sciences (NAMES 09) , Universitas Lambung Mangkurat, Banjarmasin 3-4 Juli 2009.

Abstract
The activity times in a project network are seldom precisely known, and then could be represented into the fuzzy number, that is called fuzzy activity times. This paper aims to determine the fuzzy latest completion times and fuzzy float times in the project networks with fuzzy activity times using fuzzy number max-plus algebra. The finding shows that the project network with fuzzy activity times can be represented as a matrix over fuzzy number max-plus algebra. The project networks dynamics can be represented as a system of fuzzy number max-plus linear equations. The fuzzy latest completion times for each node in the project networks is a solutions vector of the modified system. The fuzzy float times for each activity in the project networks can be determined by using some operations of matrices over interval max-plus algebra.

Keywords: max-plus algebra, fuzzy number, project network, fuzzy activity times, latest completion times,

### Artikel: Applications of Fuzzy Number Max-Plus Eigenvalues on Queuing Networks with Fuzzy Activity Times

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

Prociding International Conference on Mathematics, Statistics and Their Applications (ICMSA 2009). Unand Bukttinggi. 9-11 Juni 2009.

Abstract
The activity times in a network is seldom precisely known, and then could be represented into the fuzzy numbers. This paper aims to determine the service cycle completion time of the acyclic fork-join queuing networks with fuzzy number activity times using fuzzy number max-plus algebra. This paper is a theoretical investigation based on literature and computation using MATLAB program. The finding shows that the service cycle completion time is a eigenvalue of matrices over fuzzy number max-plus algebra in the system.

Keywords: Max-Plus Algebra, Queuing Networks, Fuzzy Number, Completion Times, Eigenvalues.

### Artikel: Penerapan Aljabar Max-Plus Bilangan Kabur pada Jaringan Antrian dengan Waktu Aktifitas Kabur

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

PROSIDING SEMINAR NASIONAL Penelitian, Pendidikan, dan Penerapan MIPA , 16 Mei 2009, FMIPA UNY bid MATEMATIKA

Abstrak. Makalah ini membahas tentang pemodelan dan waktu kabur sikel layanan jaringan antrian fork-join taksiklik kapasitas penyangga takhingga dengan waktu aktifitas kabur, dengan menggunakan aljabar max-plus bilangan kabur. Hasil pembahasan menunjukkan bahwa dinamika jaringan antrian fork-join taksiklik kapasitas penyangga takhingga dengan waktu aktifitas kabur dapat dimodelkan ke dalam suatu persamaan matriks atas aljabar max-plus bilangan kabur.

Kata-kata kunci: aljabar max-plus bilangan kabur, sistem persamaan linear, jaringan antrian fork-join dan waktu aktifitas kabur.

### Artikel : PENENTUAN WAKTU AWAL TERCEPAT PADA JARINGAN KABUR DENGAN MENGGUNAKAN ALJABAR MAX-PLUS BILANGAN KABUR

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

Prosiding seminar nasional matematika, UNEJ, Jember 28 Pebruari 2009.

ABSTRACT. The activity times in a project network are seldom precisely known, and then could be represented into the fuzzy numbers. This paper aims to determine the earliest starting times for each node in the project networks with fuzzy numbers activity times using fuzzy number max-plus algebra. The finding shows that the project networks with fuzzy number activity can be represented as a matrix over fuzzy number max-plus algebra. The project networks dynamics can be represented as a iterative system of fuzzy number max-plus linear equations. The earliest starting times for each node in the networks are the solutions of the system. Form this result, we can also determine the earliest completion for every activity in the network.

Keywords: max-plus algebra, earliest starting times, project network, fuzzy number.

(Artikel dalam bahasa Indonesia)

### Artikel: Penerapan Aljabar Max‐Plus Interval pada Jaringan Antrian dengan Waktu Aktifitas Interval

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

Prosiding Seminar Nasional Aljabar, Pengajaran dan Terapannya di UNY Yogyakarta pada tanggal 31 Januari 2009

Abstrak: Makalah ini membahas tentang pemodelan dan interval waktu periodik layanan jaringan antrian fork‐join taksiklik kapasitas penyangga takhingga dengan waktu aktifitas interval, dengan menggunakan aljabar max‐plus interval. Hasil pembahasan menunjukkan bahwa dinamika jaringan antrian fork‐join taksiklik kapasitas penyangga takhingga dengan waktu aktifitas interval dapat dimodelkan ke dalam suatu persamaan matriks atas aljabar max‐plus interval. Interval waktu sikel layanan jaringan antrian adalah nilai eigen max‐plus interval dari matriks pada persamaan tersebut.

Kata‐kata kunci: aljabar max‐plus interval, nilai eigen max‐plus interval, jaringan antrian fork‐join dan waktu aktifitas interval.

### Artikel: Determining the Earliest Starting Times in Project Networks with Interval Activity Times Using Interval Max-Plus Algebra

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

Prociding The First International Seminar on Science and Technology (ISSTEC 2009). UII Yogyakarta. 24-25 Januari 2009
.

Abstract: The activity times in a project network are seldom precisely known, and then could be represented into the interval. This paper aims to determine the earliest starting time for each node in the project networks with interval activity times using interval max-plus algebra. The finding shows that the project networks with interval activity can be represented as a matrix over interval max-plus algebra. The project networks dynamics can be represented as a iterative system of interval max-plus linear equations. The interval of earliest start time for each node in the project networks is a solutions vector of the system.

Keywords: max-plus algebra, earliest starting times, project network, interval.

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