A Fuzzy Programming Method for Optimization of Autonomous Logistics Objects

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1 A Fuzzy Programmng Method for Optmzaton of Autonomou Logtc Obect [Afhn Mehra, Klau-Deter Thoben] [Faculty of Producton Engneerng] [Unverty of Bremen] [Bremen], [Germany] [Hamd Reza Karm] [Faculty of Engneerng and Scence] [Unverty of Agder] [Grmtad], [Norway] Abtract Recently everal tude have explored the realzaton of autonomou control n producton and logtc operaton. In dong o, t ha been tred to tranmt the mert of deconmakng from central controller wth offlne decon to decentralzed controller wth local and real-tme decon makng. However, th mon ha tll ome drawback n practce. Lack of global optmzaton one of them,.e., the lot chan between the autonomou decentralzed decon at operatonal level and the centralzed mathematcal optmzaton wth offlne manner at tactcal and trategc level. Th dtncton can be reaonably olved by conderng fuzzy parameter n mathematcal programmng to meet the requred tolerance for autonomou obect at operatonal level. Th clam recommended and partally expermented n th paper. An aembly cenaro modeled by a dcrete-event mulaton, n whch autonomou pallet carry product throughout the ytem. Th cenaro optmzed wth regard to t obectve n a mulaton, whle fuzzy parameter n optmzaton programmng can conder autonomou decon done at operatonal level. Keyword- Mxed-nteger optmzaton; Manufacturng ytem; Fuzzy ytem I. INTRODUCTION Nowaday, the mportance of handlng dynamc behavor n materal and nformaton flow n producton and logtc ytem completely acknowledged by cholar and practtoner []. Regardng the dynamc charactertc of bune envronment, a broad reearch ha explored the oluton how to deal wth fluctuaton n materal flow a well a wth complexty n nformaton flow. Bede the tandard methodologe to challenge dynamc behavor n ther root, e.g., Lean Manufacturng, ome other oluton have been ntroduced to nteract wth uch behavor. Among them the decentralzed autonomou control (AC) [2] for procee and obect n logtc []. Baed on th paradgm, complexty of plannng and control n logtc operaton decreae thank to decentralzed and local decon-makng n a relatvely realtme manner. However, ntegrated coordnaton and optmzaton of decentralzed autonomou operaton crucal to the holtc body and depend on the level of autonomy for each operaton and obect [2] [3]. Realzaton of th mportance at n confguraton of the lot chan between autonomou decon-makng at operatonal level and optmzaton programmng n tactcal and trategc level. Wherea autonomou procee and obect ue local control ytem to render real-tme decon, optmzaton programmng collect all parameter and contrant n an ntegrated and offlne manner to optmze a cot functon, concernng them. Thee two heterogeneou approache gve the mpreon to be excludable wth the conventonal technque n both parte. In other word, the autonomou obect may not be fully autonomou n ther performance, but autonomou n a pecfc range at operatonal level to acheve a common conenu n the eventual effect. Smlarly, the optmzaton programmng ha to change the crp value, a nput of the global optmzaton proce, to fuzzy one wth enough tolerance for coverng the optmzed range n local dynamm. Indeed, the optmzaton reflect not pecfc value for varable, but contnuou range whch can be ued by elf-organzed procee and entte. It notceable that the recommended approach n th paper a novel method to deal wth dynamm of a global logtc ytem wth lot of local effectve player. However, ome conventonal approache already ext whch plt uch a global problem wth local dynamm nto maller problem wth local obectve, whlt they do not volate the global contrant, e.g., branch and bound []. Nonethele, thee conventonal technque are not able to ncorporate autonomou procee and obect wth ntellgent local decon. Therefore, the paper organzed a follow: frt optmzaton programmng wth crp value, called conventonal, and wth fuzzy parameter, called fuzzy optmzaton programmng, ntroduced. Next, a logtc aembly cenaro gven n detal by modelng t mathematcal optmzaton. Followng that the concept of autonomou obect n aembly ytem brefly explaned. In mulaton reult the performance of the aembly ytem wth autonomou obect dplayed. In concluon the man contrbuton and future work are explaned. II. OPTIMIZATION PROGRAMMING WITH CRISP AND FUZZY PARAMETERS A. Conventonal Optmzaton wth Crp Value Generally, an optmzaton model a mathematcal model of a problem whch ntended to be optmally olved regardng one or more obectve() and ome contrant. Th type of mathematcal model an abtract model that called mathematcal programmng n general []. Bacally, optmzaton of a problem refer to the achevement of maxmum degree of atfacton wth dong the leat amount of

2 effort for olvng the problem [6]. Intally, the optmzaton algorthm deal wth effcency and effectvene factor n achevng the optmum degree of atfacton [7]. Mathematcal programmng the core of any organzng framework package, lke advanced plannng ytem (APS) [8]. The ue of mathematcal programmng gve re to ntegraton and optmzaton of procee throughout any organzaton pannng from upply network (SN) to mall hop floor. The tak of trategc and producton plannng, nventory management, tranportaton, a well a chedulng, all can be modeled by mathematcal programmng ether olely or n an ntegrated form to optmze ther obectve. The model mut be optmzed n uch a way that all contrant get atfed. However, t not alway traghtforward to fnd optmum value for obectve, nce producton and logtc parte n SN have uually dfferent and heterogeneou target. Mathematcal programmng model are ubect to be olved by tatc olver. Optmzaton model reflect the problem of SN at a defnte occaon of tme, wthout conderng any dynamc apect [9], e.g., contrary to mulaton. Bacally, the contrant of a model bound the oluton pace of the repectve optmzaton problem. However, f ome of contrant are not avalable at a moment, the oluton pace may be nfeable to be olved. Moreover, complexty n mathematcal programmng and n ther olver make them napproprate tool for real tme decon by mean of local percepton under hghly dynamc crcumtance. Thee pecfcaton of mathematcal programmng contradct the real tme decon-makng, whch nherent to autonomou obect. Generally, a lnear mathematcal model defned a:.t. Mn Z = CX. () AX B (2) X 0. (3) where A { R } n m, B { R } n, C { R } m, and R denote a et of real number. n and m are nteger value for row and column of matrx { R } n m. B. Optmzaton wth Fuzzy Value Bacally, operaton n manufacturng and logtc envronment are expoed to dynamc and complexty, bede, human nterventon that reemble a fuzzy nature. Indeed, fuzzy nature hold true than tochatc nature; becaue n a dtrbuted envronment, ll defned nformaton and lack of all requred data for makng a decon n real tme nevtable [0] []. Applcaton of fuzzy et theory n mathematcal programmng lead to everal advantage. For ntance, ettng atfacton degree for a mult-obectve problem facltate normalzaton of that and convert t nto a ngle obectve (called a calarzed problem) wth maxmzaton of atfacton degree for all ngle obectve, a broadly dcued by Sakawa [0]. In addton, by mean of fuzzy et t poble to defne robut dmenon for fuzzy contrant wth fuzzy parameter and make a robut optmzaton model, a tated by Zhang et al. [2]. Accordng to Zmmermann et al. [3] and Shh et al. [], fuzzy mathematcal programmng can be generally clafed nto the problem wth fuzzy obectve, wth fuzzy contrant, and both. For example, f the coeffcent n contrant and obectve, ntead of crp value, belong to fuzzy et (.e., A { R } n m, B { R } n, C { R } m, wth R a a et of fuzzy parameter) then the mathematcal programmng problem get nto fuzzy pace doman wth nonconventonal oluton method. Among everal technque to convert the model from fuzzy doman nto ordnary one, α- cut baed model can be ntroduced, a explaned by []..t. Mn Z = CX. () AX B () X 0. (6) By mean of α-cut the above mathematcal programmng wth fuzzy coeffcent can be rewrtten by followng change. A, B, C are matrce wth fuzzy parameter a A = a ; =,...,n; =,...,m, B = b ; =,...,n and C = c ; =,...,m. Thu, for a certan α-degree, defned by decon-maker, the ordnary et of ( a,b,c ) α wth memberhp degree over α, can be wrtten a: ( ) ( ) { c a b } a,b,c A,B,C,,,,...,n,,...,m α = μ α μ α μ α = = (6) The α cut level defne feable nterval for each fuzzy number a, b, and c, o a the left ( l ) and the rght ( r ) boundare for them are a l α,a r α, b l α,b r α, c l α,c r α. However, by changng the obectve to Max and changng the de of nequalte, the left ( l ) and the rght ( r ) power change. Concluvely, the problem reult n everal conventonal lnear programmng wth alternatve α cut level.. l l Mnc x c m x m.t. l l r ax amxm b. (7) l l r a nx a nmxm bn x 0, =,...,m Furthermore, mult obectve programmng (MOP) facltate a optmum decon makng for multple contradctory obectve by tradeoff between alternatve [2]. One way to convert MOP nto a ngle obectve problem to mply defne a atfacton degree for each of the obectve and then, accordng to the extenon prncple [9], maxmzng the mnmum of memberhp degree n each obectve. Th broadly explored by Sakawa [0] and ued by Petrovc et al. [] and Fayad et al. [6]. For example, f we have two contradctory obectve n MOP wth one maxmzaton and two mnmzaton obectve, then the decon maker hould defne the nterval and memberhp functon of each

3 obectve. Each of thee memberhp functon, correpondng to an obectve, nterpreted a atfacton degree. Th approach explaned by followng equaton. ;fx h q x μ a = ;f h x q. (8) q h 0;fq x ;fq2 x2 x2 h μ 2 b = ;f h 2 x 2 q 2. (9) q2 h2 0;fx2 q2 ob contant wth 0 mn for the nter-arrval tme between each mlar product type. TABLE I. Product Type PROCESSING TIMES IN EVERY STATION Product Proceng Tme Staton 2 3 Stochatc ( Neg-Exp (ß)) ;fx h3 q3 x μ 3 c = ;f h 3 x q 3. (0) q3 h3 0;fq3 x3 Fgure 2. Aembly cenaro wth poble permutaton n operaton order. Fgure. a) and c) the memberhp functon of mnmzaton and b) the memberhp functon of maxmzaton obectve. where x, x 2, and x 3 can have any functon or value. Here, μa and μc reflect the atfacton degree of the Mn obectve, whle μ b for the Max obectve. Th approach for MOP the mplet technque for aggregatng mult obectve n a unque atfacton degree. Here, the aggregaton operator choen a Mn of memberhp degree (whch the mot commonly ued and a mple operator), for more nformaton ee [0]. However, the boundare for obectve (.e., h and q), concernng ther atfacton degree, have to be ubectvely elected by decon maker, ee Fg.. III. ASSEMBLY SCENARIO WITH SCHEDULING For the current tudy, an aembly cenaro condered whch ha workng taton and (un)load taton that produce three type of product, ee Fg. 2. Each product of any type ha to vt every taton once by t own technologcal contrant. There are ome poble permutaton n the order of each ob operaton. The ob, comng by tochatc nature to the aembly network, have free choce between the taton number, 2, 3, and for tartng the operaton. However, the lat operaton mut be done on taton or. In addton, f the lat operaton on taton then t prevou operaton mut be done on taton, for more nformaton ee [7]. Table I how the mean proceng tme of each taton whch dfferent for each product type and follow neg-exponental dtrbuton wth ß. There a contant number of 2 pallet ( for each product type) condered for carryng 300 product (00 each type) over the chedulng horzon. The upply rate of comng IV. OPTIMIZATION PROGRAMMING FOR THE SCENARIO In order to optmze the aembly ytem ome obectve can be condered. Mnmzaton of flow tme F of all product, maxmzaton of utlzaton U, and mnmzaton of makepan C. The unveral mathematcal programmng for the tated problem can be modeled a follow: Set: product number; {,..., 00} product type; = {,..., 3}, =, f pallet number; q, t, k round number, taton number; = {,..., }, Parameter: a appearance of product type at the entrance of aembly ytem, P proceng tme of taton for product type, W watng tme of product type at taton,? ρ load at taton, λ total arrval rate at taton, Varable: C max the latet completon tme between all product type, U utlzaton of machne at the end, F flow tme of product type, W watng tme of product type at taton,? X f f pallet f type for product ready to be elected; 0 otherwe,

4 c av t completon tme of pallet type for product th, avalablty tme of pallet type for th product, tartng tme of pallet type for aembly, Obectve: th product to.t = = =. () Mn F = F + F + F max, Max U. (2) = Mn C. (3) max ( ) { 3} { 00} C = max t + F ; =,...,, =,...,. () F = ( P + W ) ;, =. () L W = ;,,. (6) λ ρ L = ;. (7) ρ t max a,av ;,,..., ( ) { 00} = =. (8) { } { 00} t = a ;, =,...,. (9) c = t + F ;, =,...,. (20) av = X c ;, =,..., k= ( k k) { 8}. (2) 7 ( ( )) { 9 00} = (q t) q t+ q t = t= av X av F ;,q,,...,. (22) 7 q = ;. (23) X k = ;, = {,..., 8}. (2) k= X (q t) = ;,q, = { 9,..., 00}. (2) t= f X 0;,, f. (26) av,f,c,t 0;,. (27) Fgure 3. The condered memberhp functon for watng tme by memberhp value of μ W. However, th problem a nonlnear mathematcal model wth NP-hard charactertc, whch dffcult to be olved n a polynomal tme. Therefore, t requre to be reduced to a mpler model. Here, the recommended fuzzy optmzaton model ueful. Some aforementoned varable, whch tranfer the model nto dffcult doman, can ut be relaxed nto fuzzy parameter for the ake of mplcty. For example, nce the watng tme varable W not a determntc varable wth nterdependency to queung tuaton at each moment of every taton and the dtance between taton, t hould be relaxed. Addtonally, the mon of operaton equencng agned to autonomou obect n the aembly lne, whch can be freely a well a ntellgently choen by elf-organzed logtc obect. Moreover, regardng queung theory the parameter λ λ and ρ = ( ε ervce rate of a ngle erver at taton ) ε are not eay to be dtnguhed before olvng the problem wth mulaton, ee [7]. Thu, thee complexte of hnder a conventonal oluton of th problem. However, a a oluton, by convertng the watng varable nto a fuzzy et parameter W the hard contrant n () get relaxed, o that the problem become mpler to be olved. A a reult, th contrant change a follow, wherea the contrant (6) and (7) can be removed: ( ) = l r P + W F 0;,. (28) Now, by defnng everal α cut level n t memberhp functon (Fg. 3) the problem tranformed nto everal conventonal mxed nteger nonlnear programmng; to be olved to acheve the bet performance. Repectvely, f other parameter (e.g., a, P ) take vague pecfcaton they can be converted nto fuzzy parameter wth left and rght boundare a,a l r, l r p,p. However, the boundare are defned by the decon-maker and may be ubectve. Moreover, a mentoned before, th mult-obectve problem can be converted nto one maxmzaton obectve of atfacton degree a n (29). The atfacton degree of each obectve acheved from (8), (9), (0) by ubttutng the repectve varable from (30), (3), (32). Max Mn ( μa, μ, μc) x b. (29) = F. (30)

5 x2 U = =. (3) x3 = C. (32) All n all, the contrbuton of fuzzy parameter to uch an optmzaton problem provde a relatvely optmzed problem wth defnng a range of optmalty for varable ntead of a crp value for each. Th gve the prt of local optmzaton to the autonomou player n operatonal level wth real-tme decon. Of coure, the autonomou obect mut be commtted to the boundare that the optmzaton program calculate for them and they have to guarantee the feablty of the oluton pace for the program. Th fact the man contrbuton of the current paper. V. AUTONOMOUS OBJECTS IN ASSEMBLY Bede (or eparate from) the lat ecton about global optmzaton programmng wth offlne manner, autonomou obect at operatonal level can make real-tme local optmzaton. Generally, AC ntroduced to deal wth complexte n dtrbuted and decentralzed approache, by employng elf-organzaton and ntellgence mert; recently, ued by logtc [3]. To realze th, everal technque and control method are ued, e.g., thoe dtrbuted heurtc to olve complex problem n decentralzed form, lke warm ntellgence, ant colony, and mult-agent ytem [20] [2]. AC can be acheved by employng reponve control technque wth learnng and adutablty capablty [22] [23]. Among everal alternatve fuzzy control ytem ha hown t adequacy a a utable control method [8], whch can be ued a controller of autonomou obect. A. Lpallet Indeed, autonomy n logtc equpped wth the concept of autonomou obect a dtrbuted element of an entre ytem, e.g., ndvdual, contaner, machne, product, and pallet. Th mert facltate the realzaton of autonomy n practce. In th tudy, npred by artfcal ntellgence and cloed-loop ytem n Conwp materal flow, the concept of learnng pallet (Lpallet) employed, ee [7] [22]. It aumed that pallet n each round trp can experence and learn the behavor of the local taton a well a the entre ytem. Thu, Lpallet dea addree the deployment of autonomou obect n logtc. Fuzzy controller tool ntegrated nto Lpallet n order to fnd the bet operaton equence for each pallet n every moment of the mulated aembly ytem. B. Lpallet wth Fuzzy Controller Lpallet are able to make decentralzed real-tme decon at each epoch, whle defnng the operaton order for themelve. The bet order n each moment tend to mnmze the flow tme and n overall performance the makepan (completon tme of the lat product). By ntegratng fuzzy controller (ytem) nto Lpallet, each of whch ha the capablty of etmatng every taton eparately and udgng them baed on ther local tuaton. Thee udgment are the decon crtera n the next round to defne operaton equence for each Lpallet. Th udgment technque followng. It uppoed that the watng tme n a taton queue plu operaton tme on that taton reult n vague performance etmaton for the repectve taton. Th crteron, a nput of the fuzzy ytem μ A, udged by three lngutc term a: good, normal, and bad. Then, by mean of (alpha) α-cut technque, the memberhp value of thee fuzzy lngutc term are mapped nto ome other fuzzy term (fat, medum, low) n concluon of the condered fuzzy rule. Afterward, the udgment of taton, accompaned wth ther aocatve memberhp value ( μu ), are recorded n Lpallet. Later, n every decon era, thee fuzzy value are defuzzfed to a crp value for every taton. In other word, at the entrance of each taton, the fuzzy controller trggered to compare the defuzzfed (crp) value of all left operaton and then wll elect the operaton wth the bet crp value (the leat one), a the frt operaton. Th equence acheved n decendng order. Thee all occur va the correpondng fuzzy aocatve memory (FAM), ee fgure. Here, the memberhp functon n the fuzzy controller are elected to be trangular []. In order to meet dynamc of the aembly lne, the boundare of the trangular are not fxed, but employ movng average of the lat three record of every taton. Lower control lmt (LCL) and upper control lmt (UCL) are borrowed from tattcal proce control (SPC) [2], calculated by (33). Where Aavg the expectaton of mean value of mean value and R the range of the recorded data. Fgure. The employed FAM n Lpallet. VI. UCL= Aavg+ EE R. (33) LCL= Aavg EE R SIMULATION RESULTS The mulaton of the aembly problem wth AC at operatonal level expermented n th part and the oluton of the recommended optmzaton problem condered a further work. The acheved reult from AC wth no lmtaton can later be compared agant thoe wth tolerance gven by the olved optmzaton program. In contrat to the uggeted algorthm for the aembly problem here ome conventonal technque a heurtc and dpatchng rule (DR) could be appled for aembly chedulng, e.g., frt n-frt out (FIFO), hortet proceng tme (SPT), and general hftngbottleneck routne (GSBR). Table II how the performance crtera (the mentoned obectve of the mathematcal model)

6 of AC agant ome conventonal rule. For more nformaton about thee conventonal method ee [22]. TABLE II. PERFORMANCE OF THE SIMULATION Performance of AC and DR n Schedulng Problem Staton 2 3 AC Utlzaton % FIFO Utlzaton % SPT Utlzaton % AC Makepan :08:7:28 (Day:Hour:Mn:Second) FIFO Makepan :3:23:2 SPT Makepan :::3 AC Average Flow Tme 3::2 (Hour:Mn:Second) FIFO Average Flow Tme 3:8:27 SPT Average Flow Tme :0:0 Fgure. The aumed framework for combnng optmzaton programmng wth autonomou logtc obect. VII. CONCLUSION Here, frt the operaton n logtc at ndutre have been hortly addreed. The mportance of employng optmzaton programmng pecfcally n tactcal level of plannng ha been reflected concely. Then a uggetve method for embeddng ll-defned or vague parameter n optmzaton programmng wa ntroduced. Th method ha been ued later n the optmzaton programmng of an exemplary aembly cenaro. 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