Remanufacturing planning based on constrained ordinal optimization

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<ul><li><p>Front. Electr. Electron. Eng. China 2011, 6(3): 443452</p><p>DOI 10.1007/s11460-011-0162-y</p><p>Chen SONG, Xiaohong GUAN, Qianchuan ZHAO, Qing-Shan JIA</p><p>Remanufacturing planning based on constrained ordinal</p><p>optimization</p><p>c Higher Education Press and Springer-Verlag Berlin Heidelberg 2011</p><p>Abstract Resource planning for a remanufacturingsystem is in general extremely dicult in terms of prob-lem size, uncertainties, complicated constraints, etc. Inthis paper, we present a new method based on con-strained ordinal optimization (COO) for remanufactur-ing planning. The key idea of our method is to esti-mate the feasibility of plans by machine learning andto select a subset with the estimated feasibility basedon the procedure of horse racing with feasibility model(HRFM). Numerical testing shows that our method isecient and eective for selecting good plans with highprobability. It is thus a scalable optimization methodfor large scale remanufacturing planning problems withcomplicated stochastic constraints.</p><p>Keywords remanufacturing systems, constrainedordinal optimization (COO), simulation-based optimiza-tion, machine learning</p><p>1 Introduction</p><p>Planning remanufacturing system with high eciencyhas drawn considerable attention due to its signi-cant economic impact [13]. A typical remanufactur-ing process includes disassembling a module or unitwith reusable parts, cleaning and refurbishing repairableparts and replacing unusable parts by new ones. It is</p><p>Received March 18, 2011; accepted May 10, 2011</p><p>Chen SONGUbiquitous Energy Research Center, ENN Research and Develop-</p><p>ment Corporation, Langfang 065001, China</p><p>Xiaohong GUAN, Qianchuan ZHAO, Qing-Shan JIACenter for Intelligent and Networked Systems, Department of</p><p>Automation, TNLIST, Tsinghua University, Beijing 100084, China</p><p>E-mail: jiaqs@tsinghua.edu.cn</p><p>Xiaohong GUAN</p><p>State Key Laboratory for Manufacturing Systems Engineering</p><p>(SKLMS Lab) and Ministry of Education Key Laboratory forIntelligent Network and Network Security (MOE KLINNS Lab),</p><p>Xian Jiaotong University, Xian 710049, China</p><p>then reassembled to become a unit fully equivalent inits performance to a completely new unit. The reman-ufacturing system considered in this paper is based ona practical engine repair shop, as shown in Fig. 1. Theplanning problem for this system is very complicated,including discrete states, probabilistic constraints anduncertainties in cost performance described in details inour previous work [4]. In this paper, we will propose anew optimization method based on this concern withhigh generality for solving a wide range of similar prob-lems.</p><p>Fig. 1 Structure of remanufacturing system</p><p>Many methods including the eective methods ofmulti-stage stochastic programming have been devel-oped to deal with complex stochastic optimization prob-lems in manufacturing planning and scheduling [5]. How-ever, the issues of the multi-stage planning, fork-joinstructure, great computational eorts of the huge statespace, uncertainties, etc. are very challenging [6]. Theremanufacturing problem in this paper is more compli-cated since the objective function and constraints cannotbe evaluated or determined analytically without simula-tion as will be seen in Sect. 2. As a result, the multi-stagestochastic programming methods can hardly be appliedto solve this problem.</p><p>In the literature, computational intelligence methods,such as data mining, genetic algorithms, and neural net-works [710] were applied to analyze and optimize one ormore segments of a remanufacturing system such as dis-assembly [11,12]. A remanufacturing system was seldom</p></li><li><p>444 Front. Electr. Electron. Eng. China 2011, 6(3): 443452</p><p>taken as a whole and interactions among the activitiesof the segments were generally not considered.</p><p>In the current practice, time-consuming simulation isperformed to evaluate the remanufacturing plans, whichare obtained heuristically. Since simulation-basedoptimization became possible to analyze random sys-tems using computers, scientists and engineers havesought the means to optimize systems using simulationmodels[13]. In fact, simulation could be the only avail-able approach for planning complicated remanufacturingsystems.</p><p>In order to overcome the diculties in compli-cated planning problems, the ordinal optimization (OO)method oers an ecient simulation-based optimiza-tion approach [14,15]. It is based on the concept thatthe order of the planning decisions is much easier todetermine than the value and is exponentially conver-gent versus the simulation eorts [16,17], and the goalsoftening by seeking the good enough with high prob-ability will exponentially lessen the computational load[18]. OO method is particularly attractive for stochasticdiscrete optimization since it is immune to large noisewith aordable computational complexity [1921]. How-ever, incorporating constraints eciently is one of themajor challenges in applying the regular unconstrainedOO method [2224]. Based on the feasibility model gen-erated by the machine learning method for planningremanufacturing systems [4], the framework of the con-strained ordinal optimization (COO) approach for thesimulation-based optimization problems has been con-structed [25]. The simplest blind picking (BP) selectionrule is designed to solve discrete constrained optimiza-tion problems.</p><p>Clearly, the feasibility model is more time-saving thanthe penalty method commonly used in computationalintelligent approaches in terms of handling constraintssince the former makes use of the structure informationof the problem and can directly exclude infeasible plans.However, building a feasibility model needs prior knowl-edge or that obtained by machine learning. The problemstructure information can also be utilized to design theselection rule. Since BP does not use the observed per-formance, it supplies an upper bound to the size of theselected subset [14,15]. The horse racing (HR) selectionrule screens out some observed bad plans from consider-ation, and thus usually requires a smaller selected subset[26].</p><p>In this paper, the COO approach with the HRselection rule is developed to solve the remanufactur-ing planning problem. The key idea of our method is toestimate the feasibility of plans by machine learning andto select a subset with the estimated feasibility basedon the procedure of horse racing with feasibility model(HRFM). In the HRFM scheme the candidate plans arecompared by quick simulation with a small number of</p><p>replications. A number of plans with the best estimatedperformance and feasibility are selected, forming theselected subset and the best plan is then obtained withinthis set by more accurate simulation (with more repli-cations). The combination of feasibility model with theHR rule is not trivial, since there is usually a correlationbetween the feasibility and the performance of a plan.We use a correlation factor to quantify this correla-tion, and present a general procedure to determine theselected subset as a function of the correlation factor.</p><p>The numerical testing results for a practical reman-ufacturing system show that the HRFM method pre-sented in this paper is more ecient to meet the samerequired alignment probability in comparison with bruteforce simulation and the blind picking with feasibilitymodel (BPFM) selection method.</p><p>2 Problem formulation</p><p>The problem considered in this paper is motivated bythe daily planning tasks encountered in an actual enginerepair shop. Enough amount of the processing capac-ity and the spare part inventories are reserved to satisfythe probabilistic cycle time requirement. However, thereserved capacity and inventories should not be exces-sive in order to save the associated costs. The cycle timeof an asset to be repaired depends on the capacity andinventory level allocated in every quarter of a year andis random due to uncertain arrival and worn-out pre-condition.</p><p>To avoid the proprietary issue, we formulate thefollowing simplied remanufacturing planning problemwith the key characteristics preserved. To facilitate pre-sentation, the notations are given rst as : total part surplus; : non-negative cost coecients;Cc : total allocated capacity;C : the total cost consisting of capacity cost Cc</p><p>and inventory cost V (O,);V (I, ) : inventory cost related to the inventory I and</p><p>uncertain dierence ;Ccapi,j : allocated capacity for part i in quarter j;Cmini : lower bound of repairing capacity for part i;Cmaxi : upper bound of repairing capacity for part i;Ci : limit of capacity change in consecutive quar-</p><p>ters for part i;Ccapj : total available capacity in quarter j;Ccapi,j : capacity adjustment for part i in quarter j;d : required cycle time (days);Ti : the planning horizon (days);k : index for repairable asset;m : number of quarters;n : number of part types;Oi,j : order for new part i in quarter j;</p></li><li><p>Chen SONG et al. Remanufacturing planning based on constrained ordinal optimization 445</p><p>Pr() : probability;Prc : required probability of the cycle time;Tc(k) : remanufacturing cycle time of asset k;i,j : surplus of part i in quarter j, i.e., the dier-</p><p>ence between the number of the repaired parti, the number of order for new part i and thenumber of the assembly requests for part i inquarter j;</p><p>pi,j : number of repaired part i in quarter j;qi,j : number of the assembly requests for part i in</p><p>quarter j;Ii,j : inventory for part i in quarter j;k : nish time of asset k.</p><p>The remanufacturing problem is formulated as astochastic optimization problem in Eqs. (1)(6) as</p><p>minCc,I</p><p>C = Cc + EV (I, ), (1)</p><p>where E represents the expectation over ,</p><p>Cc =n</p><p>i=1</p><p>m</p><p>j=1</p><p>Ccapi,j , (2)</p><p>I =n</p><p>i=1</p><p>m</p><p>j=1</p><p>Ii,j , (3)</p><p>Ii,j = Ii,j1 + i,j1, (4)</p><p> =n</p><p>i=1</p><p>m</p><p>j=1</p><p>i,j , (5)</p><p>i,j = pi,j1 + Oi,j1 qi,j1, (6)subject to probabilistic cycle time requirement (7) andthe repairing capacity constraints (8)(11):</p><p>Pr(Tc(k) d|k &lt; Tt) Prc, (7)Ccapi,j [Cmini , Cmaxi ], (8)</p><p>|Ccapi,j | Ci, (9)n</p><p>i=1</p><p>Ccapi,j Ccapj , (10)</p><p>where Tt is a given constant deadline of the total time.The activities in the remanufacturing system are</p><p>under the following assumptions:1) the inter-arrival time of assets follows a Poisson</p><p>distribution with rate i in quarter i;2) the part repair time complies triangular distribu-</p><p>tion;3) the repairing capacity is adjustable (controllable)</p><p>for each quarter;4) the buer capacities of disassembly, repair, order</p><p>and assembly are not considered or assumed innite;5) disassembling, assembling and shipping times are</p><p>assumed deterministic but the buering time are uncer-tain, and the shipping time is neglected.</p><p>Suppose the cycle time Tc of asset k is calculated basedon its nish time k (completion of assembly, which is</p><p>related with allocated capacity, inventory and uncertain-ties in every quarter) and arrival time and the randominter-arrival time between asset l 1 and asset l is a(l)and initial time is zero. The cycle time is calculated as</p><p>Tc(k) = k(Cc, I, )k</p><p>l=1</p><p>a(l), l = 1, 2, . . . , k. (11)</p><p>One of the major diculties in solving this problemis that the expectation in the cost function and theconditional probability in the probabilistic cycle timeconstraint cannot be evaluated analytically as a func-tion of the decision variables. The main reason is thatthe model does not satisfy Markovian assumptions orregenerative assumptions required by analytical meth-ods. Furthermore, the following features of the problemadd additional dimensions of diculties:</p><p>1) presence of rotable inventories with adjustablerepairing capacities;</p><p>2) lack of event synchronization caused by the rotableinventory;</p><p>3) fork-join disassembly and assembly of the repairshop.</p><p>Clearly, the above problem is a very complicatedstochastic discrete optimization problem. In fact, dueto presence of rotable inventories, ecient simulation ofthe system is a non-trivial task. After being disassem-bled, the parts to be repaired will be processed by a setof parallel machines whose processing times have a tri-angle (non-exponential) distribution. The repaired partswill then go into the rotable inventory or buer readyfor re-assembling without sequence. That is, the partbeing disassembled rst could be re-assembled last. Thiscauses the events in the system to be interdependent.As a result, updating a state in simulation has to beperformed globally and every individual part has to betracked. The simulation on this process turns out to bevery time consuming when the inventory level and num-ber of parallel machines are large. As we reported in Ref.[4], simulation by the commercial software ED in actualindustrial application is computationally very expensive.</p><p>Given the diculties in applying stochastic program-ming methods or brute force Monte Carlo simulation, weshall develop an ecient simulation-based optimizationmethod COO to solve this problem.</p><p>3 Method of constrained ordinal optimization(COO)</p><p>3.1 Framework</p><p>For convenience of presentation, consider the followinggeneral optimization problem with stochastic cost and</p></li><li><p>446 Front. Electr. Electron. Eng. China 2011, 6(3): 443452</p><p>constraints in Eqs. (12) and (13):</p><p>min</p><p> E(L(x(t; , ))), (12)s.t. hi() E(Li(x(t; , ))) 0,</p><p>i = 1, 2, . . . ,m, (13)</p><p>where is the candidate plans in planning space , L,the performance functional of the sample path of a dis-crete dynamic state x(t), and , i.e., all the uncertaintiesor random noises of the problem [4], E is the expecta-tion operator and hi() is the constraint measure.</p><p>In OO, a selection rule or method is needed toselect subset S with certain evaluation scheme (crudealgorithms, heuristics and even BP) to guarantee thedesired degree of matching called alignment level k,and the condence of achieving a certain alignment levelwith good enough subset G is referred to as the align-ment probability PA:</p><p>PA = Pr{|G S| k}. (14)The diculty of the COO problem is that direct</p><p>selection of S in is inecient since there are manyinfeasible plans that cannot be excluded beforehand.Consequently, the alignment level cannot be estimatedby the regular OO method without going through time-consuming simulation since the quantity of infeasibleplans in S is unknown. In other words, the uncertainnumber of infeasible plans in and S is the main di-culty of COO approach.</p><p>In order to overcome the above diculty, we applythe feasibility model generated by machine learning inRef. [4] to exclude infeasible plans and obtain an approx-imate feasible set f in the search space . The feasi-bility model can be integrated in the selection methodin</p><p>minf</p><p>J() (15)</p><p>to obtain a selected subset Sf in f . It is easy to seethat applying OO to set f to get Sf will be better thanapplying regular OO to , since no time-consuming sim-ulation is needed to exclude infeasible plans. The degreeof improvement and the size of the selected subset Sfcan be quantied with respect to the accuracy of thefeasibility model.</p><p>Assume the probability that the feasibility of a planis correctly classied by the feasibility model is Pf .Although some infeasible plans in f and Sf may beclassied as feasible plans incorrectly, the densities offeasible plans in Sf (or f) are hi...</p></li></ul>

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