MODEL AND ALGORITHMS OF THE FUZZY THREE-DIMENSIONAL AXIAL ASSIGNMENT PROBLEM WITH AN ADDITIONAL CONSTRAINT

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1 MODEL AND ALGORITHMS OF THE FUZZY THREE-DIMENSIONAL AXIAL ASSIGNMENT PROBLEM WITH AN ADDITIONAL CONSTRAINT C.-J. Lin * & K.-T. Ma 2 Department of Industrial Engineering and Management Ta Hwa University of Siene and Tehnology, Taiwan lj@tust.edu.tw 2 Department of Industrial Engineering and Engineering Management National Tsing Hua University, Taiwan d @oz.nthu.edu.tw ABSTRACT This study onstruts a pratial fuzzy three-dimensional axial assignment model, and proposes two effiient algorithms to solve the model. In our ase, the model is applied to team performane management in a ompany to promote the performane of all members in a team. Two algorithms, namely the index-based branh and bound (B&B) algorithm and the f-g trade-off algorithm, whih is a hybrid of the trade-off and B&B onepts, are proposed. A numerial example is presented to illustrate these two algorithms. The omputational results show that the proposed algorithms are suffiiently effiient and aurate. Two speial ases are also disussed. OPSOMMING n Praktiese, wasige, driedimensionele aksialetoekenningsmodel word voorgehou en twee doeltreffende algoritmes om die model op te los word voorgestel. Die model word toegepas op die spanprestasiebestuur in n maatskappy met die doel om die spanlede se vertoning te verbeter. Twee algoritmes, naamlik die indeksgebaseerde vertak-en-begrens algoritme en die f-g kompromie algoritme, word voorgestel. n Numeriese voorbeeld word ter illustrasie van die twee algoritmes voorgehou. Die resultate toon dat die voorgestelde algoritmes doeltreffend en akkuraat is. Twee spesiale gevalle word ook bespreek. * Corresponding author South Afrian Journal of Industrial Engineering November 205 Vol 26(3) pp 54-70

2 INTRODUCTION The three-dimensional (3D) axial assignment problem is an extension of the linear assignment problem i.e., the 2D assignment problem. The 3D axial assignment problem is atually NP-hard []. It is usually stated as follows: assign n workers to n jobs on n mahines to minimise the total ost [2]. There should be one-to-one orrespondenes between workers and mahines, workers and jobs, and mahines and jobs. These orrespondenes make up a ubi assignment. The model an be formulated as the following 0- integer programming problem [-5]: Min x ( 0) i= j= k= n n s.t. j= k= n n i= k= n n i= j= x x x = for i =, 2,..., n = for j =, 2,..., n (.) = for k=, 2,..., n x {0, } for i, j, k=, 2,..., n where is the ost of assigning worker i to job j on mahine k, and all s onstrut a ubi matrix. If x =, then worker i is assigned to do job j on mahine k. If x = 0, then worker i is either not assigned to job j or not assigned to mahine k. In a deterministi environment, the 3D axial assignment problem is important in operations researh. Appliations of the 3D axial assignment problem arise in many areas, suh as sheduling apital investments, addressing a rolling mill, military troop assignment, and satellite overage optimisation [,6]. Algorithms for the 3D axial assignment have been developed. However, the oeffiient matrix of Equation. is not totally unimodular to ensure an integer solution [7]; thus it annot be treated as effiiently at ontinuous problem-solving for its optimum as in a 2D assignment problem. Hene, algorithms to searh for the exat solution of a 3D axial are mostly impliit enumeration methods [8], suh as the branh and bound (B&B) method [3,9]. Some papers have developed good bounds for the B&B method, espeially the lower bound for minimising the problem [6,8,0]. Hung and Lim [] proposed a loal searh geneti algorithm-based method to solve the 3D axial assignment problem, but they ould not guarantee that an optimal solution ould be obtained. Although all s are deterministi numbers in Equation., the ost of many real-world appliations may not be deterministi numbers. Many researhers have begun to study the 2D assignment problem in a fuzzy environment. These situations inlude various fuzzy assignment problem (FAP) models and algorithms [2-5], sensitivity analysis of FAP [6], quadrati FAPs [7], multi-riteria FAPs [8-20], traffi FAPs [2], two-objetive FAPs [20,22], group deision-making [23], and multi-job FAPs [24]. Solution proedures of these 2D FAPs all take advantage of the totally unimodular property for effiieny. However, no paper disusses the modelling and methods for a fuzzy 3D axial assignment problem. Therefore, this paper presents a fuzzy 3D axial assignment model, and proposes two B&Bbased algorithms to solve the model. The next setion desribes the related fuzzy theory and onstrution of the fuzzy 3D axial assignment model. Setion 3 presents two algorithms for solving this model and a numerial example. Setion 4 desribes omputational results, and Setion 5 offers a onlusion to the paper. 55

3 2 MODEL CONSTRUCTION 2. Basi onept of related fuzzy theory Fuzzy theory is useful for aessing many real situations of human life. In this setion, we first reall several related definitions of fuzzy theory, whih are used in this paper. Definition 2... A risp set A is a proper subset of a universe of disourse X suh that A X ; the membership funtion of A an be defined as [25]:, if x A ua( x) = 0, if x A where x X. Definition A fuzzy set A is a subset of a universe of disourse X, whih is haraterised by a membership funtion u ( x), x X, whih maps x to a value within the A range of [0,] [26]. Aording to Definitions 2... and 2..2., x ompletely belongs to A if the value of the membership funtion u ( x ) = ; x does not belong to A if u ( x ) = 0. A Definition The -ut of A of the universe of disourse X is defined as A = { x Xu( x) } [26]. A Therefore, the -ut is a subset of the universe of disourse X, where u ( x) A A. Definition The largest value of of the -ut of A, whih is non-empty, is alled the height of A, denoted as h( A ) [26]. Definition A fuzzy set A of the universe of disourse X is normal if h( A )=; otherwise, a fuzzy set A of the universe of disourse X is subnormal [5,26,27]. Definition A fuzzy set A of a universe of disourse X is onvex if and only if u ( µ a+ ( µ ) b) min( u ( a), u ( b)), x [ ab, ],and µ [0,] [26]. A A A Definition A fuzzy set A of a universe of disourse X is a fuzzy number or fuzzy interval if A is both normal and onvex [26]. Definition A fuzzy set A of a universe of disourse X is a subnormal fuzzy number or subnormal fuzzy interval if A is both subnormal and onvex [27]. Definition A subnormal fuzzy interval [5,27] is a subnormal fuzzy subset of the universe of disourse X, whih is both subnormal and onvex. Aordingly, a fuzzy number and fuzzy interval are a speial ase of subnormal fuzzy number and subnormal fuzzy interval, suh that the maximal height of the membership funtion u ~ A (x) is, respetively. 56

4 2.2 Model formulation Suppose a projet team onsists of n workers and a manager. The workers and the manager are responsible for performing n jobs and for minimising the total osts, respetively. Eah worker is assigned to one and only one job to perform it. In many ases, the job quality ahieved depends on the assigned worker s skill, the mahine used, and the input-ost of the job. Generally, a higher job quality indiates greater job ost. The job quality ahieved is defined as the performane of the worker. Hene, workers desire high-input job osts and a suitable mahine to perform the assigned job well. Thus the ost of worker i performing job j on mahine k, whih is denoted as ~, is positively related to the performane of worker i. Let q (0 <q ) be the highest possible quality of job j performed by worker i on mahine k. Generally, q should be between 0.5 and.0. The membership funtion of ~ is then defined as the linear monotone inreasing funtion in Equation 2. and Figure. In the funtion, α is the minimal ost for worker i to perform job j on mahine k, and β is the minimal ost required to reah the maximal job quality q. When the job input ost is between α and β, the job ost is linearly and positively related to the job quality ahieved. However, any expense exeeding β is wasteful beause quality an no longer be enhaned. Without loss of generality, it is assumed that 0 <α <β. Condition x = is added to Equation 2. beause there is no real expense if x = 0 in any feasible solution. μ ( ) = q q( α) β α 0 if if α β, x otherwise β =, x = (2.) q μ ( ) Notation <α, β > is used to denote ~ 0 α β Figure : Membership funtion of ~. The ubi matrix [ ~ ] is shown as [ ~ ] = [<α, β >]. Additionally, all α s form ubi matrix [α ], all β s form ubi matrix [β ], and all q s form ubi matrix [q ]. Define the ubi matrix [γ ] as β α [γ ]=[ ] for i, j, k=, 2,, n. (2.2) q where γ is the slope of the membership funtion. 57

5 The manager s duty is to keep the total ost ( ~ T ) as low as possible. Numbers a and b are assumed to be onstants hosen subjetively by the ompany. When the total ost is lower than a, the manager s performane is, when the total ost is greater than b, the manager s performane is 0, and when the total ost is between a and b, the total ost is linearly and negatively related to the manager s performane. Thus, the membership funtion of total ost ~ T is defined as the linear monotonially dereasing funtion in Equation 2.3), whih is shown in Figure 2. Notation <a, b> denotes the fuzzy interval ~ T. It is suggested that a should be a number less than or equal to the minimum of Equation., with α as its parameter, and b should be a number larger than or equal to the maximum of Equation., with β as its parameter. μ ( ) = μ ( T T T i= j= k= x ) = b i= j= k= b a 0 x, b T = b a, T T a, a b T b (2.3) Beause the performane of worker i (π ) is the job quality ahieved, then i π =μ ( ) for x = and i =, 2,, n (2.4) i The performane of the manager (π ) is m π =μ ( m T i= j= k= x ) (2.5) µ T ( T ) 0 a b T Figure 2: Membership funtion of Usually, the ompany takes the lowest performane of the team members, omposed of all workers and the manager, as the team performane for promoting overall performane. Thus the performane of every team member must be equally emphasised. To maximise the team performane is then to maximise the minimal performane of all team members. Therefore, the study uses Bellman-Zadeh s riterion [28], whih maximises the minimum of all membership funtions orresponding to that solution. The objetive funtion is then defined as or Max- Min (π, π ) i m i 58

6 x = (μ ( ), μ ( T i= j= k= x )) (2.6) where x is an element of feasible solution x of Equation.. The fuzzy 3D axial assignment problem model is represented as follows: Max- Min Max- Min x = (μ ( ), μ ( T i= j= k= x )) s.t. the onstraints of (.) (2.7) Using the membership funtions from Equation 2. and Equation 2.3, Equation 2.7 an be represented as the following equivalent model: Max s.t. where x b q ( α ) x for i, j, k=, 2,..., n β α x i= j= k= b a 0 x q x for i, j, k=, 2,..., n (2.8) 0 for i, j, k=, 2,..., n x {0, } for i, j, k=, 2,..., n and the onstraints of (.) is the -ut of ~ Equation 2.8, the x s, mixed nonlinear programming model. and x i= j= k= is the orresponding total ost T. In s, and are all deision variables. Hene, Equation 2.8 is a In our ase, Equation 2.8 is applied to team performane in human resoure management that equally emphasises the performane of the manager and workers. However, Equation 2.8 an also be used for situations that equally emphasise the quality and quantity, time and ost, humanity and tehnology, and so on. 2.3 Model transformation Suppose ω = {( i, j, x = } is the orresponding ubi assignment of the feasible solution x of (.) and the disussion is onfined based on ω. Then, Equation 2.8 an be simplified as Max s.t. q( β 0 q 0 b α α ( i, j, ω b a ) ; ( i, j, ω ; ( i, j, ω ; ( i, j, ω (2.9) 59

7 Theorem 2.. Let x be the optimal objetive funtion value of (2.9). Then, b x = ( Max Min ( b a + α i,j, γ, ) Min ( q ). Proof. Rewriting Equation 2.9 into a linear programming model results in Max (β α) s.t. α ; ( i, j, ω q ( i,j, ω 0 + ( b a) b min ( q 0 ; ( i, j, ω ) (2.0) The dual problem of Equation 2.0 is then Min s.t. (αhi ) + bhn+ + min ( q) y h i + h n+ 0 for i=, 2,..., n ( β α) wi + ( b a) hn + y q + ( i,j, ω h 0, h + 0, y 0 for i=, 2,..., n i n (2.) (β α) Obviously, Equation 2.0 produes α + > 0 for all (i, j, ω. Using the q omplementary slakness theorem yields the following: 60 h i + h n + = 0 for i =, 2,, n or hn + = h = h = = h. (2.2) 2 n Based on Equation 2.2 and Equation 2.2, Equation 2. an be simplified as follows: Min s.t. ( b ( b a + h α γ ) h n+, y 0 The dual problem of Equation 2.3 is Max s.t. ( b a + Rewriting Equation 2.4 as n+ ) h + n+ Min ( q + ) y y (2.3) γ) b α ω ω (2.4) min ( q ) ( i,j, 0

8 Max s.t. ompletes the proof. ( b a + b min ( q 0 α γ ) ) (2.5) Beause x is the maximal value of Equation 2.9 by giving the feasible solution x of Equation., the maximum of x must be the optimal solution of Equation 2.8. Equation 2.8 is rewritten as b (α x) i= j= k= Max Min, ( ( q) x) for i, j, k =,2,..., n ( b a + (γ x)) i= j= k= s.t. the onstraints of (.) (2.6) Obviously, Equation 2.6 is a speial multi-frational max-min 0- programming problem. 3 SOLUTION ALGORITHM This setion presents two effiient algorithms for solving the exat solution of Equation 2.6. The onstraint of Equation 2.6 is the same as that of Equation. i.e., the lak of a totally unimodular property. The methods in the literature for solving FAP are diffiult to implement for Equation 2.6. Hene, impliit enumeration methods are employed. One of the proposed algorithms is modified from the index-based B&B algorithm [8-9], whih quikly solves Equation. [9]. The other is a hybrid of the effiient trade-off onept [29] and index-based B&B algorithm. A numerial example is shown to larify these algorithms. 3. The index-based B&B algorithm This algorithm performs impliit enumeration of all feasible solutions for solving Equation 2.6. Figure 3 shows part of the tree graph of a problem used to illustrate the algorithm. The proposed algorithm proeeds from tree nodes from top to bottom and from left to right until all nodes are heked or eliminated by a fathoming test. For eah node (i, j, residing at level i, a path leads to it from the root node. The variables in the path are fixed at, and the other variables at level t (0 t i) equal 0. Thus, eah node is assoiated with a set of feasible solutions. For example, node (2, 3, 2) at level 2 in Figure 3 orresponds to the feasible solutions x = x = ; other variables at level and 2 are 0, while other variables are undetermined. Consider the following notations: α i = min (α, α,, α ) i i2 inn for i =, 2,, n (3.) γ i = min (γ, γ,, γ ) for i =, 2,, n (3.2) i i2 inn n SLA(i) = α = α i + α i+ + + α n for i =, 2,, n (3.3) = i n SLG(i) = γ = γ i + γ i+ + + γ n = i for i =, 2,, n (3.4) SLA(n+) = SLG(n+) =0 (3.5) 6

9 Level root node 0 (,,) (,,2) (,,3) (,,4) (,2,) (,2,3) (,2,4) (,3,) (,3,2) (,3,3) (,3,4) (,4,) (,4,2) (,4,3) (,4,4) (2,3,2) (2,,) (2,,2) (2,,3) (2,3,) (2,3,3) (2,4,) (2,4,2) (2,4,3) 2 (3,,) (3,,3) (3,4,) (3,4,3) 3 (4,,3) 4 Figure 3: Part of the tree graph of a problem Assume that the algorithm has urrently reahed node (p, j p, k p ) (p>0) and that nodes (, j, k ), (2, j 2, k 2 ),,(p, j p, k p ) ompose the path Path (p) from the root node to the urrent node. In addition, let TA(0)=SLA() (3.6) TG(0)=SLG() (3.7) TA(p)=TA(p ) SLA(p)+ α + SLA(p +) (3.8) TG(p) =TG(p ) SLG(p)+ pj p k p γ pj p k + SLG(p +) (3.9) p b TA(p) Theorem 3.. Funtion UPV (p) = provides an upper bound of the optimal b a + TG(p) objetive value of Equation 2.6 orresponding to Path (p). Proof. Using Equations 3.3, 3.5, 3.6, and 3.8 yields TA(p)= α jk + α j2k α 2 p jp-k + α p- p+ + α p α n (3.0) Suppose nodes (p+, j, k ), (p+2, j, k ),..., and (n, j, k ) ompose the p+ p+ p+2 p+2 n n optimal solution of Equation 2.6 orresponding to Path (p) and all it Path (n). Then, TA(n)= α jk + α j2k α 2 p jp-k + α p- p+ j p + α + k p+ p k α jp p+ 2 njnk (3.) n Using Equation 3. produes α p+ + k α p+, α jp p+ p+ 2 α jp+ 2 k p+ 2,..., and α α p+ 2 njn k n. Thus, n TA(n) TA(p). (3.2) Similarly, it an be verified that TG(n) TG(p). (3.3) Using Equation 3.2 and Equation 3.3 produes b TA(p) b TA(n) b a + TG(p) b a + TG(n) and ompletes the proof. 62

10 The following theorem is derived from Equation 2.6. Theorem 3.2 If (i, j, is a node of Path (p), then q gives an upper bound of the optimal objetive funtion value of Equation 2.6 orresponding to Path (p). Let z* represent the largest known objetive funtion value, whih is assoiated with a feasible solution. Using Theorems 3. and 3.2, the solution proedure of the index-based B&B algorithm is desribed as follows: Step : Find α i and γ i for i =, 2,, n. Step 2: Compute SLA(i) and SLG(i) for i =, 2,, n+. Let TA(0)= SLA() and TG(0)= SLG(). Step 3: Use x = x = = x = as the initial feasible solution and its objetive funtion 222 nnn value as z*. Step 4: Let p= and proeed downward to node (,, ) from the root node. Step 5: Suppose node (p, j, k ) is reahed. p p Let TA(p) = TA(p ) SLA(p)+ α pj p k p + SLA(p +) and TG(p) = TG(p ) SLG(p)+ γ pj p k + SLG(p +). p () If q pj p k z* or UPV (p) z*, then node (p, j, k ) is fathomed and moves toward p p p the immediate right node of (p, j, k ) at level p. If node (p, j, k ) is the last p p p p node of level p, then move upward to the first node right of (p, j, k ) at p p level p and update p p. (2) If q > z*, UPV (p)> z*, and p n, then move downward from (p, j, k ) to the p p pj p k p first node at level p+ and update p p+. (3) If q > z*, UPV (p)> z*, and p= n, then replae z* with UPV (p). Move upward pj p k p to the first node right of (p, j, k ) at level p and update p p. p p Step 6: If p= 0, then stop; otherwise, proeed to Step 5. The main onept of the index-based B&B algorithm is as follows. Whenever the algorithm reahes a new node (p, j, k ), the two upper bounds (UPV (p) and q p p pj p k ) are alulated p for that path before proeeding forward to any branhes leaving from node (p, j, k ). If p p both UPV (p) and q pj p k are larger than z*, then a better feasible solution may be found by p proeeding from node (p, j, k ). However, if UPV (p) or q p p pj p k are less than or equal to z*, p no better feasible solution an be obtained by proeeding forward from node (p, j, k ). p p That is, (p, j, k ) is fathomed. The algorithm loates the exat optimal solution. p p 3.2 The f-g trade-off algorithm Beause the performanes of team members are equally emphasised to maximise the team performane, the f-g trade-off algorithm is proposed. The f-g trade-off algorithm balanes the maximum of the manager's performane and minimum of the workers' performanes to maximise team performane. In the algorithm, some proedures are repeated to solve updated problems with different parameters. In eah solution, the maximum of the manager's performane dereases (or remains unhanged), and the minimum of the workers' performanes simultaneously inreases until the maximal team performane is reahed. To obtain the maximal manager's performane, Equation 3.4 is solved. b Max f (t) i = b a + t α ( ) x = j= k= i= j= k= γ x s.t. the onstraints of (.) (3.4) 63

11 where t is an index of iteration and t = 0 indiates the original problem parameter. Equation 3.4 is a variant of the 3D axial frational assignment problem. To solve Equation 3.4 effiiently when t = 0, the onept of the fration ost penalty method (FCPM) [30] is b α used to obtain a good initial feasible assignment. Cubi matrix [ψ ] = n is b a + γ n defined as the ost oeffiients of Equation., but its objetive funtion is hanged to the maximal ase. An approah of the ubi FCPM is applied to [ψ ]. In [ψ ], the penalty for eah row, olumn, and height was alulated, defined as the differene between the two largest elements of the row, olumn, and height. In the row, olumn, or height with the largest penalty, the largest ψ must be seleted and its row, olumn, and height rossed out. The proedure is repeated to obtain a feasible assignment and its orresponding objetive funtion value. The obtained objetive funtion value and assignment are z* and ω*, respetively. Other than in the ubi FCPM, z* and ω* represent the largest known objetive funtion value and assoiated assignment, respetively. To optimise Equation 3.4, the revised branh and bound (R-B&B) algorithm is defined. It is used repeatedly in the f-g trade-off algorithm. The R-B&B algorithm is redued from the index-based B&B algorithm. The R-B&B algorithm: All steps of the R-B&B algorithm are the same as those in the index-based B&B algorithm, exept that the ondition q pj p k z* is deleted from () of Step 5 and q p pj p k > z* is deleted p from (2) and (3) of Step 5. Whenever the R-B&B algorithm is performed, it produes the optimal objetive funtion value f (t) of Equation 3.4 and its assoiated assignment ω (t). Equation 3.5 is then used to ompute g (t) : g (t) = Min t ( i, j, ( ) ( q ) (3.5) where g (t) is the minimum of all workers performane at iteration t. The largest known performane of the team z* an then be updated by using f (t) and g (t). To promote z*, the ubi matrix [α ] in (3.4) is updated; the updated rule of [α ] is () if q > z*, α remains unhanged. (2) If q z*, α is replaed with M (M ) i.e., fathom out that α is useless. The rule means that f (t) dereases (or remains unhanged) and g (t) inreases at eah iteration for z* to reah the maximum. Therefore, the flowhart in Figure 4 is used to depit the proposed f-g tradeoff algorithm. 3.3 Numerial example The following numerial fuzzy 3D axial assignment problem illustrates the two proposed algorithms. Table shows the job osts and quality. Aording to Equation 2.6, the numerial example an be formulated as 62 (0x Max Min x s.t. 3 3 j= k= x 333) x x = for i =, 2, 3 333, ( 0.64)x,..., ( 0.69)x

12 3 3 i= k= 3 3 i= j= x = for j=, 2, 3 (3.6) x = for k=, 2, 3 x {0, } for i, j, k=, 2, 3 Start Let t=0. Use ubi FCPM to initialise (3.4) with [ ] and obtain z* and ω*. Use the R-B&B algorithm to solve (3.4) with [ ]. No Any optimal solution? Yes Obtain ω (t), f (t), and g (t). Print ω* Yes Is z* f (t)? No Let t=t+ and update [ ]. Stop Let z*=min(f (t),g (t) ) and ω*=ω (t). No Is z*< f (t)? Yes Figure 4: Flow hart of the f-g trade-off algorithm Using Equation 2.2, α and β are transformed into γ, as shown in Table 2, a = 309, and b = 62. The algorithms are then used to solve Equation 3.6. (I) The index-based B&B algorithm Iteration 0: Step : Find α = 0, α 2 = 04, α 3 = 04, γ = 28.7, γ 2 =24, and γ 3 = Step 2: Determine SLA() = 309, SLA(2) = 208, SLA(3) = 04, SLA(4) = 0, SLG() = 78.2, SLG(2) = 50.03, SLG(3) = 26.03, SLG(4) = 0, TA(0) = 309, and TG(0)= Step 3: Use x = x = x = as the initial feasible solution and obtain z*=

13 Step 4: Let p = and proeed downward to node (,, ) from the root node. Step 5: Identify TA() = 309 and TG() = Beause q = 0.64 > 0.6 = z* and UPV() = 0.76 > 0.6 = z*, proeed downward to node (2, 2, 2) and update p = 2. Step 6: Let p = 2 and proeed to Step 5. Iteration : Step 5: Beause q 222 = 0.6 z*, node (2, 2, 2) is fathomed. Proeed toward node (2, 2, 3). Step 6: Let p = 2 and proeed to Step 5. (0,30) 0.64 (73,93) 0.7 (68,96) 0.64 (86,22) 0.65 (05,29) 0.6 (78,96) 0.75 (42,64) 0.65 (56,85) 0.68 (26,47) 0.57 Table : The ubi matrix of job osts and quality (63,88) 0.77 (63,89) 0.62 (85,204) 0.73 (52,73) 0.6 (48,69) 0.64 (37,57) 0.60 (04,33) 0.69 (47,74) 0.6 (32,5) 0.63 Legend: (α, β ) (26,53) 0.59 (77,20) 0.59 (69,90) 0.75 (27,54) 0.59 (69,94) 0.69 (33,62) 0.57 (7,90) 0.68 (53,74) 0.69 (04,22) 0.69 q j i k Table 2: The ubi matrix of γ [γ ]=

14 Iteration 2: Step 5: Determine q 223 = 0.69 > 0.6 = z*, TA(2) = 309, and TG(2) = Beause UPV(2)= > 0.6= z*, proeed downward to node (3, 3, 2) and update p = 3. Step 6: Let p = 3 and proeed to Step 5. Iteration 3: Step 5: Determine q 332 = 0.69 > 0.6= z*, TA(3) = 358, and TG(3) = Beause UPV(3)= 0.604> 0.6= z* and p= 3= n, replae z*= and move upward to node (2, 3, 2) and to the first node right of (2, 2, 3) of level 2. Update p = 2. Step 6: Let p = 2 and proeed to Step 5. The remaining proedures repeat these iterations and obtain the optimal solution x = x 223 = x 332 = and optimal value z*= (II) The f-g trade-off algorithm When t = 0: Equation 3.4 produes Equation 3.7. Max f = 62 (0x x333) x x333 s.t. the onstraints of Equation 3.6 (3.7) After using the ubi FCPM and R-B&B program to solve Equation 3.7, ω (0) = (x, x, 222 x ), f (0) =0.662, and g (0) = 0.6. Let ω*= ω (0) and z*= Min (0.662, 0.6)= 0.6 = g (0). 333 When t = : (0) () Update [ α ] by replaing α, α, α, α, α, and α with M and obtain [ α ]. () Solving Equation 3.7 again by using [ α ] produes ω () = (x, x, x ), f () =0.604, and g () = Beause f () = > 0.6= z*, update ω*= ω () and z*= Min (f (), g () )= When t = 2: () (2) Update [ α ] by replaing α, α, α, α, α, and α with M and obtain [ α ]. (2) Solving Equation 3.7 again by using [ α ] produes ω (2) = (x, x, x ), f (2) = 0.604, and g (2) = Beause f (2) = = z*, the algorithm stops. The urrent ω*= (x, x, x ) and z*= are the optimal solution and optimal objetive funtion value, respetively. 4 COMPUTATIONAL RESULTS AND DISCUSSION The omputational test verifies the effiieny of the proposed algorithms. Both proposed algorithms were oded in Visual Basi 6.0 and ompared with the LINGO pakage by using the global optimal funtion. They were all run on an Intel Pentium III-500 CPU. Problem sizes from to were tested. For eah problem of N N N size, uniform distribution numbers between 0 and 0+20N were randomly generated as α s, and uniform distribution numbers between 6N to 0N were randomly generated as the differene between β s and α s. Uniform distribution numbers between 0.6 and.0 were generated as q s. For eah problem size, 30 problems were generated. Table 3 shows the average omputational times (in seonds) of the two proposed algorithms and the LINGO pakage. The results show that both proposed algorithms have shorter omputational times than the LINGO pakage. 67

15 Table 3: Computational results when the quality interval is [0.6, ] Problem size The index-based B&B algorithm The f-g trade-off algorithm LINGO Average time (s) Average time (s) Average time (s) T. L. a T. L. T. L. a T. L. denotes the time limit exeeds,000 CPU seonds. In general ases, the manager or any worker may be the person with the lowest performane and may determine the team s performane. Here, two speial ases are presented. In the first ase, the ompany is assumed to support the team with suffiient funds, e.g., 0 times of b. The omputational results of the three approahes are shown in Table 4. The results show that the index-based B&B algorithm beomes more effiient. The time taken by the LINGO pakage also redues. The f-g trade-off algorithm, on the ontrary, beomes ineffiient. This is beause the optimal values f (t) of Equation 3.4 approah, suh that the fathom speed of α dereases. To disuss this further theoretially, b moves the membership funtion in Figure 2 lose to the horizontal line μ ( ) = and T T 68 b ( α x ) i= j= k = limit = b, b a + ( γ x ) i j k = = = i.e., the manager s performane approahes. The manager s performane is greater than the maximal of q. Thus the manager never determines the team performane; team performane is deided by a worker. The workers also obtain suffiient expense and reah the highest quality (performane). Therefore, Equation 2.6 is simplified to ( ( ( q )x ) for i,j,k,2,...,n) Max Min = s.t. the onstraints of Equation.. (4.) Equation 4. is a variant of the 3D axial bottlenek assignment problem. Hene, a proper algorithm for the 3D axial bottlenek assignment problem will improve the effiieny if b is known in advane. Problem size Table 4: Computational results using 0b The index-based B&B algorithm The f-g trade-off algorithm LINGO Average time (s) Average time (s) Average time (s) T. L. a T. L. a T. L. denotes that the time limit exeeds,000 CPU seonds.

16 In the seond ase, b is small; that is, the team is underfinaned. For example, define the manager s performane as 0 when the total ost is greater than (a+b)/2. The omputational results of the three approahes are shown in Table 5. The results show that the f-g tradeoff algorithm is the most effiient. Theoretially, when b (Z [α] * a min(q ))/( min(q )), where Z [α] * is the optimal value of the risp 3D axial assignment problem (min) using [α ] as the ost oeffiient, the b is too small for the manager s performane to reah the minimum of q. Team performane is then determined mainly by the manager, despite the max-min objetive funtion equalising the performane of the n workers and the manager. Thus Equation 2.6 is simplified to Equation 3.4 with t=0. Hene, a proper speifi algorithm for the 3D axial frational assignment problem will improve the effiieny. Table 5: Computational results using (a+b)/2 Problem size The index-based B&B algorithm The f-g trade-off algorithm LINGO Average time (s) Average time (s) Average Time (s) a T. L. denotes that the time limit exeeds,000 CPU seonds T. L. a T. L. 5 CONCLUSIONS This paper first formulates a fuzzy 3D axial assignment problem model to overome the atual unertain environment, and then fouses on developing algorithms to obtain the exat solutions of the proposed model effiiently. The initially-onstruted fuzzy 3D axial assignment problem is a mixed nonlinear programming problem. For simpliity, it is transformed into a speial multi-frational max-min 0- programming problem. An indexbased B&B algorithm and the f-g trade-off algorithm are developed. Computational results show that the proposed methods are both more effiient than LINGO in every test. In addition, two speial ases are disussed. One is when the ompany amply supports the team with funds, while the other is when the team is underfunded. With respet to omputational time, the index-based B&B algorithm is better for the former ase. However, the f-g trade-off algorithm is better for the latter ase. These two speial ases are further onsidered from a theoretial view. For the ase with unbounded funds, the model is redued to a 3D axial bottlenek assignment problem; for the ase with rare funds, the model is redued to a 3D axial frational assignment problem. Hene, the developments of algorithms for these two problems would be interesting and pratial diretions for future researh. Moreover, a sensitivity analysis proedure for identifying the largest sensitivity range one that maintains the urrent optimal assignment is worth studying. REFERENCES [] Burkard, R., Dell'Amio, M. & Martello, S Assignment problems. st edition, SIAM. [2] Pentio D.W Assignment problems: A golden anniversary survey. European Journal of Operational Researh, 76(2), pp [3] Balas, E. & Saltzman, M.J. 99. An algorithm for the three-index assignment problem. Operations Researh, 39(), pp

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