On NP-completeness of the cell formation problem
JJanuary 10, 2019 International Journal of Production Research np˙complete
To appear in the
International Journal of Production Research
Vol. 00, No. 00, 00 Month 20XX, 1–10
On NP-completeness of the cell formation problem
Mikhail V. Batsyn a ∗ , Ekaterina K. Batsyna b , Ilya S. Bychkov a a Laboratory of Algorithms and Technologies for Network Analysis b Department of Applied Mathematics and InformaticsNational Research University Higher School of Economics,25/12 B. Pecherskaya, Nizhny Novgorod, Russian Federation, 603155 ( v1.1 released Dec 2018 ) In the current paper we provide a proof of NP-completeness for the Cell Formation Problem (CFP) withthe fractional grouping efficacy objective. For this purpose we first consider the CFP with the linearobjective minimizing the total number of exceptions and voids. Following the ideas of Pinheiro et al.(2016) we show that it is equivalent to the Bicluster Graph Editing Problem (BGEP), which is known tobe NP-complete (Amit 2004). Then we suggest a reduction of the CFP problem with the linear objectivefunction to the CFP with the grouping efficacy objective.
Keywords: cell formation problem; bicluster graph editing problem; grouping efficacy; np-complete
1. Introduction
The Cell Formation Problem (CFP) consists in optimal grouping of machines together with partsprocessed on them into manufacturing cells. The goal of such a bi-clustering (clustering of bothmachines and parts) is to minimize the inter-cell movement of parts between different cells duringthe manufacturing process and to maximize the loading of machines with parts processing insidetheir cells. The input to this problem is given by a binary machine-part matrix defining for everymachine what parts are processed on it. In terms of input matrix the objective of the CFP is topartition rows (machines) and columns (parts) of the input matrix into rectangular cells minimiz-ing the number of ones outside cells, called exceptions (representing the inter-cell movements ofparts), and minimizing the number of zeroes inside cells, called voids (reflecting the underloadingof machines). An example of the input matrix is shown in Table 1 and a feasible solution for thisinstance is shown in Table 2.A number of papers on the CFP are devoted to its simplest formulation, called Machine Par-titioning Problem (MPP), in which only machines are clustered into cells and the objective iscomputed as an explicit function from this partition and machine-part matrix (Kusiak et al. 1993;Spiliopoulos & Sofianopoulou 1998; Arkat et al. 2012). Though we are not aware of the proof ofNP-completeness for the MPP, we believe it exists in literature. It is probably present in the PhDthesis of Ballakur (1985), judging by the references to this work. Unfortunately we have failed tofind it in electronic databases. Besides Ghosh et al. (1996) states that the NP-hardness of the MPPcan be proved ”by a straightforward reduction of the clustering problem (Garey & Johnson 1979)”to the MPP.The CFP problem becomes much harder when we want to cluster machines and parts togetherinto biclusters. In spite of the fact that most papers in last decades consider the CFP in its ∗ Corresponding author. Email: [email protected] a r X i v : . [ c s . CC ] J a n anuary 10, 2019 International Journal of Production Research np˙complete biclustering formulation, there are no papers providing the proof of its NP status to the best ofour knowledge. There is a big number of papers, where authors just write that the problem isNP-hard (Mak et al. 2000; Goncalves & Resende 2004; Chan et al. 2008). Other authors includingTunnukij & Hicks (2009); Elbenani & Ferland (2012) state that the CFP is NP-hard citing thepaper of Dimopoulos & Zalzala (2000). But Dimopoulos & Zalzala (2000) only mention that ”thecell-formation problem is a difficult optimization problem”.Many papers including James et al. (2007); Chung et al. (2011); Paydar & Saidi-Mehrabad(2013); Solimanpur et al. (2010); Utkina et al. (2016) refer to Ballakur & Steudel (1987) whenwriting about the NP-hardness of the CFP. However Ballakur & Steudel (1987) present a heuristicfor the CFP with different objective functions and do not state anything about the NP status ofthese CFP formulations. Finally there are some papers citing Ballakur (1985) PhD thesis whereseveral CFP formulations are considered. However this paper is not available in any electronic pub-lication databases. According to the existing references to this thesis and other papers of Ballakurwe can only conclude that he considers the machine partitioning and machine-part partitioningproblems with some objective functions, but not with the grouping efficacy function introducedlater by Kumar & Chandrasekharan (1990). At the same time the grouping efficacy is currentlywidely accepted and considered as the best function successfully joining the both objectives ofinter-cell part movement minimization and inta-cell machine loading maximization.In the current paper we provide a proof of NP-completeness for the CFP problem with thefractional grouping efficacy objective. For this purpose we first consider the CFP with the linearobjective minimizing the total number of exceptions and voids. Following the ideas of Pinheiro etal. (2016) we show that it is equivalent to the Bicluster Graph Editing Problem (BGEP), whichis known to be NP-complete (Amit 2004). Then we suggest a reduction of the CFP problem withthe linear objective function to the CFP with the grouping efficacy objective.
2. Problem formulation
In the CFP we are given m machines, p parts processed on these machines, and m × p Booleanmatrix A in which a ij = 1, if machine i processes part j during the production process, and a ij = 0otherwise. We should cluster both machines and parts into biclusters, called cells, so that for everypart we minimize simultaneously the number of processing operations of this part on machines fromother cells and the number of machines from the same cell which do not process this part. Thuswe minimize the movement of parts to other cells (inter-cell operations) and maximize the loadingof machines with processing operations inside cells (intra-cell operations) during the productionprocess. In other words we need to choose machine-part cells in matrix A , such that the number ofones outside these cells (called exceptions) is minimal possible and at the same time the numberof zeroes inside these cells (called voids) is also minimal possible. The objective function whichprovides a good combination of these two goals and is widely accepted in literature is the groupingefficacy suggested by Kumar & Chandrasekharan (1990): f = n − en + v → max (1)Here n is the number of ones in the input matrix, e is the number of exceptions (ones outsidecells), v is the number of voids (zeroes inside cells).Below we present a straightforward fractional programming model for the CFP (Utkina et al.(2018), Bychkov et al. (2014)). Since the number of cells cannot be greater than the number ofmachines and the number of parts, then the maximal possible number of cells is equal to min( m, p ).We denote this value as c = min( m, p ). 2 anuary 10, 2019 International Journal of Production Research np˙complete x ik = (cid:40) i is assigned to cell k y jk = (cid:40) j is assigned to cell k e = n − c (cid:88) k =1 m (cid:88) i =1 p (cid:88) j =1 a ij x ik y jk (4) v = c (cid:88) k =1 m (cid:88) i =1 p (cid:88) j =1 (1 − a ij ) x ik y jk (5)Objective functions: f = e + v → min (6a) f = n − en + v → max (6b)Constraints: c (cid:88) k =1 x ik = 1 ∀ i = 1 , . . . , m (7) c (cid:88) k =1 y jk = 1 ∀ j = 1 , . . . , p (8)Objective function (6a) minimizes the number of exceptions and voids and objective function (6b)maximizes the grouping efficacy. Assignment constraints (7) and (8) provide that all machines andparts are partitioned into disjoint cells. 3 anuary 10, 2019 International Journal of Production Research np˙complete
3. NP-completeness
To prove the NP-completeness of the CFP with linear objective (6a) we use the Bicluster GraphEditing Problem (BGEP). The first authors who have noticed the closeness of the CFP and BGEPproblems are Pinheiro et al. (2016). They applied it in their exact algorithm for the CFP with thegrouping efficacy objective.
Figure 1.: BGEP instanceThe BGEP problem consists in determining the minimum number of edges which should be addedto/removed from the given bipartite graph so that it transforms to a set of isolated bicliques. Anexample of a BGEP instance is presented in Figure 1 and its solution – in Figure 2. Here dottedthick lines show the added edges and red thin lines – the removed edges. The BGEP problem isNP-complete. To be more exact – its decision version is NP-complete. The decision version of anoptimization problem with objective function f → max ( f → min) is a problem with the sameconstraints, which only answers the question, whether there exists a feasible solution with f ≥ c ( f ≤ c ) for any given constant c . Since the theory of NP-completeness is applicable only for decisionproblems in all the propositions and theorems below we will talk about the decision versions of theproblems. Theorem 1 (Amit (2004)) . The BGEP problem is NP-complete because the NP-complete 3-exact3-cover problem can be polynomially reduced to BGEP.
The 3-exact 3-cover problem is defined as follows. Given a set of elements U = { , , ..., n } anda collection C of triplets of these elements, such that each element can belong to at most 3 triplets,determine if there exists a subcollection of C with size n which covers U .Hereafter we will call CFP 1 the CFP problem with the linear objective function f = e + v (6a),and CFP 2 – the CFP problem with the grouping efficacy objective f = ( n − e ) / ( n + v ) (6b). Theorem 2.
The CFP with linear objective f = e + v (CFP 1) is NP-complete since it is equivalentto the BGEP problem. anuary 10, 2019 International Journal of Production Research np˙complete Figure 2.: BGEP solution
Proof.
There is a one-to-one correspondence between these two problems. Every machine in theCFP corresponds to a vertex in one part of the bipartite graph in the BGEP, and every part inthe CFP corresponds to a vertex in another part of this graph. The machine-part matrix in theCFP coincides with the bipartite graph biadjacency matrix in the BGEP. Every exception in asolution of the CFP corresponds to an edge which should be removed from the bipartite graph inthe BGEP in order to transform it to a set of isolated bicliques. And every void in a CFP solutioncorresponds to an edge which should be added to the bipartite graph in the BGEP.It is clear that the CFP f = e + v → min objective is equivalent to the BGEP objective ofminimizing the number of added / removed edges needed to transform the input bipartite graphto a set of isolated bicliques. Every biclique corresponds to a rectangular cell in the CFP. If weremove the added edges and return back the removed ones then every isolated clique will become anon-isolated quasi-biclique completely coinciding with a rectangular cell in a CFP solution. Thusthe CFP 1 problem is equivalent to the BGEP problem and it is NP-complete.For example, rows (machines) 2, 4, 5, 3, 1 in Table 2 correspond to vertices 2, 4, 5, 3, 1 in the leftpart of the bipartite graph in Figure 2 and columns (parts) 1, ..., 7 correspond to vertices 1, ..., 7in the right part of this graph. The solution of this BGEP instance contains 3 bicliques shown withthick lines in Figure 2. Here two dashed lines represent two edges which should be added to thegraph to form bicliques. Red thin lines show the edges which should be removed from the graphto isolate the bicliques from each other.To prove the NP-completeness of the CFP 2 problem we suggest the reduction of CFP 1 problemto it. The CFP 2 objective can be written in the following way. f = n − en + v = 1 − e + vn + v → max ⇔ e + vn + v → minThis expression is almost equivalent to the linear objective of the CFP 1, except the value of v inthe denominator. Our idea is to nullify the influence of this value by significant increasing of thenumber of ones n . We reduce the CFP 1 problem to CFP 2 by extending the original machine-partmatrix A with a big block of ones as it is shown in Table 3. For example, for the CFP 1 instanceshown in Table 1 the extended matrix ˜ A will be as shown in Table 4. Before the main theoremusing the suggested reduction and stating the NP-completeness of CFP 2 we will need to provetwo propositions first. Proposition 1.
If the machine-part matrix for the CFP 2 problem has identical rows then therewill be optimal solutions in which these rows belong to the same cell. anuary 10, 2019 International Journal of Production Research np˙complete A 01 ... 10 ... ...1 ... 1Table 3.: Extended matrix ˜ A Proof.
Let us assume that there are two identical rows which belong to different cells in an optimalsolution, the first of these rows has e exceptions (ones outside its cell) and v voids (zeroes insideits cell), the second row has e exceptions and v voids, and all other rows in this solution have intotal e exceptions and v voids. Then the objective function value for this solution is the following. f = n − e − e − e n + v + v + v If we move the second of the identical rows to the cell of the first one then these two rows willhave 2 e exceptions and 2 v voids. Otherwise, if we move the first row to the cell of the second one,we will get 2 e exceptions and 2 v voids. Without loss of generality we can assume that joining theidentical rows in the cell of the first of them gives the value of the grouping efficacy not smallerthan we get in the opposite variant: n − e − e n + v + 2 v ≥ n − e − e n + v + 2 v ⇔ v ( n − e ) − e ( n + v ) − e v ≥ v ( n − e ) − e ( n + v ) − e v Now we will prove that the first variant of joining the identical rows gives the objective functionvalue not worse than the original optimal solution has. We need to prove the following. n − e − e n + v + 2 v ≥ n − e − e − e n + v + v + v ⇔ ( v + v )( n − e ) − e ( n + v ) − e ( v + v ) ≥ v ( n − e ) − ( e + e )( n + v ) − v ( e + e ) ⇔ v ( n − e ) − e ( n + v ) − e v ≥ v ( n − e ) − e ( n + v ) − e v The last line exactly coincides with the expression we have obtained above from our assumptionthat the first variant of joining the identical rows is not worse than the second one. Thus thesolution with the joined rows is also optimal. 6 anuary 10, 2019 International Journal of Production Research np˙complete
The next proposition determines how much ones it is enough to add in the extended matrix inorder to nullify the influence of f denominator. Proposition 2.
If the number of added ones ∆ n in the extended matrix ˜ A is equal to ( mp ) thenthe maximum of f on ˜ A is obtained at the same solution (extended with the cell of added ones) atwhich f has its minimum on matrix A .Proof. According to Proposition 2 the optimal solution for CFP 2 on the extended matrix has theadded block of ones as a separate cell. This means that this block adds no voids or exceptions tothe solution and thus a CFP 1 solution and the corresponding CFP 2 solution (obtained by addingthe block of ones as an additional cell) have the same number of voids v and exceptions e . We willnow prove that if ∆ n = ( mp ) then for any two CFP 1 solutions with objective function values f and f (cid:48) and the correspoding CFP 2 solutions with objective function values f and f (cid:48) from f (cid:48) < f it follows that f (cid:48) > f . f (cid:48) = ˜ n − e (cid:48) ˜ n + v (cid:48) , f = ˜ n − e ˜ n + v , f (cid:48) = e (cid:48) + v (cid:48) , f = e + vf (cid:48) > f ⇔ ˜ n − e (cid:48) ˜ n + v (cid:48) > ˜ n − e ˜ n + v ⇔ e (cid:48) + v (cid:48) ˜ n + v (cid:48) < e + v ˜ n + v ⇔ f (cid:48) ˜ n + v (cid:48) < f ˜ n + v ⇔ f (cid:48) < f ˜ n + v (cid:48) ˜ n + v ⇔ f (cid:48) < f + f v (cid:48) − v ˜ n + v Note that in case n = 0 the CFP 1 problem becomes trivial and so we consider only the case n ≥
1. Since f ≤ mp, v − v (cid:48) ≤ mp for ∆ n = ( mp ) we have: f v − v (cid:48) ˜ n + v = f v − v (cid:48) n + ∆ n + v ≤ ( mp ) ( mp ) + 1 ⇔ f v (cid:48) − v ˜ n + v ≥ − ( mp ) ( mp ) + 1From this it follows that: f + f v (cid:48) − v ˜ n + v ≥ f − ( mp ) ( mp ) + 1 > f − f (cid:48) and f are integer, then f (cid:48) < f is equivalent to f (cid:48) ≤ f −
1. Thus we have: f + f v (cid:48) − v ˜ n + v > f − ≥ f (cid:48) ⇒ f (cid:48) < f + f v (cid:48) − v ˜ n + v ⇔ f (cid:48) > f So we get that from f (cid:48) < f it follows that f (cid:48) > f . This means that the minimum value of f gives the maximum of f on the ”extended” solution. Theorem 3.
The CFP with grouping efficacy objective f = ( n − e ) / ( n + v ) (CFP 2) is NP-complete because CFP 1 can be polynomially reduced to it.Proof. We will prove that CFP 1, which answers the question, whether there exists a solution with f = e + v ≤ c with the input matrix A can be polynomially reduced to problem CFP 2 on theextended matrix ˜ A (see Table 3), which answers the question, whether there exists a solution with f = ( n − e ) / ( n + v ) ≥ ˜ c . Here constant ˜ c can depend on constant c and other input parameters.According to Proposition 3 to get a better solution for CFP 2 we should simply extend thesolution for CFP 1 with an additional cell represented by the added block of ones in the extendedmatrix ˜ A (see Table 3). It is clear that for the considered decision version of CFP 2 we should alsotake the best possible solution with maximal value of f to guarantee the satisfaction of inequality7 anuary 10, 2019 International Journal of Production Research np˙complete f ≥ ˜ c . Thus the solution for CFP 1 and the corresponding suggested solution for CFP 2 areconnected in the following way. ˜ e = e, ˜ v = v, ˜ n = n + ∆ n f = ˜ n − ˜ e ˜ n + ˜ v = ˜ n − e ˜ n + v = 1 − e + v ˜ n + v = 1 − f ˜ n + v Let us find a value of ˜ c such, that for a CFP 1 solution with f ≤ c the corresponding solution forCFP 2 will have f ≥ ˜ c . f ≤ c ⇒ f = 1 − f ˜ n + v ≥ − c ˜ n So for ˜ c = 1 − c/ ˜ n we have f ≤ c ⇒ f ≥ ˜ c . This guarantees that if there are no solutions forCFP 2 with f ≥ ˜ c then there exist no solutions for CFP 1 with f ≤ c .Now let us prove that for this ˜ c from f ≥ ˜ c for CFP 2 solution it follows that the original CFP1 solution has f ≤ c . We have: f = ˜ n − e ˜ n + v = 1 − e + v ˜ n + v ≥ − c/ ˜ n ⇔ e + v ˜ n + v ≤ c/ ˜ n ⇔ e + v ≤ c + cv ˜ n Now we can use the fact that the number of added ones is ∆ n = ( mp ) , and so ˜ n = n + ( mp ) .We also note that the cases when n = 0 or c ≥ mp are trivial, because in such cases we do notneed to construct any CFP 2 instance and can immediately answer the CFP 1 question. Since c < mp and v ≤ mp we have. e + v ≤ c + cv ˜ n < c + ( mp ) n + ( mp ) < c + 1Since e + v is integer we can conclude that f = e + v ≤ c .Thus we have found the value of ˜ c = 1 − c/ ˜ n such that the answer for any CFP 1 instance onthe question, whether there exists a solution with f ≤ c , is ”yes”, if and only if the answer forthe corresponding CFP 2 instance on the question, whether there exists a solution with f ≥ ˜ c , isalso ”yes”. Consequently, the answer to the CFP 1 question is ”no”, if and only if, the answer tothe CFP 2 question is also ”no”. This proves that CFP 2 is at least as hard as the NP-completeproblem CFP 1. It is also clear that CFP 2 belongs to class NP, because any ”yes”-solution can beverified in polynomial time. This proves that CFP 2 is an NP-complete problem. Funding
Sections 1, 2 and Theorems 1, 2 in Section 3 were prepared within the framework of the BasicResearch Program at the National Research University Higher School of Economics (NRU HSE).Propositions 1, 2 and Theorem 3 in Section 3 were formulated and proved with the support of RSFgrant 14-41-00039. 8 anuary 10, 2019 International Journal of Production Research np˙complete
References
Amit N. (2004). The bicluster graph editing problem. Master thesis. Tel Aviv University, 50 p.Arkat, J., Abdollahzadeh, H., Ghahve, H. (2012). A new branch-and-bound algorithm for cell formationproblem. Applied Mathematical Modelling, 36, 5091—5100.Ballakur A. (1985). An Investigation of Part Family/Machine Group Formation for Designing CellularManufacturing Systems. Ph.D. Thesis, University Wisconsin, Madison.Ballakur, A., & Steudel, H. J. (1987). A within cell utilization based heuristic for designing cellular manu-facturing systems. International Journal of Production Research, 25, 639–655.Bychkov, I., Batsyn, M., Sukhov, P., Pardalos, P.M.(2013) Heuristic Algorithm for the Cell Formation Prob-lem. In: Goldengorin B. I., Kalyagin V. A., Pardalos P. M. (eds.) Models, Algorithms, and Technologiesfor Network Analysis. Springer Proceedings in Mathematics & Statistics 59, 43–69.Bychkov, I., Batsyn, M., Pardalos, P. (2014). Exact model for the cell formation problem. OptimizationLetters 8(8), 2203–2210.Chan F. T. S., Lau K. W., Chan L. Y., Lo V. H. Y. (2008). Cell formation problem with considerationof both intracellular and intercellular movements. International Journal of Production Research 46(10),2589-2620.Chung S.-H., Wu T.-H., Chang C.-C. (2011). An efficient tabu search algorithm to the cell formation problemwith alternative routings and machine reliability considerations. Computers & Industrial Engineering60(1), 7-15.Dimopoulos C., Zalzala A. (2000). Recent developments in evolutionary computation for manufacturingoptimization: problems, solutions and comparisons. IEEE Transactions on Evolutionary Computation4(2), 93-113.Elbenani, B., Ferland, J. A. (2012). An exact method for solving the manufacturing cell formation problem.International Journal of Production Research, 50(15), 4038–4045.Garey M.R., Johnson D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness,Freeman, New York.Ghosh S., Mahanti A., Nagi R., Nau D. S. (1996). Manufacturing cell formation by state-space search,Annals of Operations Research 65(1), 35–54.Goncalves, J. F., Resende, M. G. C. (2004). An evolutionary algorithm for manufacturing cell formation.Computers & Industrial Engineering 47, 247–273.James, T. L., Brown, E. C., Keeling, K. B. (2007). A hybrid grouping genetic algorithm for the cell formationproblem. Computers & Operations Research, 34(7), 2059–2079.Kumar, K. R., Chandrasekharan, M. P. (1990). Grouping efficacy: A quantitative criterion for goodness ofblock diagonal forms of binary matrices in group technology. International Journal of Production Research,28(2), 233–243.Kusiak, A., Boe, J. W., Cheng, C. (1993). Designing cellular manufacturing systems: branch-and-bound andA* approaches. IIE Transactions, 25:4, 46–56.Mak K. L., Wong Y. S., Wang X. X. (2000). An Adaptive Genetic Algorithm for Manufacturing CellFormation. The International Journal of Advanced Manufacturing Technology 16(7), 491-497.Paydar, M. M., Saidi-Mehrabad, M. (2013). A hybrid genetic-variable neighborhood search algorithm for thecell formation problem based on grouping efficacy. Computers & Operations Research, 40(4), 980–990.Pinheiro R. G. S., Martins I. C., Protti F., Ochi, L. S., Simonetti L.G., Subramanian A. (2016). On solvingmanufacturing cell formation via Bicluster Editing, European Journal of Operational Research, 254(3),769–779.Solimanpur M., Saeedi S., Mahdavi I. (2010). Solving cell formation problem in cellular manufacturingusing ant-colony-based optimization. The International Journal of Advanced Manufacturing Technology50, 1135–1144.Spiliopoulos, K., Sofianopoulou, S. (1998). An optimal tree search method for the manufacturing systemscell formation problem. European Journal of Operational Research, 105, 537–551.Tunnukij T., Hicks C. (2009). An Enhanced Grouping Genetic Algorithm for solving the cell formationproblem. International Journal of Production Research 47(7), 1989-2007.Utkina, I., Batsyn, M., Batsyna, E. (2016). A branch and bound algorithm for a fractional 0-1 programmingproblem. Lecture Notes in Computer Science, 9869, 244–255.Utkina I. E., Batsyn M. V., Batsyna E. K. (2018). A branch-and-bound algorithm for the cell formation anuary 10, 2019 International Journal of Production Research np˙complete problem. International Journal of Production Research 56(9), 3262–3273problem. International Journal of Production Research 56(9), 3262–3273