This paper deals with the allocation technique of the layout whose solution it takes as a little computation time as possible to obtain and which becomes as near to the optimal method as possible. In this method, the relative transport distance of each location and the relative transport volume of each department are calculated from the distance matrix and the volume matriX by the statistical procedures. And allocating departments to locations is determined by one to one correspondence between the arranged relative transport distances and the arranged relative transport volumes. This method was called the allocation technique by the statistical procedure ( ATSP in short ).
This method doesn't use the heuristic algorithm.
Therefore the calculation time can be reduced much in
comparison with any other methods. As the algorithm
of ATSP method is very simple, the sub-optimal layout
can be determined easily by using the desk-calculator
in cases of any layout problems

Fujiwara, Yutaka

Kikuchi, Susumu

Osaki, Hirokazu

Okayama University Digital Information Repository

Allocation Technique Using Relative Distance and Relative Volume Estimated by Statistical Procedure

Okayama University Scientific Achievement Repository

MEMOIRS OF THE SCHOOL OF ENGINEERING, OKAYAMA UNIVERSITY, Vol. 8, NO.l,JUNE 1973Allocation Technique Using Relative Distance andRelative Volume Estimated by Statistical Procedure* - **Yutaka FUJIWARA ,Hirokazu OSAKI**and Susumu KIKUCHI(Received May 30, 1973)SynopsisThis paper deals with the allocation technique ofthe layout whose solution it takes as a little compu-tation time as possible to obtain and which becomes asnear to the optimal method as possible. In this method,the relative transport distance of each location andthe relative transport volume of each department arecalculated from the distance matrix and the volumematriX by the statistical procedures. And allocatingdepartments to locations is determined by one to onecorrespondence between the arranged relative transportdistances and the arranged relative transport volumes.This method was called the allocation technique by thestatistical procedure ( ATSP in short ).This method doesn't use the heuristic algorithm.Therefore the calculation time can be reduced much incomparison with any other methods. As the algorithmof ATSP method is very sim~le, the sub-o~timal layoutcan be determined easily by using the desk-calculatorin cases of any layout problems.* Department of Production Engineering, Matsue Technical Collage.** Department of Industrial Science, School of Engin33ringOkayama University.*** Department of Industrial Science, School of EngineeringOkayama University.1516 Y. FUJIWARA, H. OSAKI, and S. KIKUCHIL IntroductionThe appropriate arrangement of workers, machines and materials intothe factory is the most important to combine these elements systemati-cally and rationally and further to improve the efficiency of theproduction. Especially allocating machines to locations much influencesthe transport distance of materials. Therefore many quantitive allo-cation techniues to solve the layout problem have been developed untilnow. Most of these techniques improve the layout by the heuristicalgorithm.In this paper, authors attempt to study the allocation techniquewhich doesn't use the heuristic algorithm. The relative transportdistance of each location and the relative transport volume of eachdepartment playa important role in this technique. These relativevalues are calculated by the statistical procedure.2. Restriction of the Layout ProblemThe various factors which influence the layout should be taken intoconsideration in determining the optimal layout planning. In generallythe factors of transport distances among locations and transport volumesamong departments are frequently used to determine the layout. Then inthis paper, the technique of allocating departments to locations isconsidered under the following restrictions.(Restriction 1)It is assumed that the number of departments is equal to that oflocations. That is,number of departments ::: number of locations ::: nn departments are equal to each other in area. n locations areequal to each other in area, too. And the area of a departmentis equal to that of a location.(Restriction 2)The distance between location Ii and I j puts as dij • And it isassumed that the distance from Ii to I j is equal to that from I jto Ii • Then:::i j::: 1,2, .•• ,nA /location Technique by Statistical Procedure 17department B. and B. is put as~ JBi to Bj and that from Bj to Bi •(Restriction 3)The transport volume v .. between. ~Jthe sum of the volume fromTherefore==i j = 1,2, •.• ,n(Restriction 4)The efficiency of the obtained layout was evaluated by the follow-ing equation.nEV = 2:""5"d."xv(.) (.)i<j j:r ~J S ~ s JsCi) and s(j) are the department numberswhich are allocated to the location I. and I.~ JThis evaluated value EV is generally used in order to approximatethe material handling cost clOsely.(l)The smaller the evaluated value becomes, the better the layout does• And layout which has minimum value of EV is called the optimallayout.3. Allocation Technique by the Statistical Procedure3.1. Optimal Producing Method (OPM in short)(2)It is assumed that n departments are assigned to n availablelocations and that any departments are compatible with any locations.Then there are n! ways of allocating departments to locations in thiscase. The allocation which has minimum EV among n! allocations is theoptimal layout.3.2. Allocation Technique by the Statistical Procedure (ATSP in short)In OPM method, as n becomes large, the huge number of times ofthe repititive calculation are necessary to determine the optimallayout. For example , when n = 10, about four million times of calcu-lation are necessary. Therefore it is impossible to determine theoptimal layout in the case of layout problem having more than tp.ndepartments. Then various types of the technique to obtain the sub-optimal layout have been developed by Buffa, etc •• (3) But the algorithmof these techniques uses mainly the heuristic algorithm.Therefore the authors attempt to study the allocation techniquei = 1,2, •••• ,n18 Y. FUJIWARA, H. OSAKI, and S. KIKUCHIwhich doesn't use the heuristic algorithm in order to obtain the sub-optimal layout. In this technique, the relative distance of eachlocation and the relative volume of each department are defined bythe statistical method. This technique is constructed of ATSP1, ATSP2and ATSP3 distinguished by the difference of the definition of therelative distance and the relative volume.3.2.1. ATSPlIn this technique, the relative distance of each location and therelative volume of each department are defined by the mean value andthe standard deviation as follows.(Step 1)The mean value of distance of each location is calculated asfollows.nDJ.' = 2: d. . I (n - 1) i = 1, 2, ••• , ni~j=l J.JThe standard deviation of distance of each location is calculatedas follows.= ff (d .. - DJ..)2 / (n - 2) , i = 1,2, ••• ,nJ i~f," J.JAnd the relative distance Pi of the location Ii is defined by thefollowing equation.Pi = Di + CXSDi ••••••• ( 1 )i=1,2, •••• ,nC is the constant which indicates thescale of the statistical distribution(step 2)The mean value and the standard deviation of volume of eachdepartment are calculated as follows.V. = ~v .. /(n-l)1 i~j=1 lJSVi =Ji~~ (vij - Vi )2/ (n - 2)And the relative volume Qi of the department Bi is defined by thefollowing equation.= Vi + CXSVi ••••••• ( 2 )i = 1,2, .... ,nC is the same value in the equation ( 1 )(Step 3)The relative distances Pi ' i=1,2, ••• ,n are arranged in order fromA llocation Technique by Statistical Procedurethe smallest of them. And the relative volumes Qi ' i=1,2, •.• ,n arearranged in order from the largest of them. It is assumed thatthese results are as follows.19~>o >0 > ... >0 >1 ~2 ~3 ~j >~n• . . .. ( 3 )• • . .• ( 4 )From thePk . andJrelation (3) and (4) , it becomes possible to make~. ' j = 1,2, •• ,n correspond one to one.JTherefore the department Bm. is allocated to the location I k . 'j=1,2, ••• ,n. J JThe sub-optimal layout can be determined from this allocation. Andthe evaluated value of EV of this allocation is calculated.3.2.2. ATSP2The standard deviation used in ATSPI needs the complex calculation.Therefore the range is used to estimate the standard deviation. Therange RDi of distance of the location Ii is calculated by the followingequation.= max d ..h'~n-l ~ J min d",~n.l ij i = 1,2, ••• ,nAnd the relative distance is defined as follows.= + • • • . • .• ( 1 )'Similary RVi is put as the range of volume of the department Bi •And the reiative volume is defined as follows.= + • • • • • •• ( 2 )'weighted average,PERT. (4), (5)ATSP2 is the technique in which the equation ( 1 )' and ( 2 )' areused in step ( 1 ) and ( 2 ) in ATSPI instead of equation ( 1 ) and( 2 ).3.2.3. ATSP3The mean value is calculated by the statisticallyincorporating the maximum, most likely and minimum inThe relative distance is defined as follows.( 1 )"AD; = max d .. , MD; = mode d .. , ID; = min diJ·.... l.:Jgl-l ~J .... lijSl1-1 ~J .... 1'1511-120 Y. FUJIWARA, H. OSAKI, and S. KIKUCHISimilary the relative volume is defined as follows.AVi = 1~;n:l{1 Vij , MVi =f~?n~r Vij , IVi =1~{~!i vijATSP3 is the technique in which the equation ( 1 )" and ( 2 )" areused in step ( 1 ) and ( 2 ) in ATSPI instead of equation ( 1 ) and( 2 ).4. ResultsFirst, OPM and ATSP method were applied to the layout problem(6) ofn = 4 having the distance matrix D = ( d .. ) in Table 1 and the volumelJmatrix V = ( vij ) in Table 2.4.1. The Optimal Layout by OPM MethodThere were four departments and four available locations in thisproblem. Therefore the optimal layout was selected among 41 = 24 waysof the allocation. And the result was the allocation of the departmentB2 to the location II' Bl to 12 , B4 to 13 , and B3 to 14 , The evaluatedvalue of EV of this optimal layout was 11048.4.2. The Sub-optimal Layout by ATSP MethodThe mean value, the standard deviation, the range and the weightedaverage of each location and department were calculated from thedistance matrix and the volume matrix. These values were shown in theright half of Table 1 and Table 2.The mean value or the sum of mean value and standard deviation, ortwo times of it, or three times of it was used generally in the sta-tistical method. Then the constant C in equation ( 1 ) and ( 2 ) inATSPI was determined as follows.C = 0 , C =1 , C = 2 and C = 3The solution by ATSPI in the case of C = 0 was shown in accordancewith its algorithm.The relative distance of each location was as follows from Table 1.PI = 26.00 , P2 = 30.67 , P3 = 18.00 , P4 = 17.33The relative volume of each department was as follows from Table 2.~l = 80.00 , ~2 =77·77 , ~3 = 120.00 , ~4 = 87.33A llocation Technique by Statistical ProcedureThe results of the above mentioned arrangement of Pi and ~ becameas follows.P4 <P3 <Pl <P2Q3 > Q4 > Q1 > Q2Therefore the correspondence between P4 and Q3' P3 and Q4' Pl andQl' and P2 and Q2 were made. Then the sub-optimal layout was the allo-cation of the department B3 to the location 14 , B4 to 13 , Bl to 11 , andB2 to 12 , And the evaluated value of this layout was 11640. This valuewas larger by about 500 than that of the optimal layout by OPM method.The results of the sub-optimal layout by ATSPl in cases of C = 1,~ 11 12 13 14 mean s.d. range w.m.I, 0 42 14 22 26.00 14.43 28 18.6712 42 °30 20 30.67 11.02 22 29·4413 14 30 0 10 18.00 10.5820 10.6714 22 20 10 0 17.33 6.43 1222.22s.d. = standard deviationw.rn. weighted meanTable 1, Distance matrix, mean value, standard deviation,range and weighted mean~ B1 B2 B3 B4 mean s.d. range w.rn.Bl 0 55 135 50 80.00 47.70 85 34.17B2 55 0 95 82 77.33 20.40 40 85.89B3 135 95 0 130 120.00 21.79 40 138.33B4 50 82 130 0 87.33 40.27 80 77.56s.d. standard deviationw.m. = weighted meanTable 2, Volume matrix, mean value, standard deviation,range and weighted mean2122 Y. FUJIWARA, H. OSAKI, and S. KIKUCHIC = 2 and C = 3 wereshown ~n Table 3.From these results,the layout by ATSPIin the case of C = 0had the smallestevaluated value.Then this solutionwas the most ef-ficient among othersolutions by ATSP.~ I 12 I~ 14 EVAT 5 P 1A'l'SPl (C = 1) £',4 ]J2 D1 B3 11904(C = 2) B2 3 3 B4 B1 13448(C = 3) B2 :33 B4 B1 13448A'rSP2 B2 B3 B1 B4 13714ATSP3 B2 B1 B3 B4 11794Table 3, Results by ATSP method4.3. Comparison of Various TechniquesSecond, OPM and ATSP method were applied to the layout problems inthe paper written by Nugent, Vollamnn and RUml(2). The evaluated valueof the determined layout and the computation time and the results byNugent etc.(2) were shown in Table 4 and Table 5.It was impossible to obtain the optimal layout by OPM in the case ofthe layout problem having more than twelve departments,as the hugecomputation time was needed. The minimum evaluated value by ATSP1, ATSP2and ATSP3 was shown by the symbol II * II in Table 4.It made clear that the evaluated value became minimum almost inn 5 6 7 8 12 15 20 30H63 27.6 44.2 78.8 114.4 317·4 632.6 1400.4 3267.2HC 29·4 44·2 78.4 110.2 310.2 600.2 1345.0 3206.863-66CRAFT 28.2 44.2 79.6 113.4 296.2 600.6 1339.0 3189.6B.S. 26.8 43.6 74.8 107.0 293.0 480.2 1313.0 3124.0OPM 25 43 74 107A c=o 26* 57* 86* 141 383 776 1483* 3756T C=l 37 57 96 141 384 706* 1602 3698*SP C=2 30 57 97 138 357* 721 1620 37841 C=3 34 57 98 137 374 751 1623 3825ATSP2 30 57 86 141 397 738 1573 3773ATSP3 38 61 103 123* 365 828 1536 3872Table 4, Evaluated values by various allocation techniquesA !location Technique by Statistical Procedure 23useful tomore thanalgorithmTable 5, Computation time of eachallocation technique (sec)HCn H63 63-66 CRAFT B.S. OPM ATSP5 6 10 1 11 27 476 7 9 2 21 40 457 15 12 5 57 146 458 14 14 10 109 1372 4812 55 19 70 658 4615 78 40 160 2192 4720 168 75 528 6915 4930 398 285 3150 44224 56M~~r G E 2 6 5 NEAC2200-500was the most usefulto solve the layoutproblem.Further fromthe results ofNugent etc., itmade clear that the layout by Biased Sampling method ( B.S. in short)was more efficient than that by CRAFT. But the computation time was muchlonger than that by CRAFT. The smallest evaluated values by ATSP wereaveragely larger by 17 per cent than those by OPM in cases of n = 5,6,7and 8. And they were averagely larger by 20 per cent than those by B.S ••Further they were averagely larger by 13 per cent than those by CRAFT.On the other hand, the computation time by ATSP was not influenced bythe number of department till n = 30. And it was almost equal to 50 sec •• But the computation time by CRAFT and B.S. extremly increased with theincrease of the number of departm-mt.From these results, it made clear that ATSP method wasobtain the sub-optimal layout in the layout problem havingtwelve departments. Further the sub-optimal layout by thiscould be obtained easily by using the desk-calculator.cases of C = 0, C =1 and C = 2 ofATSPl except thelayout problem ofeight departments.Therefore it madeclear that ATSPlin cases of C = 0,C = 1 and C = 25. ConclusionFor the purpose of the determination of the efficient layout in thelayout problem, the allocation technique which didn't use the heuristicalgorithm was examined in this paper. The following results wereobtained.1) The relative distance of each location was estimated by the meanvalue or the sum of mean value and standard deviation, or two times ofit, or three times of it calculated from the distance matrix. The rela-tive volume of each department was estimated by the above mentioned24 Y. FUJIWARA, H. OSAKI, and S. KIKUCHIvalues calculated from the volume matrix.The allocation of departments to locations was determined by one toone correspondence between the arranged relative distances and thearranged relative volumes. This allocation technique was called theallocation technique by the statistical procedure ( ATSP in short ).2) The layout determined by using the relative distance and the rela-tive volume estimated from the mean value or the sum of mean value andstandard devia~ion was more useful than any other layouts determined byusing other relative distances and relative volumes in ATSP. And thecomputation time of ATSP was not influenced by the number of department.3) It was possible to determine the optimal layout by the Optimalproducing method to the layout problem haVing less than eight de-partments. And the sub-optimal layout could be determined economicallyby ATSP to the layout problem haVing more than eight departments.References(1) E.S. Buffa: Models for Production and Operations Management,(1963), p.44-51, p.532-539, Wiley & Sons.(2) C.E. Nugent, T.E. Vollmann and J. Ruml : Operations Research,16-1(1963), 150-152, 158-170.(3) E.S. Buffa, G.C. Armour and T.E. Vol1mann : Harvard BusinessReview, ~(1964), 136-158.(4) M.G Kendall and A. Stuart : The Advanced Theory of StatisticsVolume 1, (1958), p.39,179, Charles Griffin.(5) R.W. Miller: Schedule, Cost and Profit Control with PERT,(1963), p.219, Mcgraw-Hill.(6) R.Jr. Reed: Plant Location, Layout and Maintenance, (1970),p. 84-129 , Irwin.

Allocation Technique Using Relative Distance and Relative Volume Estimated by Statistical Procedure

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