machine layout

8/12/2019 machine layout

1/6

Optimal Plant Layout Designbased on MASS Algorithm

Mohammad KhoshnevisanSchool of Accounting and Finance

Griffith [email protected]

Sukanto BhattacharyaSchool of Business / Inform. Tech.

Bond [email protected]

Abstract In this paper we have proposed a semi-heuristic optimization algorithm for designing optimal plant layouts in process-focused manufacturing/service facilities. Our proposed algorithm marries the well-known CRAFT (Computerized Relative Allocation of Facilities Technique) with the Hungarian assignment algorithm. Being a semi-heuristic search, our algo-rithm is likely to be more efficient in terms of com-

puter CPU engagement time as it tends to converge on the global optimum faster than the traditional CRAFT algorithm - a pure heuristic. We also present a nu-merical illustration of our algorithm. We also suggest an extension to the problem under study through the incorporation of principles of neutrosophic statistics.

Keywords: Assignment, entropy, neutrosophic statis-tics, Dezert-Smarandache combination rule.

1 IntroductionThe fundamental integration phase in the design

of productive systems is the layout of productionfacilities. A working denition of layout may be givenas the arrangement of machinery and ow of mate-rials from one facility to another, which minimizesmaterial-handling costs while considering any physicalrestrictions on such arrangement.

Usually this layout design is either on considera-tions of machine-time cost and product availability;thereby making the production system product- focused ; or on considerations of quality and exibility;thereby making the production system process-focused .

It is natural that while product-focused systems arebetter off with a line layout dictated by availabletechnologies and prevailing job designs, process- focused systems, which are more concerned with job organization, opt for a functional layout. Of

course, in reality the actual facility layout often liessomewhere in between a pure line layout and a purefunctional layout format; governed by the specicdemands of a particular production plant. Since ourpresent paper concerns only functional layout designfor process-focused systems, this is the only layoutdesign we will discuss here.

The main goal to keep in mind is to minimize ma-terial handling costs - therefore the departments thatincur the most interdepartmental movement shouldbe located closest to one another. The main type of design layouts is Block diagramming , which refers tothe movement of materials in existing or proposedfacility. This information is usually provided witha from/to chart or load summary chart, which givesthe average number of units loads moved betweendepartments. A load-unit can be a single unit, a palletof material, a bin of material, or a crate of material [4].The next step is to design the layout by calculatingthe composite movements between departments andrank them from most movement to least movement.Composite movement refers to the back-and-forthmovement between each pair of departments. Finally,trial layouts are place on a grid that graphicallyrepresents the relative distances between departments.This grid then becomes the objective of optimizationwhen determining the optimal plant layout. We givea visual representation of the basic operational con-siderations in a process-focused system schematicallyin Figure 1.

In designing the optimal functional layout, the fun-

damental question to be addressed is that of relativelocation of facilities. The locations will depend onthe need for one pair of facilities to be adjacent (orphysically close) to each other relative to the need forall other pairs of facilities to be similarly adjacent (or

1365
http://mail%20to:[email protected]/http://mail%20to:[email protected]/http://mail%20to:[email protected]/http://mail%20to:[email protected]/http://mail%20to:[email protected]/http://mail%20to:[email protected]/


2/6

physically close) to each other. Locations must be al-located based on the relative gains and losses for thealternatives and seek to minimize some indicative mea-sure of the cost of having non-adjacent locations of fa-cilities. Constraints of space prevents us from goinginto the details of the several criteria used to deter-mine the gains or losses from the relative location of facilities and the available sequence analysis techniquesfor addressing the question; for which we refer the in-terested reader to any standard handbook of produc-tion/operations management [5, 11, 12, 13, 14, 15].

Figure 1: Block diagram of a process-focused system

2 The CRAFT algorithmCRAFT (Computerized Relative Allocation of

Facilities Technique) [1] is a computerized heuristicalgorithm that takes in load matrix of interdepartmen-tal ow and transaction costs with a representation of

a block layout as the inputs. The block layout couldeither be an existing layout or; for a new facility, anyarbitrary initial layout. The algorithm then computesthe departmental locations and returns an estimateof the total interaction costs for the initial layout.The governing algorithm is designed to compute theimpact on a cost measure for two-way or three-wayswapping in the location of the facilities. For eachswap, the various interaction costs are computedafresh and the load matrix and the change in cost(increase or decrease) is noted and stored in the RAM.The algorithm proceeds this way through all possiblecombinations of swaps accommodated by the software.

The basic procedure is repeated a number of timesresulting in a more efficient block layout every time tillsuch time when no further cost reduction is possible.The nal block layout is then printed out to serve asthe basis for a detailed layout template of the facilities

at a later stage [1]. Since its formulation, morepowerful versions of CRAFT have been developedbut these too follow the same, basic heuristic routineand therefore tend to be highly CPU-intensive [6, 7, 8].

The basic computational disadvantage of a CRAFT-type technique is that one always has got to start withan arbitrary initial solution. This means that thereis no mathematical certainty of attaining the desiredoptimal solution after a given number of iterations.If the starting solution is quite close to the optimalsolution by chance, then the nal solution is attainedonly after a few iterations. However, as there is noguarantee that the starting solution will be close tothe global optimum, the expected number of iterationsrequired to arrive at the nal solution tend to be quitelarge thereby straining computing resources [2].

In our present paper we propose and illustrate theModied Assignment (MASS) algorithm as an exten-sion to the traditional CRAFT, to enable faster con-vergence to the optimal solution. This we propose todo by marrying CRAFT technique with the Hungar-ian assignment algorithm. As our proposed algorithm

is semi-heuristic, it is likely to be less CPU-intensivethan any traditional, purely heuristic CRAFT-type al-gorithm.

3 The Hungarian algorithmA general assignment problem may be framed as a

special case of the balanced transportation problemwith availability and demand constraints summing upto unity. Mathematically, it has the following generallinear programming form:

Minimize

C ij X ij

Subject to X ij = 1 , for each i, j = 1 , 2, . . . , n

In words, the problem may be stated as assigningeach of n individuals to n jobs so that exactly oneindividual is assigned to each job in such a way asto minimize the total cost. To ensure satisfaction of the basic requirements of the assignment problem, thebasic feasible solutions of the corresponding balancedtransportation problem must be integer valued.However, any such basic feasible solution will contain(2n 1) variables out of which ( n 1) variables will bezero thereby introducing a high level of degeneracy inthe solution making the usual solution technique of a

transportation problem very inefficient [10].

This has resulted in mathematicians devising analternative, more efficient algorithm for solving thisclass of problems, which has come to be commonly

1366


3/6

known as the Hungarian assignment algorithm . Basi-cally, this algorithm draws from the following simpletheorem in linear algebra:

Theorem : If a constant number is added to any rowand/or column of the cost matrix of an assignment-type problem, then the resulting assignment-typeproblem has exactly the same set of optimal solutionsas the original problem and vice versa.

Proof : Let Ai and B j (i, j = 1 , 2, . . . , n ) be addedto the ith row and/or jt h column respectively of the cost matrix. Then the revised cost elements areC ij = C ij + A i + B j . The revised cost of assignmentis given by C ij X ij = (C ij + A i + B j )X ij =C ij X ij + A i X ij + B j X ij . But bythe imposed assignment constraint is X ij = 1 (foreach i, j = 1 , 2, . . . , n ), we have the revised cost as

C ij X ij + A i + B j i.e. the cost differs fromthe original by a constant. As the revised costs differfrom the originals by a constant, which is indepen-dent of the decision variables, an optimal solution toone is also optimal solution to the other and vice versa.

This theorem can be used in two different waysto solve the assignment problem. First, if in anassignment problem, some cost elements are negative,the problem may be converted into an equivalentassignment problem by adding a positive constant toeach of the entries in the cost matrix so that they allbecome non-negative.

Next, the important thing to look for is a feasiblesolution that has zero assignment cost after adding

suitable constants to the rows and columns. Sinceit has been assumed that all entries are now non-negative, this assignment must be the globally optimalone.

Given a zero assignment, a straight line is drawnthrough it (a horizontal line in case of a row and avertical line in case of a column), which prevents anyother assignment in that particular row/column. Thegoverning algorithm then seeks to nd the minimumnumber of such straight lines, which would cover all thezero entries to avoid any redundancy. Let us say that

k such lines are required to cover all the zeroes. Thenthe necessary condition for optimality is that numberof zeroes assigned is equal to k and the sufficient con-dition for optimality is that k is equal to n for an n ncost matrix.

3.1 The MASS algorithm

The basic idea of our proposed algorithm is to de-velop a systematic scheme to arrive at the initial inputblock layout to be fed into the CRAFT program so thatthe program does not have to start off from any initial(and possibly inefficient) solution. Thus, by subject-ing the problem of nding an initial block layout to amathematical scheme, we in effect reduce the purelyheuristic algorithm of CRAFT to a semi-heuristic one.Our proposed MASS algorithm follows the followingsequential steps:

Step 1 : We formulate the load matrix such thateach entry l ij represents the load carried from fa-cility i to facility j .

Step 2 : We insert lij = M , where M is a largepositive number, into all the vacant cells of theload matrix signifying that no inter-facility loadtransportation is required or possible between theith and j th vacant cells.

Step 3 : We solve the problem on the lines of astandard assignment problem using the Hungarianassignment algorithm treating the load matrix asthe cost matrix.

Step 4 : We draft the initial block layout trying tokeep the inter-facility distance dij between the i thand j th assigned facilities to the minimum possiblemagnitude, subject to the available oor area andarchitectural design of the shop oor.

Step 5 : We proceed using the CRAFT programto arrive at the optimal layout by iteratively im-proving upon the starting solution provided bythe Hungarian assignment algorithm till the over-all load function L = l ij d

ij subject to anyparticular bounds imposed on the problem.

The Hungarian assignment algorithm will ensurethat the initial block layout is at least very close to theglobal optimum if not globally optimal itself. Thereforethe subsequent CRAFT procedure will converge on theglobal optimum much faster starting from this near-optimal initial input block layout and will be muchless CPU-intensive that any traditional CRAFT-type

algorithm. Thus MASS is not a stand-alone optimiza-tion tool but rather a rider on the traditional CRAFTthat tries to ensure faster convergence to the optimalblock layout for process-focused systems, by makingthe search semi-heuristic.

1367


4/6


5/6

the above solution is optimal. The optimal assignmenttable (subject to the 2 meters of aisle between adjacentfacilities) is:

F 1 F 2 F 3 F 4 F 5 F 6F 1 - - - - -F 2 - - - - -F 3 - - - - -F 4 - - - - -F 5 - - - - - F 6 - - - - -

Initial layout of facilities as dictated by the Hun-garian assignment algorithm is as follows: The above

Figure 2: Hungarian assignment

layout conforms to the rectangular oor plan of theplant and also places the assigned facilities adjacentto each other with an aisle of 2 meters width betweenthem. Thus F 1 is adjacent to F 2 , F 3 is adjacent to F 4and F 5 is adjacent to F 6 .

Based on the cost information provided in the load-matrix the total cost in terms of load-units for theabove layout can be calculated as follows:

L = 2(20 + 10) + (50 + 30) + (10 + 15)+ (44 25) + (22 40) + (22 15) = 2580

By feeding the above optimal solution into theCRAFT program the nal, the global optimum isfound in a single iteration. The nal, optimal layoutas obtained by CRAFT is shown on gure 3.

Based on the cost information provided in the load-matrix the total cost in terms of load-units for the op-timal layout can be calculated as follows:

L = 2(10 + 20) + (15 + 10) + (5 + 30)+ (22 25) + (44 15) + (22 40) = 2360

Therefore the nal solution is an improvement of just220 load-units over the initial solution! This showsthat this initial solution fed into CRAFT is indeed nearoptimal and can thus ensure a faster convergence.

Figure 3: Optimal CRAFT assignment

6 ConclusionIn this paper we have proposed and developed a

semi-heuristic computational algorithm based on theHungarian assignment technique for the optimal lay-out design of process-focused plants, in an attemptedextension of the well-known CRAFT-type algorithms.We have further proposed a non-deterministic versionof the assignment problem through the incorporationof neutrosophic logic.

We have also provided a numerical illustration of the deterministic (non-neutrosophic) MASS algorithmin the Appendix by designing the optimal block layoutof a small, single-storied, process-focused manufactur-ing plant with six different facilities and a rectangularshop oor design. Our model can however be extendedto cover bigger plants with more number of facilities.Also the MASS approach we have advocated here caneven be extended to deal with the multi-oor version of CRAFT by constructing a separate assignment tablefor each oor subject to any inter-facility predecessor-successor relationships [9]. However the practicabilityof the extended version of the model incorporating neu-

trosophic statistics along our suggested lines is open tofurther analytical exploration.

References[1] Buffa, Elwood S., Armour G. C. and Vollmann

T. E., Allocating Facilities with CRAFT , Har-vard Business Review, Vol. 42, No.2, pp.136-158,March-April 1964.

[2] Carrie A. S., Computer-Aided Layout Planning The Way Ahead , International Journal of Produc-tion Research, Vo. 18, No. 3, pp. 283-294, 1980.

[3] Dezert J., Optimal Bayesian Fusion of Multi-ple Unreliable Classiers , Proceedings of the 4thInternational Conference on Information Fusion(Fusion 2001), Montreal, August 7 10, 2001.

1369


6/6

[4] Driscoll J., Sangi N. A., An International Survey of Computer-aided Facilities Layout The Devel-opment And Application Of Software , PublishedConference Proceedings of the IXth InternationalConference on Production Research, Anil Mital(Ed.), Elsevier Science Publishers B. V., N.Y.U.S.A., pp. 315-336, 1988.

[5] Heizer J., Render B., Production and Operations Management: Strategies and Tactics , Allyn andBacon Inc., Boston, U.S.A., pp. 377-386, 1988.

[6] Hicks P. E., Cowan T. E., CRAFT-M for Lay-out Rearrangement , Industrial Engineering, Vol.8, No. 5, pp. 30-35, 1976.

[7] Hillier F. S., Quantitative Tools for Plant Layout Analysis , Journal of Industrial Engineering, Vol.14, No. 1, pp. 33-40, 1963.

[8] Hillier F. S., Connors M. M., Quadratic Assign-ment Problem Algorithm and the Location of In-visible Facilities , Management Science, Vol. 13,No. 1, pp. 42-57, 1966.

[9] Johnson R. V., SPACECRAFT for Multi-Floor

Layout Planning , Management Science, Vol. 28,No. 4, pp. 407-417, 1982.

[10] Musta C. K., Operations Research: Methods and Practice , New Age International Ltd., New Delhi,India, 3rd Ed., pp124-131, 1996.

[11] Riggs J.L., Production Systems: planning, anal-ysis and control , John Wiley & Sons Inc., N.Y.,U.S.A., 4th Ed., pp. 306-313, 1987.

[12] Schmenner R.W., Production/Operations Man-agement: Concepts and Situations , MacmillanInc., N.Y., U.S.A., 4th Ed., pp. 287-295, 1990.

[13] Sule D. R., Manufacturing Facilities Location,Planning and Design , PWS-KENT Publishingcompany, Boston U.S.A., pp. 337-339, 1988.

[14] Walley B. H., Production Management Handbook ,Gower Publishing Company, Vermont, U.S.A.,2nd Ed., pp. 367-375, 1986.

[15] Wickham S., The Focused Factory , Harvard Busi-ness Review Vol. 52, No. 3, pp. 113-121, May-June1974.

1370

machine layout

Documents