Papers - Seminar (1)

7/23/2019 Papers - Seminar (1)

http://slidepdf.com/reader/full/papers-seminar-1 1/164

ENVIRONMENT ORIENTED INVENTORY MODEL AND BENEFITS OFINCINERTION AS WASTE DISPOSAL METHOD

A.Shanthi

Idhaya College of Arts and Science forWomen, Pondicherry

Abstract

It is well known that industrial sectors are the backbone of any nation’s economy. Themotive of the establishment of any industry is to satisfy the demands of the needyinstantaneously, which explicitly states that it has social responsibilities. To make thefulfillment of the demands in right proportion and at the right time, inventory models( !"# $" % conomic order#production &uantity' were formulated. !n the other hand ina production # inventory situation items which are, received or produced are not of perfect&uality. Thus the presence of defects which is inevitable in a produced#ordered lot issorted out by the process of screening. The imperfect items cannot be discarded as waste,since the industries possess environment responsibilities, remanufacturing tactics andwaste management techni&ues have been implemented. )everal literatures have discussedabout inventory models with production, remanufacturing and waste disposal, butdisposal methods and techni&ues were not discussed in particular. In this paper aninventory model is formulated in which the defective items in a lot are made to undergoseperate screening which in turn are categorised based on the *ife +ycle Assessment(*+A methodology as reusable, recyclable and final residues are sub ected to

incineration one of the waste disposal methods which is both economically andenvironmentally beneficial.

-. Introduction

Inventory models have been a great support for the successful run of the industries whichundergo a number of modifications to act as the remedial measure to the problems thatarise in the production #inventory situations. emands of the customers must be satisfiedinstantaneously for which the production#ordering rate is fostered to avoid shortages. It isvivid that &uantity of items is focussed on, but in addition to it the &uality of the itemsmust also be given significance as customer’s consistency depends on it. The &uality of the produced # received lot is determined by the proportion of defective items present.The process of screening is carried out to isolate the perfect items from the imperfectones. To transform the imperfect items as perfect, remanufacture is undertaken.

/emanufacture accounts to recycle, repair and rework. The items that areimperfect have defects, but the percentage of defects present in each item is distinct. Afew numbers of defect can be mended manually within a short span of time so that they





%75'/osenblatt and *ee in their paper have formulated a model where the defective itemsare reworked%5-'.As the concept of repair of items gained much significance it stimulatedmany researchers to work on it. )chrady %55'developed an !" model withinstantaneous production and repair rates and no disposal. As environmental issuesstarted surfacing and becoming a public concern, /ichter %4;,4>' picked up the problemof )chrady %55' and turned it into a production, remanufacturing and waste disposal problem. The industrial sectors must bear the cost of disposal in addition to the cost of production and remanufacture.The items that cannot be remanufactured are disposed. A number of disposal methods areavailable, but the proper method has to be selected to minimi3e the cost of disposal andthe environmental effects for which *+A is preferred due to its systematic approach andthe frame work of waste hierarchy which is widely accepted by the decision makers inindustries%4'.The international standard I)! -7?7?:75 defines *+A as a compilation andevaluation of the inputs, outputs and the potential environmental impacts of a product

system throughout its life cycle %4'.*ife:cycle assessments were initially developed for the purpose of analysing products, although recently, it has also been applied to thetreatment of waste. *+A studies the environmental aspects and potential impactsthroughout a product’s life. The structure of *+A is discussed briefly in %-4'

)alameh in his paper has formulated a simple inventory model in which thedefective items which are expressed as the percentage (p of lot si3e y with a known probability density function are sold as a single batch at a discounted price after -??@ percent of screening and also the total profit per unit time is determined. In this paper thismodel is modified by the inclusion of screening cost of the defective items, cost of remanufacture and the cost of incineration. Also in addition, it is assumed that defectives

of a known proportion were present in produced # received lot. In this paper an integratedinventory model which includes production, remanufacture and waste disposal method incineration is formulated.

5. Incineration

2aste is considered as fuel %-7' and 2aste to energy has become the ma or goal to beachieved in the implementation of any waste disposal activity. Amidst a number of wastedisposal options incineration is udged to be desirable after the analysis of social cost.

Incineration is the most important way of utili3ing the energy content of the waste whichis the conclusion made by 0ernadette %5' in his paper and also he has explained about theincineration method briefly as follows. Incineration is a controlled process whichinvolves oxidative conversion of combustible solid material to harmless gases suitable for atmospheric release. It converts a waste to a less bulky, less toxic, or less noxiousmaterial. It also reduces sludge to an ash residue which can be more easily handled for ultimate disposal. Incineration has its own environmental effects which have also been



explained by 0ernadette, but these effects could be minimi3ed by proper installation of mitigating techni&ues to treat the emissions. The cost of setting an incineration plant andits associated cost parameters such as operating, maintenance, residue disposal,transportation waste haulage and fuel surcharge are included, and the revenue earned outof it is briefed in the literature. After the analysis of the various disposal methodsincineration rank first, because even though the incineration operational tasks areexpensiveB it yields revenue in terms of energy and materials. This advantage of incineration has encouraged many industrial sectors to construct small incinerators withintheir firms for waste management to promote environment sustainability.

7. 1odel Cormulation7.- $roblem escription

+onsider the case where a lot of si3e D is delivered instantaneously with a purchasing price of c per unit and an ordering cost of E. It is assumed that each lotreceived contains a known percentage of defectives p, after -??@ percent of screeningimperfect items is separated from the good items. To make profit out of the defectiveitems, they are made to undergo second screening i.e the *+A methodology is used torank the defective items as reusable, reworkable and the residues to be incinerated basedon their proportion which is expressed as the percent of imperfect items. It is classified based on the degree and nature of defects present in each imperfect item.7.4 Fotations

The following notations will be used throughout the paper when developing themathematical model.

D order si3e

c unit variable cost

E fixed cost of placing an order

p percentage of defective items in D

& percentage of defective items those are suitable for reuse.

r percentage of defective items those are suitable for recycle.

i percentage of defective items that are sub ected to incineration.(-:&:r

s unit selling price of items of good &uality.

v unit selling price of defective items that are categori3ed as reusable.

(v G s



/ unit cost of recycling

/ - unit selling price of recycled items.(/ - H /

I unit cost of disposal by the method of incineration

I- revenue earned per unit.

x screening rate

d unit screening cost

d- unit screening cost of the defective items.

mission caused by incineration mitigating cost

T cycle length.

5.5 Assumptions

-. The demand is constant.

4. Fo shortages are allowed.

5. The items to be reused, reworked and incinerated are expressed as the percentageof imperfect items.

7. The cost of mending the items that are categori3ed as reusable is neglible.

8. *ead time is assumed to be <ero.. $lanning hori3on is infinite.

7.7 1athematical 1odel.

As in )alameh %5?' model to avoid shortages it is assumed thatt J F(D,p , where F(D,p is the number of good items in each order, also in

addition to that p, the percentage of defective items in y is restricted to-: i.e p J -: .

*et T/(D and T+(D be the total revenue and the total cost per cycle,respectively. T/(D is the sum total sales volume of good &uality, reusable items,reworked items and the revenue from the incineration of non: repairable units.T/(D K sD(-:p L v&pD L / -rpD L I-ipD



T+(D is the sum of procurement cost per cycle, screening cost per cycle, holdingcost per cycle,)creening cost of defective items per cycle, /emanufacturing cost per cycle,cost of incineration per cycle.T+(D K ELcDLdDLd- pDL LhM

The total profit per cycle is the total revenue per cycle less the total cost per cycle,T$(D K T/(D T+(D , and it is given asT$(D K sD(-:p L v&pD L / -rpD L I-ipD N ELcDLdDLd- pDL LhM

The total profit per unit of time is given by dividing the total profit per cycle bythe cycle length, T$O(D K B and can be written as

T$O(D K M (s v&: L

L M

(-

M (s v&:

L L

M



T$OP(D K

(4

The ob ective is to determine the optimal &uantity. The necessary condition is

(5

Fote that when p K ?, &.(5 reduces to the traditional !" model.

The reason why vanishes is, if there is no defective items present in

the produced lot then there is no need of the waste disposal method incineration so thecosts associated with emission mitigation is excluded.

8. Fumerical results.

To illustrate the usefulness of the model developed in )ection 7, let us consider theinventory situation where a stock is replenished instantly with D units of which not all areof the desired &uality.The parameters needed for analy3ing the above inventory situation are given belowQ K 8?,??? unit # yearB c K R 48 # unitB E K R-?? # cycleB s K R 8?#unitB vKR7?#unitB h#unit # yearB /KR8#unitB K R 8?? # cycle / - K R -8 # unit B I K R 4 # unit B I- K R 8 # unit B xK - # unit # time B d K R?.8 #unitB d- K R?.8 # unitB p K?.?8B & K ?.8B r K ?.5, i K ?.4.Assume that the inventory operation operates on an > hours#day, for 5 8 days a year, thenthe annual screening rate, xK-M ?M>M5 8K-;8 4?? units#year.

Then the optimum value of D that maximi3es &.(- is given from &.(4

K 5-8>5 units.



)ubstituting units in &.(4 the maximum profit per

unit year is given asT$O(D K -49?54-.9#year.

. +onclusionThis paper presents an environmental oriented inventory model which discusses

about the benefits that are gained out of using incineration, the waste disposal method.This paper also presents a numerical example to validate the above model also it discussabout the waste disposal method in particular. This model also assists the decision makersin industries to expel the waste more economically and also enables the industries infulfilling its environmental responsibilities.

/eferences.

%-' A +!1. (4??9 . 1anagement of 1unicipal )olid 2aste in 1etro Sancouver A +omparative Analysis of !ptions for 1anagement of 2aste after /ecycling. httpQ##www.metrovancouver.org#services#solidwaste#planning#Thenextsteps# ) 5 A +!1 CO** / $!/T.pdf.

%4'Arena, O., 1astellone, 1.*., $erugini, C., 4??5. The environmental performance of alternative solid waste management optionsQ a life cycle assessment study. +hemical ngineering Uournal 9 , 4?; 444.

%5' 0ernadette and Duri.,The environment comparison of landfilling vs.

Incineration of 1)2 accounting for waste diversion., 2aste 1anagement

(4?--

%7' ):). +hung, +:). $oon, Accounting for the shortage of solid waste disposalfacilities in )outhern +hina, nviron. +onserv. 4> (4 (4??- 99 -?5.

%8' I. obos, E. /ichter, A production#recycling model with stationary demand and return rates, +entral ur. U. !per. /es. -- (- (4??5 58 7 .

% ' I. obos, E. /ichter, An extended production#recycling model with stationary demand and return rates, Int. U. $rod. con. 9? (5 (4??7 5--

545.%;' I. obos, E. /ichter, A production#recycling model with &uality consideration, Int. U. $rod. con. -?7 (4 (4?? 8;- 8;9.%>' A.1.A. l )aadany, 1.D. Uaber, A production, repair and waste disposal inventory model when returns are sub ect to &uality and price



considerations,+omput. Ind. ng. 8> (5 (4?-? 584 5 4.%9' A.1.A. l )aadany, 1.D. Uaber, The !" repair and waste disposal model

with switching, +omput. Ind. ng. 88 (- (4??> 4-9 455.%-?' A.1.A. l )aadany, 1.D. Uaber, $roduction#remanufacture !" model

with returns of subassemblies managed differently, Int. U. $rod. con. -55(- (4?-- --9 -4 .

%--' A.1.A. l )aadany, 1.D. Uaber, 1. 0onney, 6ow many times to

remanufactureV Int. U. $rod. con., doiQ-?.-?- # .i pe.4?--.--.?-;.

%-4'=oran.et.al.,*ife cycle assessment of energy from solid waste: part I Q

general methodology and results.,Uournal of +leaner $roduction -5 (4??8

4-5: 449.

%-5'Cinnveden, =., kvall, T., -99>. *ife cycle assessment as a decision support toolWthe case of recycling vs. incineration of paper./esource,+onservationand /ecycling 47, 458 48 .

%-7'Cinnveden, =., Uohansson, U., *ind, $., 1oberg, A X ., 4???. *ife +ycleAssessments of nergy from )olid 2aste. )tockholms Oniversitet, )weden.

%-8' 1. Cleischmann, U.1. 0loemhof:/waard, /. ekker, . vander *aan, U.A. . . van Funen, *.F. van 2assenhove, "uantitative models for reverse

logistics, ur. U. !per. /es. -?5 (- (-99; - -;.

%- 'Cri riksson, =. 0., Uohnsen, T., 0 arnasoY ttir, 6. U., )lentnes, 6., 4??4. =uidelines for the use of *+A in the waste management sector. Fordtest

$ro ect no. -85; ?-.

%-;'6arris C.2., -9-5.6ow many parts to make at once, factoryV,1ag 1anage.-?, -58 -5 (p. -84 .

%->'6arrison, E.2., umas, /. ., 0arla3, 1.A., 4???. *ife:cycle inventory model of municipal solid waste combustion. Uournal of the Air and 2aste 1anagement Association 8?, 995 -??5.

%-9' 1.D. Uaber, 1.A. /osen, The economic order &uantity repair and waste

disposal model with entropy cost, ur. U. !per. /es. ->> (- (4??> -?9

-4?.



%4?' 1.D. Uaber, A.1.A. l )aadany, The production, remanufacture and waste

disposal with lost sale, Int. U. $rod. con. -4? (- (4??9 --8 -47.

%4-' I. Eonstantaras, ). $apachristos, *ot:si3ing for a single:product recovery

system with backordering, Int. U. $rod. /es. 77 (-? (4?? 4?5- 4?78.

%44' I. Eonstantaras, E. )kouri, 1.D. Uaber, *ot si3ing for a recoverable productwith inspection and sorting, +omput. Ind. ng. 8> (5 (4?-? 784 7 4.

.%45' 1.+. 1abini, *.1. $intelon, *.C. =elders, !" type formulations for

controlling repairable inventories, Int. U. $rod. con. 4> (- (-994 4- 55.

%47'1iranda, 1.*., 6ale, 0., -99;. 2aste not, want notQ the private and social costs ofwaste:to:energy production. nergy $olicy 48,8>; ??.

%48'1orte3a /asti 0ar3oki, 1orte3a Uahanba3i, 1ehdi 0i ari., ffects of imperfect products on lot si3ing with work in process inventory., Applied 1athematics and+omputation 4-; (4?-- >54> >55 .

%4 ' F. Fahmias, 6. /ivera, A deterministic model for a repairable iteminventory system with a finite repair rate, Int. U. $rod. /es. -; (5 (-9;94-8 44-.

%4;' E. /ichter, The extended !" repair and waste disposal model, Int. U.$rod. con. 78 (- 5 (-99 775 77;. %4>' E. /ichter, $ure and mixed strategies for the !" repair and waste

disposal problem, !/ )pectrum -9 (4 (-99; -45 -49.%49' .). /ogers, /.). Tibben:*embke, =oing 0ackwardsQ /everse *ogistics

Trends and $ractices, /everse *ogistics xecutive +ouncil $ress,$ittsburgh, $A,-999.

%5?' A. /oy, E. 1aity, ). Ear, 1. 1aiti, A production inventory model withremanufacturing for defective and usable items in fu33y:environment,

+omput.Ind. ng. 8 (- (4??9 >; 9 .%5-'1.E )alameh and 1.D.Uaber., conomic production &uantity model for

items with imperfect &uality.,International Uournal of $roduction conomics

7 (4??? 89: 7



%54' *.A. )an:Uose, U. )icilia, U. =arcia:*aguna, An economic lot:si3e modelwith partial backlogging hinging on waiting time and shortage period,Appl. 1ath. % ' /.6. Teunter, *ot:si3ing for inventory systems with product recovery, +omput. Ind. ng. 7 (5 (4??7 75- 77-.

%55' .A. )chrady, A deterministic inventory model for repairable items, Faval/es. *og. "uart. -7 (5 (-9 ; 59- 59>.

%57'T I, 4??5. Cinal /eport for the $ro ect on *ife +ycle Assessment for Asian+ountriesW$hase III. Thailand nvironmental Institute, Thailand.

%58' /.6. Teunter, conomic order &uantities for stochastic discounted costinventory systems with remanufacturing, Int. U. *og. 8 (4 (4??4 - -

-;8.1odel. 5- (- (4??; 4-79 4-89.

%5 'OF $, 4??5. nvironmental management toolsQ life cycleassessment,available fromQ #httpQ##www.uneptie.org#pc#pc#tools#lca.htm).

%5;' O) $A. (4??8 . Cirst:!rder Einetic =as =eneration 1odel $arameters for2et*andfills. +ontract $:+:?7:?45, O) nvironmental $rotectionAgency.2ashington, +.

%5>'O) $A. (4?? . )olid 2aste 1anagement and =reenhouse =ases. A *ife:

cycle Assessment of missions and )inks, 5rd ed. O) nvironmental$rotection Agency. 2ashington, +.

%59'O) $A. (-99 . A$ 74, Cifth dition, Solume I. +hapter 4Q )olid 2asteisposalQ/efusecombustion.GhttpQ##www.epa.gov#ttn#chief#ap74#ch?4#final#

4s?-.pdfH.

%7?'O) $A. (4??> . 0ackground Information ocument for Opdating A$74)ection 4.7 for stimating missions from 1unicipal )olid 2aste *andfills.GhttpQ##www.epa.gov#ttn#chief#ap74#ch?4#draft#db?4s?7.pdfH.

%7-'2anichpongpan, 2., =heewala, ).6., 4??;. *ife cycle assessment as adecision support tool for landfill gas:to energy pro ects. Uournal of +leaner$roduction -8,->-9 ->4 .

%74' . 2aters, Inventory +ontrol and 1anagement, 2iley, Few Dork, 4??5.



%75' <hang, D. =erchak, Uoint lot si3ing and inspection policy in an !" model withrandom yield, II Transactions 44(- (-99? 7-.

ENVIRONMENTALLY RESPONSIBLE REPAIR AND WASTE

DISPOSAL INVENTORY MODELS

K.Elavarasan

SCMS School of engineering and technology, Kerala -683 58

Abstract : Traditional inventory models involve different decisions that attempt to

optimi3e material lot si3es by minimi3ing total annual switching costs. 6owever, the

increasing concern on environmental issues stresses the need to treat inventory

management decisions as a whole, by integrating economic and environmental

ob ectives. /ecent studies have underlined the need to incorporate additional criteria intraditional inventory models in order to design Zresponsible inventory systemsZ. This

paper explores the problem of determining the optimal batch si3es for production and

recovery in an !" (economic order#production &uantity repair and waste disposal

model context. This paper assumes that a first shop is manufacturing new products as

well as repairing products used by a second shop. The used products can either be stored

in the second shop and then be brought back to the first shop in an approach used to

reduce inventory costs, or be disposed outside the system. The works available in the

literature assumed a general time interval and ignored the very first time interval where

no repair runs are performed. This assumption resulted in an over estimation of the

average inventory level and subse&uently the holding cost. These works also have not

accounted for switching costs when alternating between production and recovery runs,

which are common when switching among products or obs in a manufacturing facility.



This paper addresses these two limitations. 1athematical models are developed with

numerical examples presented and results discussed.

Keyw r!s : !" model, $roduction#recovery, /euse, 2aste disposal. )witching cost.

I" INTROD#CTION :

In competitive markets products are pushed faster and faster to customers along

supply chains resulting in faster generation of waste and depletion of natural resources.

These environmental issues that many governments around the world are confronted with

led their legislators to devise laws that re&uire manufacturers to initiate product recovery

programs that collect used items of products from their customers once these products

reach their economic or useful lives. This gave rise to reverse logistics (/* as a business

term (e.g., %l,4' . +arter and llarm (-99> have collected a number of definitions of

reverse logistics. I will cite one of the definitions. The more general definition is

[/everse logistic is such activity which helps to continue an environmental effective

policy of firms with reuse of necessary materials, remanufacturing, and with reduction of

amount of necessary materials.Z This efficiency touches the personal in productionsupply and consumption process. +arter and llarm (-99> approach reverse logistics

from point of view of environmental protection. nvironmental consciousness occurs at

three level of activity of firms Q governmental regulation, social pressure and voluntary

self restriction. /everse *ogistic xecutive +ouncil (/* + has given a more general

definition of reverse logistics which summari3es the above definitions Q /everse logistics

is a movement of materials from a typical final consumption in an opposite direction in

order to regain value or to dispose of wastes. This reverse activity includes take back of damaged products, renewal and enlargement of inventories through product take back

remanufacturing of packaging materials, reuse of containers, and renovation of products,

and handling of obsolete appliances. /eduction of use of raw materials, decrease of costs

of waste disposal, and value added through reuse.



/everse logistics is an extension of logistic, which deals with handling and reuse

of reusable used products withdrawn from production and consumption process. )uch a

reuse is eg recycling or repair of spare parts. An environmental conscious materialsmanagement and # or logistics can be achieved with reuse. It has an advantage from

economic point of view, as reduction of environmental load through return of used items

in the manufacturing process, but the exploitation of natural resources can be decreased

with this reuse that saves the resources from extreme consumption for the future

generation.

/everse logistics (/* extends the traditional forward flow of raw materials,

components, finished products to account for activities such as reusing, recycling,

refurbishing or recovering of these products, components and raw materials. /ogers and

Tibben:*embke (4??- defined /* as the process of planning, implementing, and

controlling the efficient and cost effective flow of raw materials, in:process inventory,

finished goods, and related information from the point of consumption to the point of

origin for the purpose of recapturing value or proper disposal. Cleischmann et al. (-99;

provided a survey addressing the logistics of industrial reuse of products and materialsfrom an !perational /esearch perspective. They subdivided /* into three main areas,

namely distribution planning, inventory control, and production planning. Inventory

management of returned items (including repaired#recovered items will be the focus of

this paper. The importance of the repairable#recoverable inventory problem was

recogni3ed back in the -9 ?s. )chrady (-9 ; and )herbrooke (-9 > reported that

repaired#recovered items accounted for more than 8?@ of the dollars invested in

inventory. )chrady (-9 ; was the first to address the inventory problem for repairable(recovered items. 6e developed an !" model for repaired items which assumes that

the manufacturing and recovery rates are instantaneous with no disposal cost. Fahmias

and /ivera (-9;9 extended



)chrady\s model to allow for a finite repair rate with the assumption of limited

storage in the repair and production shops.

Along the same line of research, /ichter (-99 a, -99 b assumed the production

and repair system to consist of two shops. The first shop manufactures new items of a

product and repairs used items of the same product collected by the second shop. The

used items are collected at a repair rateβ, ? ≤ β G -. /ichter (-99 a, -99 b also assumed

that not all the collected items are repairable, and therefore some are disposed as waste

outside the system at a rateα K - : β, ? ≤ β G -, which may display the ecological

behavior of the system. This assumption is different from that adopted in )chrady (-9 ;

who assumed a continuous flow of used items to the manufacturer. /ichter extended his

work cited above in several directions. 6e has done so either individually or in

collaboration with obos. /ichter (-99; investigated the cost analysis developed in

/ichter (-99 a, -99 b for extreme waste disposal rates. 6e showed that the pure (bang:

bang policy of either no waste disposal (total repair or no repair (total waste disposal

dominates a mixed waste disposal and repair strategy. /ichter and obos (-999 further

examined the bang:bang policy where they showed that the properties of the minimal

cost function and the optimal solution known for the continuous !" repair and wastedisposal problem (/ichter, -99 a, -99 b, -99; could be extended to the more realistic

integer problem. The characteristics of the cost function developed in /ichter and obos

(-999 were further studied in obos and /ichter (4??? . They showed that the minimum

cost function is partly piecewise convex and partly piecewise concave function of the

waste disposal rate. In a follow:up paper, obos and /ichter (4??5 extended their earlier

work ( obos ] /ichter, 4??? assuming finite repair and production rates with a single

repair cycle and a single production cycle per time interval. In a subse&uent paper, obos

and /ichter (4??7 generali3ed the work they presented in obos and /ichter (4??5 by

assuming that a time interval to consist of multiple repair and production cycles. )teering

their earlier work into a new direction, obos and /ichter (4?? investigated the

production:recycling model in obos and /ichter (4??7 for &uality considerations.

In this work, they assumed that the &uality of used collected (returned items is not



always suitable for further recycling, i.e., not all used items can be reused. obos and

/ichter (4?? showed that when considering &uality, a mixed strategy is more

economical than the pure strategies of their earlier work. Apart from the above surveyed

works, other researchers have developed models along the same lines as )chrady and

/ichter and obos, but with different assumptions. )ome of the recent works, but not

limited to, are those of Teunter (4??-, 4??7 , Inderfurth, *indner, and /achaniotis (4??8 ,

Eonstantaras and $apachristos (4?? , and Uaber and /osen (4??> . The work of /ichter

(-99 a, -99 b has limitations. Two of these limitations are addressed in this paper.

Cirst, this paper modifies the method that /ichter (-99 a, -99 b adopted for

calculating the holding costs. /ichter assumed that collected items are transferred fromshop I to shop 4 in m batches to be repaired at the start of each time interval. /ichter

considered a general time interval ignoring the effect of the first time interval. The first

time interval has no repair batches, as there is nothing manufactured before that is to

be collected and repaired in this interval. This assumption results in a residual inventory

and thus overestimates the holding cost. )econd, this paper accounts for switching costs

(e.g., production loss, deterioration in &uality, additional labor when alternating between

production and recovery runs. 2hen shifting from producing (performing one product

( ob to another in the same facility, the facility may incur additional costs referred to as

switching costs (e.g., =la3ebrook, -9>?B $aul, *aw, ] *eong, -9>?B Eella, -99-B Dan ]

<hang, -99;B Teunter ] Clapper, 4??5B 6a i, =haribi, ] Eenne, 4??7B )ong, 2ang, ]

*i, 4??7B Ehou a ] 1ehre3, 4??8 .

$" NOTATION AND ASS#MPTIONS :

In this section, the present study develops !" repair and waste disposal

inventory model. The following notations and assumptions are used throughout this

paper.



N tat% & :

n Q number of newly manufactured batcher in are interval of length T.

m Q number of repaired batches in an interval of length T

d Q demand rate (unit per unit of time

h Q holding cost per unit per unit of time for shop -.

u Q holding cost per unit per unit of time for shop 4.

α Q waste disposal rate, where ? Gα G l

β Q repair rate of used items, whereαLβ K I and ? Gβ G -.

" Q batch si3e for interval T which includes n newly manufactured and m

repaired batches.

r Q repair setup cost per batch

s Q manufacturing set up cost per batch

a Q fixed cost per trip(monetary unit b Q variable cost per unit transported per distance travelled

Q distance travelled

µ Q proportion of demand returned (? Gµ G -

θ Q social cost from vehicle emission

v Q average velocity (km#h

γ Q cost to dispose waste to the environment(mu#unit

δ Q proportion of waste produced per lot ".

Ass'()t% &s :



-. Infinite manufacturing and recovery rates.

4. /epaired items are as good as new.

5. emand is known, constant and independent.

7. *ead time is 3ero.

8. )ingle product case.

. Fo shortage.

;. Infinite planning hori3on.

>. Onlimited storage capacity is available.

*" MODEL FORM#LATION

/itcher (-99 a, -99 b introduced repair and waste disposal model in which

demand is satisfied by manufacturing [new^ and repairing [used^ items of a certain

product. There are n batches of newly manufactured items and m batches of repaireditems in some collection time interval T. /itcher assumed instantaneous manufacturing

and repair rates and a considered repaired items to be as good as new. As a /itcher

(-99 a, -99 b , this paper assumes demand is supplied by n newly manufactured batches

and m repaired batches in some collection time interval of length T. /itchers work, as

well as the other studies in the literature did not account for T, when there are no items to

be repaired, perhaps because all these studies assumed an infinite planning hori3on.

1odelling the total cost expression for shops - and 4 and considering the cost to dispose

waste to the environment, the optimal total cost consists of the following elements.

)etup cost for shop - and shop 4 is



/epair setup cost per batch K r. Therefore setup cost per m repair batches is mr.

1anufacturing setup cost per batch K s, setup cost for n manufacturing batches is ns.

Total setup cost K mr L ns

6olding cost for the first shop is K

h_" _T h `" `T . . n L . . m

4 n n 4 m m

÷ ÷

K

h_" _T h `" `T . . n L . . m

4 n n 4 m m

÷ ÷

K

4 4 4 4h_ " ` " " L since T K4 dn dm d

6olding cost for the second shop is K

"`" - `" `"u : (m - m

d 4 4 m dm −

K

4 44u` "

`" : (m -4d 4

−

K

4u`" `- : (m -

4d m −

Transportation cost per cycle is

+ t(" K 4a L b " L b µ"

mission cost from transportation per cycle is

e+ (" K 4S

The number 4 refers to a roundtrip waste produced by the inventory system per cycle is



+w(" K γ ? L γ "( δ L µ

whereγ ? is the fixed cost per waste disposal activity.

The total cost E(" K setup cost L holding cost L transportation cost L emission cost

L waste disposal cost.

K

( )4 4 4 4 4h_ " ` " u`" `

mr ns L L L - (m -4 dn dm 4d m

+ − − ÷

L %4a L b " L b µ" L

4S

L γ ? L γ "( δ L µ '

The total cost per unit of time is given by dividing the above expression by T.

E(" K

( )4 4 4 4 4- h_ " ` " u`" `

mr ns L L L - (m -T 4 dn dm 4d m

+ − − ÷

L?4a b " b " 4 "(

S+ + + + + +

where T K

"d

E(" K

( )4 4 4 4 4d d h_ " ` " u`" d `

mr ns L . L L - (m -" " 4d dn dm 4d " m

+ − − ÷

?4ad d d d d

L b " L b " 4 (" " " S " "

+ + + +



E(" K

( )4 4d "_ ` u` ` 4ad

mr ns L h L L -`" 4 n m 4 m "

+ − + + ÷ ÷

?d dL bd L b 4 ( dS " "+ + + +

. . . (-

Fow diff (- , w.r.t ",

(ie

4

4

d E d"

for every " H ? and it has a uni&ue minimum and derived by setting its first

derivative e&ual to 3ero (ie

dE K ?.

d"

dE d"

K

( )4 4

4 4

d h_ ` u` ` 4admr ns L L L -` ? ?

" 4 n m 4 m " − + − + − + + ÷ ÷

?4 4

4 d d: ? ?S" "− + =

K

( )4 4

?4 4

- 4 d - _ ` `d mr ns L 4ad L d L h L u` - ` K ?

" S" 4 n m m

− + − + − + ÷ ÷

. . . (4

( )4 4

?4- 4 d - _ ` `d mr ns L 4ad L d K h L u` - `

" S 4 n m m + + + − + ÷ ÷

( )4 4

4?

4 d _ ` `4 d mr ns L 4ad L d K " h L u` - `

S n m m

+ + + − + ÷ ÷



( ) ?4

4 4

7 d4d mr ns L 7ad L 4 d

S" K_ ` `

h L u` - `n m m

+ + + − + ÷ ÷

( ) ?

4 4


S" K_ ` `

h L u` - `n m m

+ + + − + ÷ ÷

. . . (5

Fow diff (4 w.r.t. ",

4

4d E d"

K

( ) ?5 5 5 54d 7ad 7 d dmr ns L L 4" " S" "

+ +

K

( ) ?5

- 7 d4d mr ns L7ad L 4 d

" S + +

. . . (7

)ubs the value of " in (5 , we get

4

4

d E d"

K

( )

( )

?

5# 4

?

4 4


S H ?


S_ ` `

h L u` - `n m m

+ +

+ + + − + ÷ ÷

The optimal value of "? is

" ? K

( ) ?

4 4


S_ ` `

h L u - `n m m

+ + + − + ÷ ÷



)ubs the value of " ? in (-

E(" K

( )? 4 4

? ?

d "_ ` u` ` 4admr ns L h L L -`

4 n m 4 m" "

+ − + + ÷ ÷

?? ?

d dL bd L b 4 ( d

S " "+ + + +

K

( )? 4 4

??

- d "_ ` u` `d mrLns L4adL4 d L h L - `L Lbd Lbd

" S 4 n m 4 m

+ − ÷ ÷

E(" K l - L l4 L l5 . . . (8

where l- K

( ) ??

- dd mr L ns L 4ad L 4 d

" S +

l4 K

? 4 4"_ ` `h L L u` - `

4 n m m

− + ÷ ÷

l5 K bd L bdµ

l- K

( )

( )

?

?

4 4

dd mr L ns L 4ad L 4 d

S7 d

4d mr ns L 7ad L 4 dS

_ ` `h L u` - `n m m

+

+ +

+ − + ÷ ÷







0rahimi, F., au3ere:$eres, )., Fa id, F. 1., ] Fordii, A. (4?? . )ingle item lot si3ing

problems. uropean Uournal of !perational /esearch, - >(- , -:- .

de 1atta, /., ] 1iller, T. (4??7 . $roduction and inter:facility transportation

scheduling for a process industry. uropean Uournal of !perational /esearch, -8>(- , ;4:>>.

obos, I., ] /ichter, E. (4??? . The integer !" repair and waste disposal model :

Curther analysis. +entral uropean Uournal of !perations /esearch, >(4 , -;5:-97.

obos, I., ] /ichter, E. (4??5 . A production#recycling model with stationary demand

and return rates. +entral uropean Uournal of !perations /esearch, --(- , 58 .

obos, I., ] /ichter, E. (4??7 . An extended production#recycling model with stationarydemand and return rates. International Uournal of $roduction conomics, 9?(5 , 5--:545.

obos, I., ] /ichter, E. (4?? . A production#recycling model with &uality consideration.

International Uournal of $roduction conomics, -?7(4 , 8;-:8;9.

Cleischmann, 1., 0loemhof:/uwaard, U. 1., ekker, /., San er *aan, ., San Funen,

U. A. . ., ] San 2assenhove, *. F. (-99; . "uantitative models for reverse logisticsQ A

review. uropean Uournal of !perational /esearch, -?5(- , -:-;.

=ascon, A., ] *eachman, /. +. (-9>> . A dynamic programming solution to the

dynamic, multi:item, single:machine scheduling problem. !perations /esearch, 5 (- ,

8?:8 .

=alvin, T. 1. (-9>; . conomic lot scheduling problem with se&uence dependent setup

costs. $roduction and Inventory 1anagement Uournal, 4>(- , 9 :-?8.

=la3ebrook, E. . (-9>? . !n stochastic scheduling with precedence relations and

switching costs. Uournal of Applied $robability, -;(7 , -?- :-?47.

6a i, A., =haribi, A., ] Eenne, U. $. (4??7 . $roduction and set:up control of a failure:

prone manufacturing system. International Uournal of $roduction /esearch, 74( , --?;:

--5?.



6all, /. 2. (-9>> . =raphical techni&ues for planning changes in production capacity.

International Uournal of $roduction /esearch, 4 (7 , ;8: >9.

6an, D.6., <hou, + 0ras, 0., 1c=innis, *., +armichael, +., ] Fewcomb, $. (4??5 .

$aint line color change reduction in automobile assembly through simulation. In$roceedings of the 4??5 winter simulation conference, ecember ;:-?, Sol. 4, pp. -4?7:

-4?9.

6ur, )., Eim, U., ] Eang, +. (4??5 . An analysis of the 1#=#I system with F and T

policy. Applied 1athematical 1odelling, 4;(> , 8: ;8.

Inderfurth, E., *indner, =., ] /achaniotis, F. $. (4??8 . *ot si3ing in a production

system with rework and product deterioration. International Uournal of $roduction

/esearch, 75(; , -588:-5;7.

Inman, /. /. (-999 . Are you implementing a pull system by putting the cart before the

horseV $roduction and Inventory 1anagement Uournal, 7?(4 , ;:;-.

Uaber, 1. D., ] /osen, 1. A. (4??> . The economic order &uantity repair and waste

disposal model with entropy cost. uropean Uournal of !perational /esearch, ->>(- ,

-?9:-4?.

Eamrad, 0., ] rnst, /. (4??- . An economic model for evaluating mining and

manufacturing ventures with output yield uncertainty. !perations /esearch, 79(8 , 9?:

99.

Eella, !. (-99- . !ptimal manufacturing of a two product mix. !perations /esearch,

59(5 , 79 :8?-.

Ehou a, 1. (4??8 . The use of minor setups within production cycles to improve product

&uality and yield. International Transactions in !perations /esearch, -4(7 , 7?5:7- .

Ehou a, 1., ] 1ehre3, A. (4??8 . A production model for a flexible production system

and products with short selling season. Uournal of Applied 1athematics and ecision

)ciences, 4??8(7 , 4-5:445.



Eim, U. 6., ] 6an, +. (4??- . )hort:term multiperiod optimal planning of utility systems

using heuristics and dynamic programming. Industrial ] ngineering +hemistry

/esearch, 7?(> , -94>:-95>.

Eim, ., ] San !yen, 1. $. (-99> . ynamic scheduling to minimi3e lost sales sub ect toset:up costs. "ueuing )ystems, 49(4:7 , -95:449.

Eonstantaras, I., ] $apachristos, ). (4?? . *ot:si3ing for a single:product recovery

system with backordering. International Uournal of $roduction /esearch, 77(-? , 4?5-:

4?78.

Eoulamas, +. (-995 . !peration se&uencing and machining economics. International

Uournal of $roduction /esearch, 5-(7 , 98;:9;8.

Eumar, /. *. (-998 . An options view of investments in expansion:flexible

manufacturing systems. International Uournal of $roduction conomics, 5>(4:5 , 4>-:

49-.

*ahmar, 1., rgan, 6., ] 0en aafar, ). (4??5 . /ese&uencing and feature assignment on

an automated assembly line. I Transactions on /obotics and Automation, -9(- , >9:

-?4.

*arsen, +. (4??8 . The economic production lot si3e model extended to include more

than one production rate. International Transactions in !perational /esearch, -4(5 , 559:

585.

*iaee, 1. 1., ] mmons, 6. (-99; . )cheduling families of obs with setup times.

International Uournal of $roduction conomics, 8-(5 ,

Teunter, /. 6., ] Clapper, ). . $. (4??5 . *ot:si3ing for a single:stage single:product

production system with rework of perishable production defectives. !/ )pectrum, 48(- ,>8:9 .

Tsubone, 6., ] 6orikawa, 1. (-999 . A comparison between machine flexibility and

routing flexibility. International Uournal of Clexible 1anufacturing )ystems, --(- , >5:

-?-.



2atkins, 6. /. 2. (-98; . The cost of re ecting optimum production runs. !perations

/esearch, >(7 , 4??:4?8.

2eber, A. (4?? . )ome change is goodB too much is bad. Assembly, 79(4 , 8>: 7.

2olsey, *. (-99; . 1I$ modelling of changeovers in production planning and scheduling

problems. uropean Uournal of !perational /esearch, 99(- , -87:- 8.

2u, /., )ong, +., ] *i, $. (4??7 . /eceding hori3on control of production systems with

aging of e&uipment. In Cifth world congress on intelligent control and automation

(2+I+A 4??7 , Uune -8:-9, Sol. 5, pp. 4; 5:4; ;.

Dan, 6., ] <hang, ". (-99; . A numerical method in optimal production and setup

scheduling of stochastic manufacturing systems. I Transactions on Automatic+ontrol, 74(-? , -784:-788.s

An analysis of Help-Seeking Behaviour amongWomen with Urinary Incontinence using u!!y

"ognitive #ap #o$el!r."ose#h,

Christ $niversity,%angalore

Abstract . Cemale urinary incontinence is a ma or health problem and has a significantdetrimental impact &uality of life. Ceelings of low self:esteem, embarrassment and helplessnessare common. 0etween 4?:8?@ of women suffer from incontinence at some time in their lives andthe prevalence of symptoms and demand for services is rising substantially. The present studywas designed with the ob ectives to study the factors of not seeking a help of a doctor forurinary incontinence. 6ence, this research investigates the most contributing # impactful factor of not seeking help of a doctor due to urinary incontinence using Cu33y +ognitive 1aps 1odel.Cu33y +ognitive 1ap 1odel is a fu33y:graph modeling approach based on expert’s opinion. Thisis the non:statistical approach to study the problems with imprecise information.

Keyw r!s: Cu33y cognitive maps (C+1s , Orinary Incontinence , 6elp:seeking 0ehaviour.

+" INTROD#CTIONOrinary Incontinence (OI is a common but neglected problem of women. =lobally,

urinary incontinence affects the &uality of life of at least one third of women. Also, OI affectsgeneral well:being, self:esteem, and social functioning. 1any women are too embarrassed totalk about it and some believe it to be untreatable even in western countries. This problem ismore pronounced in India, where women usually do not seek treatment for their reproductive



health problems and do not vocali3e their symptoms. There is a [culture of silence^ and lowconsultation rate among Indian women regarding such problems. 2omen in India have beenreported to have a high tolerance threshold for seeking treatment. The condition can beeffectively treated with significant improvement of clinical and &uality of life parameters%8'.

$revious research has indicated that many women with OI do not consult a doctor abouttheir condition %8,;,>'. /easons for not consulting a doctor includeQ seeing OI as a minor problem which can be self:managed%9', not regarding symptoms as abnormal or serious%-?:-4',hoping the condition improves by itself%-5', regarding incontinence as a normal part ofaging%-7', being embarrassed to discuss it with a doctor%-5', and lack of knowledge of possibletreatments%-?,--,-5'.

$revious research has shown that lack of money#time, fear of surgery or hospital and pain are usually the reasons given by women for non:consultation%4,5, '.

OI is a condition that poses considerable human and social complications, bringing physical discomfort, economic burden, shame, and loss of self:confidence, all diminishing&uality of life. Appropriate management can reduce the suffering that accompanies OI. 1anystudies have reported that conservative treatments (therapies that do not involve pharmacologicalor surgical intervention can be helpful in managing the condition (2ilson et al., 4??4 .6owever, most women accept OI as an ailment connected with childbearing and age, and believe symptoms should not be considered serious (6agglund, 2alkwers, *arsson, ] *eppert,4??5 . Although urinary symptoms are extremely common, with as many as one:third of the population over 7? years of age experiencing a clinically significant disorder ($erry et al., 4??? ,and the impact on &uality of life can be substantial, relatively few women seek medical help()haw, 0rittain, Tansey, ] 2illiams, 4??>B )haw, as, 2illiams, Assassa, ] 1c=rother, 4?? "!nly one of four eventually seeks help from a physician (Addis, 4??> .

In a population:based prevalence study conducted in +anada, =ermany, Italy, )weden,and ngland, the vast ma ority (>?@ who had OI for at least one year, as well as 79@ who hadOI for three years, had been suffering from symptoms. !nly ?@ of females with OI haveconsulted with a doctor, and only 4;@ have had continuous treatment (1ilsom et al., 4??- . Inmany studies, it has been determined that females with OI seek help for its symptoms in verylow percentagesQ -7@ (6agglund, 2alker: ngstr m, *arsson, ] *eppert, 4??- , 4;.>@ (Du,2ong, +hen, ] +hic, 4??5 , and 5>@ (Einchen et al., 4??5 .This paper attempts to identify some important factors for not seeking a doctor for OI.After analy3ing the factors using C+1 tool, the predominant factors are found. The resultantfactors are the more impactful factors for not consulting a doctor due to OI.

This paper is organi3ed as followsQ )ection 4 presents C+1s and their foundations onfu33y logic. )ection 5 introduces the C+1 methodology developed. )ection 7 presents theimportant factors for the task. )ection 8 gives the calculation of C+1 model for the given data.)ection discusses the results in a detailed way. )ection ; gives the limitation of the study in brief. Cinally, )ection > outlines the conclusions.$" F#,,Y CO-NITIVE MAPS .FCMs/

A Cu33y +ognitive 1ap (C+1 is a graphical representation consisting of nodesindicating the most relevant factors of a decisional environment and links between these nodes

representing the relationships between those factors %4'. C+1 is a modeling methodology forcomplex decision systems, which has originated from the combination of fu33y logic and neuralnetworks. A C+1 describes the behavior of a system in terms of conceptsB each conceptrepresenting an entity, a state variable, or a characteristic of the system %5'.

Eosko(-9> introduces the concept of Cu33y +ognitive 1aps (C+1s which areweighted cognitive maps with fu33y weights % '.

Cu33y logic considers that everything is a matter of degree. The variables in a system arenot true or false in a -??@B they can be true or false to some extent under certain conditions. Amembership of a variable to certain universe. The mathematical models provided by fu33y logic



allow the study in a systematic manner of the linguistic descriptions and explanations that humanobserves provide. Cu33y modeling transforms linguistic description into mathematical models %4?'.

!ne characteristic of fu33y logic that makes itself very interesting for its application toC+1 is that fu33y logic allows a membership of more than one set of concepts and conse&uently,sets of statements overlap and merge with one another.

*" FCM METHODOLO-Y

*"+" I&tr !'ct% & t t0e (et0 ! 1 2yThe methodology developed uses four matrices to represent the results that the

methodology provides in each one of its stages. These are Initial 1atrix of )uccess (I1) ,Cu33ified 1atrix of )uccess (C<1) , )trength of /elationships 1atrix of )uccess ()/1) andCinal 1atrix of )uccess (C1) %-'.

The methodology uses data gathered from interviews with people whose knowledgeand background enable them to identify and evaluate under solid criteria those factors thataccording to their understanding. The data are then transformed into numerical vectors associatedwith each one of the factors identified. ach component in a vector represents the value that the

corresponding factor has for each one of the individuals interviewed. This generates an Initial1atrix of )uccess (I1) . Thresholds are applied to the I1) in order to adapt the informationcontained to the real world and maintain the logical integrity of data.

Fext, the numerical vectors associated with each factors are fu33ified and treated asa fu33y set. The fu33ification of the I1) produces the Cu33ified 1atrix of )uccess (C<1)containing the grades of membership of each component of a vector to a fu33y set. The factorsare treated as variables, sometimes also referred as concepts. The degree of similarity betweenany two variables is evaluated to estimate the strength of the relationship between them and the polarity of that relationship. The strength of the relationship is given by a fu33y weight preceded by a positive or negative sign indicating whether that relationship is direct or inverse respectively.

The result of these computations is a new matrix called )trength of /elationships1atrix of )uccess ()/1) . At this point, an expert determines whether there is a relationship of

causality between each pair of variables analysed or not. The result is a Cinal 1atrix of )uccess(C1) which contains only those numerical fu33y components which represent relationships ofcausality between the factors. The graphical representation of the Cinal 1atrix of )uccess in theform of a C+1 produces the targeted C+1 for mapping factors for 6elp:seeking behavior ofwomen with urinary incontinence. !nce the graphical representation of the factors is produced,an in depth analysis of the map should be undertaken in order to extract all information containedinto it.*"$" C &str'ct%&2 t0e FCM

The procedure for creating the targeted C+1 is shown inCig.-.

Cu33y +ognitive 1ap (C+1+onstruction )teps

=raphical/epresentationof the C1)

+ausalityevaluation

egree ofsimilarity betweenconcepts and polarity evaluation

ata fu33ificationQThresholds ]grades ofmemberships

ata ac&uisitionfrom domainexperts



Cigure.-3.2.1. Initial Matrix of Success (IMS)The I1) is an %n x m' matrix, where n’’ is the number of factors identified, also

referred as concepts or variables, and m’’ is the number of people interviewed to obtain the data.ach element ! i of the matrix represents the importance that an individual ’’ gives to a certainfactor(concept i’’ within a scale %?,-??'. The elements !i-, ! i4, . . . ,! im are the components ofthe vector Si associated with the factors belonging to row i’’ of the matrix.3.2.2. Fuzzified Matrix of Success (FZMS)

The numerical vectors Si are transformed into fu33y sets, where each element of thefu33y set represents the degree of membership of component !i of vector Si to the own vectorS i. The numerical vectors are converted into fu33y sets with values within the interval %?, -' inthe following way.

C<1) K % i ' nxm, wherei i K -, for the maximum value in Si i.e., 1ax(! i&

jj (-ii i K ?, for the minimum value in Si

i.e., 1in(! ip

jj (4iii Cor the other values in Si ,

jj (5where i(! i is the degree of membership of the element !i to the vector Si.

irectly pro ecting values into the interval %?, -' may result in assigning grades ofmembership that do not reflect the real world and are unsupportable by common sense reasoning.In those cases, the introduction of an upper and#or lower threshold value by the expert analysingthe data is re&uired.

Therefore, if Si is the numerical vector of m elements associated to concept i’’, and !i ,with K -, 4, . . . ,m, are the components of Si , the upper and lower threshold values (_u and_l respectively are as followsQ

j.. (7

j. (8

The remaining elements of the vector are pro ected into the interval %?, -' proportionally.All threshold values introduced during the process to generate the C<1) are further explainedand ustified in the discussion of the case study included in this paper.

C1))/1)C<1) C+1I1)







4"APPLICATION OF FCM METHODOLO-Y

$.1. he Initial Matrix of Success (IMS)The identified proposed factors mentioned above are given to the panel of experts

consisting of 9 experts. )o I1) matrix is of 9x9 type matrix, as the number of factors K numberof people interviewed K 9.

ach element ! i of the matrix represents the importance that individual [ ^ gives tofactor [i^, within the scale from ? ( the factor does not contribute at all to the success of the pro ect to -?? ( the contribution of that factor in very much essential for the task . The I1)1atrix is given by

Individuals

I#S K Cactors -??-??-??-??;?5?8?>?7?

?7?5?8?-???8?7?7?

-??-??5?-??-??>?>?5?8?-??>?>?8?;?-??-??>?8?

9?-??-??7?-??-??-??7?-??

;??-??>?;?;?5?-??;?

>?;?;?5?;?5?7?-??9?

9?;??7?8?7??-???

;?;?-??4?8??7??9?

%&%

$.2. Fuzzified Matrix of Success (FZMS)The fu33ification of numerical vectors in I1) can be done using the e&uations (- to (8 .Cor example,

.

)imilarly for all the vectors, i values calculated.Therefore, C<1) 1atrix is obtained as

Individuals



C<1) K Cactors ---->5.?-;.?8.?-55.?

;.?55.?-;.?8.?-;.?8.?55.?55.?

---;.?-----;.?8.?

---8.?>5.?---8.?---55.?---55.?-

>5.?;.?-->5.?>5.?-;.?->5.?

->5.?>5.?-;.?>5.?-;.?55.?--

->5.?;.?55.?8.?55.?;.?-;.?

>5.?>5.?-?8.?;.?55.?;.?-

4"* Stre&2t0 5 Re1at% &s0%) Matr%6 .SRMS/

Osing the e&uations ( (9 , the )i values are calculated from i values for each i, .

Cor example, for )-4 ,d values are ?.55 ?.55 ?.57 ?.57 ? ?.55 ?.55 ? ?.-;

*ikewise all the values )i have been calculated. The resultant )/1) matrix is as followsQ - 4 5 7 8 ; > 9

)/1) K89.?5.?;>.?-.?;>.?;>.?;.?5.?

89.?;?.?89.?-.?;.?8.?-.?89.?

5.?;?.?;7.?;.?89.?7>.?8;.?7>.?

;>.?89.?;7.?>5.?;7.?;?.?8;.?;.?

-.?-.?;.?>5.?8.?>.?;.?;4.?

;>.?;.?89.?;7.?8.?;7.?9.?;7.?;>.?8.?7>.?;?.?>.?;7.?>7.?>-.?

;.?-.?8;.?8;.?;.?9.?>7.?;.?

5.?89.?7>.?;.?;4.?;7.?>-.?;.?

4"3 T0e F%&a1 Matr%6 5 S'ccess .FMS/

!nce the )/1) 1atrix is found, it has been sorted out according to domain expert withthe fact that only the factors are included which have causal relationship mathematically andlogically.

According to the xpert’s opinion ( octor , The C1) 1atrix is formed in the followingwayQ

- 4 5 7 8 ; > 9

- ?.; ?.>-



4 ?.; ?.>75 ?.>- ?.>7 ?.;7 ?.;>

C1) K 7 ?.;7 ?.;>8 ?.>5 ?.;

?.>5; ?.;>9 ?.;> ?.;>

4"4 Tar2ete! FCM -ra)0 Re)rese&tat% &

7"8+ ?.>7

?.; ?.;7

?.;>

?.>5 ?.;>

?.;

Cigure 4. C+1 =raph

Cig.4. shows the strong influence that certain factors have over others, according to the

data supplied by the expert in the relevant field.It should be mentioned here that there is no negative fu33y value in the directed arc. Itshows that no factor influences negatively other factor. The reason is that the factors mentionedin this papers are well analysed recommendations for the help:seeking behaviour of women withurinary incontinence.

9"DISC#SSION

The results achieved are logically coherent and according to human sense, nocontradictions are observed. The analysis of the C+1 not only identifies the factors but alsoshows how far the causal relation exists between them clearly and gives overall concrete picture before our eyes. The discussion is mainly based on the C+1 graph.

The factor that [Fot regarding symptoms as abnormal or serious^ directly influences thethird factor that [6oping the condition improve by itself^ with the degree L?.>7 . It means that people not considering symptoms of urinary incontinence as a serious one may think that thecondition can be improved by itself.

The fifth factor [0eing embarrassed to discuss OI with a doctor^ greatly influences [*ack of knowledge of possible treatments^ with the degree L?.>5. That is, a person with OIembarrassed to discuss it with a doctor may not know the possible treatments of the urinaryincontinence and the way in which it must be treated effectively.

'

%

)

*

+

,

.



The aim of this paper is to identify high impactful factors for the help:seeking behaviorof the OI. This aim is fulfilled successfully here by ust looking at the C+1 graph. In the graph ,the factors 4,5,and 8 have outgoing arcs with high arc weight namely above L?.>?.

Also, the second, fourth and ninth factor influences the third factor directly with highdegree. )o, a person not considering symptoms of OI as a serious one, a person considering OI asa normal part of aging, and a person with a financial problems may hope the condition of urinaryincontinence improve by itself and they don’t want to consult a doctor for their condition. 6encethe most impactful factor for the help:seeking behaviour for not to consult a doctor is that [ 6opethe condition improve by itself ^.

" LIMITATIONS OF THIS ST#DY

In the case of this study, when transforming data into fu33y sets, a variation of 4?@ has been considered. It has been understood that those degrees of importance between >? and -??mean that the interviewee considers the contribution of that factor very important for this pro ect,and that those degrees of importance between ? and 4? mean that the interviewee considers thecontribution of that factor irrelevant for this study. In this point of the methodology application,different experts analysing the data could have different criteria, all of them valid if accompaniedwith a logical explanation and ustification.

!ne of the main drawback of this methodology is that the )/1) matrix contains plentyof misleading data. Fot all indicators of success represented in the matrix are related, and muchless there is a relationship of causality between them. They may present a close mathematicalrelation while being logically totally unrelated. To clean up the )/1) matrix and keep only thosecausal relations in the C1) matrix, the opinion of an expert is re&uired. The perceptions of theexpert could be not -??@ accurate and could lead to erroneous results. In addition, differentexperts may have different perceptions working with the same data, which will lead to differentCu33y +ognitive 1aps and different conclusions. In most cases, the opinion of more than oneexpert should be considered.

The solution described in this paper is only one of the several solutions available. Afurther analysis of the data may lead to a different, more complex and perhaps more accuratesolution.

8" CONCL#SION

A fu33y model, like C+1, represents a system in a form that corresponds closely to theway humans perceive it. Therefore, the model is easily understandable, even by a non: professional audience and each parameter has a perceivable meaning.

)everal researchers have attempted research on 6elp:seeking 0ehaviour among

women with urinary incontinence’. 6owever, they identified to assess the proportion of womenwho consult their doctor about urinary incontinence, and explore factors associated with help:seeking in all countries.

In this study, the most impactful factor for not seeking a help of a doctor with urinaryincontinence and relation between the factors with corresponding degree were found. This proposed study believes that it has been taken one more step forward in this perspective research.

REFERENCES



%-' *uis /odrigue3:/episo, /ossit3a )etchi , Uose *. )almeron, 1odelling IT pro ectssuccess with Cu33y +ognitive 1aps, xpert )ystems with Applications 54 (4??; 875 889.

%4' Ahmed ) l:A3ab-, !mar 1 )haaban, 1easuring the barriers against seekingconsultation for urinary incontinence among 1iddle astern women , 01+ 2omen’s6ealth 4?-?, -?Q5 .

%5' Fe3ihe Ei3ilkaya 0e i, Ayfer !3bas, rgul Aslan, ilek 0ilgic, 6abibe Ayyildi3 rkan,!verview of the )ocial Impact !f Orinary Incontinence with A Cocus on Turkish2omen, 4?-? )ociety of Orologic Furses and Associates Orologic Fursing, pp. 54;:558.

%7' Teunissen , van 2eel +, *agro:Uanssen T. Orinary incontinence in older people living inthe communityQ examining help:seeking behaviour. 0r U =en $ract 4??8B88Q;; :;>4.%$ub1edQ - 4-4>85'.

%8' 1. !’ onnell a,, =. *ose b, . )ykes c, ). Soss c, ). 6unskaar a, ,6elp:)eeking 0ehaviourand Associated Cactors among 2omen with Orinary Incontinence in Crance, =ermany,)pain and the Onited Eingdom , uropean Orology 7; (4??8 5>8 594 .

% ' )haw +. A review of the psychosocial predictors of help:seeking behaviour and impacton &uality of life in people with urinary incontinence. U +lin Furs 4??-B-?Q-8. %$ub1edQ-->4?454'.

%;' 6annestad D), /ortveit =, 6unskaar ). 6elp:seeking and associated factors in femaleurinaryincontinence. )cand U $rim 6ealth +are 4??4B4?Q-?4 ;.

%>' Einchen E), 0urgio E, iokno A+, et al. Cactors associated with women’s decisions toseek treatment for urinary incontinence. U 2omens 6ealth (*archmt 4??5B-4Q >;.%$ub1edQ -78>5-?9'.

%9' 6a gglund , 2alker: ngstro m 1*, *arsson =, *eppert U. /easons why women withlong:term urinary incontinence do not seek professional helpQ a cross:sectional population:based cohort study. Int Orogynecol U 4??5B-7Q49 5?7.

%-?' 6olst E, 2ilson $ . The prevalence of female urinary incontinence and reasons for notseeking treatment. F< 1ed U -9>>B-?-Q;8 >.

%--' 6unskaar ), *ose =, )ykes , Soss ). The prevalence of urinary incontinence in womenin four uropean countries. 0UO Int 4??7B95Q547 5?.%-4' /. Axelrod, )tructure of ecision, the cognitive maps of $olitical lites, $rinceton, FUQ

$rincetonOniversity $ress, -9; .

%-5' 0art, Cu33y +ognitive 1aps Int. U. 1an:1achine )tudies 47 (-9> 8 ;8.%-7' =erald U. +alais, Cu33y +ognitive 1aps TheoryQ Implications for Interdisciplinary

/eading, Fational$ublications C!+O) !n +olleges 4 (4??> - -8.

%-8' Eosko, 0., Uanuary, -9> , [Cu33y +ognitive 1aps^, International ournal of man:machinestudies, pp. 4:;8.

%- ' 2.0. Sasantha Eandasamy and Clorentin )marandache, Cu33y +ognitive 1aps and Feutrosophic+ognitive 1aps, +opyright (4??5 , 8-? , Townley Ave, O)A.

%-;' Eardaras, ., ] Earakostas, 0. (-999 . The use of fu33y cognitive maps to simulate theinformation systems strategic planning process. Information and )oftware Technology,7-, -9; 4-?.

%->' Einchen E), 0urgio E, iokno A+, et al. Cactors associated with women’s decisions toseek treatment for urinary incontinence. U 2omens 6ealth (*archmt 4??5B-4Q >;.%$ub1edQ -78>5-?9'.





Inventory an idle resource, which is maintained by every business

system whose ultimate aim is profit maximi3ation. To run a healthy business

inventory planning and execution must have high synchroni3ation, but

inspite of well and organi3ed planning the business sectors face thechallenge of handling the problems of shortage and overage. The problem

of shortage can be managed by the maintenance of safety stocks and other

suggestive measures are also proposed by several researchers. To mention a

few 6arris (-9-5 , 2ilson (-957 , 1urdoch (-9 8 , *ewis (-9;? ,=ardner

and annenbring (-9;9 , )ilver.et.al(-9;9 , 0rown(-9>4 , )hah (-994 have

researched the effective management of inventory at times of shortage. The

other situation where the management attention is re&uired is at times of

overage. !verage is any inventory above the maximum planned stock level

=eoff ( 4??5 . )hah (-994 was the first person who used the term overage

in his classification when describing excess inventory. 0ocking (-99 and

6adley (-999 in their work demonstrated that overage have a positive

effect. =eoff in his paper has argued that overage is important because there

is evidence that, even in well:managed business a significant proportion of

the inventory is in overage at any timeB evaluated twenty inventory profiles

from companies in different business sectors which showed that inventory

values in overage is between -?@ and 9>@B described causes and

measurement of overage The remaining of the paper is organi3ed as follows.

)ection 4 discusses the reasons for incompatible transport services which

lead to overage. )ection 5 elucidates the novelty approach and customer ac&uisition. )ection 7 describes the model formulation. )ection 8 presents a

numerical example. )ection presents the sensitivity analysis and the last

section concludes the proposed work.





that transport systems on the whole perform worse under adverse and

extreme weather conditions. This is especially true in densely populated

regions, where one single event may lead to a chain of reactions that

influence large parts of the transport system. In current days the effects of climate change are numerous in number which causes trouble in

transporting.

.%%/" Occ'rre&ces 5 ca1a(%ty

Fowadays the occurrences of calamities are &uite common. arth&uakes,

floods, storm are some of the calamities that affect the normal life of the

people. +onsider the situation where a calamity (flood has affected a place,this calamity will certainly disturb the transportation of goods between the

supplier and the customer. Though the supplier tries his level best to

transport, the goods will not reach its destination at right time due to some

reasons as follows.

-. The transport network would have ruined. 4. The transporter might have

been forced to take a route longer than the usual one. 5. ither the originlocation or the destination location is flooded. 7. The persons engaged in

charge of transportation might have faced direct effect of flood. These are

some of the reasons discussed by $ablo.et.al.

.%%%/" I()r )er tra&s) rt ser;%ces

$resently the impact of calamities causes road damage, breakage of bridges

which highly interrupt the flow of transportation. +onsider the happening of

landslides in hilly regions, felling of huge trees on the roads during storm

that widely causes degradation of the &uality of the transport services. As it



takes time to restore back into normal condition transportation stoppage

takes place. This is also responsible for stagnation which leads to overage.

.%;/" D%s)'tes a&! Str%<es

In many countries strikes hit the transport sector from time to time. )trikes

are the outcome of disputes and disagreements. The reasons for strikes are

numerous in number like increase of prices of commodities, political

problems, economic crisis, social problems and so on. The magnitude of the

effect of these strikes on transportation varies strongly according to the

circumstances of action. +arrying out transportation at times of strike is a

risky task. Therefore happenings of strikes perturb transportation.

*" T0e N ;e1ty A))r ac0 a&! C'st (er Ac='%s%t% &"

To maintain steady business status, the manufacturers should employ various

tactics to reduce overage to maximum stock level. 0ut practically many

manufacturers feel that it is not possible to exercise control over overage as

it burdens them. A new approach to make benefit out of overage is proposedin this paper. The products that raise the maximum stock to overage must be

segregated and then should be channeli3ed by the manufacturers to make

profit out of it. !ne of the ways to yield gain from the products added to

overage is by using as a tool to drag new customers. Increasing the number

of customers is very significant as it enable to expand the business. To run

any sort of business, customers are very important. Attracting the customers

is in fact an art as it re&uires many skills. The manufacturers therefore

segregate the products that add to overage from the maximum stock and

maintains as a special inventory to sell it to the new customers at a feasible

cost. This renders two benefits to the manufacturers, one is to avoid the





items that increase the maximum stock to overage in each cycle are

segregated from the maximum stock and they are maintained as special

inventory (fig -.b .This special inventory is utili3ed in ac&uiring new

customers who are worthier more than money. To determine the expected profit of the enterprise which channeli3es the overage to customer

ac&uisition a mathematical model is developed with the following notations

and assumptions.

Cig - a 1aximum stock level to !verage.

3"+ Ass'()t% &s .

-. The maximum stock level reaches overage due to any one of the causes

mentioned above. 4. The ac&uisition cost is the cost of ac&uiring new

customer.5. The products are not of deteriorating type. 7. $lanning hori3on isinfinite.

3"$ M !e1 F r(at% &

3verag

#a&2Sto

#in2Sto

SafetyStock

/lanne$ inventory

121

12*

)21

12.

12,

4



*et us define T/(" and T+(" as the total revenue and the total cost per

cycle, respectively. T/(" is the sum of total sales volume of items in

maximum stock and overage items.

T/(" K s(-:p " L vp" jjjjj( -

The total revenue per unit of time is obtained by multiplying the total

revenue by the number of cycles per year we obtain the following

expression for total revenue per unit of time.

T/O(" K s(-:p L vp

$+(" is the sum of procurement cost per cycle, Ac&uisition cost per cycle,

A+(" /enovating cost /+(" and holding cost.

0ased on the shape of the areas in Cig.-.b. the holding cost per cycle for the

items in maximum stock is h times the area of the lower trape3oid with

parallel sides e&ual to .It is given by

K K



The holding cost for the items in special inventory times the area of the

triangle of Cig - is 1ultiplying the sum of the

above holding costs by the number of cycles per year, , we obtain the

following expression of the total holding costs per unit of time.

Therefore, the total inventory cost per unit time is

T+O(" K

The total profit per unit time T$O(" is obtained by T/O(" : T+O("

T$O(" K s(-:p L vp :

(-





jjjjjjjj. (7

4"N'(er%ca1 E6a()1e

To illustrate the usefulness of the model developed in section 7, let us

consider the inventory situation where a stock is replenished instantly with

" units. The parameters needed for analy3ing the above inventory situation

are given belowQ

emand rate, K 8?,??? units# year ,!rdering cost, E K -??# cycle ,6oldingcost for maximum stock, h K 7# unit#year ,6olding cost for special inventory,

h? K 8# unit # year , )elling price of items in maximum stock, s K 8?# unit ,

)elling price of items in overage, v K 5?# unit ,+ost of ac&uiring new

customers, A K 5?# customer

Fumber of new customers ac&uired, say nK 4 ,/enovating cost per item in

overage , r K 8 # item 2e assume that p, is uniformly distributed with its p.d.f as

f(p K

Therefore, we get (p as follows from (8

(p K





items in " that adds to overage in the next cycle in the above problem we

get " MK 5-? and we get total profit per unit to be negative.

9"Se&s%t%;%ty A&a1ys%s

The total costs per unit in the case where the overage is not taken into

consideration are only the ordering costs per unit and the holding costs per

unit, as the renovation costs , ac&uisition costs and holding costs of the

special inventory are excluded as p becomes 3ero. The optimal order

&uantity is greater in the case where the overage is included (first case than

the optimal order &uantity obtained in the case where the overage is not

taken into consideration (second case . Also the total profit per unit in the

first case is positive where as in the second case it is negative . Therefore we

conclude that this inventory model with overage is applicable in the case

when the enterprises face the problem of overage. This model is uni&uely

designed to deal the issues of overage. If the overage is not taken into

consideration then one can employ the classical method to determine the

optimal order &uantity.

/eferences

-. 0ocking, +., -99 . Inventory management at 6usky

computers.Institute of !perations 1anagement Annual +onference

-99 $roceedings, Uune -99 .

4. 0rown, /., -9>4. Advanced )ervice $arts Inventory +ontrol.1aterials

1anagement )ystems, Inc., Sermont, O)A.

5. =ardner, ., annenbring, ., -9;9. Osing optimal policy surfaces to

analyse aggregate inventory tradeoffs. 1anagement )cience 48 (> .



7. =eoff /elph., $eter 0arrar.,4??5., !verage inventoryWhow does it

occur and why is it importantV Int. U. $roduction conomics >- >4

(4??5 - 5 -;-

8. 6adley, /., -999. Inventory management control. The Uournal of theInstitute of !perations 1anagement 48 (4 .

. 6arris, C., -9-5. 6ow many parts to make at once, CactoryWThe

1aga3ine of 1anagement -? (4 , -58 -5 , -84.

;. Ibrahim, A. and 6all, C. (-997 ffect of Adverse 2eather +onditions

on )peed:pow:occupancy /elationship, Transportation /esearch

/ecord -78;. Transportation /esearch 0oard, 2ashington, +.

>. Uones, 0., Uanssen, *. and 1annering, C. (-99- Analysis of the

fre&uency and duration of freeway accidents in )eattle. Accident

Analysis and $revention 45(7 , 459:488.

9. Ehattak, A., )chafer, U. and 2ang, 1. (-998b A simple procedure for

predicting freeway incident duration. IS6) Uournal 4(4 , --5:-5>.

-?.*ewis, +., -9;?. +overage Analysis, )cientific Inventory +ontrol.

0utterworths, *ondon.

--.1urdoch, U., -9 8. +overage analysisWnew techni&ue for optimising

the stock ordering policy. $roceeding from !ne day +onference,

+ranfield, OE.

-4.$ablo )uare3, 2illiam Anderson, Si ay 1ahal , T./.

*akshmanan .,4??8., Impacts of flooding and climate change on urban

transportationQ A systemwide performance assessment of the 0oston1etro Area Transportation /esearch $art -? , 45- 477

-5.)hah, )., -994. )etting parameters in 1/$ for effective management

of bought out inventory in a UIT assembly environment, $h. . Thesis,

Aston Oniversity, 0irmingham,Uanuary -994, unpublished.





1eningitis is inflammation of the meninges. The meninges are the collective

name for the three membranes that envelope the brain and spinal cord (central nervous

system , called the dura mater, the arachnoid mater, and the pia mater. The meninges\

main function, alongside the cerebrospinal fluid is to protect the central nervous system.

1eningitis is generally caused by infection of viruses, bacteria, fungi, parasites, and

certain organisms. Anatomical defects or weak immune systems may be linked to

recurrent bacterial meningitis. In the ma ority of cases the cause is a virus. 6owever,

some non:infectious causes of meningitis also exist %7, 9'.

0acterial meningitis is generally a serious infection. It is caused by three types of

pneumonia bacteria. 1eningitis caused by Feustria meningitides is known as

meningococcal meningitis, while meningitis caused by )treptococcus pneumonia is

known as pneumococcal meningitis. $eople become infected when they are in closecontact with the discharges from the nose or throat of a person who is infected. About

>?@ of all adult meningitis are caused by F. meningitides and ). pneumonia. $eople over

8? years of age have an increased risk of meningitis caused by *. monocytogenes %-?'.

1eningitis is not always easy to recogni3e. In many cases meningitis may be

progressing with no symptoms at all. In its early stages, symptoms might be similar to

those of flu. 6owever, people with meningitis and septicemia can become seriously ill

within hours, so it is important to know the signs and symptoms. arly symptoms of

meningitis broadly includeQ Somiting, Fausea, 1uscle pain, 6igh temperature (fever ,

6eadache, +old hands and feet, a stiff neck, severe pains and aches in your back and

oints, sleepiness or confusion, a very bad headache (alone, not a reason to seek medical

help ,a dislike of bright lights, very cold hands and feet, shivering, rapid breathing , a

rash that does not fade under pressure. This rash might start as a few small spots in any

part of the body : it may spread rapidly and look like fresh bruises. This happens because

blood has leaked into tissue under the skin. The rash or spots may initially fade, and then

come back. -?:-4@ of meningitis cases in the industriali3ed countries are fatal.

4?@ of meningitis survivors suffer long:term conse&uences, such as brain damage,

kidney disease, hearing loss, or limb amputation. There are 4,5?? cases of meningitis and

meningococcal septicemia in the OE each year. ;?@ of meningitis patients are aged

fewer than 8 or over ? %9, -5'.

http://www.medicalnewstoday.com/articles/15107.php









A number of studies have shown that the diagnosis and treatment management of

meningitis is a complex and challenging problem for government and health care

agencies re&uiring novel approaches to its management and intervention %-5, -7'. In this

paper, we are proposing a modeling approach to understanding meningitis which focuses

on capturing the various symptoms associated with the disease. The main scope of this

work is the construction of a knowledge based tool for modeling meningitis diagnosis for

adult. This paper proposes a T!$)I) based methodology for ranking C+1 based

scenarios.

This paper is structured as followsQ )ection 4 presents the methodology. )ection 5

gives the selected risk factors of meningitis. )ection 7 explains the calculation of the

methodology for the given data. )ection 8 discusses the experimental result. Cinally,

)ection outlines the conclusion.$" Pr ) se! Met0 ! 1 2y

)cenarios describe events and situations that would occurred in the future real:

world. The whole methodology proposal is composed of three blocks %--'.

-. 0uilding C+1 models using experts’ opinion.

4. )cenarios simulation. It is composed of two stages. The first one is the

scenarios definition and the second one is the C+1 inference.

5. /anking the scenarios with T!$)I). The closer scenario to the positive:ideal

scenario is the best solution.

$"+" F'??y C 2&%t%;e (a)s

$"+"+" FCM F'&!a(e&ta1s

+ognitive maps (Axelrod, -9; are a signed digraph designed to capture the

casual assertions of an expert with respect to a certain domain and then use them to

analy3e the effects of alternatives. A fu33y cognitive map (C+1 is a graphical

representation consisting of nodes indicating the most relevant factors of a decisional

environment and links between these nodes representing the relationships between those

factors (Eosko, -9> . C+1 has two significant characteristics. The first one, casual

relationships between nodes have different intensities. These are represented by fu33y

numbers. The second one, the system is dynamic, it evolves with time. It involves



feedback, where the effect of change in a concept node may affect other concept nodes,

which in turn can affect the node initiating the change.

After an inference process, the C+1 reaches either one of two states following a

number of iterations. It settles down to a fixed pattern of node values, the so:called

hidden pattern or fixed:point attractor. Alternatively, it keeps cycling between several

fixed states, known as a limit cycle. Osing a continuous transformation function, a third

possibility known as a chaotic attractor exists. This occurs when, instead of stabili3ing,

the C+1 continues to produce different results (known as state:vector values for each

cycle. The relationships between nodes are represented by directed edges. An edge

linking two nodes models the causal influence of the causal variable on the effect variable

(e.g. the influence of the price to sales . )ince C+1s are hybrid methods mixing fu33y

logic (0ellman ] <adeh, -9;?B <adeh, -9 8 and neural networks (Eosko, -994 , each

cause is measured by its intensity wi %?, -', where i is the cause node and the effect

one.

$"+"$" FCM !y&a(%cs

An ad acency matrix A represents the C+1 nodes connectivity. C+1s measure

the intensity of the causal relation between two factors and if no causal relation exists it is

denoted by ? in the ad acency matrix.

:::::::::::::::::::::: (-

C+1s are dynamical systems involving feedback, where the effect of change in a node

may affect other nodes, which in turn can affect the node initiating the change. The

analysis begins with the design of the initial vector state (? , which represents the value

of each variable or concept (node . The initial vector state with n nodes is denoted as

::::::::::::::::::::: (4

where is the value of the concept i K - at instant t K ?.



The new values of the nodes are computed in an iterative vector:matrix

multiplication process with an activation function, which is used to map monotonically

the node value into a normali3ed range %?, -'. The sigmoid function is the most used one

(0ueno ] )almeron, 4??9 when the concept (node value maps in the range %?, -'. The

vector state tL- at the instant t L - would be

::::::::::::::::::

::: (5

where t is the vector state at the t instant, is the value of the i concept the t

instant, f(x is the sigmoid function and A the ad acency matrix. The state is changing

along the process.

The sigmoid function is defined as

:::::::::::::::::::::: (7

where is the constant for function slope (degree of normali3ation . The value of K 8

provides a good degree of normali3ation (0ueno ] )almeron, 4??9 in %?, -'.

The C+1 inference process finish when the stability is reached. The final vector

state shows the effect of the change in the value of each node in the C+1. After theinference process, the C+1 reaches either one of two states following a number of

iterations. It settles down to a fixed pattern of node values, the so:called hidden pattern

or fixed:point attractor.

$"$ TOPSIS (et0 !

$"$"+" C &ce)t

The techni&ue for order performance by similarity to ideal solution (T!$)I) is a

multicriteria method to detect the best alternative from a finite set of one’s (6wang ]

Doon, -9>- . The chosen alternative should has the shortest distance from the positive

ideal solution and the farthest distance from the negative ideal solution. The positive ideal

solution is composed of all best values attainable from the criteria, whereas the negative

ideal solution consists of all worst values attainable from the criteria (2ang ] lhag,

4?? .



=eneral T!$)I) process is briefly explained in the next section.

$"$"$" TOPSIS )r cess

*et us define the set of alternatives as and the set of

criteria as Curthermore, let us assume a decision matrix,

and be defined as

:::::::::::::::::::: (8

where is composed of n alternatives (scenarios in this proposal and m attributes

(nodes’ values in this proposal B xi denotes the value of the ith alternative with respect to

the th criterion or attribute.

The procedure of T!$)I) techni&ue can be expressed in the following stages.

)tage -. etermine the normali3ed decision matrix (/ K %r i ' . The raw decision matrix is

normali3ed for criteria comparability. The normali3ed value of xi , r i , can be

obtained by

, K-,4,j,m, iK-,4,j,n

:::::::::::::::::::: (

)tage 4. +ompute the weighted normali3ed decision matrix (S K %vi ' . The weighted

normali3ed value of r i will be denoted by vi and can be computed by

:::::::::::::::::::: (;

Fote that w is the weight of the th criterion and

.





, iK-, 4j n

::::::::::::::: (-?

In addition, the distance to the negative:ideal alternative, , is

denoted as

, iK-, 4j n

::::::::::::::: (--

)tage 8. +ompute the relative closeness to the ideal alternative and rank the preferenceorder. The relative closeness of the ith alternative, , , is

defined as

, iK-, 4j n

::::::::::::::: (-4

)ince and , then

A set of alternatives then can be preference ranked according to the

descending order of then larger means better alternative.

*" Se1ecte! R%s< Fact rs 5 Me&%&2%t%s

$revious studies on predicting 1eningitis were focused either on rules to classifya patient to a group of risk of getting pneumonia or data mining techni&ues that extract

rules from data to predict 1eningitis risk %9'. Artificial neural networks and machine

learning techni&ues were investigated to predict the outcomes of patients with meningitis.

6owever, the previous works that have been done to predict meningitis state using C+1s

%9'.



Fow we illustrate the dynamical system by a very simple model from the

symptoms of meningitis for adults. At the first stage we have taken the following ten

arbitrary attributes (concepts (+-, +4... +-? . It is not a hard and fast rule we need to

consider only these ten attributes but one can increase or decrease the number of

attributes according to needs. The following attributes are taken as the main nodes for

study. An expert system spells out the ten ma or concepts relating to the meningitis. !f

the -? concept nodes, 9 represent a list of the symptoms and risk factors considered by

the experts (physicians and the central node 1eningitis is the basic decision concept

which gathers the cause:effect interactions from all other input nodes. The selected nodes

for C+1 are as followsQ

+- : Cever

+4 : Somiting+5 : 6eadache

+7 : /ash

+8 : )tiff neck

+ : islike of bright colours

+; : Sery sleepy

+> : +onfused#delirious

+9 : )ei3ures

+-? : $ossibility of 1eningitis

3" I()1e(e&tat% & 5 t0e (et0 ! 1 2y t t0e st'!y

0ased on the expert’s opinion, the directed diagram (figure-. is drawn with ten

nodes and twelve edges. The corresponding connection matrix A is given as followsQ



")1

")

",

"'

"

"*"+

".

"

"%

F%2"+ F'??y C 2&%t%;e Ma)

The ad acency matrix A is given by

? ? ?.7 ? ? ? ? ? ?. ?.;

? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ?.5 ? ?.

? ? ? ? ? ? ? ? ? ?.8

? ? ? ? ? ? ? ? ? ?.>4? ? ? ? ? ? ? ? ? ?.;

? ? ? ? ? ? ? ? ? ?.;

? ? ? ? ? ? ? ? ? ?.;8

? ? ? ? ? ? ? ? ? ?.>-

? ? ? ? ? ? ? ? ? ?

A

=

Curthermore, five initial stimuli have been defined as follows (Table - . ach of initial

stimuli vector is used for generating C+1:based scenarios. Fext stage is the C+1dynamics. The results are shown in Table 4.

In addition, the final scenarios are represented graphically at Cigure.4. Fote

that the figure suggests the fifth scenario as the best one, but there is not more

information about the preference between the different scenarios.



After C+1 dynamics, the next stage is to rank scenarios with T!$)I). The

normali3ed decision matrix / is given in Table 5. The weighted normali3ed decision

matrix S is given in Table 7.

Tab1e + Tab1e $

I&%t%a1 St%('1% FCM !y&a(%c res'1ts

N !es

Sce&ar% s.S%/

C% S+ S$ S* S3 S4

N

!

e

s

I&%t%a1

st%('1

% .1%/

C

%

1 1 1 1 1

C

+

+ 7 + + +

C

$

+ 7 7 7 +

C

*

+ 7 7 7 +

C

3

7 + 7 7 +

C4

7 + 7 7 +

C

9

7 7 7 7 +

C + 7 + 7 +

C

8

7 + 7 7 +

C

@

7 7 7 7 +

C

+

7

7 7 + 7 +



C+ +"7 7"++ +"7 +"7 +"7C$ +"7 7"++ 7"778 7"77@9 +"7C* 7"3 7"++ 7"37$+ 7"3748 7"37$C3 7"7+38 +"7 7"778 7"7+ 3 +"7C4 7"7+38 +"7 7"778 7"7+ 3 +"7

C9 7"7+38 7"++ 7"778 7"77@9 +"7C +"7 7"++ +"7 7"7738 +"7C8 7"+$3* 7"778* 7"+$94 7"+$ 7 7"+$37C@ 7"97+* 7"7+@ 7"97+@ 7"97 4 7"97+*C+7 +"7 +"7 +"7 +"7 +"7

Tab1e *

?.7 97 ?.7 97 ?.->;> ?.?? 9 ?.?? 9 ?.?? 9 ?.7 97 ?.?8>7 ?.4>45 ?.7 97

?.? > ?.? > ?.? > ?.8;?> ?.8;?> ?.? > ?.? > ?.??7; ?.?--4 ?.8;?>

?.85-8 ?.??75 ?.4-5; ?.??75 ?.??75 ?.??75 ?.85-8 ?.? ;4 ?.5-99 ?.85-8

?. 4 - ?.?? ? ?.487-

R =?.?-?9 ?.?-?9 ?.?? ? ?.??5? ?.?;98 ?.5>?7 ?. 4 -

?.5 74 ?.5 74 ?.-7 7 ?.5 74 ?.5 74 ?.5 74 ?.5 74 ?.?784 ?.4-9? ?.5 74

[ ]?.8 ; ?.; ?.78 ?.8 ?.>4 ?.; ?.; ?.;8 ?.>- ?w =

Tab1e3

?.4 ? ?.54> ?.?>78 ?.??58 ?.??8; ?.??7> ?.54> ?.?75> ?.44>; ?

?.?5;9 ?.?7 > ?.?5?- ?.4>87 ?.7 >- ?.?7 > ?.?7 > ?.??58 ?.??9- ?

?.5?-4 ?.??5? ?.?9 4 ?.??44 ?.??58 ?.??5? ?.5;4- ?.?8?7 ?.489- ?

?.587> ?.??74 ?.--75 ?.??88 ?.??>9 ?.?

V =?74 ?.??4- ?.?89 ?.5?>- ?

?.4? 7 ?.-?48 ?.? 89 ?.->4- ?.49> ?.4879 ?.4879 ?.?559 ?.-;;7 ?

According to the T!$)I) methodology, the positive:ideal scenario ($I) is

calculated by the higher scores of each node and the negative:ideal scenario (FI) is

calculated by the lower scores of each node. After applying T!$)I) algorithm, the results

are shown in table 8.



Tab1e 4

TOPSIS res'1ts

% S+ S$ S* S3 S4?.8749 ?.7;;4 ?. >44 ?.;;?> ?.5>4?.8 57 ?.8797 ?.847; ?.77;5 ?.8 ->

C % ?.8?95 ?.8584 ?.757> ?.5 ;4 ?.8979Ra&< * $ 3 4 +

4" D%sc'ss% &

Cinally the simulated scenarios are ranked as )8 )4 )- )5 )7.

Crom this analysis, the 8th scenario, that is, when all the symptoms are present, possibility

of getting the meningitis is very high. The second rankings is for the second scenario, that

is, when the symptoms corresponding to rashes, stiff neck and confused states are present

then the possibility of getting meningitis disease is high. *ikewise, we can conclude that

third ranking is for the first scenario that corresponds to the symptoms of fever, vomiting,

headache and very sleepy. If these symptoms are present, the risk of getting the disease is

moderate.

9" C &c1's% &





%--' )almeron, U.*., /osario Sidal, ] Angel 1ena, (4?-4 . /anking fu33y cognitive

map based scenarios with T!$)I). xpert )ystems with Applications, 59,4775:

478?.

%-4' 2ang, D. 1., ] lhag, T. 1. ). (4?? . !n the normali3ation of interval and

fu33y weights. Cu33y )ets and )ystems, -8;(> , 478 47;-.

%-5' 2orld 6ealth !rganisation, /eport on global surveillance of epidemic:prone

infectious disease. Tech:rep. 26!#+ )#+)/#I)/#4???--4;, 4???.

%-7' 2orld 6ealth !rganisationQ 1eningococcal meningitis. Tech:rep. 4?--,

%http##www.who.int#mediacentre#factsheets#fs-7-#en#'.

%-8' <adeh, *.A. (-9 8 . Cu33y )ets. Information and +ontrol, >,55>:585.

OPTIMI,ATION OF A SIN-LE>VENDOR SIN-LE>B#YER F#,,YINVENTORY MODEL WITH DISCRETE DELIVERY ORDER

RANDOM MACHINE #NAVAILABILITY TIME AND LOST SALES

A.&eorge Mary

Marivanios College, 'ir$vanantha#$ra(-6)5 *+5 , ,Kerela

Abstract : )upply chain management is concerned with the coordination of

material and information flows in multi:stage production systems. Inventory

management has long been treated as an isolated function solely focused on

individual entities, taking into account concerned with single vendor:single

buyer and single:vendor multiple buyer models. Integrated single:vendor

single:buyer inventory model with multiple deliveries has proved to result in

less inventory cost. 6owever, many researchers assumed that the production

run is perfect and there is no production delay. In reality, production delay is



prevalent due to random machine unavailability and shortages. This study

considers lost sales, and two kinds of machine unavailability distributions

uniformly and exponentially distributed. A classical optimi3ation techni&ue

is used to device an optimal solution and a numerical example is provided toillustrate the theory. In this model we consider the fu33y total cost under

crisp order &uantity or fu33y order &uantity in order to extend the traditional

inventory model to the fu33y environment. 2e use Cunction $rinciple as

arithmetic operators of fu33y total cost and use the =raded 1ean Integration

/epresentation 1ethod to defu33ify the fu33y total cost. Then we use the

Euhn:Tucker 1ethod to find the optimal order &uantity of the fu33y order

inventory model. The results show that delivery fre&uency has significant

effect on the optimal total cost and a higher lost sales cost will result in a

higher delivery fre&uency.



+" INTROD#CTION

ue to unreliable production system, vendors may not deliver

products to the buyers when needed, resulting in lost sales. 6owever

excessive supplies to fulfill customer’s re&uirement results in higher

inventory cost. The inventory cost is one of the dominant costs for many

industries. Industries should plan their strategy to provide products and

services to the customers at a minimum cost. The order &uantity and the time

to order are critical decisions for both the manufacturing and the service

industries. )ome industries implement Uust In Time (UIT )ystem to reduce

their inventory cost. In order to support an efficient UIT system, it isimportant to ensure the reliability of the vendor’s production system.

0ut in reality, there are possibilities that the production process is

delayed due to machine unavailability and shortages of materials and

facilities. Abboud et al. (4??? developed $" models by considering

random machine unavailability with backorders and lost sales. the models

were extended by Uaber and Abboud (4??- who assumed learning andforgetting in production. later +hung et al (4?-- by considering

deteriorating items In this study, we assume a UIT system where the buyer

who pays the transportation cost, decides the order &uantity si3e of items and

re&uests items delivery in multiple shipments. The vendor products the items

using an conomic $roduction "uantity ( $" model. Ideally, the machine

starts a production run when the inventory level is e&ual to 3ero. In some periods, there is a possibility that the machine may not be available. If this

situation occurs, the vendor cannot deliver the predetermined &uantity

ordered by the buyer, resulting in the buyer’s lost sales. 2e consider two

distribution models for the random machine unavailability case. The



distribution models represent two different types of distribution. Oniformly

distributed means constant number of machine unavailability over a period

of time while exponentially distributed means machine unavailability may

increase with time. 0oth cases can occur in real life. )imilar distributiontypes were used by Abboud et al. (4??? and =iri and ohi (4??8 .

In section 4, the methodology is introduced. In section 5, discuss with

fu33y !" inventory models with different situation. A numerical example

is shown in section 7 and section 8 concludes.

$" METHODOLO-Y

$"+" -ra!e! Mea& I&te2rat% & Re)rese&tat% & Met0 !

+hen and 6sieh introduced =raded mean Integration /epresentation

1ethod based on the integral value of graded mean h:level of generali3ed

fu33y number for defu33ifying generali3ed fu33y number. 6ere, we fist

define generali3ed fu33y number as follows Q

)upposeA

is a generali3ed fu33y number as shown in Cig.-. It is

described as any fu33y subset of the real line /, whose membership function

Ae

satisfies the following conditions.

-.( )xA

is a continuous mapping from / to %?, -',

4.( )xA

K ?, :∞ G x J a- ,



5.( )xA

K *(x is strictly increasing on %a-, a4',

7.

( )xA

K 2 A, a4 J x J a 5,

8.( )xA

K /(x is strictly decreasing on %a5, a7',

.( )xA

K ?, a7 J x G ∞,

where ? G 2A J - and a - , a4, a5 and a7 are real numbers.

This type of generali3ed fu33y numbers are denoted asA

K (a- , a4, a5,

a7 BωA */ whenωAK -, it can be formed asA

K (a- , a4, a5, a7 BωA */ . )econd,

by =raded 1ean Integration /epresentation 1ethod, * :- and / :- are the

inverse functions of * and / respectively and the graded mean h:level value

of generali3ed fu33y numberA

K (a- , a4, a5, a7 B ωA */ is given by

( ):- :-h* (h L / (h .

4(see fig.4 . Then the graded 1ean Integration

/epresentation of $(A

with grade wA, where

$(A

K

( )A

A

:- :-

?

?

h* (h L / (h dh

4

h dh

∫ ∫

. . .(-



where ? G h≤ wAand ? G wA ≤ -.

Throughout this paper, we only use popular trape3oidal fu33y number

as the type of all fu33y parameters in our proposed fu33y production

inventory models. *et0

be a trape3oidal fu33y number and be denoted as

0 K (b-, b4, b5, b7 . Then we can get the =raded 1ean Integration

/epresentation of0

by the formula (- as

$(0

K

( ) ( )-

- 7 4 - 7 5?

-

?

h b Lb h b : b b b dh

4

h dh

+ − + ∫ ∫

- 4 5 7 b 4b 4b b K

+ + +

. . .(4

F%2"+" T0e 2ra!e! (ea& 0>1e;e1 ;a1'e 5 2e&era1%?e! 5'??y &'(berA

.a + a $ a * a 3 : w A/LR"

)

WA

H

1

A

(&6 7(&6

( ):- :-h* (h L / (h .

4



$"$" T0e F'??y Ar%t0(et%ca1 O)erat% &s '&!er F'&ct% & Pr%&c%)1e

Cunction $rinciple is introduced by +hen (-9>8 to treat the fu33y

arithmetical operations by trape3oidal fu33y numbers. 2e will use this

principle as the operation of addition, multiplication, subtraction and

division of trape3oidal fu33y numbers, because (- the Cunction $rinciple is

easier to calculate than the xtension $rinciple, 94 the Cunction $rinciple

will not change the shape of trape3oidal fu33y number after the

multiplication of two trape3oidal fu33y numbers, but the multiplication of

two trape3oidal fu33y numbers will become drum:like shape fu33y number

by using the xtension $rinciple (5 if we have to multiply more than four trape3oidal fu33y numbers then the xtension $rinciple cannot solve the

operation, but the function principle can easily find the result by pointwise

computation. 6ere we describe some fu33y arithmetical operations under the

Cunction $rinciple as follows.

)upposeA

K (a- , a4, a5, a7 ]0

K (b- , b4, b5, b7 are two trape3oidalfu33y numbers. Then

(- The addition ofA

and0

is

0e Ae

K (a- L b-, a4 L b4, a5 L b5, a7 L b7

where a- , a4, a5, a7, b- , b4, b5 and b7 are any real numbers.

(4 The multiplication ofAe

and0

is

A 05

K (+- , +4, +5, +7





(ii α q ?, α A

K (αa7, αa5, αa4, αa-

$"*" T0e K'0&>T'c<er C &!%t% &s

Taha (-99; discussed how to solve the optimum solution of nonlinear

programming problem sub ect to ine&uality constraints by using the Euhn

Tucker conditions. The development of the Euhn Tucker conditions is based

on the *agrangean method.

)uppose that the problem is given by

1inimi3e y K f(x

)ub ect to g i(x ≥ ?, i K -, 4, . . . , m.

The nonnegativity constraints x≥ ?, if any, are included in the m

constraints.

The ine&uality constraints may be converted into e&uations by using

nonnegative surplus variables. *et4i) be the surplus &uantity added to the ith

constraint gi(x ≥ ?. *et λ K (λ- , λ4, . . . , λm , g(x K (g-(x , g4(x , . . . gm(x .

. . and )4 K( )4 4 4

- 4 m) , ) , . . . , ) .

The Euhn Tucker conditions need x andλ to be a stationary point of

the minimi3ation problem, which can be summari3ed as follows Q







5

5 5

5v

4v 5 t 55 5 5

h & E(- : L - pA +A h &

,&E 4 &E & 4

÷ + + +

7

7 7

7v

-v 7 t 77 7 7

h & E(- : L - pA +A h &

&E 4 &E & 4

÷ + + +

which implies

F* K ((Av (& E L ((h v & #4 (E (- (

$ -

((A (& E ((+ t & L

(h & #4

where and are the fu33y

arithmetical operators under function principle.

)uppose

vA5

K (Av- , Av4, Av5, Av7



K ( - , 4, 5, 7

vh5

K (hv- , hv4, hv5, hv7

$ K ($- , $4, $5, $7

t+5

K (+t- , +t4, +t5, +t7

h K (h- , h4, h5, h7

A K (A- , A4, A5, A7

are non negative trape3oidal fu33y numbers. Then we solve the optimal

order &uantity as the following steps. )econd, we defu33ify the fu33y total

inventory cost, using the =raded 1ean Integration /epresentation 1ethod.

The result is

/9 F* ' K

-

- -

-v

7v - t -- - -

h & E(- : L -$A +- A h &

&E 4 &E & 4

÷ + + +

4

4 4

4v

5v 4 t 44 4 4

h & E(- : L -$A +A h &

&E 4 &E & 4

÷ + + + +



5

5 5

5v

4v 5 t 55 5 5

h & E(- : L -$A +A h &

&E 4 &E & 4

÷ + + + +

7

7 7

7v

-v 7 t 77 7 7

h & E(- : L -$A +A h &

&E 4 &E & 4

÷ + + + +

Third we can get the optimal order &uantity

M F*& ,

when $% F* ' is

minimi3ation. In order to find the minimi3ation of $% F* ' the

derivative of $% F* ' with &M is

F*$ TO+

&

∂ ∂

5

K ?.

6ence we find the optimal order &uantity



M&5 K

{ } { }

{ }- 4 5 7

- 4 5 7

- 4 5

v - v 4 v 5 v 7 - - 4 4 5 5 7 7

t - t 4 t 5 t 7

5- 4

v v v7 5 4

A 4A 4A A A 4A 4A A

+ E 4+ E 4+ E + E 4

E h E - - 4h E - - 4h E - -$ $ $

+ + + + + + +

+ + + +

− + + − + + − + ÷ ÷ ÷

{ }7

7v - 4 5 7

-

h E - - h 4h 4h h$

+ − + + + + + ÷

*"$" F'??y I&;e&t ry EO M !e1 w%t0 F'??y Or!er 'a&t%ty

In this section, we introduce the fu33y inventory !" models by

changing the crisp order &uantity&

be a trape3oidal fu33y number

& K (&- , &4, &5, &7 with ? ≤ &- ≤ &4 ≤ &5 ≤ &7. Then we get the fu33y total cost

function as

F* K

-

- -

-v -

7v - t -- - - -

7 7 7

h & E(- : L -$A +A h &

,& E 4 & E & 4

÷ ÷ ÷ + + + ÷ ÷ ÷

4

4 4

4v 4

5v 4 t 44 4 4 4

5 5 5

4h & E(- : L -$4A 4+4A h &

,& E 4 & E & 4

÷ ÷ ÷ + + + ÷ ÷ ÷







-

7

7

-v

v 77- 7 7t 7 7 -4

-

h E(- : L -A$- h - A

+ K ?4 4 & E E

÷ + − + + − + ÷

4

5

5

4v

5 v 54 5 5t 5 - 44

4

h E(- : L -$ A4 h - A

+ K ?4 4 & E E

÷ + − + + − + ÷

5

4

4

5v

v 445 4 4t 4 4 54

5

h E(- : L -A$4 h - A

+ K ?4 4 & E E

÷ + − + + − + ÷

7

-

-

7v

v --7 - -

t - 547

h E(- : L -A$- h - A

+ K ?4 4 & E E

÷ + − + + − ÷

λ-(&4 &- K ?

λ4(&5 &4 K ?

λ5(&7 &5 K ? λ7&- K ?



0ecause &- H ? andλ7 &- K ? thenλ7 K ?. Ifλ- Kλ4 Kλ5 K ? then &7 G &5 G &4 G

&- , it does not satisfy the constraints ? G &- ≤ &4 ≤ &5 ≤ &7. Therefore &4 K &- ,

&5 K &4 ] & 7 K &5. (ie &- K &4 K &5 K &7 K &M.

6ence from we find the optimal order &uantityM F*&

as

M F*&

K

{ } { }

{ }- 4 5 7

- 4 5 7

- 4 5

v - v 4 v 5 v 7 - - 4 4 5 5 7 7

t - t 4 t 5 t 7

5- 4v v v

7 5 4

A 4A 4A A A 4A 4A A

+ E 4+ E 4+ E + E 4

E h E - - 4h E - - 4h E - -$ $ $

+ + + + + + +

+ + + + − + + − + + − + ÷ ÷ ÷

{ }

7

7v - 4 5 7

-

h E - - h 4h 4h h$

+ − + + + + + ÷



3" N#MERICAL E AMPLES

In this section, a numerical example is shown to illustrate the model.

*et the

$roduction rate, $ K -9,4?? units#year,

emand /ate, K 7,>?? units#year,

Sendor setup cost, Av K R ??#cycle,

!rdering cost of buyer, A K R48#order,

Sendor holding cost, hv KR #unit#year,

0uyer holding cost, h K R;#unit#year,

0uyer’s transportation cost, +t K R8?#delivery,

Fumber of delivery, E K

The optimal solution is & K -94.587

TO+ F* K ; 97.-8

)uppose Cu33y production rate $ is [more or less than -94??^

$ K (->>??, -9???, -97??, -9 ??

Cu33y annual demand rate is [more or less than 7>??^

K (74??, 78??, 8-??, 87??

Cu33y setup cost Av for vendor per cycle is [more or less than ??^





In this paper, the machine unavailability time is assumed to be

uniformly distributed. The numerical example illustrates how the multiple

deliveries result in a lower cost than the single delivery model. The

stochastic machine time model results in a higher cost and more deliveryfre&uencies when compared to a perfect machine model. The proposed

model helps enterprises to optimi3e their profit by coordinating the number

of deliveries for various machine unavailability time and lost sales cost. Cor

each fu33y model, a method of defu33ification namely the =raded 1ean

Integration /epresentation is employed to find the estimate of total cost

function in the fu33y sense and then the corresponding optimal order lotsi3e

is derived from Euhn:Tucker 1odel.

REFERENCES

-. Abboud, F. ., Uaber, 1.D., Foueihed, F.A., 4???. conomic lot

si3ing with consideration of random machine unavailability time.

+omputers ] !perations/esearch 4; (7 , 558:58-.4. Abboud. F. ., 4??-. A discrete:time 1arkov production:inventory

model with machine breakdowns. +omputers ] Industrial

ngineering 59, 98:-?;.5. Aghe33af, .6., Uamali, 1.A., Ait:Eadi, ., 4??;. An integrated

production and preventive maintenance planning model. uropean

Uournal of !perational /esearch ->-, ;9: >8.7. 0aner ee, A., -9> . Ad oint economic lot si3e model for purchaser and

vendor. ecision )ciences -;, 494:5--.8. 0en: aya, 1 6ariga, 1 4??7. Integrated single vendor single buyer

model with stochastic demand and variable lead time. International

Uournal of $roduction conomics 94,;8:>?.



. 0en: aya, 1., arwish, 1., ] rtogral, E. (4??> . The oint

economic lotsi3ing problem. /eview and extensions. uropean

Uournal of !perational /esearch, ->8, 4 :;74.;. +ampos, *., Serdegay, U.*., -9>9. *inear $rogramming problems and

ranking of fu33y numbers. Cu33y sets and systems. -:--.>. +hen, ).6., -9>8. !perations on fu33y numbers with function

principle. Tarnkang Uournal of 1anagement )ciences (- -5:48.9. +hen, ).6., 6sieh, +.6., -99>. =raded 1ean Integration

/epresentation of =enerali3ed fu33y numbers. In the sixth conference

on Cu33y Theory and its applications (+ /!1 , filename Q ?5-.wdl,

+hinese Cu33y Association, Taiwan -: .-?. +hen, D.+., 4?? . !ptimal inspection and economical production

&uantity strategy for an imperfect production process. International

Uournal of )ystems )cience 5; (8 , 498:5?4.--. +hiu, D.).$, 2ang, ).), Ting, +.E., +huang, 6.U *ien, D.*, 4??>.

!ptimal run time for 1". model with backordering, failure:in:

rework and breakdown happening in stock:piling time. 2) A)

Transaction on Information )cience ] Applications 7, 7;8:7> .-4. +hristoph =lock. A multiple:vendor single:buyer integrated inventory

model with a variable number of vendors. +omputer sand Industrial

ngineering ?(4?- - -;5:->4.-5. +hung, +.U., 2idyadana, =.A., 2ee, 6.1., 4?--. conomic

production &uantity model for deteriorating inventory with random

machine unavailability and shortage. International Uournal of

$roduction /esearch 79, >>5:9?4.-7. l:Cerik. )., 4??>. conomic production lot:si3ing for an unreliable

machine under imperfect age:based maintenance policy. uropean

Uournal of !perational /esearch -> .-8?:- 5.



-8. rtogral. E., arwish, 1., 0en: aya. 1., 4??;. $roduction and

shipment lot si3ing in a vendor:buyer supply chain with transportation

cost. uropean Uournal of !perational /esearch -; , -894:- ? .- . =iri, 0.+.. ohi, T., 4??8. +omputational aspects of an extended

1". model with variable production rate. +omputers ] !perations

/esearch 54, 5-75:5- -.-;. =lock, +.6. (4?-?a . )upply chain coordination via integrated

inventory models. A review, working paper. Oniversity of 2uer3burg.->. =oyal., ).E., Febebe, C., 4???. etermination of economic

production:shipment policy for a single:vendor:single:buyer system.

uropean Uournal of !perational /esearch -4-, -;8:-;>.-9. 6ill, /.1 (-99; . The single:vendor single:buyer integrated

production:inventory model with a generali3ed policy. uropean

Uournal of !perational /esearch. 9;, 795:799.4?. 6uang, +.E., 4??7. An optimal policy for a single:vendor single:

buyer integrated production:inventory problem with process

unreliability consideration. International Uournal of $roduction

conomics 9-, 9-:9>.4-. 6ui 1ing 2ee, =ede Augs 2idyadana, 4?--. )ingle vendor )ingle

0uyer Inventory 1odel with discrete delivery order, random machine

unavailability time and lost sales, International Uournal of $roduction

conomics doi Q -?.-?- # .i pe.4?--.--.?-9.44. Uaber, 1.D., !sman, I.6 4?? . +oordinating a two:level supply chain

with delay in payments and profit sharing. +omputers ] Industrial

ngineering 8?, 5>8:7??.45. Eaufmann, A., =upta, 1.1., -99-. Introduction to fu33y arithmetic

theory and applications. San Fostrand /einhold.47. *ee, 6.1., Dao, U.)., -99>. conomic production &uantity for fu33y

demand &uantity and fu33y production &uantity. uropean Uournal of

!perational /esearch, -?9, 4?5:4--.







9)'@ use$ trape!oi$al fu!!y num;er to fu!!ify the or$er cost0 inventorycost an$ ;ackor$er cost in the total cost of the inventory mo$elwithout ;ackor$er2 4hen0 they foun$ the estimate of the total cost inthe fu!!y sense ;y functional principle2 #irko u:osevicD et al2 9))@ use$trape!oi$al fu!!y num;er to fu!!ify the or$er cost in the total cost of the inventory mo$el without ;ackor$er2 4hen0 they got fu!!y total cost2

4hey o;taine$ the estimate of the total cost through centroi$ to$efu!!ify2 In 9%@0 Eao an$ 8ee fu!!i=e$ the or$er >uantity q as thetrape!oi$al fu!!y num;er for the inventory without ;ackor$er an$o;taine$ the fu!!y total cost2

In the crisp inventory mo$els0 all the parameters in the total costare known an$ have $e=nite values without am;iguity0 as well as thereal varia;le of the total cost is positive2 But in the reality0 it is not sosure2 Hence it is essential to consi$er the fu!!y inventory mo$els2 Inor$er to simplify the calculation of trape!oi$al fu!!y num;er0 we use"henDs unction /rinciple 9 @ instea$ of <&tension /rinciple to calculatethe fu!!y :oint total cost of our planne$ mo$el2 unction /rinciple ispro:ecte$ as the fu!!y arithmetical operations of fu!!y num;ers in)% *2

Also the principle is making that it $oes not change the type of mem;ership function un$er fu!!y arithmetical operations of fu!!ynum;er2 In the fu!!y sense0 it is reasona;le to $iscuss the gra$e of each point of support set of fu!!y num;er for representing fu!!ynum;er2 4herefore0 "hen an$ HsiehDs Fra$e$ #ean Integration7epresentation metho$ 9'@ a$opte$ gra$e as the important $egree of each point of support set of generali!e$ fu!!y num;er2 With thisreason0 we use it to $efu!!ify the trape!oi$al fu!!y :oint total cost2 Infu!!y three-echelon :oint mo$el for crisp :oint shipment >uantity0 the=rst $erivative of fu!!y :oint total cost is use$ to solve the optimal :ointshipment >uantity2 urthermore0 the algorithm of <&tension of the8agrangean metho$ 9)*@ is use$ to solve ine>uality constraints in fu!!y

:oint mo$el for :oint shipment >uantity2



<conomic lot si!e mo$els have ;een stu$ie$ e&tensively2 SinceHarris 9 @presente$ the eminent <3G formula in )%)'2 ive years later0the economic pro$uction >uantity (</G6 inventory mo$el wasanticipate$ ;y 4aft 9) @2 However0 in recent years0 ;oth aca$emiciansan$ researchers have shown an increasing level of interest in =n$ingalternative ways to solve inventory mo$els2 In this paper0 we intro$ucethe fu!!y all :oint optimal mo$el in which fu!!y parameters an$ fu!!y

:oint shipment >uantity are all trape!oi$al fu!!y num;ers2

()( sym&ols +se!

4he retailerDs $eman$ an$ cost parameters are

D

fu!!y annual $eman$ >uantity of the retailer~ A

7 fu!!y retailerDs or$ering cost per contract

~

h7 fu!!y stock hol$ing cost per unit per year for the retailer

~

Z 7 fu!!y =&e$ cost of receiving a shipment from the $istri;ution

center

7 fu!!y cost of losing e&i;ility per unit per year

4he $istri;ution centerDs cost parameters are~ A

J fu!!y $istri;ution centerDs or$ering cost per contract

hJ fu!!y stock hol$ing cost per unit per year for the $istri;ution

center~

Z )J fu!!y =&e$ cost of receiving a shipment from the manufacturer

Z ,J fu!!y cost of a shipment from the $istri;ution center to the

retailer



J fu!!y cost of losing e&i;ility per unit per year

4he manufacturerDs pro$uction an$ cost parameters are

P fu!!y annual pro$uction rate of the manufacturer

~ A

# fu!!y =&e$ pro$uction setup cost per lot

~

Z # fu!!y cost of a shipment from the manufacturer the $istri;ution

centre~

h# fu!!y stock hol$ing cost per unit per year for the manufacturer

n num;er of shipments from the $istri;ution cener

m , num;er of shipments per lot from the manufacturer

>7 shipment >uantity using the retailerDs optimum mo$el

: 4hree echelon in$e& (: )0,0'6~

q : fu!!y shipment >uantity using the all-:oint optimum mo$el

~

q : L fu!!y optimal value of > :

~

T 7" : (n0 m , 6 :oint total cost of the three echelon mo$el

/M~

T 7" : (n0m , 6N Jefu!!i=e$ value of fu!!y total cost /9~

T 7" : (n0m , 6@

for three echelon mo$el

(), Assumptions'

Jeman$ for the item is constant over time2Shortages are not allowe$2

4ime hori!on is in=nite2





this type of generali!e$ fu!!y num;er ;e $enote$ asA

(a ) 0 a , 0 a ' 0a R wA687 2When

w A )0it can ;e simpli=e$ as

A

(a ) 0 a , 0 a ' 0 a 687 2

,),) Trape$oi!al #u$$y um&er

4he fu!!y num;erA

(a ) 0 a , 0 a ' 0 a 60 where a ) Q a , Q a ' Qa an$ $e=ne$

on 7 is calle$ the trape!oi$al fu!!y num;er0 if the mem;ership

functionA

is given ;y

A(x5

- 7

-- 4

4 -

4 5

75 7

5 7

? B x G a or x H a

(x : a B a x a

(a : a

- B a x a

(x : a B a x a

(a : a

≤ <

≤ <

≤ ≤

,).) The #unction %rinciple

The function principle was familiari3ed by +hen %7' to treat fu33y arithmeticaloperations. This principle is used for the operation of addition, multiplication, subtraction

and division of fu33y numbers.)upposeA

K (a- , a4, a5, a7 and0

K (b- , b4, b5, b7 are twotrape3oidal fu33y numbers. Then

?

-

a -

a 4

a 5

a 7

Ab (x5



)24he a$$ition ofA

an$0

isA

0

(a ) ; ) 0 a , ; , 0 a ' ; ' 0 a; 60

where a ) 0 a, 0 a ' 0 a 0 ; )0 ; , 0 ; ' an$ ; are any real num;ers2

,24he multiplication ofA

an$0

isA

0

(c ) 0 c , 0 c ' 0 c 60where4 Ma) ; ) 0 a) ; 0 a ; ) 0

a ; N0 4) Ma , ; , 0 a , ; ' 0 a ' ; , 0 a ' ; ' N0 c) min 40c , min 4 ) 0 c' ma& 4 ) 0 c ma& 42

If a ) 0 a , 0 a ' 0 a 0 ; ) 0 ; , 0 ; ' an$ ; are all non !ero positive real

num;ers0 thenA

0

Ma) ; ) 0 a , ; , 0 a ' ; ' 0 a ; N

'2-0

(-; 0 -; ' 0 -; , 0 -; ) 60 then the su;traction ofA

an$0

isA

0 (a ) - ; 0 a, - ; ' 0

a ' - ; , 0 a T ; ) 60 where a ) 0 a, 0 a ' 0 a 0 ; ) 0 ; , 0 ; ' an$ ; are any realnum;ers2

2

-

0

0-)

7 5 4 -

- - - -, , ,

b b b b

÷

0 where ; ) 0 ; , 0 ; ' an$ ; are all positive realnum;ers2 If

a ) 0 a , 0 a ' 0 a 0 ; ) 0 ; , 0 ; ' an$ ; are all non-!ero positive real num;ers0then the $ivision of

A an$ 0 is A 0

5- 4 7

7 5 4 -

aa a a, , ,

b b b b ÷

*2 8et k 7 then k A

- 4 5 7

7 5 4 -

(ka , ka , ka , ka , if k ?

(ka , ka , ka , ka , if k G ?

≥



,)/) Gra!e! ean *ntegration 0epresentation etho!

If

A

(a ) 0 a , 0 a ' 0 a R wA687 is a generali!e$ fu!!y num;er then the

$efu!!i=e$ value( )$ A5

;y gra$e$ mean integration representationmetho$ is given ;y

( )$ A5

A Aw w:- :-

? ?

* (h L/ (hh dh h dh

4

÷ ∫ ∫ with 1 Q h ≤ w A an$ 1 Q w A ≤ )2

IfA

(a ) 0 a, 0 a ' 0 a 6 is a trape!oi$al num;er0 then the gra$e$

mean integration representation ofA

;y a;ove formula is

( )$ A5

- -- 7 4 - 7 5

? ?

a L a L (a : a a L a hh dh h dh

4−

÷ ∫ ∫

( )$ A5

- 4 5 7a L 4a L 4a L a

,)1) E2tension of the 3agrangean etho!

In 9)%@0 4aha $iscusse$ how to solve the optimum solution of non-linear programming pro;lem with e>uality constraints ;y using8agrangean metho$D an$ showe$ how the 8agrangean metho$ may ;ee&ten$e$ to solve ine>uality constraints2 4he general i$ea of e&ten$ing

the 8agrangean proce$ure is that if the unconstraine$ optimumpro;lem $oes not satisfy all constraints0 the constraine$ optimum mustoccur at a ;oun$ary point of the solution space2

Suppose that the pro;lem is given ;y

#inimi!e y f(&6



Su;:ect to g i(&6≥ 10 i )0 ,02 2 2 m2

4he non-negativity constraints & ≥ 10 if any0 are inclu$e$ in the mconstraints2 4hen the proce$ure of <&tension of the 8agrangeanmetho$ involves the following steps2

Step 4(5 '

Solve the unconstraine$ pro;lem2 #inimi!e y f(&62 If theresulting optimum satis=es all the constraints0 stop ;ecause allconstraints are re$un$ant2 3therwise0 set k ) an$ go to step ,2

Step 4,5 '

Activate any k constraints (ie2 convert them into e>uality6 an$optimi!e f(&6 su;:ect to the k active constraints ;y the 8agrangeanmetho$2 If the resulting solution is feasi;le with respect to theremaining constraints0 stop R it is a local optimum2 3therwise0 activateanother set of k-constraints an$ repeat the step2 If all sets of activeconstraints taken k at a time are consi$ere$ without encountering afeasi;le solution0 go to step '2

Step 4.5 '

If k m0 stop no feasi;le solution e&ists2 3therwise0 set k k ) an$ go to step ,2

.) #u$$y athematical o!el

In this metho$ retailers $eman$0 manufactures pro$uction ratean$ the cost parameters are represente$ ;y trape!oi$al fu!!ynum;ers2 8et the fu!!y num;ers J0 A 7 0 h7 0 7 0 θ7 0 AJ 0 hJ 0 )J 0

,J 0 θJ 0/0A# 0 # 0 h# an$ > : are represente$ as

($ ) 0 $, 0 $' 0 $ 60/ A5

(a r) 0 ar, 0 ar ' 0 ar 60/ h5

(h r) 0 hr, 0 hr ' 0 hr 60/ <5

(! r) 0 ! r, 0 ! r ' 0 ! r 60





7 7 7

7 7 7 7

-

m d r m 7 -d 4d 7 r 7

4 4 4

a a a-3 d L3 Ln3 d n3 d

& m m m

+ + + + ÷ ÷ ÷

7 77 7 7

7

d d 4m 4 r r 47 7

- 4 - 4

n:-h L m

h m h L nmd - 4d n& - L4 p m p m 4 4n

+ − − + + ÷

By Fra$e$ #ean Integration0 solve the unconstraine$ pro;lem

#inimi!e

( )( ) 4T/+$ n,m5

- - -

- - - -

7

m d r m - -d 4d - r -

, 4 4 4

a a a- -3 d L3 Ln3 d n3 d

I & m m m

+ + + + ÷ ÷ ÷

- -- - -

-

d d 4m 4 r r 4- -

,7 4 7 4

n:-h L mh m h L nmd - 4d n& - L

4 p m p m 4 4n

+ − − + + ÷

4 4 4

4 4 4 4

5

m d r m 4 -d 4d 4 r 4

4 4 4

a a a-4 3 d L3 Ln3 d n3 d

& m m m

+ + + + + ÷ ÷ ÷

4 44 4 4

4

d d 4m 4 r r 44 4

5 4 5 4


4 p m p m 4 4n

÷ + − − + + ÷ ÷

÷ ÷

5 5 5

5 5 5 5

4

m d r m 5 -d 4d 5 r 5

4 4 4

a a a-4 3 d L3 Ln3 d n3 d

& m m m

+ + + + + ÷ ÷ ÷



5 55 5 5

5

d d 4m 4 r r 45 5

4 4 4 4

n:-h L mh m h L nmd 4d- n& - L

4 p m p m 4 4n

÷ + − − + + ÷ ÷

÷ ÷

7 7 7

7 7 7 7

-

m d r m 7 -d 4d 7 r 7

4 4 4

a a a-3 d L3 Ln3 d n3 d

& m m m + + + + + ÷ ÷ ÷

7 77 7 7

7

d d 4m 4 r r 47 7

- 4 - 4


4 p m p m 4 4n

+ − − + + ÷

V22 ()6 With 1 >: )

>: , >: ' >: 2

JiXerentiate ()6 partially with respect to >: ) 0 >:, 0 >:' 0 >: an$ all the

partial $erivatives e>ual to !ero0 an$ solve them2 4hose are

7 7 7

7 7 7 7

-

- -- - -

m d r m 7 -d 4d 7 r 7

4 4 4

d d 4m 4 r r 4- -

7 4 7 4

a a a3 d L3 Ln3 d n3 d

m m m& K

n:-h L mh m h L nmd - 4d n- L

4 p m p m 4 4n

+ + + + ÷ ÷ ÷

− − + + ÷

5 5 5

5 5 5 5

4

4 44 4 4

m d r m 5 -d 4d 5 r 5

4 4 4

d d 4m 4 r r 44 4

5 4 5 4

a a a4 3 d L3 Ln3 d n3 d

m m m& Kn:-

h L mh m h L nmd - 4d n4 - L4 p m p m 4 4n

+ + + + ÷ ÷ ÷

− − + + ÷



4 4 4

4 4 4 4

5

5 55 5 5

m d r m 4 -d 4d 4 r 4

4 4 4

d d 4m 4 r r 45 5

4 4 4 4


m m m& K

n:-h L mh m h L nmd 4d- n4 - L

4 p m p m 4 4n

+ + + + ÷ ÷ ÷ − − + + ÷

- - -

- - - -

7

7 77 7 7

m d r m - -d 4d - r -

4 4 4

d d 4m 4 r r 47 7

- 4 - 4


m m m& K

n:- h L mh m h L nmd - 4d n- L4 p m p m 4 4n

+ + + + ÷ ÷ ÷

− − + + ÷

4he a;ove show that with >: ) Y >: , Y >: ' Y >: 0 it $oes not satisfy

the constraint with 1 >: ) >: , >: ' >: 0 set k ) an$ go to step

,2

"onvert the ine>uality constraint into e>uality constraint >: , - >: )

1 an$ optimi!e ()6 su;:ect to >: , - >: ) 1 ;y the 8agrangean

metho$2 We have 8agrangean function is 8(>: ) 0 >:, 0 >:' 0 >: 0 λ6

( )( ) 4T/+$ n,m5

- λ(>: , - >: ) 62 8et all the partial $erivatives e>ual to !ero

an$ solve >: ) 0 >:, 0 >:' an$ >: 0 then we get



5 5 5

5 5 5 5

7 7 7

7 7 7 7

- 4

-

m d r m 5 -d 4d 5 r 5

4 4 4

m d r m 7 -d 4d 7 r 7

4 4 4

m 4 - -

7 4 7 4


m m m

a a a 3 d L3 Ln3 d n3 d

m m m& & n:-

h m d - 4d n-4 p m p m

+ + + + ÷ ÷ ÷ + + + + + ÷ ÷ ÷

= = − − + + ÷

- -- -

4 44 4 4

d d 4r r 4

d d 4m 4 r r 44 4

5 4 5 4

h L m h L nmL

4 4n

n:-h L mh m h L nmd - 4d n L 4 - L

4 p m p m 4 4n

− − + + ÷

an$

4 4 4

4 4 4 4

5

5 55 5 5

m d r m 4 -d 4d 4 r 4

4 4 4

d d 4m 4 r r 45 5

4 4 4 4


m m m& K ,

n:- h L mh m h L nmd 4d- n4 - L4 p m p m 4 4n

+ + + + ÷ ÷ ÷ − − + + ÷

- - -

- - - -

7

7 77 7 7

m d r m - -d 4d - r -

4 4 4

d d 4m 4 r r 47 7

- 4 - 4

a a a3 d L3 Ln3 d n3 dm m m

& Kn:-

h L mh m h L nmd - 4d n- L4 p m p m 4 4n

+ + + + ÷ ÷ ÷

− − + + ÷

4he a;ove show that >: ' Y >: 0 it $oes not satisfy the constraint 1 Q

>:) >: , >: ' >: 2 4herefore it is not a local optimum2 Set k , an$

go to step '2

Step . '



"onvert the ine>uality constraint >:, - >:) Z 1 an$ >: ' - >: , Z

1into e>uality constraints >:, - >:) 1 an$ >: ' - >: , 12 We optimi!e

( )( ) 4T/+$ n,m5

su;:ect to >: , - >: ) 1 an$ >: ' - >: , 1 ;y the8agrangean metho$2 4hen the 8agrangean function is 8(>: ) 0 >:, 0 >:' 0

>: 0λ1, λ26

( )( ) 4T/+$ n,m5

- λ1(>: , - >: ) 6 - λ2(>: ' - >: , 62 8et all the

partial $erivatives e>ual to !ero an$ solve >: ) 0 >:, 0 >:' an$ >: 0 then

we get

7 7 7

7 7 7 7

5 5 5

5 5 5 5

4 4

4 4 4

- 4 5

m d r m 7 -d 4d 7 r 7

4 4 4

m d r m 5 -d 4d 5 r 5

4 4 4

m d r m 4 -d 4d 4

4 4


m m m


m m m

a a a L4 3 d L3 Ln3 d

m m& & &

+ + + + ÷ ÷ ÷ + + + + + ÷ ÷ ÷

+ + + ÷ ÷ = = =4

4

- -- - -

4 44 4 4

5

r 44

d d 4m 4 r r 4- -

7 4 7 4

d d 4m 4 r r 44 4

5 4 5 4

m 4 5 5

4 4

n3 dm


n:- h L mh m h L nmd - 4d n L 4 - L4 p m p m 4 4n

h m d 4d- 4 -4 p m

+ ÷

− − + + ÷

− − + + ÷

+ − − + 5 55 5

d d 4 r r 4

4 4

n:-h L m h L nmn L

p m 4 4n

+ ÷



an$

- - -

- - - -

7

7 77 7 7

m d r m - -d 4d - r -

4 4 4

d d 4m 4 r r 47 7

- 4 - 4


m m m&


4 p m p m 4 4n

+ + + + ÷ ÷ ÷ = − − + + ÷

4he a;ove result show that >: ) Y >: 0 it $oes not satisfy the

constraint 1 >: ) >: , >: ' >: 0 therefore it is not a local optimum2

Set k ' an$ go to step 2

Step / '

"onvert the ine>uality constraints >: , - >: ) ≥ 10 >: ' - >: , ≥ 1 an$

>: - >: ' ≥ 1 into e>uality constraints >: , - >: ) 10 >: ' - >: , 1 an$ >: -

>: ' 12 We optimi!e

( )( ) 4T/+$ n,m5

su;:ect to >: , - >: ) 10 >: ' - >: ,

1 an$ >: - >: ' 1 ;y the 8agrangean metho$2 4he 8agrangean

function is given ;y 8(>: ) 0 >:, 0 >:' 0 >: 0 λ1, λ2, λ36

( )( ) 4T/+$ n,m5

-

λ1(>: , - >: ) 6 - λ2(>: ' - >: , 6 - λ3(>: - >: ' 62 In or$er to =n$ the

minimi!ation of 8(>: ) 0 >:, 0 >:' 0 >: 0 λ1, λ2, λ36 we take the partial

$erivatives of 8(>: ) 0 >:, 0 >:' 0 >: 0 λ1, λ2, λ36 with respect to >: ) 0 >:, 0 >:' 0

>: 0 λ1, λ2an$ λ3an$ let all the partial $erivatives e>ual to !ero to solve

>:) 0 >:, 0 >:' an$ >: then we get



- - -

- - - -

4 4 4

4 4 4 4

5 5

5 5 5

- 4 5 7

m d r m - -d 4d - r -

4 4 4

m d r m 4 -d 4d 4 r 4

4 4 4

m dm 5 -d 4d 5

4 4


m m m

a a aL4 3 d L3 Ln3 d n3 d

m m m

a a4 3 d L3 Ln3 d

m m

& & & &

+ + + + ÷ ÷ ÷

+ + + + ÷ ÷ ÷ + + + ÷ ÷

= = = =

5

5

7 7 7

7 7 7 7

- -- - -

4

r r 5

4

m d r m 7 -d 4d 7 r 7

4 4 4

d d 4m 4 r r 4- -

7 4 7 4

m 4 4 4

5 4 5 4

an3 d

m


m m m


4 p m p m 4 4n

h m d - 4d L 4 -4 p m p m

+ + ÷ + + + + + ÷ ÷ ÷

− − + + ÷

− − +

4 44 4

5 55 5 5

7 77 7 7

d d 4r r 4

d d 4m 4 r r 45 5

4 4 4 4

d d 4m 4 r r 47 7

- 4 - 4

n:-h L m h L nm

n L4 4n

n:-h L mh m h L nmd 4d- n L4 - L

4 p m p m 4 4n

n:-h L mh m h L nmd - 4d n - L

4 p m p m 4 4n

+ ÷

− − + + ÷

+ − − + + ÷

2 2 2

(,6

Because the a;ove solution &

(> :) 0 > :, 0 > :' 0 > : 6 satis=es all

ine>uality constraints0 the proce$ure terminates with & as a local

optimum solution to the pro;lem2 Since the a;ove local optimumsolution is the only one feasi;le solution of gra$e$ mean integrationformula0 so it is an optimum solution of the inventory mo$el with fu!!y

:oint shipment >uantity accor$ing to e&tension of the 8agrangeanmetho$2



8et > :) > :, > :' > : > :0 then the optimal fu!!y :oint shipment

>uantity is( )M M M M M

& K & , & , & , & , , where5

/) umerical E2ample

37J manufacturing company pro$uces commercial car units in ;atch2 4he =rm estimate$ that the fu!!y annual $eman$ >uantity (

~

D 6 of

the retailer is )*11 units0 the fu!!y retailerDs or$ering cost (~ A

7 6 per

contract is a;out [ 110 the fu!!y stock hol$ing cost( h7 6 per unit per

- - -

- - - -

4 4 4

4 4 4 4

5 5 5

5 5 5 5

m d r m - -d 4d - r -

4 4 4

m d r m 4 -d 4d 4 r 4

4 4 4

m d r m 5 -d 4d 5 r

4 4 4

M

a a a3 d L3 Ln3 d n3 dm m m

a a aL4 3 d L3 Ln3 d n3 d

m m m

a a a4 3 d L3 Ln3 d n3

m m m

&

+ + + + ÷ ÷ ÷

+ + + + ÷ ÷ ÷ + + + + + ÷ ÷

=5

7 7 7

7 7 7 7

- -- - -

44

5

m d r m 7 -d 4d 7 r 7

4 4 4

d d 4m 4 r r 4- -

7 4 7 4

dm 4 4 4

5 4 5 4

d


m m m


n:-h Lh m d - 4d n L 4 -

4 p m p m

÷ + + + + + ÷ ÷ ÷

− − + + ÷

− − + + ÷

44 4

5 55 5 5

7 77 7 7

d 4r r 4

d d 4m 4 r r 45 5

4 4 4 4

d d 4m 4 r r 47 7

- 4 - 4

m h L nmL

4 4n

n:-h L mh m h L nmd 4d- n L4 - L

4 p m p m 4 4n

n:-h L mh m h L nmd - 4d n - L

4 p m p m 4 4n

− − + + ÷

+ − − + + ÷



year for the retailer is a;out [, 10 the fu!!y =&e$ cost ( Z 7 6 of

receiving a shipment from the $istri;ution center is a;out [)*0 the

fu!!y cost of losing e&i;ility (~

7 6per unit per year is a;out [) 0 the

fu!!y $istri;ution centreDs or$ering cost ( A J 6 per contract is a;out

[ 110 the fu!!y stock hol$ing cost( hJ 6 per unit per year for the

$istri;ution centre is a;out [ 2+0 the fu!!y =&e$ cost( Z )J 6 of

receiving a shipment from the manufacturer is a;out ['10 the fu!!y

cost of a shipment(~

Z ,J 6 from the $istri;ution centre to the retailer is

a;out [*10 the fu!!y cost of losing e&i;ility (~

J 6per unit per year is

a;out [),2+%0 the fu!!y annual pro$uction rate ( P 6 of the

manufacturer is ,111 units0 the fu!!y =&e$ pro$uction setup cost( A

# 6 per lot is a;out [)+110 the fu!!y cost of a shipment ( Z # 6from the

manufacturer the $istri;ution centre is a;out ['110 the fu!!y stock

hol$ing cost ( h# 6 per unit per year for the manufacturer is a;out

[ 2+0 the num;er of shipments (n6 from the $istri;ution centre is '0the num;er of shipments (m , 6 per lot from the manufacturer is 2Howmany car units shoul$ 37J manufacturing company pro$uce in each;atchC

Here we use a general rule to transfer the linguistic $ata0 \a;out ]^0into trape!oi$al fu!!y num;ers as \a;out ]^ (12%*]0 ]0 ]0 )21*]62Byusing the a;ove rule0 the fu!!y parameters in this e&ample can ;etransferre$ as follows

D () ,*0 )*110 )*110 )*.*6R A 7 (' 10 110 110 ,16R





propose$ integrate$ mo$el is suita;le for either an in$epen$ent :ust intime (_I46 system or pro=t centers from the main o ce2 4he result of the e&ample applying this mo$el is >uite feasi;le an$ satisfactory2 4hismo$el coul$ ;e applie$ in more practical an$ sophisticate$ cases2 or

a special case that all varia;les are set as real num;ers0 the result will;e the same as the tra$itional non-fu!!y mo$el2

References

[1] Apte, U. M., !is"anathan, S. #$%%%&. '(ecti)e cross doc*ing forimpro)ing

distri+ tion e-ciencies. International o rnal of /ogistics, 0, 12

0%$.

[$] Arora, !., Chan, 3. 4. S., 4i"ari, M. 5. #$%% &. An integratedapproach for logistic

and )endor managed in)entory in s pply chain. '6pert Systems "ith Applications,7,7. doi8 1%.1%19:;.es"a.$%% .%<.%19.

[0] S.=.Chen,C.=.=sieh, >raded Mean Integration ?epresentation of>enerali@ed f @@y

n m+er, o rnal of Chinese 3 @@y Systems. <#$& #1 & 1 B.

[ ] S.=. Chen, Dperations on f @@y n m+ers "ith f nction principle,4am*ang o rnal

of Management Sciences 9#1& #1 7<&, 10 $9.

[<] S.=. Chen, C.C. Wang, A. ?amer, Eac*order f @@y in)entory modelnder



f nction principle, Information Sciences, an International o rnal <#1 9& B12B .

[9] S.=. Chen, C.=. =sieh, Dptimi@ation of f @@y simple in)entorymodels,In8 I''' International 3 @@y System Conference Froceedings,Seo l, 5orea,)ol. 1,1 , pp. $ %2$ .

[B] Chih =s n =sieh, #$%%$&. Dptimi@ation of f @@y prod ctionin)entory models,

Information, Sciences, 1 9. F.G.$ %.

[7] 3.W. =arris, =o" many parts to ma*e at once, factoryH, MagManage. 1% #1 10& 10<2109 #p. 1<$&.

[ ] ing Shing ao and = ey Ming /ee, 3 @@y In)entory "ith or "itho tEac*order for

3 @@y Drder J antity "ith 4rape@oid f @@y n m+er,3 @@y sets and

Systems 1%<,#1 & 011 00B

[1%] 5reng, E.!, Chen, 3. 4. #$%%B&. 4hree echelon + yer2s pplierdeli)ery policy 2 A

s pply chain colla+oration approach. Frod ction Flanning andControl, 17# &,0072 0 .

[11] Mir*o ! ;ose)ic, Kofrila Fetro)ic and ?adi)o; Fetro)ic, 'DJ3orm la

"henIn)entoryCost is f @@y, Int. . Frod ction 'conomics <#1 9&<%



[1$] 5.S. Far*, 3 @@y set theoretic interpretation of economic orderq antity, I'''

4ransactions on Systems, Man, and Cy+ernetics SMC 1B #1 7B&1%7$21%7 .

[10] Shan = o Chen and Chien Ch ng Wang,Eac*order 3 @@y In)entory Model nder

3 nction Frinciple, Information Science <,#1 9& B1 B

[1 ] '.W.4aft, 4he most economical prod ction lot, 4he Iron Age1%1

#1 17&1 1%2 1 1$.[1<]

[1<]=.A. 4aha in 8 Dperations ?esearch, Frentice 2 =all, 'ngle"oodCli(s, G , USA,1 B, pp.B<0BBB.

[19] 4oni, A. K., Gassim+eni, >. #$%%%&. st in time p rchasing8 anempirical st dy of

operational practices, s pplier de)elopment and performance.Dmega, $7,

[1B] !an =o t m, >. ., Inderf rth, 5., Li;m, W. =. M. #1 9&.Materials coordination

instochastic m lti echelon systems. ' ropean o rnal ofDperational ?esearch, <,1 $0.

[17] !ictor E. 5reng a , 3ang 4@ Chen + #$%11& 4he +ene ts of a threele)el coordinated

distri+ tion policy in the )al e chain, e6pert systems "ithapplications , 07,70< 7 $



[1 ] Wagner, S. M. #$%%9&. A rmNs responses to de cient s ppliers andcompetiti)e

ad)antage. o rnal of E siness ?esearch, < , 97929 <.90129<1.

[$%] O , 5., Kong, ., ')ers, F. 4. #$%%1&. 4o"ards +etter coordinationof the s pply

chain. 4ransportation ?esearch Fart ', 0B, 0<2< .

[$1] Limmer, 5. #$%%$&. S pply chain coordination "ith ncertain ; stin time deli)ery.

International o rnal of Frod ction 'conomics, BB, 121<.7 $.

AC6* E 0E%A*0 %0O73E + 8E0 + CE0TA* E 9*0O E T

Sreelekha menon.B**

::8epartment of athematics; SC S School of engineering an! technology;

Kerala -<=. 1=,

A&stract

#achine repair is an important pro;lem fre>uently encountere$ in pro$uctionan$ manufacturing operations such as semicon$uctor manufacturing an$maintenance operations2 Jue to uncontrolla;le factors0 parameters in themachine repair pro;lem may ;e fu!!y2 Uncertainty associate$ with the input

parameter are solve$ using u!!y set theory 24his paper proposes a integernon linear programming 1 - ) approach to construct the mem;ershipfunction of the performance measure of the machine repair pro;lem with themachine ;reak$own rate an$ the service rate ;eing fu!!y num;ers2 Usinge&tension principle an$ ` cuts 0 a pair of mathematical programs isformulate$ to calculate the lower an$ upper ;oun$s of the fu!!y performancemeasures2 By enumerating $ifferent values of `0 the mem;ership function of the system performance measure is constructe$2



Key "or!s u!!y set theory0 machine interference system0 integer nonlinear programming 0 e&tension principle 0 ` cuts2

() *ntro!uction

#achine repair mo$els have wi$e applications in many practical situations0such as pro$uction line systems0 clientTserver computing0maintenanceoperations0 an$ manufacturing systems "onsi$er a manufacturing systemconsisting of K machines su;:ect to ;reak$owns from time to time2 When amachine ;reaks $own0 it is repaire$ ;y one of a crew of 7 repair persons0 thusthis repair person cannot repair other ;roken machines for a perio$ of time2

4hus0 $uring this ;usy perio$ of time0 it is possi;le that there are ;rokenmachines have to wait an$ are interfere$ with ;y the machine ;eingrepaire$2 #any stu$ies have ;een pu;lishe$ on the machine interferencepro;lem an$ its costOpro=t analysis recent stu$ies inclu$e 0 an$ so on2

< cient metho$s have ;een $evelope$ for analysing the machineinterference pro;lem when its parameters0 such as the ;reak$own rate an$service rate0 are known e&actly2 3ne commonly use$ type of solutionmetho$s is the >ueuing theory approach in that the machine interferencepro;lem is mo$ele$ as a =nite calling population >ueuing system2 4hemachine ;reak$owns are treate$ as the customers an$ the repair personsare servers in the system2 3n the ;asis of tra$itional >ueuing theory we can$erive the system performance measures (e2g20 the e&pecte$ num;er of ;roken machines0 the e&pecte$ time a machine spen$s waiting for repair0 an$the fraction of the time a particular repair

person is i$le6 of the machine interference pro;lem an$ its variants whentheir parameters are known e&actly2 However0 there are cases that theseparameters may not ;e presente$ precisely $ue to uncontrolla;le factors2Speci=cally0 in many practical applications0 the statistical $ata may ;eo;taine$ su;:ectivelyR e2g20 the ;reak$own pattern an$ repair pattern aremore suita;ly $escri;e$ ;y linguistic terms such as fast0 mo$erate0 or slow0rather than ;y pro;a;ility $istri;utions ;ase$ on statistical theory2 Impreciseinformation of this kin$ will un$ermine the >uality of $ecisions viaconventional machine interference mo$els2 4o $eal with impreciseinformation in making $ecision0 Bellman an$ a$eh an$ a$eh intro$uce$the concept of fu!!iness24o$ay0 fu!!y set theory is a well-known means formo$elling imprecision or uncertainty arising from mental phenomena2Speci=cally0 fu!!y >ueues have $iscusse$ ;y several researchers2 Since themachine interference pro;lem with fu!!y parameters is more realistic thanthe conventional machine interference pro;lem0 it $eserves furtherinvestigation2 However0 little research has ;een pu;lishe$ on fu!!y machineinterference pro;lems2 Buckley investigate$ multiple-channel >ueuing S2-/2"hen systems with =nite or in=nite waiting capacity an$ calling population0 in



that the arrivals an$ $epartures are followe$ ;y some possi;ility $istri;utions27ecently0 Buckley et al applie$ the previous results to the machine servicingpro;lem for $etermining the optimal si!e of crews2 "learly0 when the machine;reak$own rate or service rate are fu!!y0 the system performance measuresof the machine interference pro;lem will ;e fu!!y as well2 4o conserve the

fu!!iness of input information completely0 the fu!!y performance measureshoul$ ;e $escri;e$ ;y a mem;ership function rather than a crisp value2

,) #+>>? AC6* E 0E%A*0 O8E3

"onsi$er a conventional machine interference mo$el that consists of K machines

an$ 7 repair crews2 At any instant in time0 a particular machine is in eithergoo$ or ;a$ con$ition2 When a machine ;reaks $own0 one of 7 availa;lerepair crews is calle$ upon to $o the repair2 ?ormally a repair person is incharge of more than one machine (i2e20 7 Q K60 ;ut a repair person can repairat most only one machine at one time2 When a machine ;reaks at the perio$of time that all repair persons are ;usy0 it has to wait an$ is interfere$ with ;ythe machine ;eing repaire$2 3nce a machine is repaire$0 it returns to goo$con$ition an$ is again suscepti;le to ;reak$own2

4he length of time that a machine remains in goo$ con$ition follows ane&ponential $istri;ution with ;reak$own rate & (or the rate of ;reak$own permachine is & ;reak$owns per unit time60 an$ the time it takes to completerepairs on a ;roken machine is assume$ e&ponential with service rate y (ormean repair time is )Oy62 In fact0 the machine interference pro;lem is atypical e&ample of the >ueuing mo$els with =nite call ing population0 which is$enote$ as #O#O7OFJOKOK using Ken$allT8ee s notation Suppose the;reak$own rate & an$ service rate y are appro&imately known an$ can ;e

represente$ ;y conve& fu!!y setsλ

an$ respectively2

K

x(xx,

KDy(yy,

where ] an$ Eare the crisp universal sets of the ;reak$own rate an$ theservice

rate0 respectively2 Without loss of generality0 in this mo$el the ;reak$own



rate an$ the service rate are assume$ to ;e fu!!y num;ers0 as crisp valuescan

;e represente$ $egenerate$ mem;ership functions that only one value intheir

$omain2 4his mo$el will hereafter ;e $enote$ ;y #O #O7OFJOKOK0 where

$enotes fu!!y time an$ # $enotes fu!!i=e$ e&ponential time2

"learly0 when the ;reak$own rate an$ the service rate are fu!!y0 thesystem

performance measures in stea$y state are fu!!y as wellR thus0 they shoul$ ;e

$escri;e$ ;y mem;ership functions to conserve the fu!!iness of inputinformation

completely2 If the o;taine$ results are crisp values0 then it may lose some

useful information2

M%6e! %&te2er )r 2ra((%&2 a))r ac0

!ne approach to construct the membership function,, (f

is on the basis of

deriving theα cuts of

, enote the α cut of

, are

λα K ( O_*_ x,x K

{ } { }≥≥ _(xxx

max ,_(xx

x

min

o o

jjj. (-

µα K( O

_*_ y,y

K

{ } { }≥≥ _(yyDy

max ,_(yy

Dy

mino

///. 0 1These interval indicate where the batch arrival rate, service rate and vacation rate

lie at possibility level _. Fote that _, _, are crisp sets rather than fu33y sets. 0y theconcept of a cuts, the imbedded fu33y 1arkov +hain in C1#C1#/#= #E#E the can bedecomposed into the family of ordinary 1arkov +hains with different transition probability matrices, which are also parametrised by _. +onse&uently, theC1#C1#/#= #E#E &ueue can be reduced to a family of crisp &ueue 1#1#/#= #E#E

&ueues with different _ level sets{ }-_?o _ <<

and{ }-_?_ <<

. These sets causenested structures for expressing the relationship between ordinary sets and fu33y sets andthey represent sets of shifting boundaries.



0y the convexity of fu33y number the boundaries of these interval are function of _ andcan be obtained as

_(minx -

*_ =

O_x K _(max

-

*_y

K_(min -

O_y

K_(max -

respectively.

+learly defined membership function of,, (f

is also parameterised by _.+onse&uently we can use it’s _ cut to construct the corresponding membership function.

,, (f is also parameterised by _. 2e can use it as _ cut to construct the corresponding

membership function.

( )µλ=µλ=µµ=µ µλ

,*,(f <y(,x(

min)up3(*

jjj.( 5

3(*µis the minimum of

(x ,

(y we need either

_(x =

_(y ≥B oror

_(x ≥

_(y =BB

such that to satisfy3(*µ

K. _

To find the membership function3(*µ

, it will suffice to find the left shape function and

right shape function of3(*µ

, which is e&uivalent to find the lower boundary*

*

α and the

upper boundary*

O

α

of the _ cuts of

3(*µ.

To find the membership function3(*µ

it suffices to find the left shape function and right

shape function of3(*µ

, which is e&uivalent to find the lower bound*

*

α and upper

bound*

O

α of the _ cuts of

3(*µ.



)ince the re&uirement of_(x =

represented by

*_xx =

or

O_xx =

this

can be formulated as the constraint of

O_-

*_- x` (-x`x +=

whereβ- K ? or -.

)imilarly,

y K

O_4

*_4 y` (-y` +

, β4 K ? or -

1oreover, from the definition ofλα, µα in (- and x λ α y µ α can be replaced

by

O_

*_ x,xx

,[ ]O

_*_ y,yy

. +onse&uently, considering both of these two cases above,

the membership function3(*µ

can be constructed via finding the lower bound*

*

αand

upper bound*O

α of α cuts of * . In that we see

**

α K min

*,*4- **

αα

*O

α K max

*,*4- OO

αα respectively where

( )-*_*

K min,, (f

such that x Kt-

*_x

L(- t -

O_x

*

_y

≤ y ≤O

_y

t-K ? or - jjj ( 7

( )*4_*

Kmin,, (f

such that y K t4

*_y

L(- t 4

O_y

*_x ≤ x ≤

O_x

t4 K ? or - jjj ( 8

( ) -O_*

K max,, (f



such that x Kt5

*_x

L(- t 5

O_x

*_y

≤ y ≤O_y

t5K ? or - jjj (( )O4

_* K max

,, (f

such that y K t7

*_y

L(- t 7

O_y

*_x

≤ x ≤O_x

t7 K ? or - jjjj..(;

Crom the knowledge of calculus, a uni&ue minimum and a uni&ue maximum of the

ob ective models ( 7 ( 8 ( ( ; are assured which shows that the lower bound*

*

α

and upper bound*

O

α of the _ cuts of

* can be found by solving these six models. In fact

these four models are 1IF*$ with ?:- variables. There are several effective and efficientmethods for solving these problems .1oreover, they involve the systematic study of how

the optimal solutions change as

*_x

,O_x

,*_y

,O_y

, vary over the interval _ %?,-'B theyfall into the category of parametric programming .

Therefore, for two values _- and _4 such that ? G _4 G _- G - we have**

*

4

*

- αα ≥ and

**O

4

O

- αα ≤ in other words

**

α is non:decreasing with respect to _ and

*O

α

is non:increasing with respect to _.

If both*

*

α and

*O

α are invertible with respect to _, then the left shape function

( ) 2+-

S 3- 041 5 - and a right shape function

( ) 2+6

S 37 041 5 - can be obtained.

1embership function3(*µ

is constructed as



3(*

µ

≤ ≤≤ ≤≤ ≤

- -S35* 35+

- 635+ 35+

6 6S35+ 35*

- 041 0-1 4 0-1

+ 0-1 4 0-1

7 041 0-1 4 0-1

!ther membership functions of performance measures can be derived in the similar manner./) umerical e2ample

4o $emonstrate the vali$ity of the propose$ approach0 a numerical e&ample

inspire$ ;y Fross an$ Harris is solve$2

4he W2< 2 inish machine shop company has four turret lathes24hese machines;reak $own perio$ically an$ the company has two repair man to service thelathes when they ;reak $own2When a lathe is =&e$ 0 the time until the ne&t;reak$own is e&ponentially $istri;ute$ with a fu!!y rate that can ;erepresente$ ;y a trape!oi$al fu!!y num;er

λ

)0 , 0 ' 0 @ per $ay2 4he service time for each service is e&ponentially$istri;ute$ with a fu!!y rate that can ;e represente$ ;y a trape!oi$al fu!!ynum;er

µ +0 .0 %0)1@ per $ay2 4he shop manager wants to know the average

num;er of lathes operational at any time3(*µ

2 "learly0 this system can ;e$escri;e$ ;y the #O #O,OFJO O mo$el0 an$ the performance measures can;e constructe$ ;y using the propose$ approach2

The interest performance measures C1#C1#4#= #7#7 are given as

8(&0 y6

4 7

:/ K ? K 5

7

4 K 5

:/ K ?

7 x

7 y x i

y 4 4

7 x

7 y x

y 4 4

÷ ÷ + ÷ ÷ ÷ ÷ + ÷ ÷

∑ ∑

∑∑



8>

( ) 7

K 5

4 7

K ? K 5 :4

x : 4 7

y

x x 7 7 y y

(7 : (7 : 4 4

÷

÷ ÷ +

∑

∑ ∑

W *7(x

*−

*7(x

*2

&

&& −=

8et the ;reak $own an$ service rate ;eλ

)0 , 0 ' 0 @µ

+0 .0 %0)1@

%

*_x

,

O_x

' K %α L - , 7 : α '

%*_y

,O_y

' K % Lα , -? : α ]

8(&0 y6

4 5 7

4 5 7

x x x x7 L -4 L -> L -4

y y y y

x x x x- 7 L L L 5

y y y y

÷ ÷ ÷ ÷

+ ÷ ÷ ÷ ÷



4hus following mi&e$ integer non linear parametric 1 T ) programming is

constructe$ $eriving the mem;ership function of*

*_*

#in 9

5 4 4 5 7

7 5 5 4 4 7

7xy L -4x y L ->x y L -4xy 7xy L x y L x y L 5x+

@

s2tα−≤≤α+ 7x-

α−≤≤α+ -?y

O_*

#a& 9

5 4 4 5 7

7 5 5 4 4 7

7xy L -4x y L ->x y L -4xy 7xy L x y L x y L 5x+

@

s2tα−≤≤α+ 7x-

α−≤≤α+ -?y

4he shortest* α

occurs when & α ) an$ y )1- α

*_*

744557

75445

-(5-?(-(-?(-(-?(-(7-?(-(-4-?(-(->-?(-(-4-?(-(7

+α+α−+α+α−+α+α−+α+α−+α+α−+α+α−+α+α−+α

In the ma&imum response 0on the contrary 0 & - α an$ y + α



O_*

744557

75445

7(5(7((7((7(7(7(-4(7(->(7(-4(7(7 α−+α+α−+α+α−+α+α−+α+ α−+α+α−+α+α−+α+α−

It is a challenging task to =n$ their inverse functions analytically2 3wing tothe complicate$ form of the o;:ective function0it is impossi;le to represent

the optimal solution*

*

α an$

*O

α interms of α 2"onse>uentely0a close$ form

mem;ership function for 3(*µ cannot ;e o;taine$2 It However we

enumerate3(*µ

$iXerent values of α . Τa;le presents the α cuts at ))$istinct values : 1012)012,022222222)2124he mem;ership functions for thee&pecte$ >ueue length 0the e&pecte$ system waiting time an$ the e&pecte$>ueue waiting time can ;e $erive$ similarly2

8 >

5 7

7 5 5 4 4 7(yx L xy 7xy L x y L x y L 5x+

Similarly 0 the lower ;oun$s an$ upper ;oun$s of

*&*

744557

75

-(5-?(-(-?(-(-?(-(7-?(-(-?(-( +α+α−+α+α−+α+α−+α+α− +α+α−+α



O&*

744557

75

7(5(7((7((7(7(7(7((

α−+α+α−+α+α−+α+α−+α+α−+α−+α

*_2

*7(-(

**

*

α

α

−+α

O

_2

*7(7(

*O

O

α

α

−α−

_

*&2

*&7(-(

*&*

*

α

α

−+α

_

O&2

*&7(7(

*& O

O

α

α−α−

4a;le lists the a-cuts of their values at )) $istinct a values 10 12)0 12,0 2 2 2 0)212

α*_*

O_*

*&*

O&*

*_2

O_2

*&2

O&2

0 12'+.., )2 111211*

12' 1 121)1)

12,1*

1211)),

121,'

12)12 1*' )2.11

1211**

12')1

12)1,*

12)%%*

1211),+

121,,)



12,12 ',, )2.1'

1211..

12,%++

12)1' 12)%*)

1211)+)

121)%+

12'12 )1) )2+*'

1211% 12,.*%

12)1*)

12)%1'

1211)%1

121)%1

1212*)% + )2+1'

121),1

12,*+1

12)1++

12) *.

1211,) 121)%

12*12** ** )2**1

121)*1

12,'+%

12)1)

12) 1.

1211,*1

121).%

12+12*%.* )2*1

121)'

12,)+

12)1%.

12)..,

1211, 121).1

12.12+'+.) )2 *

121,)%

12,1))

12)))'

12).'1

1211',*

121)+1

1212+.+', )2 1*

121,+1

12)*

12))'1

12)+%)

1211'+'

121)*)

12%12.)+,, )2'*+

121'1+

12)++

12)).

12)+* 1211 1*

121))

)12.*+11 )2'1.

121'*+

12)*'.

12))+*

12)+) 1211 121)''

Conclusions

#achine interference mo$els play an important role in many real worl$situations0

such as maintenance operations an$ operationsOmanufacturing systems2 Inreal worl$ applications0 parameters in the machine interference pro;lem may;e fu!!y such that its system performance measures are fu!!y as well2 4hispaper proposes a mi&e$ integer non linear mathematical programmingapproach to =n$ the mem;ership function of the fu!!y performance measure

of the machine interference pro;lem with ;reak$own rate an$ service rate;eing fu!!y num;ers2 4he i$ea is ;ase$ on a$ehDs e&tension principle2 4helower an$ upper ;oun$s of the `-cuts of the fu!!y performance measure arecalculate$ via a pair of mathematical programs parametri!e$ ;y `2 rom$ifferent values of `0 the o;taine$ lower an$ upper ;oun$s of these a-cutsare a$opte$ to construct the correspon$ing mem;ership function2



0eferences

9)@ K2<2 Stecke0 _2<2 Aronson0 ?e)ie" of operator:machine interference

models 0 International _ournal of /ro$uction 7esearch ,' ()% *6

),%T)*)2

9,@ J2I2 "ho0 #2 /arlar0 A s r)ey of maintenance models for m lti nit

systems, <uropean _ournal of 3perational 7esearch *) ()%%)6 )T

,'2

9'@ J2 Fross0 "2#2 Harris0 3 ndamentals of J e eing 4heory 0 thir$ e$20

_ohn Wiley0 ?ew Eork0 )%% 2

9 @ <2A2 <lsaye$0 An optim m repair policy for the machine

interference pro+lem 0 _ournal of the 3perational 7esearch Society

', ()% )6 .%'T 1)2

9*@ J2 Fross0 _22 Ince0 4he machine repair pro+lem "ith

heterogeneo s pop lations 0 3perations 7esearch ,% ()% )6 *',T

* %2

9+@ K2H2 Wang0Fro t analysis of the machine repair pro+lem "ith a

single ser)ice station s +;ect to +rea*do"ns, _ournal of the

3perational 7esearch Society ) ()%%16 ))*'T))+12

9.@ K2H2 Wang0 Fro t analysis of the M:M:? machine repair pro+lem

"ith spares and ser)ice +rea*do"ns 0 _ournal of the 3perational

7esearch Society * ()%% 6 *'%T* 2

9 @ F2I2 alin0 A m ltiser)er retrial q e e "ith a nite n m+er of

so rces of primary calls 0 #athematical an$ "omputer #o$elling

'1 ()%%%6 ''T %2

9%@ _2"2 Ke0 K2H2 Wang0Cost analysis of theM:M:? machine repair

pro+lem "ith +al*ing, reneging, and ser)er +rea*do"ns 0 _ournal of

the 3perational 7esearch Society *1 ()%%%6 ,.*T, ,2



9)1@ _2B2 Atkinson0 Some related parado6es of q e ing theory8 ne"

cases and a nifying e6planation, _ournal of the 3perational

7esearch Society *) (,1116 %,)T%'*2

9))@ E22 Hung0 I2H2 "hen0 Kynamic operator assignment +ased on

shifting machine loading 0 International _ournal of /ro$uction

7esearch ' (,1116 ' 1'T' ,12

9),@ W2K2 "hing0 Machine repairing models for prod ction systems,

International _ournal of /ro$uction <conomics .1 (,11)6 ,*.T,++2

9)'@ #2_2 Armstrong0 Age repair policies for the machine repair

pro+lem, <uropean _ournal of 3perational 7esearch )' (,11,6

),.T) )2

9) @ "272 Schult!0 Spare parts in)entory and cycle time red ction 0

International _ournal of /ro$uction 7esearch , (,11 6 .*%T..+2

9)*@ 82A2 a$eh0 3 @@y sets as a +asis for a theory of possi+ility 0 u!!y

Sets an$ Systems ) ()%. 6 'T, 2

9)+@ 72_2 8i0 <2S2 8ee0 Analysis of f @@y q e es 0 "omputers an$

#athematics with Applications ). ()% %6 )) 'T)) .2

9).@ _2_2 Buckley0 'lementary q e eing theory +ased on possi+ility

theory 0 u!!y Sets an$ Systems '. ()%%16 'T*,2

9) @ J2S2 ?egi0 <2S2 8ee0 Analysis and sim lation of f @@y q e es 0 u!!y

Sets an$ Systems + ()%%,6 ',)T''12

9)%@ _2B2 _o0 E2 4su:imura0 #2 Fen0 F2 Eama!aki0 Ferformance e)al ation

of net"or* models +ased on f @@y q e ing system 0 _apanese

_ournal of u!!y 4heory an$ Systems ()%%+6 '%'T 1 2

9,1@ "2 Kao0 "2"2 8i0 S2/2 "hen0 Farametric programming to the analysis

of f @@y q e es, u!!y Sets an$ Systems )1. ()%%%6 %'T)112



9,)@ _2_2 Buckley0 42 euring0 E2 Hayashi0 3 @@y q e eing theory

re)isited 0 International _ournal of Uncertainty0 u!!iness an$

Knowle$ge-Base$ Systems % (,11)6 *,.T*'.2

9,,@ 7272 Eager0 A characteri@ation of the e6tension principle 0 u!!y

Sets an$ Systems ) ()% +6 ,1*T,).2

LINEAR PRO-RAMMIN- APPROACH FOR TWO ECHELON S#PPLY CHAINWITH M#LTI>PROD#CT AND M#LTI>TIME PERIOD

!. sharani ,

College of Engineering , Kidangoor,Kerala

Abstract :

In today’s rapidly evolving business dynamics, the study of the supply chain model is gaining

phenomenal importance around the globe. This work presents a linear programming approach to make

strategic resource planning decisions in multi:product and multi:time period two:echelon supply chains.

The ob ective of the proposed model is to provide an optimal inventory level for the warehouses and

distribution centers and also, minimi3ing the total cost related to transportation, inventory, packing and

toll fees of the entire supply chain for a finite planning hori3on. An industrial case demonstrates the

feasibility of applying the proposed model to real world problem in a two:echelon supply chains. In

addition to that, if the efficiency of the transportation will be increased by using the light weight eco:friendly plastic pail rather than the traditional one, then the reduction of the total cost in the supply chain is

also analy3ed.

e"*ords + )upply chain management, linear programming, multi:product, multi:time period, two:echelon

supply chains

+" I&tr !'ct% &



)upply +hain 1anagement ()+1 issues have long attracted interest from both practitioners and

academics because of its ability to reap more benefits by efficiently managing it. (Anamaria ] /akesh,

-999 B 0eamon, -99> B 0ilgen ] !3karahan, 4??7, +hen, *in ] 6uang, 4?? , rengiic, )impson ]

Sakharia, -999 B =en ] )yanf, 4??8, $etrovic, 4??-, $etrovic, /oy ] $etrovic, -99> . )upply +hain

1anagement is a set of synchroni3ed decisions and activities utili3ed to efficiently integrate suppliers,

manufacturers, warehouses, transporters, retailers and customers so that the right product or service is

distributed at the right &uantities, to the right locations and at the right time, in order to minimi3e system:

wide costs while satisfying customer service level re&uirements. The ob ective of )+1 is to achieve

sustainable competitive advantage. To stay competitive, organi3ations should improve customer service,

reduction of costs across the supply chain and efficient use of resources available in the supply chain.

In this field, numerous researches are conducted. 2illiams, -9>-, developed seven heuristic

algorithms to minimi3e distribution and production costs in supply chain. +ohen and *ee, -9>9, presented

a deterministic, mixed integer, non:linear programming with economic order &uantity techni&ue to develop

global supply chain plan. $yke and +ohen, -995, developed a mathematical programming model by using

stochastic sub:models to design an integrated supply chain involving manufacturers, warehouses and

retailers. 3damar and Da3ga , -99;, developed a distribution#production system involving a manufacturer

centers and its warehouses. They try to minimi3e total costs such as inventoryB transportation costs etc

under production capacity and inventory e&uilibrium constraints. $etrovic et al., -999, modeled supply

chain behaviors under fu33y constraints. Their model showed that, uncertain customer demands and

deliveries play a big role about behaviors. )yarif et al., 4??4, developed a new algorithm based genetic

algorithm to design a supply chain distribution network under capacity constraints for each echelon. Dan et

al., 4??5, tried to contrive a network which involves suppliers, manufacturers, distribution centers and

customers via a mixed integer programming under logic and material re&uirements constraints. Dilma3,

4??7, handled a strategic planning problem for three echelon supply chain involving suppliers,

manufacturers and distribution centers to minimi3e transportation, distribution, production costs. +hen and

*ee, 4??7, developed a multi:product, multi:stage, and multi:period scheduling model to deal with multiple

incommensurable goals for a multi:echelon supply chain network with uncertain market demands and

product prices. The uncertain market demands are modeled as a number of discrete scenarios with known

probabilities, and the fu33y sets are used for describing the sellers’ and buyers’ incompatible preference on

product prices. The supply chain scheduling model is constructed as a mixed:integer nonlinear

programming problem to satisfy several conflict ob ectives, such as fair profit distribution among all participants, safe inventory levels, maximum customer service levels, and robustness of decision to

uncertain product demands, therein the compromised preference levels on product prices from the sellers

and buyers point of view are simultaneously taken into account. Fagurney and Toyasaki, 4??8, tried to

balance e:cycling in multi tiered supply chain process. =en and )yarif, 4??8, developed a hybrid genetic

algorithm for a multi period multi product supply chain network design. $aksoy, 4??8, developed a mixed



integer linear programming to design a multi echelon supply chain network under material re&uirement

constraints. *in et al., 4??;, compared flexible supply chains and traditional supply chains with a hybrid

genetic algorithm and mentioned advantages of flexible ones. 2ang, 4??;, explained the imbalance

between echelons with pecans supply chain by changing chain’s perfect balanced. 6e used ant colony

techni&ue to minimi3e costs in pecans imbalanced supply chains. A3aron et al., 4??>, developed a multi:

ob ective stochastic programming approach for supply chain design under uncertainty. emands, supplies,

processing, transportation, shortage and capacity expansion costs are all considered as the uncertain

parameters. Their multi:ob ective model includes (i the minimi3ation of the sum of current investment

costs and the expected future processing, transportation, shortage and capacity expansion costs, (ii the

minimi3ation of the variance of the total cost and (iii the minimi3ation of the financial risk or the

probability of not meeting a certain budget. Dou and =rossmann, 4??>, addressed the optimi3ation of

supply chain design and planning under responsive criterion and economic criterion with the presence of

demand uncertainty. 0y using a probabilistic model for stock:out, the expected lead time is proposed as the

&uantitative measure of supply chain responsiveness. )ch t3 et al., 4??>, presented a supply chain design problem modeled as a se&uence of splitting and combining processes. They formulated the problem as a

two:stage stochastic program. The first:stage decisions are strategic location decisions, whereas the second

stage consists of operational decisions. The ob ective is to minimi3e the sum of investment costs and

expected costs of operating the supply chain. Tu3kaya and n t, 4??9, developed a model to minimi3e

holding inventory and penalty cost for suppliers, warehouse and manufacturers based a holononic

approach. )ourira an et al., 4??9, considered a two:stage supply chain with a production facility that

replenishes a single product at retailers. The ob ective is to locate distribution centers in the network such

that the sum of facility location, pipeline inventory, and safety stock costs is minimi3ed. They used genetic

algorithms to solve the model and compare their performance to that of a *agrangian heuristic developed inearlier work. Ahumada and Sillalobos, 4??9, reviewed the main contributions in the field of production and

distribution planning for agri:foods based on agricultural crops. Through their analysis of the current state

of the research, they diagnosed some of the future re&uirements for modeling the supply chain of agri:

foods. =unasekaran and Fgai, 4??9, have developed a unified framework for modeling and analy3ing

0T!:)+1 and suggested some future research directions. u and Fo3ick, 4??9, formulated a two:stage

stochastic program and a solution procedure to optimi3e supplier selection to hedge against disruptions.

Their model allows for the effective &uantitative exploration of the trade:off between cost and risks to

support improved decision:making in global supply chain design. )hin et al., 4??9, provided buying firms

with a useful sourcing policy decision tool to help them determine an optimum set of suppliers when anumber of sourcing alternatives exist. They proposed a probabilistic cost model in which suppliers’ &uality

performance is measured by inconformity of the end product measurements and delivery performance is

estimated based on the suppliers’ expected delivery earliness and tardiness. Cahimnia, 0.B *ee *uongB

1arian, /, 4??9, developed a mixed integer formulation for a two:echelon supply network considering

the real:world variables and constraints. A multi:ob ective genetic algorithm (1!=A is then designed for



the optimi3ation of the developed mathematical model. $. )ubramanian, F. /amkumar , T.T. Farendran,

4?-?, formulated a generalised multi:echelon, single time:period, multi:product, closed loop supply chain

as an integer linear programme (I*$ . E.0ala i /eddy et al.,4?--,developed single:echelon supply chain

two stage distribution inventory optimi3ation model.

)upply +hain 1anagement is a field that is usually been studied from a market and product perspective rather than from transport point of view. Transport process are essential parts of the supply

chain. They perform the flow of materials that connects an enterprise with its suppliers and with its

customers. The integrated view of transport, production and inventory holding process is the characteristic

of the modern supply chain management concept (0. Cleischmann, 4??8 . !nly a good coordination

between each component would bring the benefits to a maximum.

Transportation occupies one:third of the amount in the logistics and transportation systems

influence the performance of logistics system hugely. 2ithout well developed transportation systems,

logistics could not bring its advantages into full play.

)ome authors have recently worked in the development of the supply chain and transport

relationship. ()tank and =oldsby, 4???, $otter and *alwani, 4??8 , isney and Uowill, 4??5 , +hildhouse $.,

Towill . /., 4??5 , 0ask A.6., 4??- . 6owever, )upply +hain 1anagement and transport are areas that

should be discussed more in depth, in order to do so it is necessary to develop a framework that allow

holistic analysis from a system perspective.

$ackaging is a co:ordinated system of preparing goods for safe, efficient, cost effective transport,

distribution, storage, retailing, consumption and recovery reuse or disposal combined with maximi3ing

consumer value, sales and hence profit proper packaging is re&uired by all freight carriers to ensure safetyshipment. In supply chain packaging costs represents a significant part.

A toll is one of the fairest revenue sources. Cor the use of elevated road, user fee shall be collected

from all vehicles. avid Uackson of 0usiness ay, in an article published on -9th 1arch 4?--, said that, the

new tolling system put strain on business. *ogistics costs will increase by about -@ following the increased

toll road tariffs, companies may have to review their entire distribution network strategy as they seek viable

and cost efficient alternative to the toll road system.

According to our survey on the current literature, none of the previous models has considered the

ma or cost elements packing and toll fees. while taking this characteristic into consideration makes the

developed model adaptable to a wider manufacturing and distribution scenarios

This paper presents a linear programming approach to solve multi:product and multi:time period

two:echelon supply chain models.The ob ective of the proposed model is to provide an optimal inventory

http://www.inderscience.com/search/index.php?action=basic&wf=author&year1=1995&year2=2007&o=2&q=P.%20Subramanian



http://www.inderscience.com/search/index.php?action=basic&wf=author&year1=1995&year2=2007&o=2&q=%20N.%20Ramkumar


http://www.inderscience.com/search/index.php?action=basic&wf=author&year1=1995&year2=2007&o=2&q=%20T.T.%20Narendran







level for the warehouses and distribution centers and also minimi3ing the total cost of the entire supply

chain for a finite planning hori3on.

Also this study utili3es one of the leading paint company as a case study to demonstrate the

practicality of the proposed approach. In addition to that, if the efficiency of the transportation will be

increased by using the light weight eco:friendly plastic pail rather than the traditional one, then thereduction of the total cost in the supply chain is also analy3ed.

This paper is organi3ed as follows Q )ection 4, describes the problem, details the assumptions and

notations. )ection 5 formulates the multi:product and multi:time period two:echelon supply chain model.

)ection 7 represents an industrial case for implementing the feasibility of applying the proposed approach

to real situations. Cinally, section 8 presents the conclusion.

$" Pr b1e( Descr%)t% & Ass'()t% &s a&! N tat% &s

The proposed model is a two:echelon supply chain model with Fw warehouses, F distribution

centers and Fr retailers with limited capacities. In this model multi:product is being distributed from the

warehouse to distribution centers in the system. The demand for the product will be forecasted before

beginning of every period and will be used as the reference for warehouses to transfer stocks from them to

distribution centers in a particular period h. Curther the multi:product is being distributed from the

distribution centers to retailers according to their demand for the product before beginning of every period

and will be used as the reference for distribution centers to transfer stocks from them to retailers in a particular period h. This paper focuses on presenting a linear programming approach that optimi3es a two:

echelon supply chain models with multi:product and multi:time period .The aim of this approach is to

minimi3e the total cost associated with inventory holding cost, transportation cost, packing charge and toll

fees.

The following list of assumptions is the basics for this study’s mathematical programming model.

-. All the ob ective functions and constraints are linear e&uations.

4. Transportation costs on a given route are directly proportional to the units shipped.

5. $acking charges vary considerably depending on the item and how much material it takes to get it

packed.

7. Toll fees considered per shipment.



8. +apacity of the warehouse and distribution centers in each period cannot exceed their maximum

available levels.

Assumptions - to 7 indicate that linearity, proportionality properties must be technically satisfied

as a standard *$ form. Assumption 8 represents the limit on the maximum available warehouse and

distribution centers capacities in a normal business operation.

This study uses the following notation.

I&!e6 Sets

i 2arehouses i K -, 4, . . . , Fw.

istribution centers K -, 4, . . . , F .

E /etailers E K -, 4, . . . , F r .

l $roduct Type l K -, 4, . . . , F pr .

h Time period h K -, 4, . . . , 6.

) )tages ) K -, 4.

Ob ect%;e F'&ct% &s

h-T+Total cost in the first stage of period h

h4T+Total cost in the second stage of period h

hT+Total cost in period h, is e&ual to sum of

h-T+ and

h4T+

Dec%s% & Var%ab1es

I2 hli inventory level of product l by warehouse i in period h.

T2 hli units distributed of product l from warehouse i to distributor in period h.

I hl Inventory level of product l by distribution centers in period h.



T hl k units distributed of product l from distribution centers to retailer k in period h.

Para(eters

hliI+2unit inventory carrying cost of product l by warehouse i in period h.

hliT+2unit transportation cost of product l from warehouse i to distribution centers in period h.

hli$+ packing charge of product l by warehouse i in period h.

hiTC2Toll fees from warehouse i to distribution centers in period h.

hi F)2

Fumber of shipments form warehouse i to distribution centers in period h is e&ual to

-

--?

Npr

hlij

l

X TW =∑

for traditional pail,

-

--4

Npr

hlij

l

X TW =∑

for light weight pail

hlI+unit inventory carrying cost of product l by distribution centers in period h.

hl k T+unit transportation cost of product l from distribution centers to retailer k in period h.

hiTCToll fees from distribution centers to retailer k in period h.

h k F)

Fumber of shipments form distribution centers to retailer k



in period h is e&ual to

-

--?

Npr

hljk

l

X TD=∑

for traditional pail,

-

--4

Npr

hljk

l

X TD=∑

for light weight

pail

lhCCorecasted demand of product l of distribution centers in period h.

lhAActual demand of products of distribution centers in period h.

lhk

emand of product l of retailer k in period h.

+hi +apacity of ith warehouse in period h.

+h +apacity of th distribution centers in period h.

+hk +apacity of k th retailer in period h.

*" Pr b1e( F r('1at% &

*"+" M'1t%>)r !'ct a&! ('1t%>t%(e )er% ! tw >ec0e1 & s'))1y c0a%& ( !e1

The proposed model deals with optimi3ing the inventory levels of the two:echelon multi:product

and multi:time period supply chain models. The diagrammatic representation of the proposed model for the

period h is shown in Cig.-.

2arehouse - 4 . . . F w

istribution centers - 4 . . . F



/etailers - 4 . . . F r

F%2're + : Tw >ec0e1 & S'))1y C0a%& M !e1

F%rst Sta2e Pr b1e(

The ob ective function for the first stage is given below Q

h-T+K

pr pr pr w w w F F F F F F F F

hli hl ihli hl i hl ii K - l K - i K - K - l K - i K - K - l K -

I2 x I+2 L T2 x T+2 L T2∑∑ ∑∑∑ ∑∑∑w F F

hli

i K - K -

-x $+ L x

-?hij hij NSW TFW ∑∑

wherehli hl i hli hiI+2 , T+2 , $+ , TC2

denote the cost coefficients.

In the first stage we have the following constraints

- The sum of the units of product l transferred from a warehouse to all distribution centers should be

less than or e&ual to the warehouse inventory for a particular period h.

F

hl i hli w pr K -

T2 I2 , i K -, 4, . . . F , l K -, 4, . . . , F≤∑ . . . (4.4.4

4.

F

hli hil K -

I2 + , pr

≤∑ ie. Inventory at the warehouse should be less than or e&ual to the

warehouse capacity in period h. . . . (4.4.5



5. The sum of the units of product l transferred from all warehouses to a particular distribution

centers should be greater than or e&ual to the forecasted demand of that particular distribution

centers in period h.

w F

lhhl i prl K -

T2 C , l K -, 4, . . . F , K -, 4, . . . , F≥∑ . . . (4.4.7

7. The total distribution centers forecasted demand for a period h should be less than or e&ual to all

warehouse inventory of product l, in that particular period

w F F

lhhli pr l K - K -

I2 C , l K -, 4, . . . F≥∑ ∑ . . . (4.4.8

8. The total number of units transferred from all warehouse to a particular distribution centers should

be less than or e&ual to that distribution centers capacity

pr w F F

hl i hi K - l K -

T2 + , K -, 4, . . . F≤∑∑ . . . (4.4.

Sec &! Sta2e Pr b1e(

The ob ective function for the second stage is given below Q

h4T+K

pr pr r F F F F F F

hl hl k hl hl k K - l K - K - k K - l K - K - -

I x I+ L T x T+ L x Nr

jk jk

k

NSD TFD=

∑∑ ∑∑∑ ∑∑

. . . (4.4.;

wherehhl hl k I+ , T+ , TC

denote the cost coefficients.

In the second stage we have the following constraints

- The inventory of product l at a particular distribution centers at the end of stage one is e&ual to the

total number units received from all warehouse by that distribution centers minus the actual

demand of that particular distribution centers in period h.



w F

lhhl hl i pr i K -

I K T2 : A , l K -, 4, . . . F∑ . . . (4.4.>

(If this value is negative, I hl K ?

4 The sum of the units of product l transferred from all distribution centers to a particular retailer

should be greater than or e&ual to the demand of that particular retailer in period h.

F

hlk hl k pr r K -

T C , l K -, 4, . . . F , E K -, 4, . . . F≥∑ . . . (4.4.9

5. The sum of the units of product l transferred from a particular distribution centers to all retailers

should be less than or e&ual the actual demand of that particular distribution centers in period h.

r F

hlhl k pr E K -

T A , K -, 4, . . . F , l K -, 4, . . . , F≤∑ . . . (4.4.-?

7. The total number of units transferred from all distribution centers to a particular retailer should be

less than or e&ual to that retailer capacity

pr F F

hl k hk r K - l K -

T + , E K -, 4, . . . F≤

∑∑ . . . (4.4.--

3" IMPLEMENTATION AND COMP#TATIONAL ANALYSIS

3"+" Case Descr%)t% &

The company chosen for the application of the proposed methodology in this work is a leading

paint company located in the )outhern part of India. This study focuses only on the emulsion paints which

is the fast moving paint product. mulsion paints are coming in 5 different packs such as small, retailer,

bulk. The forecasted demand of these three products are different in three different time periods. Thecompany has planned to build a mathematical model to minimi3e the total cost of the supply chain. The

total cost can be minimi3ed by optimi3ing the inventory levels at warehouse and distribution centers. Also

optimi3ing the transportation cost packing charge and toll fees in the supply chain. This can be done by

transporting the product with proper packaging through the optimum route and by maintaining an optimum

inventory level at warehouse and distribution centers. This study assumes the holding cost, transportation

cost, toll fees for warehouse and distribution centers and packing charge for warehouse are same for all the



three products in three different time periods. Osually a truck can carry maximum load of -? tons per

shipment. 0ut if - @ eco friendly light weight plastic pail is used rather than the traditional one, then a

truck can carry maximum load of -4 tons per shipment. )o the number of shipments can be reduced. In

this study, the reduced cost in the supply chain due to this is also analysed. The following data is collected

for validating the above proposed model.

Fumber of warehouse Fw : -

Fumber of distribution centers F : 4

Fumber of /etailers F r : 7

Fumber of $roducts : 5

Fumber of Timeperiods : 5

The input data for warehouse distribution centers and retailers are given in Tables - >.

Tab1e + : I&)'t Data 5 r Ware0 'se

Inventory carrying cost per unit in /s. /s.5?#ton

2arehouse +apacity in Onits 8? ton

$acking +harge)mall $ack /s.5#pack

/etail $ack /s.8#pack

Tab1e $ : Tra&s) rtat% & C st 5r ( Ware0 'se t D%str%b't% & Ce&ters )er '&%t %& Rs"

D%str%b't% & Ce&ters

D+ D $

Transportation +ost in /upees from 2arehouse to -??? -7??

Tab1e * : T 11 5ees 5r ( Ware0 'se t D%str%b't% & Ce&ters )er s0%)(e&t %& Rs"


D+ D $

Toll Cees from 2arehouse to istribution +enters -;;? 4?9?

Tab1e 3 : D%str%b't% & Ce&ters De(a&!

Per% ! Pr !'ct D%str%b't% & Ce&ters

D+ D$



F recaste! Act'a1 F recaste! Act'a1

h-

) -48 -4? -8? -74

/ -?? 9? -4? -?>

0 48 -8 5? 4?

h4

) ;8 ? 9? ;>

/ ->? -;4 4?? ->9

0 48 -> 5? -8

h5

) 58 4> 78 57

/ -7? -5? - 8 -8>

0 9? >7 -?? 95

Tab1e 4 : I&)'t Data 5 r D%str%b't% & Ce&ters


D+ D $

Inventory carrying cost per unit in /s. 4? 48

+apacity in Onits 5?? 7??

Tab1e 9 : Tra&s) rtat% & c st 5r ( D%str%b't% & Ce&ters t Reta%1ers )er '&%t %& Rs"

D%str%b't% & Ce&tersReta%1ers

R + R $ R * R 3

D+ -8? 457 7 > 87?

D$ 777 85; -4? ->9

Tab1e : T11 5ees 5r ( D%str%b't% & Ce&ters t Reta%1ers )er s0%)(e&t

%& Rs"

D%str%b't% & Ce&tersReta%1ers

Nr + Nr $ Nr * Nr 3

D+ -;? 48? 75? 8 ?



D$ 8-? 898 -;? 57?

Tab1e 8 : Reta%1er De(a&!

Per% ! Pr !'ctReta%1er

R + R $ R * R 3

h-

) > 84 ;> 7

/ 84 5> ? 7>

0 -? 8 -4 >

h4

) 5? 5? 7; 5-

/ -?? ;4 --5 ;

0 -? > -? 8

h5

) -- -; 45 --

/ 89 ;- -?4 8

0 59 78 ? 55

Tab1e 8a : I&)'t Data 5 r Reta%1ers

Reta%1ers

R + R $ R * R 3

/etailers +apacity in Onits -8? -7? -9? -5?

3"$" Res'1ts

After solving the model for two stages, usingL%&! )oftware, the optimal solutions for the case

study are obtained. These results are tabulated in tables 9 -8.

Tab1e @ : O)t%(a1 Ware0 'se St c<

Per% ! Pr !'ct N'(ber 5 #&%ts

h- ) I2 --- K 4;8

/ I2 -4- K 44?



0 I2 -5- K 88

h4

) I2 4-- K - 8

/ I2 44- K 5>?

0 I2 45- K 88

h5

) I2 5-- K >?

/ I2 54- K 5?8

0 I2 55- K -9?

Tab1e +7 : O)t%(a1 &'(ber 5 '&%ts tra&s5erre! 5r ( ware0 'se t !%str%b't% & ce&ters

Per% ! Pr !'ctN'(ber 5 #&%ts

D+ D $

h-

) T2 ---- K -48 T2 ---4 K -8?

/ T2 -4-- K -?? T2 -4-4 K -4?

0 T2 -5-- K 48 T2 -5-4 K 5?

h4

) T2 4--- K ;8 T2 4--4 K 9?

/ T2 44-- K ->? T2 44-4 K 4??

0 T2 45-- K 48 T2 45-4 K 5?

h5

) T2 5--- K 58 T2 5--4 K 78

/ T2 54-- K -7? T2 54-4 K - 8

0 T2 55-- K 9? T2 55-4 K -??

Tab1e ++ : O)t%(a1 &'(ber 5 '&%ts tra&s5erre! 5r ( D%str%b't% & Ce&ters t Reta%1ers

D%str%b't% &

Ce&ters

Per%!

Pr !'ct

Reta%1er

R + R $ R * R 3

- h- ) T ---- K > T ---4 K 84 T ---5 K ? T ---7 K ?

/ T -4-- K 84 T -4-4 K 5> T -4-5 K ? T -4-7 K ?



0 T -5-- K -? T -5-4 K 8 T -5-5 K ? T -5-7 K ?

h4

) T 4--- K 5? T 4--4 K 5? T 4--5 K ? T 4--7 K ?

/ T 44-- K -?? T 44-4 K ;4 T 44-5 K ? T 44-7 K ?

0 T 45-- K -? T 45-4 K > T 45-5 K ? T 45-7 K ?

h5

) T 5--- K -- T 5--4 K -; T 5--5 K ? T 5--7 K ?

/ T 54-- K 89 T 54-4 K ;- T 54-5 K ? T 54-7 K ?

0 T 55-- K 59 T 55-4 K 78 T 55-5 K ? T 55-7 K ?

4

h-

) T --4- K ? T --44 K ? T --45 K ;> T --47 K 7

/ T -44- K ? T -444 K ? T -445 K ? T -447 K 7>

0 T -54- K ? T -544 K ? T -545 K -4 T -547 K >

h4

) T 4-4- K ? T 4-44 K ? T 4-45 K 7; T 4-47 K 5-

/ T 444- K ? T 4444 K ? T 4445 K --5 T4447 K ;

0 T 454- K ? T 4544 K ? T 4545 K -? T4547 K 8

h5

) T 5-4- K ? T 5-44 K ? T 5-45 K 45 T 5-47 K --

/ T 544- K ? T 5444 K ? T 5445 K -?4 T 5447 K 8

0 T 554- K ? T 5544 K ? T 5545 K ? T 5547 K 55

Tab1e +$ : O)t%(a1 %&;e&t ry %& '&%ts at !%str%b't% & ce&ters

Per% ! Pr !'ct D%str%b't% & Ce&tersD+ D $

h-

) I --- K 8 I --4 K >

/ I -4- K -? I -44 K -4

0 I -5- K -? I -54 K -?

h4

) I 4-- K -8 I 4-4 K -4

/ I 44- K > I 444 K --

0 I 45- K ; I 454 K -8

h5

) I 5-- K ; I 5-4 K --

/ I 54- K -? I 544 K ;

0 I 55- K I554 K ;

Tab1e +* : O)t%(a1 &'(ber 5 s0%)(e&ts 5r ( ware0 'se t !%str%b't% & ce&ters



Ty)e Per% !D%str%b't% & Ce&ters

D+ D $

Traditional

h- 48 5?

h4 4> 54

h5 4 .8 5-

*ight 2eight

h- 4?.;8 47.9?

h4 45.47 4 .8

h5 4-.99 48.;5

Tab1e +3 : O)t%(a1 &'(ber 5 S0%)(e&ts 5r ( D%str%b't% & Ce&ters t Reta%1ers

D%str%b't% &

Ce&tersTy)e Per% ! Reta%1er

R + R $ R * R 3

-

T r a d i t i o n a l h- -5 9.8

h4 -4 --

h5 -?.9 -5.5

* i g h t w e i g h t h- -?.;9 ;.>>8

h4 9.9 9.-5

h5 9.?7; --.?59

4

T r a d i t i o n a l h- -8 -4

h4 -; --.4

h5 ->.8 -?

* i g h t w e i g h t h- -4.78 9.9

h4 -7.-- 9.49

h5 -8.588 >.499



Tab1e +4 : O)t%(a1 T ta1 C st %& Rs"

Per% !Ty)e

Tra!%t% &a1 L%20t We%20t

h- >,>>,;5;.7? >, >, 79.>8

h4 9, -,575.7? 9,59,898.;7

h5 9,49,;7 .4? 9,?>,; 8.-9

3"*" C ()'tat% & A&a1ys%s

The !ptimal Total +ost in Table -8 shows that there is 4.48@ significant decrease in total cost of

the two echelon, multi:product, multi:time period supply chain due to the usage of eco:friendly light weight

plastic pail rather than the traditional one. The following graph proves that the total cost incurred while

using light weight pail is comparatively less than the traditional one. )o it is better to adopt eco: frielndly

light weight pail in future.

) , '(

(2*

%

%2*

)1

totalcost comparision &et"een tra!itional an! light "eight pail

4ra$itional8ight Weight

perio!

totalcost

4" CONCL#SION

This work presents a linear programming approach for solving multi:product, multi:time periodtwo:echelon supply chain models. The proposed model provided an optimal inventory level for warehousesand distribution centers and minimi3ing the total costs related to inventory, transportation, toll fees and packing charge of the entire supply chain. An industrial case study is utili3ed to demonstrate the feasibilityof applying the proposed approach to practical problems in a supply chain. The proposed approach yieldsmore efficient solution and several significant managerial implications for practical applications. Thereduction of the total cost in the supply chain due to the usage of eco:friendly lightweight plastic pail,



rather than the traditional plastic pail is also analy3ed in the present work. In particular, the proposedcomputational methodology can be easily extended to any other situation and can handle real life decisionmaking problems in supply chains. In future, uncertainty of costs and demands can be considered in themodel and new solution methodologies including uncertainty can be developed via fu33y models.

REFERENCES

). Ana 1aria, F. /akesh, -999,A review of integrated analysis of production and distribution systems, II Transactions5- -? - -?;7.

Ahumada !., Sillalobos U./., 4??9, Application of planning models in the agri:food supply chainQ a review, uropeanUournal of !perational /esearch -9 (- , pp. -:4?.

A3aron, A., 0rown, E.F., Tarim, ).A. and 1. 1odarres (4??> . A multi ob ective stochastic programming approach for supply chain design considering risk. International Uournal of $roduction conomics++9 .+/ , pp. -49:-5>.

E.0ala i /eddy, ).Farayan and $.$andian,4?--,)ingle: chelon supply chain two stage distribution inventoryoptimi3ation model for the confectionery industry,Applied 1athematical sciences,vol.8,no.8?,479-:48?7

0ask, A. 6. ,4??-, /elationships among T$* providers and members of supply chainsQ a strategic perspective. Uournalof 0usiness ] Industrial1arketing, 4??-, - ( , 7;?:7> .

0.1. 0eamon, 4??7, )upply chain design and analysis modelsQ models and methods, International Uournal of$roduction conomics 88 (-99> 4>- 497

+hen, 0. 0ilgen, I. !3karahan, )trategic tactical and operational production:distribution modelsQ a review, InternationalUournal of Technology 1anagement 4> -8- -;-.

+hildhouse $, Towill / ,4??5, )implified material flow holds the key to supply chain integration, !1 =A, 5-(- Q-; 4;.

+.T. +hen, +.T. *in, ).C. 6uang, 4?? A fu33y approach for supplier evaluation and selection in supply chainmanagement, International Uournal of $roduction conomics -?4 4>9 5?-.

+ohen 1.A., *ee 6.*., -9>9, /esource deployment analysis of global manufacturing and distribution networks,Uournal of 1anufacturing and !perations 1anagement 4, pp. >-:-?7.

isney ).1. and Towill ./. ,4??5, Sendor:managed inventory andbullwhip reduction in a two:level supply chain,International Uournal of!perations and $roduction 1anagement, Sol. 45 Fo. , pp. 48: 8-

).). reng c, F.+. )impson, A.U. Sakharia, -999Integrated production#distribution planning in supply chainsQ aninvited review, uropean Uournal of !perational /esearch --8, 4-9 45 .

Cahimnia, 0.B *ee *uongB 1arian, /., 4??9,!ptimi3ation of a Two: chelon )upply Fetwork Osing 1ulti:ob ective =enetic Algorithms +omputer )cience and Information ngineering, 2/I 2orld +ongress

0.Cleischmann,4??8, istribution and transport planning ,supply chain management and advanced planning ,449:477

=en 1., )yarif A., 4??8, 6ybrid genetic algorithm for multi:time period production distribution planning, +omputersand Industrial ngineering 7>, pp. ;99:>?9.

=unasekaran A., Fgai ., 4??9, 1odeling and analysis of build:to:order supply chains, uropean Uournal of !perational/esearch -98 (4 , pp. 5-9:557.

http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5170260





*., *ee, 2., 4??7, 1ulti ob ective optimi3ation of multi echelon supply chain networkswith uncertain product demandsand prices, +omputers and +hemical ngineering 4>, pp. --5-:--77.

*in *., =en 1., 2ang ., 4??;, A hybrid genetic algorithm for logistics network design with flexible multistagemodel, International Uournal of Information )ystems for *ogistics and 1anagement 5 (- , pp. -:-4.

Fagurney A., Toyasaki C., 4??8, /everse supply chain management and electronic waste

recyclingQ a multitired network e&uilibrium framework for e:cycling, Transportation /esearch $art , pp. -:4>.

!3damar *., Da3ga T., -99;, +apacity driven due date settings in make:to:order production systems, InternationalUournal of $roduction conomics 79 (- , pp. 49:77.

$aksoy T., 4??8, istribution network design and optimi3ation in supply chain managementQ under materialre&uirements constraints a strategic production:distribution model, Uournal of )elcuk Oniversity )ocial )ciencesInstitute -7, pp. 758:787, in Turkish.

. $etrovic, 4??- )imulation of supply chain behavior and performance in an uncertain environment, InternationalUournal of $roduction conomics ;-,749 75>.

. $etrovic, /. /oy, /. $etrovic, -99>, 1odelling and simulation of a supply chain in an uncertain environment,uropean Uournal of !perational /esearch -?9,499 5?9.

$etrovic ., /oy /., $etrovic /., -999, )upply chain modeling using fu33y sets, International Uournal of $roductionconomics 89, pp. 775:785.

$otter, A. and *alwani, +. ,4??8, )upply chain dynamics and transport managementQ A review, $roceedings of the -?th*ogistics /esearchFetwork +onference, $lymouth, ;th:9th )eptember.

$yke .C., +ohen 1.A., -995, $erformance characteristics of stochastic integrated production distribution systems,uropean Uournal of !perational /esearch > (- , pp. 45:7>.

)ch t3 $., Tomasgard A., Ahmed )., 4??>, )upply chain design under uncertainty using sample average approximationand dual decomposition, uropean Uournal of !perational /esearch, doiQ-?.-?- # .e or.4??>.--.?7?.

)hin 6., 0enton 2.+., Uun 1., 4??9, "uantifying suppliers’ product &uality and delivery performanceQ A sourcing policy decision model, +omputers ] !perations /esearch 5 , pp. 47 4:47;-.

F.+. )impson, A.U. Sakharia, -999,Integrated production#distribution planning in supply chainsQ an invited review,uropean Uournal of !perational /esearch--8, 4-9 45 .

)tank T and =oldsby T U ,4???, A framework for transportation decision making in an integrated supply chain. )upply+hain 1anagementQ AnInternational Uournal, Sol 8Q Fo 4

)ourira an E., !3sen *., O3soy /., 4??9, A genetic algorithm for a single product network design model with lead timeand safety stock considerations, uropean Uournal of !perational /esearch -9; (4 , pp. 899: ?>.

$. )ubramanian, F. /amkumar , T.T. Farendran, 4?-?,1athematical model for multi:echelon, multi:product, singletime:period closed loop supply chain,International Uournal of 0usiness $erformance and )upply +hain 1odelling,Sol. 4, Fo.5#7 pp. 4- : 45

)yarif A., Dun D., =en 1., 4??4, )tudy on multi:stage logistics chain networkQ a spanning tree:based genetic algorithmapproach, +omputers and Industrial ngineering 75 (- , pp. 499:5-7.

Tu3kaya O., n t )., 4??9, A holonic approach based integration methodology for transportation and warehousingfunctions of the supply network, +omputers and Industrial








http://www.inderscience.com/browse/index.php?journalID=341&year=2010&vol=2&issue=3/4











ngineering 8 , pp. ;?>:;45.

2ang 6.)., 4??;, A two:phase ant colony algorithm for multi echelon defective supply chain network design, uropeanUournal of !peration /esearch, doiQ -?.-?- # .e or.4??;.?>.?5;.

2illiams U.C., -9>-, 6euristic techni&ues for simultaneous scheduling of production and distribution in multi:echelonstructuresQ theory and empirical comparisons, 1anagement )cience 4; (5 , pp. 55 :584.

u F., Fo3ick *., 4??9, 1odeling supplier selection and the use of option contracts for global supply chain design,+omputers ] !perations /esearch, 5 , pp. 4;> :4>??.

Dan 6., Du <., +heng T.+. , 4??5, A strategic model for supply chain design with logical constraintsQ formulation andsolution, +omputers ] !perations /esearch 5? (-7 , pp. 4-58:4-88.

D lma3 $., 4??7, )trategic level three:stage production distribution planning with capacity expansion, Onpublished1aster Thesis, )abanc Oniversity =raduate )chool of ngineering and Fatural )ciences, pp. -:4?, in Turkish.

Dou, C. and . =rossmann, 4??>, esign of responsive supply chain under demand uncertainty.Computers & Chemical !"i!eeri!" 54 (-4 , pp.5?9?:5---.

EP MODEL WITH IMPERFECT #ALITY ITEMS WITH ALLOWABLE

SHORTA-ES #SIN- F#,,Y N#MBERS

!r.Senthil 'hila9 and !.S:athi,

;' ,%angalore

This chapter i!#esti"ates pro$uctio! i!#e!tor% mo$el with imper ect qualit% items' allowi!" shorta"es( Ce!t

perce!t scree!i!" process is per orme$ $uri!" the pro$uctio! sta"e itsel so that imper ect qualit% items are

picket up a!$ are sol$ as a si!"le )atch $ue to eco!omies o sale( The o)jecti#e is to $etermi!e the optimal

pro$uctio! qua!tit% a!$ shorta"e qua!tit% to maximi*e the total pro it( +ol$i!" cost' Setup cost' ,ro$uctio!

rate a!$ Suppl% rate' $e ecti#e rate are take! as tria!"ular u**% !um)ers to $e#elop a u**% mo$el to maximi*e

the total pro it a!$ "ra$e$ mea! i!te"ratio! represe!tatio! metho$ is use$ to $e u**i % the result( -ptimal lot

si*e' optimal shorta"e qua!tities are $eri#e$( Numerical examples are "i#e! to illustrate the results o the

propose$ mo$el(

9"+" I&tr !'ct% &

In production inventory scenario, the manufactured products are assumed to be, rather expected to be cent percent perfect.

0ut in real life situations, imperfect &uality items are inevitable. The producers wish to





9"$" Ass'()t% &s

-. Inventory is building up at a constant rate of E r (- : p units per unit of time during T-- .

4. Fo replenishment during time T-4 and inventory is decreasing at a rate of r units per unit time.

5. )hortage is building up at a constant rate of r units per unit of time during T4-.

7. )hortages are being filled immediately at the rate of E (- : p r units per unit of time during T44.8. *ead Time is <ero.

9"*" N tat% &s

E : $roduction rate

r : emand rate

p : $ercentage of defective

x : )creening rate

d : )creening cost per item

s : selling price of perfect &uality item

v : selling price of imperfect &uality item s H v.

y : production &uantity

y4 : shortage &uantity

w : shortage cost # unit

h : inventory holding cost # unit # unit time

E ′ : set up cost

c : production cost # unit

T : cycle length

h: fu33y inventory holding cost

E : fu33y production rate

r : fu33y demand rate

p: fu33y defective rate

b: fu33y shortage cost

E ′5: fu33y set up cost





y4 K

h (E : (- : p ry

(b L h E . . . ( .7

substituting the value of y4 in ( .5 and differentiating ( .5 with respect to y and e&uating to 3ero we have

y K

( )

E Er hpEr (E : (- : p r

E L pr bhx(E L pr 4E(b L h

′ +

( )

4 -?

4

d E Er (T$ G ? y K y K

dy hpEr (E : (- : p rE L pr bh

x(E L pr 4E(b L h +

. . . ( .8

? ?4

h (E : (- : p ry K y

bLE E ÷

. . . ( .

Va1%!%ty : If we put p K ? in ( .8 we get

? 4(b L h E Er y K

bh(E : r′

which is the value of y? in production inventory (with all perfect items with shortages and

?4y

K

4E hr(E : r(h L b bE

′

9"4" F'??y Mat0e(at%ca1 M !e1

*et- 4 5h K (h , h , h5

- 4 5E K (E , E , E5- 4 5E K (E , E , E′ ′ ′ ′5

- 4 5 b K (b , b , b5

- 4 5 p K (p , p , p5

- 4 5r K (r , r , r 5

bT$(y K

4

44 4

(s(- : p L vp : c : d Er E E : (- : p r : h y : y L by

E L pr 4y(E : r(- : p E

÷ ÷ ÷ ÷ ÷

5 5 55 5 5 555 55 5 555 5 5



E E r hpEry(E L pr y x(E L pr

′− −555 55 5 5

5 555 55

.. ( .;

4

45 - - - 5 5 - 55 4 5 4

5 5 5 - 5 - -

(s(- : p L vp : c : d E r E E :r (-:p: h y:y L b yE L p r 4y(E :r (-:p E ÷ ÷ ÷ ÷

- - - - - - -

5 5 5 5 5 5

E E r h p E r y(E L p r y x(E L p r

− − ,

4

44 4 4 4 4 4 4 44 4 4 4

4 4 4 4 4 4 4

(s(- : p L vp : c : d E r E E : r (- : p: h y : y L b y

E L p r 4y(E : r (- : p E

÷ ÷ ÷ ÷

4 4 4 4 4 4 4

4 4 4 4 4 4


′− − ,

4

4- 5 5 5 - 5 --- 4 - 4

- - - 5 - 5 5

(s(- : p L vp : c : d E r E : r (- : pE : h y : y L b yE L p r 4y(E : r (- : p E

÷ ÷ ÷ ÷

5 5 5 5 5 5 5

- - - - - -


′ − −

2 22

(+2 6

b( )$ T$(y

K



4

45 - - - 5 5 - 55 4 5 4

5 5 5 - 5 - -

(s(- : p L vp : c : d E r E E : r (- : p-: h y : y L b y

E L p r 4y(E : r (-:p E

÷ ÷ ÷ ÷

- - - - - - -

5 5 5 5 5 5


′− −

4

44 4 4 4 4 4 4 44 4 4 4

4 4 4 4 4 4 4

(s(- : p L vp : c : d E r E E : r (- : p7 : h y : y L b y

E L p r 4y(E : r (- : p E

+ ÷ ÷ ÷ ÷

4 4 4 4 4 4 4

4 4 4 4 4 4


′− −

4

4- 5 5 5 - 5 --- 4 - 4

- - - 5 - 5 5

(s(- : p L vp : c : d E r E : r (- : pE : h y : y L b y

E L p r 4y(E : r (- : p E

+ ÷ ÷ ÷ ÷

5 5 5 5 5 5 5

- - - - - -


′ − −

. . ( .9

ifferentiating ( .9 partially with respect to y4 and e&uating to 3ero we get

y4 K

5 5 - 5 5 - - - 5 -4

- 5 - - 5 5 - 5

5 5 5 4 4 4 - - -

- 5 - 4 4 4 5 - 5

h (E : r (- : p E E h (E : r (- : p7h(E r (- p E E (E r (- p

y(b h E 7(h b E (b h E

(E r (- p (E r (- p (E r (- p

+ + − − − − + + + + + − − − − − − Kcy (say ( .-?



b( )4

44

$ T$(yy∂∂

is negative.

? ?4y K cy

To determine y?, we substitute the value of

?4y

in ( .9 and differentiate with respect to y and e&uate to 3ero.

( .9 becomes

b( )$ T$(y

K

4

4 45 - - - 5 5 - 55 5

5 5 5 - 5 - -

(s(-:p Lvp :c:d E r E E :r (-:p-: h y:cy L b c y

E L p r 4y(E :r (-:p E

÷ ÷ ÷ ÷

- - - - - - -

5 5 5 5 5 5


′− −

4

4 44 4 4 4 4 4 4 44 4

4 4 4 4 4 4 4

(s(- : p L vp : c : d E r E E : r (- : p7 : h y:cy L b c y

E L p r 4y(E : r (- : p E

+ ÷ ÷ ÷ ÷

4 4 4 4 4 4 4

4 4 4 4 4 4


′− −



4

4 4- 5 5 5 - 5 --- -

- - - 5 - 5 -

(s(- : p L vp : c : d E r E : r (- : pE : h y : cy L b c y

E L p r 4y(E : r (- : p E

+ ÷ ÷ ÷ ÷

5 5 5 5 5 5 5

- - - - - -


′ − −

. . .( .--

ifferentiating ( .-- with respect to y and e&uating to 3ero we get

y4 K

5 5 5- - - 4 4 4

5 5 5 4 4 4 - - -

445 5 - 5 4

5 5- 5 - - 4 4 4

4

44 4 4 -4 4

4 5 - 5

E E r E E r 7E E r

E p r E Lp r E Lp r

E (E : r (- p E h c Lb c 7

4(E :r (- p E 4(E r (- p

(E :r (-:p E h c b c

E 4(E :r (-:p

′′ ′+ ++

− − + ÷ ÷ ÷ ÷− − − − + + ÷ ÷

4

4- 5 -- -

5

5 5 5 5- - - - 4 4 4 4

5 5 5 4 4 4 - - -

(E :r (-:ph c b c

E

h p E r h p E r 7h p E r - L

x E p r E p r E p r

− + ÷ ÷ ÷

+ + + + +

Also

b

( )4

4$ T$(y

y∂

∂ is negative.

y? K

5 5 5- - - 4 4 4

5 5 5 4 4 4 - - -4

45 5 - 5 45 5

- 5 - - 4 4 4

4

44 4 4 -4 4

4 5 - 5

E E r E E r 7E E r

E p r E Lp r E Lp r

E (E : r (- p E h c Lb c 7

4(E :r (- p E 4(E r (- p

(E :r (-:p E h c b cE 4(E :r (-:p

′′ ′+ ++ − − + ÷ ÷ ÷ ÷− − −

− + + ÷ ÷

4

4- 5 -- -

5

5 5 5 5- - - - 4 4 4 4

5 5 5 4 4 4 - - -

(E :r (-:ph c b cE

h p E r h p E r 7h p E r - L

x E p r E p r E p r

− + ÷ ÷ ÷ + + + + +

2 2 2 (+2),6



? ?4y K cy

Va1%!%ty

If we take

- - - -- 4 5 - 4 5 - 4 5E K E K E K E B E K E K E K E B h K h K h K h B

- 4 5 - 4 5 - 4 5 b K b K b K b B r K r K r K r B p K p K p K p B

e&uation ( .-? becomes y 4 K

h(E : r(- : py

(b L h E

and e&uation ( .-4 becomesy? K

E Er hpEr (E : (- : p r

E L pr L bhx(E L pr 4E(b L h

′

which are exactly the results of the crisp model.

9"9" N'(er%ca1 E6a()1e

+onsider a situation with the following parameters.

$roduction rate, E K 5???? units#year.

)upply rate, r K - ??? units#year.

)etup cost, E\ K /s.-???#setup

6olding cost, h K /s.5.48#unit#year.

)hortage cost, b K /s.5?#unit#year.

)creening rate, x K /s.-;84??#unit#year.

)creening cost, d K /e.?.8#unit.

$roduction cost, c K /s.48#unit.

)elling price of good &uality item, sK/s.8?#unit.

)elling price of imperfect &uality item, v K/s.4?#unit.

Craction of imperfect &uality item, p K ?.?4



The optimum order &uantity is found to be 7>49 units. In fu33y model ifE

K (4998?, 5????, 5??8? ,S

K (- 4??, - 5??,

- 7?? ,

$K (?.?-8, ?.?4, ?.?48 ,

E ′5

K (99?, -???, -?-? ,

h

K (5.47, 5.48, 5.4 then optimum order &uantity y?

K 7>4>. There isa difference of only one unit between crisp and fu33y optimal order &uantity. )ensitivity analysis on the above parameters is presented

in Table .-.



Tab1e 9" +

K 0 K r ) b y7.5'??y/ y7.cr%s)/ !%55ere&ce7$y

99?,-???,-?-? 5.47,5.48,5.4 4998?,5????,5??8? - 4??,- 5??,- 7?? .?-8,.?4,.?48 49,5?,5- 7>4> 7>49.- - 44-

99?,-???,-?-? 5.47,5.48,5.4 4998?,5????,5??8? - 4??,- 5??,- 7?? .?47,.?48,.?4 49,5?,5- 7>?5 7>?5 ? 44-

99?,-???,-?-? 5.47,5.48,5.4 4998?,5????,5??8? - 4??,- 5??,- 7?? .?49,.?5,.?5- 49,5?,5- 7;;> 7;;9 - 44-

99?,-???,-?-? 5.47,5.48,5.4 4998?,5????,5??8? - 4??,- 5??,- 7?? .?57,.?58,.?5 49,5?,5- 7;85 7;87 - 44-

99?,-???,-?-? 5.47,5.48,5.4 4998?,5????,5??8? - 4??,- 5??,- 7?? .?57,.?58,.?5 -7,-8,- 79; 79;; - 744

99?,-???,-?-? 5.-8,5.4,5.48 499??,5????,5?-?? - 4??,- 5??,- 7?? .?-8,.?4,.?48 47,48,4 79?; 79?9 4 4 ?

99?,-???,-?-? 5.-,5.4,5.5 499??,5????,5?-?? - -??,- 4??,- 5?? .?-8,.?4,.?48 47,48,4 7>;7 7>;> 7 4 ?

99?,-???,-?-? 7.9,8,8.- 4998?,5????,5??8? 479??,48???,48-?? .?59,.?7,.?7- -7,-8,- ;;78 ;;84 ; 5>;

99?,-???,-?-? 7.9,8,8.- 4998?,5????,5??8? 479??,48???,48-?? .?59,.?7,.?7- -9,4?,4- ;8-8 ;844 ; 5>;

99?,-???,-?-? 5.9,7,7.- 4998?,5????,5??8? - 7??,- 8??,- ?? .?59,.?7,.?7- -9,4?,4- 77>; 77>> - 585



Crom Table .-, we infer the following (i when percentage of defective items increases, there is decrease in optimum order &uantity,

keeping all parameters fixed (ii when backorder cost decreases, there is an increase in the optimum and back order &uantity keeping other values

fixed (iii when holding cost increases and backorder cost decreases, the optimum order &uantity and backorder &uantity increases (iv when

holding cost, rate of supply, percentage of defective items increase and backorder cost decreases keeping the other values fixed, the optimal order

&uantity and backorder &uantity get almost doubled (v when percentage of defective items and holding cost increase and backorder costdecreases, keeping the other parameters unchanged, there is considerable increase in optimum order &uantity and backorder &uantity.

0ut in all the cases we see that the optimal values of order &uantity and backorder are very much closer to the crisp values.



Documents

Papers - Seminar (1)