International Journal of Industrial Engineering Computations

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1 Inernaional Journal of Indusrial Engineering Compuaions 5 (214) Conens liss available a GrowingScience Inernaional Journal of Indusrial Engineering Compuaions homepage: A deerminisic invenory model for deerioraing iems wih selling price dependen demand and hree-parameer Weibull disribued deerioraion Asoke Kumar Bhunia a and Ali Akbar Shaikh b* a Deparmen of Mahemaics, he Universiy of Burdwan, Burdwan 71314, Wes Bengal, India b Deparmen of Mahemaics, Krishnagar Gov. College, Krishnagar-74111, Wes Bengal, India C H R O N I C L E A B S R A C Aricle hisory: Received November Received in Revised Forma 1 January 214 Acceped February Available online February Keywords: Invenory Deerioraion Weibull disribuion Variable demand Parially backlogged shorage Non-linear programming In his paper, an aemp is made o develop wo invenory models for deerioraing iems wih variable demand dependen on he selling price and frequency of adverisemen of iems. In he firs model, shorages are no allowed whereas in he second, hese are allowed and parially backlogged wih a variable rae dependen on he duraion of waiing ime up o he arrival of nex lo. In boh models, he deerioraion rae follows hree-parameer Weibull disribuion and he ransporaion cos is considered explicily for replenishing he order quaniy. his cos is dependen on he lo-size as well as he disance from he source o he desinaion. he corresponding models have been formulaed and solved. wo numerical examples have been considered o illusrae he resuls and he significan feaures of he resuls are discussed. Finally, based on hese examples, he effecs of differen parameers on he iniial sock level, shorage level (in case of second model only), cycle lengh along wih he opimal profi have been sudied by sensiiviy analyses aking one parameer a a ime keeping he oher parameers as same. 214 Growing Science Ld. All righs reserved 1. Inroducion According o he exising lieraure of invenory conrol sysem, mos of he invenory models have been developed under he assumpion ha he life ime of an iem is infinie while i is in sorage i.e., an iem once in sock remains unchanged and fully usable for saisfying fuure demand. In real life siuaion, his assumpion is no always rue due o he effec of deerioraion in he preservaion of commonly used physical goods like whea, paddy or any oher ype of food grains, vegeables, fruis, drugs, pharmaceuicals, ec. A cerain fracion of hese goods are eiher damaged or decayed or vaporized or affeced by some oher facors, ec. and are no in a perfec condiion o saisfy he demand. As a resul, he loss due o his naural phenomenon (i.e., he deerioraion effec) can be ignored in he analysis of he invenory sysem. Ghare and Schrader (1963) firs developed an invenory model for exponenially decaying invenory. hen Emmons (1968) proposed his ype of model wih variable deerioraion, which follows wo-parameer Weibull disribuion. hese models were exended and improved by several researchers, viz. Cover and Philip (1973), Giri e al. (23),Ghosh and * Corresponding auhor. bhuniaak@rediffmail.com (A. A. Shaikh) 214 Growing Science Ld. All righs reserved. doi: /j.ijiec

2 498 Chaudhari (24). On he oher hand, Chakrabary e al. (1998), Giri e al.(1999), Sana e al. (24), Sana and Chaudhari (24) and ohers developed invenory models for deerioraing iems wih hereparameer weibull disribued deerioraion. Misra (1975) developed an EOQ model wih weibull deerioraion rae for perishable produc wihou considering shorages. hese invesigaions were followed by several researchers like, Deb and Chaudhari (1986), Giri e. al. (1996), Goswami and Chaudhari (1991), Mandal and Phaujdar (1989a), Padmanabhan and Vra (1995), Pal e al.(1993), Mandal and Maii (1997), Goyal and Gunasekaran (1995), Sarkar e al.(1997), Bhunia and Maii (1998a,1998b), Pal e al. (25, 26), Mishra and ripahy (21), Kawale and Bansode (212), Bhunia e al. (213a, 213b), Sharma and Chaudhary (213), Amuha and Chandrasekaran (213) ec., where a ime-proporional deerioraion rae was considered. In he presen compeiive marke, he effec of markeing policies and condiions such as he price variaions and he adverisemen of an iem change is demand paern amongs he public. he propaganda and canvassing of an iem by adverisemen in he well-known media such as Newspaper, Magazine, Radio,. V., Cinema, ec. and also hrough he sales represenaives have a moivaional effec on he people o buy more. Also, he selling price of an iem is one of he decisive facors in selecing an iem for use. I is commonly observed ha lower selling price causes increase in demand whereas higher selling price has he reverse effec. Hence, i can be concluded ha he demand of an iem is a funcion of displayed invenory in a show-room, selling price of an iem and he adverisemen expendiures frequency of adverisemen, Very few OR researchers and praciioners sudied he effecs of price variaions and he adverisemen on he demand rae of iems. Koler (1971) incorporaed markeing policies ino invenory decisions and discussed he relaionship beween economic order quaniy and pricing decision. Ladany and Sernleib (1974) sudied he effec of price variaion on selling and consequenly on EOQ. However, hey did no consider he effec of adverisemen. Subramanyam and Kumaraswamy (1971), Urban (1992), Goyal and Gunasekaran (1995), Abad (1996) and Luo (1998), Pal e al. (27), Bhunia and Shaikh (211) developed invenory models incorporaing he effecs of price variaions and adverisemen on demand rae of an iem. When a purchasing manager places an order of iems, i is imporan o realize ha differen ypes of coss including he ransporaion cos of he iems should be aken ino accoun. herefore, he invenory and ransporaion coss are dependen o each oher when he ransporaion cos is involved in invenory replenishmen, I is well known ha differen ransporaion models have differen speed, reliabiliy and cos characerisics. In mos of he exising research, he ransporaion cos is no considered separaely, i is included in he replenishmen cos which is independen of he order quaniy. In realisic siuaion, he ransporaion cos is no independen of he ordered quaniy. As a resul, his cos canno be ignored in he analysis of invenory sysem. Recenly, very few researchers considered his cos ino he analysis of lo-size deerminaion. Baumol and Vinod (197) firs considered an invenory model of freigh ranspor where by ordered quaniy and ransporaion alernaive can joinly be deermined. Consable and Whybark (1978) assumed ha he ransporaion cos per uni is independen of he order quaniies. Buffa and Munn (1989) developed he model considering he ransporaion per uni o be a negaive exponenial funcion of he order quaniy. Anily and Federgruin (199) have considered he ransporaion cos in erms of ruck loading cos. Krishnawamy e al. (1995) developed an EOQ model considering he ransporaion cos boh for deerminisic and sochasic demand cases. In hese models, ransporaion cos per uni for he mode of ranspor (ranspor vehicle) is dependen on he quaniy of he produc. Recenly, Pal e al. (26) and Mondal e al. (27) considered his cos under he assumpion ha he ransporaion cos is consan for ranspor vehicle ( of a given capaciy) even if he quaniy shipped is less han a ranspor vehicle load by some quaniy. In his paper, we have developed wo invenory models for deerioraing iems wih variable demand dependen on he selling price of iems and frequency of adverisemen of iems. In he firs model, sock-ou siuaion is no allowed whereas in he second, i is allowed and parially backlogged wih a

3 A. K. Bhunia and A. A. Shaikh / Inernaional Journal of Indusrial Engineering Compuaions 5 (214) 499 variable rae dependen on he duraion of waiing ime up o he arrival of nex lo. In boh he models, he deerioraion rae follows a hree-parameer Weibull disribuion and he ransporaion cos is considered explicily for replenishing he order quaniy. he corresponding models have been formulaed and solved by considering he ransporaion cos for replenishing he iems. wo numerical examples have been given o illusrae he resuls and he significan feaures of he resul are discussed. Finally, based on hese examples, he effecs of differen parameers on he iniial sock level, shorage level (in case of second model only), cycle lengh along wih he opimal profi, sensiiviy analyses have been performed considering one parameer a a ime keeping oher parameers a heir original values. 2. Assumpions and Noaions he following assumpions and noaions are used o develop he proposed model: (i) Replenishmens are insananeous wih a known and consan lead ime. (ii) he enire lo is delivered in one bach. (iii) he invenory planning horizon is infinie and he invenory sysem involves only one iem and one socking poin. (iv) he deerioraion occurs when he iem is effecively in-sock and is rae follows a hreeparameer Weibull disribuion. (v) he deerioraed unis are no replaced. (vi) he replenishmen cos (ordering cos) is consan and does no include he ransporaion cos for replenishing he iem. (vii) A be he frequency of adverisemen in he cycle lengh. (viii) he invenory carrying cos, C 1 per uni per uni ime, shorage cos, C 2 per uni per uni ime, he purchase cos, C 3 per uni, he ordering cos, C 4 per order, he adverisemen cos C 5 per adverisemen and he selling price, p are known and consans. (ix),, be he parameers of he Weibull disribuion whose probabiliy densiy funcion is 1 f ( ) exp (x) he insananeous rae of deerioraion of he on-hand in any ime is ( ) which obeys he hree parameer Weibull disribuion. f ( ) 1 So, ( ) 1 F( ) where 1,, and F ( ) is he disribuion funcion of Weibull disribuion. For 1, ( ) is consan and he Weibull disribuion reduces o exponenial disribuion 1, ( ) is an increasing funcion of, 1, ( ) is a decreasing funcion of. For, he hree-parameer Weibull disribuion reduces o wo parameer Weibull disribuion. (xi) he demand rae D( A, p ) is dependen on selling price (p) of an iem and he frequency of adverisemen (A). We assume i as follows: D( A, p) A ( a bp), a, b,. (xii) R be he maximum shorage level. (xiii) m(>1) be he mark-up i.e., p=mc 3. (xiv) he capaciy of a ransporaion vehicle is k unis. (xv) L be he disance beween he shop and he source of he iems/commodiies from where iems/commodiies o be ranspored. (xvi) C be he ransporaion cos for full load of he ranspor vehicle and C F be he ransporaion cos per uni iem.

4 5 (xviii) U be he upper break poin, some quaniy loss han k bu more hanu, he ransporaion cos for whole quaniy is C. C C Hence U ( k) where represens he greaes ineger value which is less hen or equal C C F F o C / C. F 3. Invenory model wihou shorage Iniially, an enerprise purchases an amoun of sock of S unis of he iem. his amoun will be depleed due o deerioraion of iem and also o mee up he cusomers demand. Le q be he insananeous invenory level a any ime he differenial equaion as follows:. hen he invenory level q a any ime saisfies dq( ) ( ) q( ) D A, p, d wih he boundary condiions (1) q and S a (2) q a. (3) Using he condiion (3), he soluion of he differenial Eq. (1) is given by q D A p e e d (4), From Eq. (2), we have, S D A p e e d (5) During he invenory cycle, deerioraed unis are o be separaed. he amoun of invenory ha has deerioraed during he cycle can be derived by inegraing he produc of he deerioraion rae and he invenory level over he enire cycle i.e., he oal number of deerioraed unis is given by D ( ) q( ) d. However, D can be derived from he difference beween he iniial sock and he oal selling amoun during he cycle, i.e., D S D( A, p) d S D( A, p) (6)

5 A. K. Bhunia and A. A. Shaikh / Inernaional Journal of Indusrial Engineering Compuaions 5 (214) 51 Now, he oal invenory holding cos 1 Chol C q d where 2 C hol is given by q d (2 1) (2 1) neglecing he powers of higher han wo as 1. he oal adverisemen cos is he produc of he number of adverisemen and he cos per adverisemen i.e., Cadv C5A When he ordered quaniy is greaer han one inegral ranspor vehicle load, he ordered quaniy Q can be expressed as Q nk k1q where n or any posiive ineger ; k 1 or 1 and q k. In ha case, wo siuaions may arise: (ii) 1 (i) nk Q nk U nk U Q n k Hence he oal ransporaion cos is given by, where 1, where 1 C nc Q nk C nk Q nk U ran F n C nk U Q n k he oal cos ( C ) of he sysem is given by C = <ordering cos> + <purchasing cos> + <invenory holding cos> + <adverisemen cos> + <ransporaion cos> C C. S C q d C A C ran he ne profi ( X ) for he enire sysem is he difference beween he sale revenue per cycle and he oal cos of he sysem i.e., X pa ( a bp) C (8) (7)

6 52 herefore he profi funcion Z( m, n, A, ) (average profi per uni ime for he enire cycle) of he invenory sysem is given by X Z( m, n, A, ) i.e., Z( m, n, A, ) pa ( a bp) C4 C3. S C1 q d C5A Cran / (9) Here he profi funcion is a funcion of wo coninuous variables m, and wo ineger variables n, A. Proposiion: he profi funcion Z is an increasing funcion of m for any feasible values of n, A and. dz Proof: Clearly dm for a 2 bp. his condiion gives he increasing propery of he profi funcion Z. Hence our problem is o deermine he opimal values of n, A, and S by solving he following mixed ineger nonlinear opimizaion problem. Maximize Z( n, A, ) subjec o n, A, > and n, A are inegers C nc ( S nk) C, when nk S nk U ran F n 1 C, when nk U S ( n 1) k he above problem can be solved by using he well-known LINGO 8. sofware. However, he opimal soluion of he above problem can be obained wih he help of he following algorihm. Algorihm 1 Sep-1: Inpu all he parameers. Sep-2: Calculae he value of U. Sep-3: Solve he problem (1) by aking he ransporaion cos for firs siuaion ( nk S nk U ) Sep-4: If S lies in he inerval( nk S nk U ), hen his is he opimal policy and go o Sep-7. Oherwise, go o Sep-5. Sep-5: Solve he problem (1) by aking he ransporaion cos for second siuaion ( nk U S ( n 1) k). Sep-6: If S lies in he inerval ( nk U S ( n 1) k), hen his is he opimal policy. Sep-7: Sop. 4. Numerical Example o illusrae he developed model, an example wih he following daa has been considered. C1 $1.5 per uni per uni ime, C3 $8 per uni, C4 $25 per order, C5 $5 per adverisemen, C $1 per ranspor vehicle, CF $1.25 per uni,.5, 2, 2.5, a = 25, b =.3, k = 1 unis. L=5 km, =.1. hough he values of differen parameers are no colleced from any case sudy, he values considered here are realisic. According o our developed algorihm (i.e., Algorihm 1) of he proposed invenory sysem, he opimal soluion has been obained wih he help of well known LINGO sofware for differen values of m. he opimum values of n, A,, S along wih he maximum average profi are displayed in able 1. (1)

7 A. K. Bhunia and A. A. Shaikh / Inernaional Journal of Indusrial Engineering Compuaions 5 (214) 53 able 1 Opimal soluion for differen values of mark-up rae m m A S Z Sensiiviy Analysis For he given example menioned earlier, sensiiviy analysis has been performed o sudy he effec of changes (under or over esimaion) of differen parameers like demand, deerioraion parameers and mark-up rae on maximum iniial sock level, cycle lengh, frequency of adverisemen along wih he maximum profi of he sysem. his analysis is carried ou by changing (increasing and decreasing) he parameers from 2% o + 2%, aken one or more parameers a a ime and making he oher parameers a heir original values. he resuls of his analysis are shown in able 2. able 2 Sensiiviy analysis wih respec o differen parameers % changes in Parameer % changes of parameers % changes in Z * A * S * * C C C C a b v α γ

8 54 6. Invenory model wih shorages In his model, shorages, if any, are allowed and parially backlogged. During he shorage period, he backlogging rae is dependen on he lengh of he waiing ime up o he arrival of fresh lo.,. In his model, i is assumed ha afer fulfilling he backorder quaniy, he on-hand invenory level is S a = and i declines coninuously up o he ime 1 when i reaches he zero level. he decline in invenory during he closed ime inerval 1 occurs due o he cusomer s demand and deerioraion of he iem. Afer Considering his siuaion, he rae is defined as 1 shorage occurs and i accumulaes a he rae 1 he ime, 1 1, ( ) up o he ime when he nex lo arrives. A ime, he maximum shorage level is R. his enire cycle hen repeas iself afer he cycle lengh. Le q be he insananeous invenory level a any ime. hen he invenory level ime saisfies he differenial equaions as follows, 1 q a any dq( ) ( ) q( ) D A, p, d dq( ) D( A, p), 1 d 1 ( ) wih he boundary condiions 1 (11) (12) q and q S a, ( ) q a 1. (13) R a. (14) Also, q( ) is coninuous a 1. Using he condiions (3) and (4), he soluions of he differenial equaions (11)-(12) are given by 1 ( ) ( ) q( ) D( A, p) e e d, D( A, p) log 1 ( ) R, 1 1 From Eq. (12), we have hen, 1 q S D A, pe e d S a =. From he coninuiy condiion, we have D( A, p) R log 1 ( 1 ) Now he oal invenory holding cos for he enire cycle is given by (15) (16)

9 A. K. Bhunia and A. A. Shaikh / Inernaional Journal of Indusrial Engineering Compuaions 5 (214) Chol C q d where q d (2 1) (2 1) (17) Again, he oal shorage cos C Sho over he enire cycle is given by D( A, p) D( A, p) Csho R ( 1) log 1 ( 1) Hence he oal invenory cos ( C ) of he sysem is given by C = <ordering cos> + <purchasing cos> + <invenory holding cos> + <adverisemen cos> + <ransporaion cos>+< invenory shorage cos> 1 C C ( S R) C q d C A C C ran Now, he ne profi (X) for he enire sysem is given by sho X pa ( a bp) C. (18) herefore, he profi funcion Z( m, n, A, 1, ) (average profi per uni ime for he enire cycle) of he X invenory sysem is given by Z( m, n, A, 1, ) 1 i.e., Z( m, n, A, 1, ) pa ( a bp) C4 C3( S R) C1 q d C5 A Cran C sho / (19) Here he profi funcion is a funcion of hree coninuous variables m, 1, and wo ineger variables n, A. Clearly, he above funcion is an increasing funcion wih respec o m. Hence our problem is o deermine he opimal values of n, A, 1, and S by solving he following mixed ineger nonlinear opimizaion problem. Maximize Z( n, A, 1, ) subjec o 1, n, A, 1> and n, A are inegers (2)

10 56 where C nc ( S R nk) C when nk S R nk U ran f n 1 C when nk U S R ( n 1) k his is a non-linear opimizaion problem. he opimal soluion of he his problem can be obained wih he help of he following algorihm using LINGO sofware. Algorihm 2 Sep-1: Inpu all he parameers. Sep-2: Calculae he value of U. Sep-3: Solve he problem (2) by aking he ransporaion cos for firs siuaion ( nk S R nk U ) Sep-4: If S+R Saisfies he condiion nk S R nk U, hen his is he opimal policy and go o Sep-7. Oherwise, go o Sep-5. Sep-5: Solve he problem (2) by aking he ransporaion cos for second siuaion ( nk U S R ( n 1) k). Sep-6: If S+R Saisfies he condiion nk U S R ( n 1) k, hen his is he opimal policy. Sep-7: Sop. 7. Numerical Example o illusrae he model wih parially backlogged shorages, a numerical example wih he following daa has been considered. C1 $1.5 per uni per uni ime, C 2 =$3 per uni per uni ime, C3 $8 per uni, C4 $25 per order, C5 $5 per adverisemen, C $1 per ranspor vehicle, CF $1.25 per uni,.5, 2, 2.5, a = 25, b =.3, k = 1 unis. L=5 km, 1.5,.1. Like firs model (no-shorage case), he values of differen parameers considered here are realisic, hrough hese are no aken from any case sudy. According o he soluion procedure (Algorihm-2), he opimal soluion has been obained wih he help of LINGO sofware for differen values of m. he opimum values of n, A, 1,, S and R along wih maximum average profi are displayed in able 3. able 3 Opimal soluion for differen values of mark-up rae m m A S R 1 Z Sensiiviy Analysis For he given example menioned earlier, sensiiviy analysis has been performed o sudy he effec of changes (under or over esimaion) of differen parameers like demand, deerioraion, invenory cos parameers and mark-up rae on maximum iniial sock level, shorage level, cycle lengh, frequency of adverisemen along wih he maximum profi of he sysem. his analysis has been carried ou by changing (increasing and decreasing) he parameers from 2% o + 2%, aken one or more

11 A. K. Bhunia and A. A. Shaikh / Inernaional Journal of Indusrial Engineering Compuaions 5 (214) 57 parameers a a ime making he oher parameers a heir more parameers a a ime and making he oher parameers a heir original values. he resuls of his analysis are shown in ables 4. able 4 Sensiiviy analysis wih respec o differen parameers Parameer C 1 C 2 C 3 C 4 C 5 a b 9. Concluding Remarks % changes of % changes in % changes in parameers Z * * A * * * * R S In his paper, wo deerminisic invenory models for deerioraing iems wih variable demand dependen on he selling price and he frequency of adverisemen of he iem have been developed wih and wihou shorages. In he formulaion of he model, ransporaion cos is considered explicily for replenishing he order quaniy. In mos of he siuaions, i is he buyer s cos who mus bear he

12 58 ransporaion cos for ransporaion of he goods purchased from he supplier. Such coss are eiher assumed o be fixed and are herefore included in he ordering cos or variable and include in he uni cos of he iem. In real-life siuaions ransporaion coss of goods are fixed for a finie capaciy of a ranspor mode such as a ruck. A fixed cos is incurred when a ruck is deployed wheher i is uilized fully or parially for a ceiling quaniy or more. For quaniies less han he ceiling quaniy, a uniform rae per uni is charged. In boh models, he demand rae is aken as D A, p A a bp D A, p a bp for fixed A. Bu, why should we ake D A, p. I is well known ha for fixed values of p? Generally, he demand of iems varies due o he adverisemen in he well known media such as Radio,.V., Newspaper, Magazine, Cinema, ec. he demand of iems increases wih he increase of frequency of adverisemen and is direcly proporional o he number of adverisemen. Hence, we ake D A, p for fixed p. A he presen model is also applicable o he problems where he selling prices of he iems as well as he adverisemen of iems affec he demand. I is applicable for fashionable goods also. he problem of invenory and ransporaion ineracions is a poenial field of research. he subjec offers a lo of scope for addiional work based on wha has been presened in his paper. he following can be poenial problems ha can be sudied: (i) he possibiliy of single ruck supplying he producs o more han one go down. (ii) he possibiliy of iner-depo ransporaion can be incorporaed from he following poins of view. (a) (b) he invenory disribuion coss can be minimized. his provides anoher alernaive o deal wih he space consrain ha is operaive a some of he go downs. (iii)he possibiliy of mixing up more han one produc in he same ruck when each facory produces more han one produc. References Abad, P.L. (1996). Opimal pricing and lo-sizing under of condiions of perishabiliy and parial backordering, Managemen Science, 42, Amuha, R., & Chandrasekaran, E. (213). An invenory model for deerioraing iems wih hree parameer weibull deerioraion and price dependen demand, Journal of Engineering Research & echnology, 2(5), Anily, S., & Federgruen, A. (199). One warehouse muliple reailer sysems wih vehicle rouing coss, Managemen Science, 36, 32. Baumol, W. J., & Vinod, H. C. (197). An invenory heoreic model of freigh ranspor demand, Managemen Science,16, Bhunia, A.K., & Maii, M. (1998). Deerminisic invenory model for deerioraing iems wih finie rae of replenishmen dependen on invenory level, Compuers and Operaions Research, 25, Bhunia, A.K., & Maii, M. (1998). An invenory model of deerioraing iems wih lo-size dependen replenishmen cos and a linear rend in demand, Applied Mahemaical Modelling, 23, Buffa, F., & Munn, J. (1989). A recursive algorihm for order cycle ha minimizes logisic cos, Journal of Operaional Research Sociey, 4, 357. Bhunia, A.K., & Shaikh, A.A. (211). A deerminisic model for deerioraing iems wih displayed invenory level dependen demand rae incorporaing markeing decision wih ransporaion cos. Inernaional Journal of Indusrial Engineering Compuaions, 2(3), A

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