Masoud Rabbani 1*, Leila Aliabadi 1

Transcription

1 Joural of Idustral ad Systems Egeerg Vol. 11, No.2, pp Sprg (Aprl) 2018 Mult-tem vetory model wth probablstc demad fucto uder permssble delay paymet ad fuzzy-stochastc budget costrat: A sgomal geometrc programmg method Masoud Rabba 1*, Lela Alabad 1 1 School of Idustral Egeerg, College of Egeerg, Uversty of Tehra, Tehra, Ira mraba@ut.ac.r, Leyla.alabad@ut.ac.r Abstract Ths study proposes a ew mult-tem vetory model wth hybrd cost parameters uder a fuzzy-stochastc costrat ad permssble delay paymet. The prce ad marketg expedture depedet stochastc demad ad the demad depedet the ut producto cost are cosdered. Shortages are allowed ad partally backordered. The ma objectve of ths paper s to determe sellg prce, marketg expedture, credt perod, ad varables of vetory cotrol smultaeously for maxmzg the total proft. To solve the problem, frst some trasformatos are appled to covert the orgal problem to a mult-objectve olear programmg problem, of whch each objectve has sgomal terms. The, the mult-objectve olear programmg problem s solved by frst covertg t to a sgle objectve problem ad the by usg global optmzato of sgomal geometrc programmg problems. At the ed, several umercal examples ad sestvty aalyss are doe to test model ad soluto procedure ad also obta maageral sghts. Keywords: Sgomal geometrc programmg, delay paymet, fuzzystochastc recourse, prce ad marketg depedet stochastc demad, EOQ. 1- Itroducto By chagg market treds ad creasg competto busess world, the trade credt s gag popularty amog may retal establshmets. Uder ths polcy, sellers offer a specfed perod to buyers to pay ts paymets wthout pealty order to stmulate sales ad decrease the cost of holdg vetory. I practce, a permssble delayed paymet reduces the holdg cost because uder ths polcy the amout of captal vested vetory durg the credt perod decreases. Moreover, durg the credt perod, buyers ca accumulate reveue o sales ad ear terest o that reveue by bakg busess or share marketg vestmet. I today s competto market, most compaes use the trade credt strategy to crease the sales ad attract more customers. Therefore, the trade credt strategy plays a ma role moder busess operatos. I recet years, a substatal amout of research has bee dedcated to model *Correspodg author ISSN: , Copyrght c 2018 JISE. All rghts reserved 207

2 vetory polces volvg trade credt polcy. For the frst tme, Goyal (1985) developed a EOQ model uder permssble delay paymet. The, Aggarwal ad Jagg (1995) exteded ths model for deteroratg tems. Jamal et al. (1997) frst formulated a EOQ model wth allowable shortages ad permssble delayed paymets. Chug ad Huag (2003) geeralzed the model of Goyal (1985) from the EOQ model to the EPQ model. Huag (2007) supposed the suppler would suggest partally permssble delayed paymet f the order quatty s smaller tha a pre specfed quatty. Lag ad Zhou (2011) proposed a two-warehouse vetory model for deteroratg tem wth allowable delay paymets. Talezadeh et al. (2013) cosdered a EOQ problem wth partal delay paymets ad partal backorderg. Sarkar et al. (2015) developed a vetory model for deteroratg tems uder two level trade credt ad tme - depedet determato rate. I all above cted artcles, t s assumed that demad rate ad producto cost s costat whle these cosderatos are ot true real world markets. Some researchers cosdered ut producto cost as a fucto of demad (Islam ad Roy 2006; Pada et al. 2008) or order quatty (Samad et al. 2013; Tabatabae et al. 2017), or qualty (Cheg 1991). Moreover, real stuato, demad rate depeds o dfferet parameters such as sellg prce ad marketg expedture. Prcg s a mportat strategy for compaes to ehace ther proft. I fact, there s a egatve correlato amog sellg prce ad demad rate. That s, demad rate decreases as sellg prce creases. Ho et al. (2008) aalyzed a tegrated vetory model wth prce depedet demad uder permssble delay paymet. They determed the optmal orderg, prcg, paymet perod, ad shppg to maxmze the total proft. So (2013) formulated a vetory model wth assumpto that demad rate s a multvarate fucto of sellg prce ad vetory ad delay paymet s permtted. Other works that cosdered prce depedet demad ad trade credt smultaeously are as follows: So ad Patel (2012), Maham ad Abad (2012), Chug et al. (2015), Maham et al. (2017) ad etc. Apart from the sellg prce, most codtos, marketg expedture s also mportat fluecg demad. A compay ca stmulate demad by creasg advertsg, hrg more sales people, provdg attractve space, ad etc. All of those actvtes are costly. There are a lot of works that have bee cosdered demad rate as a fucto of marketg expedture; for example He et al. (2009), Pag et al. (2014), Samad et al. (2013), De ad Saa (2015), Tabatabae et al. (2017), ad etc. Recetly, to better demostrate the real stuato, some researches formulated ther models wth stochastc demad. He et al. (2009) vestgated the ssue of supply cha coordato by cosderg prce ad marketg depedet stochastc demad. Maham ad Karm (2014) proposed a EOQ model wth prce depedet stochastc demad ad partal backorderg for o-stataeous deteroratg tems. Maham et al. (2017) developed a prcg vetory model for o-stataeous deteroratg tems wth cosderg partal backorderg, prce depedet stochastc demad uder two- level trade credt polcy. Oe of the extesos of the vetory models that has receved more academc atteto the recet years, s mprecso defg put parameters. I geeral, the exstg formato ca be determstc, fuzzy or probablstc. Pramak et al. (2017) developed a vetory model wth fuzzy cost parameters uder three level trade credt polcy ad prce depedet demad. Das et al. (2004) formulated mult-tem stochastc ad fuzzy-stochastc vetory models uder space ad budgetary costrats. I the both models, demad ad budgetary resource are cosdered radom. They cosdered space resource as fuzzy umber fuzzy-stochastc model. But may real stuatos, a orgazato may face stuato that several cost parameters may chage such way that a part s radom ad aother part s fuzzy. These cost parameters are called hybrd cost parameters. Pada et al. (2008) proposed two vetory models wth hybrd cost parameters. I model 1: They cosdered resource parameters as fuzzy umber; model 2: some resource parameters were cosdered as fuzzy stochastc ad some as fuzzy. They provded a framework for a EOQ model fuzzy- stochastc evromet ad solved ther problem by usg Geometrc Programmg (GP) method. GP problem s a class of o-lear optmzato problems that has partcular objectve fuctos ad costras. Ths method has very useful computatoal ad theoretcal propertes to solve complex optmzato problems dfferet felds such as egeerg, maagemet, scece, etc. Ths techque 208

3 was exteded rapdly by researchers, especally egeerg desgers. Sgomal Geometrc Programmg (SGP) problem was the frst exteso of GP problems. SGP problems are categorzed class of o- covex optmzato problems ad NP- hard problems. SGP techque s well used for solvg vetory models lterature (Madal et al. 2006; Samad et al. 2013; Sadjad et al. 2015). I ths techque degree of dffculty (DD 2 ) has a mportat role. Whe DD 2, may researchers have appled dual geometrc programg for solvg vetory models. But f DD 3,, solvg vetory models wll be dffcult. Sce, the mportat secto SGP s the method used. A comparso of metoed papers s llustrated Table 1. From the Table 1, some of the major shortcomgs of prevous papers the formulato of vetory models ca be summarzed as follows: Most vetory models wth delayed paymets have faled to cosder ucerta demad. Most prevous studes have assumed the ut cost s costat. No vetory model wth delayed paymets s developed a fuzzy-stochastc evromet. No vetory model wth delayed paymets has cosdered the prce ad marketg cost depedet demad. Icorporatg all pheomea metoed above, ths paper develops a mult-tem EOQ model uder budgetary costrat wth cosderg the probablstc demad ad permssble delay paymet a fuzzy-stochastc evromet. Shortages are allowed ad partally backordered. We cosder the prce ad marketg expedture depedet stochastc demad fucto. We also adopt the demad depeded ut producto cost. The cost parameters are represeted by hybrd umbers ad the total budget to purchase vetory s cosdered as fuzzy-stochastc quatty. The ma objectve of ths paper s to determe sellg prce, marketg expedture, credt perod, ad varables of vetory cotrol smultaeously for maxmzg the total proft. For solvg our problem, we frst covert out model to a mult-objectve olear programmg (MONP) problem, of whch each objectve has sgomal terms, wth usg the methods to tur the fuzzy- radom parameters to crsp oes. The, we solve the MONP problem by frst covertg t to a sgle objectve problem ad the by usg global optmzato method dscussed by Xu (2014) for solvg SGP problems. The rest of ths paper s bee orgazed as follows: assumptos ad otatos that are requred to model the proposed problem are gve secto 2. The mathematcal formulato of the problem s preseted Secto 3. Secto 4 provdes the soluto method. Numercal examples ad sestvty aalyss are doe to test model ad soluto method ad also obta maageral sghts sectos 5 ad 6. Fally, coclusos wth future research are gve secto

4 Table 1. Bref revew of metoed studes Studes Ut cost Demad DP FSC Shortage C P-M O D S F Full Partal Huag Costat * * (2007) Pada et al Demad depedet * * * Lag ad Costat * * * Zhou (2011) Talezadeh et al. (2013) Costat * * * * Samad et al. Order quatty * * * 2013 Maham ad Karm (2014) De ad Saa (2015) Tabatabae et al Maham et al. (2017) Pramak et al. (2017) Ths study depedet Costat * * * Costat * * * * Order quatty depedet * * Costat * * * Costat * * * Demad depedet * * * * * Note: Costat (C), Prce-Marketg depedet (P-M), Other (O), Determstc (D), Stochastc (S), Fuzzy(F), Delay Paymet (DP), Fuzzy-Stochastc Costrat (FSC). 1 DD = the umber of decso varables + the umbers of terms objectve fuctos ad costrats Notato ad assumpto We formulate our problem by followg otatos ad assumptos: 2-1- Notatos dces: Sets of product types = Crsp parameters: I e Iterest eared rate ($/year) I p Iterest charged rate ($/year) β The percetage of shortages that wll be backordered for each tem C Ut purchasg cost of a tem ($/ut) α Prce elastcty to demad χ Marketg expedture elastcty to demad γ Demad elastcty to purchasg cost M 0 Upper lmt of credt perod Hybrd parameters: A Orderg cost ($/order) π Backorderg cost ($/ut/year) g Goodwll loss for ut lost sales h Holdg cost ($/ut/year) Fuzzy-stochastc parameter: y Total avalable producto cost 210

5 Decso varables: P The porto of demad that wll be satsfed from warehouse T The legth of a vetory cycle tme S The ut sellg prce of tem G Marketg expedture per ut of tem M The perod of permssble delay paymet of tem (credt perod) Idepedet decso varable: λ Demad rate of tem Q The order quatty of tem Partal backordered amout at tme T B Note: ~ ad deote radomzato ad fuzzfcato of the parameters, y ad b deote that y ad b are fuzzy-stochastc parameter ad hybrd parameter, respectvely. 2-2-Assumptos The demad rate of tem, λ = λ (S. G ) + ξ, cotas two parts: λ (S. G ): a power fucto of sellg prce ad marketg expedture as follows: α λ (S. G ) = V S χ G (1) where V s scalg factor ad α 1 ad χ 0 are sellg prce elastcty ad marketg elastcty, respectvely. ξ : a cotuous radom varable by specfed ad tme depedet dstrbuto fucto E(ξ ) = μ. Ut cost s a decreasg fucto of demad rate whch s calculated as follows: γ C = U λ (2) Shortages are allowed ad are as combato of lost sales ad backorders. There s o deterorato. Repleshmet rate s stataeous ad lead tme s zero. The tme horzo s fte. There s a lmtato o the total producto cost wth fuzzy- stochastc quatty. For each tem, orderg cost, holdg cost, ad shortage costs (A. h. π. g ) are cosdered as hybrd umbers. I the preseted supply cha, the retaler purchases the tems each cycle uder the trade credt strategy provded by the suppler. It meas the suppler gves a full credt perod of M years for each tem to the retaler. Durg the credt perod M, the retaler sells the products ad collects the sale reveue ad obtas terest at a rate I e ; the retaler must settle the accout at tme M for each tem ad pays for terest charges o goods stock wth rate I p. 3- Model formulato The behavor of the cosdered vetory system wth prce ad marketg expedture depedet stochastc demad ad demad depedet ut cost uder permssble delayed paymet s show Fg 1. Accordg to Fg 1, the order quatty of tem, = , s obtaed as: Q = P T λ + β λ (1 P )T = (V S α G χ + ξ )(β + P (1 β ))T (3) 211

6 Ivetory level λ P T λ P T - λ M Q β λ (1-P )T M P T T B tme (1-β ) λ (1-P )T Fg 1. Ivetory dagram The ma goal of the problem s to determe the sellg prce (S ), marketg expedture (G ), credt perod (M ), cycle tme (T ), ad the porto of demad that wll be satsfed from stock (P ) so that the total average proft of the vetory system s maxmzed. So, the followg are compoets of the total aual proft: The expected sales reveue (SR ) for the the tem per cycle s: SR = E(S Q ) = (V S α G χ + μ )(β + P (1 β ))S T (4) The expected marketg expedture (CM ) for the the tem per cycle s : CM = E(G Q ) = (V S α G χ + μ )(β + P (1 β ))G T (5) The expected holdg cost (CH ) for the the tem per cycle s : λ P P T α CH = E (h ) = 0.5h (V 2 S χ G + μ )P 2 2 T (6) Where h = (h 1. h 2. h 3 )(+) (μ h + σ 2 h ) The expected producto cost (CP ) for the the tem per cycle s : CP = E(C Q ) = U (V S α G χ + μ ) 1 γ (β + P (1 β ))T (7) The orderg cost (CO ) for the the tem per cycle s : CO = A Where A = (A 1. A 2. A 3 )(+) (μ A + σ 2 A ) The expected backorder cost (CB ) for the the tem per cycle s : β λ (1 P )T (1 P )T α CB = E (π ) = 0.5π β 2 (V S χ G + μ )(1 P ) 2 2 T Where π = (π 1. π 2. π 3 )(+) (μ π + σ 2 π ) The expected lost sale cost (CL ) for the the tem per cycle s: (8) (9) 212

7 CL = E (g (1 β )λ (1 P )T ) = g (1 β )(V S α G χ + μ )(1 P )T (10) Where g = (g 1. g 2. g 3 )(+) (μ g + σ 2 g ) The terest payable per cycle ad the terest eared per cycle are calculated by the relatoshp of credt perod (M ) ad the legth of tme whch o vetory shortage happes( P T ), hece we cosder the followg two cases: Case 1- M P T I ths case, the expected terest payable (IP 1 ) per cycle for the tems ot sold after the tme M s as follows (see Fg 2): λ (P T M ) (P T M ) α IP 1 = E (C I p ) = 0.5CU 2 I p (V S χ G + μ ) 1 γ (P T M ) 2 (11) The expected terest eared (IE 1 ) per cycle durg the postve vetory s as follows (see fgure 2): IE 1 = E (I e S (β λ (1 P )T M + λ 2 M )) (12) 2 = I e S (β (1 P )T M + 0.5M 2 )(V S α G χ + μ ) Case 2- P T M M 0 I ths case, the expected terest eared (IE 2 ) per cycle durg [0. M ] s (see Fg 2): IE 2 = E (I e S (β λ (1 P )T M + λ P 2 2 T + λ 2 P T (M P T ))) (13) = I e S (β T M 0.5P 2 T 2 + (1 β )P T M )(V S α G χ + μ ) I ths case, the retaler does ot eed to pay ay terest, that s IP 2 = 0. Therefore, the average total proft per year for tems for case 1 (ATP 1 ) ad case 2 (ATP 2 ) s : ATP j = [ 1 T (SR CM CH CP CO CB CL IP j + IE j )] =1 After smplfcato, the followg results are obtaed: j = 1.2 (14) ATP 1 (x) = (N X S N X G 0.5(h + θ 1 π )X P 2 T + θ 1 π X P T 0.5θ 1 π X T (15) =1 θ 2 g X + θ 2 g X P θ 3 N X 1 γ θ 4 X 1 γ P 2 T θ 4 X 1 γ M 2 T 1 + 2θ 4 X 1 γ P M +θ 5 X S M θ 5 X S M P + θ 6 X S M 2 T 1 A T 1 ) ATP 2 (x) = (N X S N X G 0.5(h + θ 1 π )X P 2 T + θ 1 π X P T 0.5θ 1 π X T (16) =1 θ 2 g X + θ 2 g X P θ 3 N X 1 γ + θ 5 X S M θ 6 X S M P 2 T + θ 7 X S M P A T 1 ) 213

8 Where α X = V S χ G + μ (17-1) N = β + P (1 β ) (17-2) θ 1 = β 0 (17-3) θ 2 = 1 β 0 (17-4) θ 3 = U 0 (17-5) θ 4 = 0.5U I p 0 (17-6) θ 5 = β I e 0 (17-7) θ 6 = 0.5I e 0 (17-8) θ 6 = (1 β )I e 0 (17-9) x = (S. T. G. M. P. X. N ) 0 (17-10) Wht, h = (h 1. h 2. h 3 )(+) (μ h + σ h ), π = (π 1. π 2. π 3 )(+) (μ π + σ π ), g = (g 1. g 2. g 3 )(+) (μ g + σ g ), A = (A 1. A 2. A 3 )(+) (μ A + σ A ), ad = As explaed above, we cosder a lmtato o the total budget for purchasg vetory wth fuzzy stochastc quatty as follows: 1 γ CP y θ 3 N X T y (18) =1 =1 Where y = (((y 1 1. y 1 ). q 1 ); ((y 2 1. y 2 ). q 2 ) ; ((y 3 1. y 3 ). q 3 )). Therefore, the mathematcal model of the problem s: Max ATP j j = 1.2 (19) 1 γ s.t. θ 3 N X T y (20) =1 α X = V S χ G + μ (21) N = β + P (1 β ) (22) x = (S. T. G. M. P. X. N ) 0 (23) M P T for j = 1 (24) P T M M 0 for j = 2 (25) Where, y = (((y 1 1. y 1 ). q 1 ); ((y 2 1. y 2 ). q 2 ) ; ((y 3 1. y 3 ). q 3 )) ad =

9 Iterest payable Iterest eared λ P T λ P T λ P T - λ M λ M λ PT B B M P T T B T P T M T B Case 1: M P T Case 2: P T M Fg 2. Ivetory dagram for cases 1 ad 2 4- Soluto method I ths secto, we frst covert out model to a mult-objectve olear programmg (MONP) problem, of whch each objectve has sgomal terms, wth usg the methods of covertg the fuzzyradom parameters to crsp oe. The, we solve the MONP problem by frst covertg t to a sgle objectve problem ad the by usg global optmzato method dscussed by Xu (2014) for solvg SGP problems. Case 1- M P T Followg example-1 Luhadjula (1983), we frst covert the fuzzy-stochastc costrat (20) to the followg determstc form: ( 1 γ θ 3 N X 1 =1 q T ) y 1 ( 1 γ θ 3 N X 1 = q T ) y 2 ( 1 γ θ 3 N X 1 =1 y 1 y q T ) y 3 1 y 2 y 3 1 α 2 y 3 y 3 After smplfcato, we have: ( q 1 y 1 y q 2 y 2 y q 3 y 3 y 3 1) ( q 1 1y 1 y 1 y1 + q 1 2y 2 1 y 2 y1 + q 1 3y γ ( θ 3 N X T ) (27) y 3 y1 + α) =1 3 The, we rewrte the costrat (21) as follows: (26) α X = V S χ G + μ { X α V S χ G + μ 1 α X V S χ G + μ 2 So, we have: (28) 1 X V S α G χ + μ X V S α G χ μ μ 1 X μ 1 V S α G χ 1 (29) 215

10 2 X V S α G χ + μ V S α G χ X 1 + μ X 1 1 (30) Followg the same maer as descrbed for costrat (21), we covert costrats (22) ad (24) to the followg form: N = β + P (1 β ) { β 1 N β 1 (1 β )P 1 β N 1 + (1 β )P N 1 (31) 1 M P 1 T 1 1 (32) The objectve fucto of the problem s maxmzg the total proft ad s wrtte as: Max ATP 1 (x). Sce, Max ATP 1 (x) s equvalet M ( ATP 1 (x)), thus, the problem (19)-(24) ca be rewrtte as follows: M Z 1 (x) (33) s.t. μ 1 X μ 1 V S α G χ 1 (34) V S α G χ X 1 + μ X 1 1 (35) β 1 N β 1 (1 β )P 1 (36) β N 1 + (1 β )P N 1 1 (37) ( q 1 y 1 y q 2 y 2 y q 3 y 3 y 3 1) ( q 1 1y 1 y 1 y1 + q 1 2y 2 1 y 2 y1 + q 1 3y 3 2 Z 1 (x) 1 γ ( θ 3 N X T ) (38) y 3 y1 + α) =1 3 x = (S. T. G. M. P. X. N ) 0 (39) M P 1 T 1 1 (40) Accordg to the hybrd umbers theory as explaed by Pada et al. (2008) the problem (33)-(40) reduces to: M EVZ 1 (x) = EZ 01 (x)(+) (0. V 1 (x)) (41) s.t. Costrats (34)-(40) Where EZ 01 (x) = (EZ 11 (x). EZ 21 (x). EZ 31 (x)) wth EZ k1 (x) = ( N X S + N X G (h k + μ h + θ 1 (π k + μ π )) X P 2 T (42) =1 θ 1 (π k + μ π )X P T + 0.5θ 1 (π k + μ π )X T +θ 2 (g k + μ g )X θ 2 (g k + μ g )X P +θ 3 N X 1 γ + θ 4 X 1 γ P 2 T + θ 4 X 1 γ M 2 T 1 2θ 4 X 1 γ P M θ 5 X S M +θ 5 X S M P θ 6 X S M 2 T 1 + A T 1 ) k = V 1 (x) = (0.25(σ 2 h + θ 2 1 σ 2 π )X 2 P 4 T 2 + θ 2 1 σ 2 π X 2 P 2 T θ 2 1 σ 2 π X 2 T 2 + θ 2 2 σ 2 2 g X =1 +θ 2 2 σ 2 g X 2 P 2 + σ 2 A T 2 ) (43) 216

11 ad = , h = (h 1. h 2. h 3 )(+) (μ h + σ 2 h ), π = (π 1. π 2. π 3 )(+) (μ π + σ 2 π ), g = (g 1. g 2. g 3 )(+) (μ g + σ 2 g ), ad A = (A 1. A 2. A 3 )(+) (μ A + σ 2 A ). Referrg to Kauffma ad Gupta (1991), the approxmated value of tragular fuzzy umber b = (b 1. b 2. b 3 ) s calculated as b = b 1+2b 1 +b 3. Therefore, a approxmated value of EZ 0(x) s as follows: 4 AEZ 01 (x) = EZ 11 (x) + 2EZ 21 (x) + EZ 31 (x) 4 (44) = ( N X S + N X G (ĥ + μ h + θ 1 (π + μ π )) X P 2 T θ 1 (π + μ π )X P T =1 +0.5θ 1 (π + μ π )X T +θ 2 (g k + μ g )X θ 2 (g k + μ g )X P + θ 3 N X 1 γ +θ 4 X 1 γ P 2 T + θ 4 X 1 γ M 2 T 1 2θ 4 X 1 γ P M θ 5 X S M + θ 5 X S M P θ 6 X S M 2 T 1 + A T 1 ) So, problem (33) -(40) s reduced to the followg mult-objectve olear programmg problem, of whch each objectve has sgomal terms: M EVZ(x) = [AEZ 01 (x). V 1 (x)] (45) s.t. Costrats (34)-(40) I what followg, we solve the mult-objectve olear programmg problem (34) -(40) ad (45) by frst covertg t to a sgle objectve problem by the followg steps ad the usg global optmzato approach dscovered by Xu (2014) for solvg SGP problems. Step 1: Solve the problem (34) -(40) ad (45) wth cosderg oly objectve fucto AEZ 01 (x) ad solve t usg the SGP algorthm of Xu (2014). Let x (1) = (S (1). T (1). G (1). M (1). P (1). X (1). N (1) )be the optmal solutos for decso varables ad so the optmal amout of objectve fucto s AEZ 01 (x (1) ). Next calculate the amout of the secod objectve fucto V 1 (x) x (1), say V 1 (x (1) ). Step 2: Cosder just the secod objectve fucto V 1 (x) ad solve t usg SGP approach sad Step 1 ad obta the optmal solutos for decso varables ad objectve fucto as x (2) = (S (2). T (2). G (2). M (2). P (2). X (2). N (2) ) ad V 1 (x (2) ), respectvely. Next compute the amout of the frst objectve fucto AEZ 01 (x) x (2), say AEZ 01 (x (2) ). Step 3: There are the followg relato amog objectve fuctos: AEZ 01 (x (1) ) < AEZ 01 (x) < AEZ 01 (x (2) ) ad V 1 (x (2) ) < V 1 (x) < V 1 (x (1) ). Step 4: Formulate the membershp fuctos for the objectve fuctos of (45) as follows: μ AEZ 0 (x) = { 1 AEZ 01 (x (2) ) AEZ 01 (x) AEZ 01 (x (2) ) AEZ 01 (x (1) ) 0 AEZ 01 (x) (x) AEZ 01 (x (1) ) AEZ 01 (x (1) ) AEZ 01 (x) AEZ 01 (x (2) ) AEZ 01 (x (2) ) AEZ 01 (x) (46) 217

12 μ V1 (x) = { 1 V 1 (x (1) ) V 1 (x) V 1 (x (1) ) V 1 (x (2) ) 0 V 1 (x) V 1 (x (2) ) V 1 (x (2) ) V 1 (x) V 1 (x (1) ) V 1 (x (1) ) V 1 (x) Step 5: Accordg to Twar et al. (1987), the membershp fuctos are maxmzg by max-covex combato operator through followg equatos : Max MZ 1 (x) = f 1 μ AEZ 01 (x) + f 2 μ V1 (x) (48) s.t. Costrats (34)-(40) Where the weghts f 1 ad f 2 are correspodg to the member fuctos μ AEZ 01 (x) ad μ V1 (x), respectvely. So, the problem (34) -(40) ad (48) ca be rewrtte as the followg costraed SGP problem: f 1 M Z 1(x) = AEZ 01 (x (2) ) AEZ 01 (x (1) ) AEZ 01 (x) + f 2 V 1 (x (1) ) V 1 (x (2) ) V 1 (x) (49) s.t. Costrats (34) -(40) Now problem (34) -(40) ad (49) ca be solved usg global optmzato of SGP problem dscussed Appedx. Case 2- P T M M 0 The mathematcal model for case 2 s: Max ATP 2 (50) s.t. Costrats (20)-(23) ad (25) All procedure to solve the above problem s smlar to the procedure used to solve case 1. Followg the same procedure used for case 1, the costraed SGP problem for case 2 s: (x) f 1 M Z 2 = AEZ 02 (x (2) ) AEZ 02 (x (1) ) AEZ 02 (x) + f 2 V 2 (x (1) ) V 2 (x (2) ) V 2 (x) (51) s.t. P T M 1 1 (52) M 1 0 M 1 (53) Ad costrats (34) -(39) 5- Numercal example I ths Secto, a example s desged to demostrate the applcato of the model ad soluto procedure proposed above for a partcular retaler that orders three types of products from the suppler ( = 3). The retaler has a lmtato o the total budget for purchasg uts whch s fuzzy stochastc. The budget amout here les wth $(232, 280) wth probablty 0.5; wth $(245, 320) wth probablty 0.35; wth $(255, 310) wth probablty 0.4. Accordg to the past reorders, the aual demad rate of three tems are calculated as 10 6 S G ξ 1, S G ξ 2, ad S G ξ 3. The crsp parameters for all tems are I e = 0.05, I p = 0.1, β 1 = 0.6, β 2 = 0.65, β 3 = 0.7, α = 0.85, γ 1 = 1.6, γ 2 = 1.5, γ 3 = 1.7, ξ 1 N (2.1), ξ 2 N (3.1), ξ 3 N (1.1), ad the hybrd parameters are lsted table 2. (47) 218

13 Table 2. Hybrd parameters for each tem h π A g 1 (0.8, 0.9,0.95) (+)' (0.85,0.06) (2, 2.5, 3) (+)' (2.5, 1) (100, 112, 115) (+)' (100, 25) (1, 1.5, 2) (+)' (2.5, 1) 2 (0.85, 0.93, 1) (+)' (0.9, 0.065) (2.5, 3, 3.5) (+)' (3, 1) (105, 112, 117) (+)' (100, 25) (1.5, 2, 2.5) (+)' (3, 1.5) 3 (1, 1.2,1.5) (+)' (1,0.07) (3, 3.2, 3.5) (+)' (3,1) (109, 115, 120) (+)' (100, 25) (2, 2.2, 2.5) (+)' (3,1) The payoff matrx of problem (19) -(24), whch s eeded to trasform problem (19) -(24), to problem (34) -(40) ad (49), s as followg: [ AEZ 01(x (1) ) V 1 (x (1) ) AEZ 01 (x (2) ) V 1 (x (2) ] = [ ) ] Smlarly, the payoff matrx of case 2 s: [ AEZ 02(x (1) ) V 2 (x (1) ) AEZ 02 (x (2) ) V 2 (x (2) ] = [ ) ] Calculatg these pay off matrxes ad cosderg the weghts 0.9 ad 0.1 plus the provded data, t s possble to solve the problem (34) -(40) ad (49) for case 1 ad the problem (34) -(39) ad (51) -(53) usg global optmzato method. The proposed algorthm s coded MATLAB R2014b software ad mplemeted o a Itel Core 5 PC wth CPU of 1.4 GHz ad 4.00 GB RAM usg GGPLAB solver (Mutapcc et al. 2006). The optmal values of decso varables alog wth the optmal values of mea proft fucto(eatp) ad the optmal values of varace proft fucto (VATP) for the both cases ad all tems are reported tables 3-5. Table 3. Optmal solutos of tem 1 for the both cases Case S 1 G 1 M 1 T 1 P 1 Q 1 B 1 EATP VATP Table 4. Optmal solutos of tem 2 for the both cases Case S 2 G 2 M 2 T 2 P 2 Q 2 B 2 EATP VATP Table 5. Optmal solutos of tem 3 for the both cases Case S 2 G 2 M 2 T 2 P 2 Q 2 B 2 EATP VATP

14 6- Sestvty aalyss Sestvty aalyses for the proposed problem are doe to aalyze the mpacts of chages the key parameter values o the optmal solutos. For smplcty, we assume there s a tem (tem 1) wth P 1 T 1 M 1. We frst cosder the effect of chages values of α 1 ad χ 1 o the sellg prce, marketg expedture, order quatty, ad mea proft fucto. The calculated results are show Fgs 3-6. We observe from fgures 3 ad 4 that whe the amout of α 1 crease, sellg prce, marketg expedture, order quatty, ad mea proft fucto decrease. Moreover, whe the amout of χ 1 creases, other parameters lke the sellg prce, marketg expedture, order quatty ad mea proft fucto also crease (see fgures 5 ad 6). Ths s because whe the prce elastcy to demad crease, demad rate ad order quatty decrease; thus, the mea proft fucto decreases. I cotrast, whe the amout of χ 1 crease, demad rate ad order quatty crease; thus, the mea proft fucto creases, whch agrees wth realty G1 S G S α 1 Fg 3. The effect of chage of α 1 o the sellg prce ad marketg expedture 220

15 152 Q1 EATP Q EATP α 1 Fg 4. The effect of chage of α 1 o the order quatty ad mea proft fucto G1 S χ 1 Fg 5. The effect of chage of χ 1 o the sellg prce ad marketg expedture 221

16 164 Q1 EATP Fg 6. The effect of chage of χ 1 o the order quatty ad mea proft fucto We also vestgate the sestvty aalyses o the optmal solutos due to the parameters I p, I e, ad β 1. The mpact of the chages s reported Table 6 ad the followg results ca be vewed: Whe the parameter I p creases, the amout of S 1 ad G 1 wll crease, whereas the amouts of M 1, T 1, P 1, Q 1, ad EATP 1 wll decrease. Whe the parameter I e creases, the amout of G 1 ad EATP 1 wll crease, whereas the amouts of M 1, T 1, P 1, Q 1, ad S 1 wll decrease. Whe the parameter β 1. creases, the amout of M 1, P 1, Q 1, ad EATP 1 wll crease, whereas the amouts of T 1, G 1, ad S 1 wll 222

17 Table 6. Sestvty aalyss o the parameters I p, I e, ad β 1 Parameters S 1 G 1 M 1 T 1 P 1 Q 1 EATP 1 I p = I p = I p = I p = I p = I e = I e = I e = I e = I e = β 1 = β 1 = β 1 = β 1 = β 1 = Fally, the chages mea ad varace proft fucto wth respect to weght parameter f 1 (= 1 f 2 ) are llustrated fgure (7). From ths fgure, whe f 1 creases, the mea proft fucto wll decrease, whle, the varace proft fucto wll crease. Ths s because f f 1 creases, f 2 decreases, therefore, the varace proft fucto ad the mea proft fucto cotradcts each other. That s, f oe decreases, ext the other creases. EATP VATP f 1 Fg 7. The effect of weght parameter f 1 o the mea ad varace proft fucto 7- Cocluso I ths study, for the frst tme a mult-tem EOQ model has bee developed wth prce ad marketg cost depedet stochastc demad uder permssble delay paymet. We cosdered some cost parameters as hybrd umber. Moreover, a lmtato o the total budget to purchase vetory was cosdered wth fuzzy-stochastc quatty. Shortages are permtted ad partally backordered. We solved our problem wth usg the methods of covertg fuzzy- radom parameters to crsp oe ad obtag the global optmum of SGP problems. Fally, several umercal examples ad a sestvty aalyss of the ma parameters were provded to demostrate the formulated model. Our study ca be exteded for 223 EATP VATP 1

18 deteroratg tems. Moreover, a mult- tem EOQ model wth varable lead tme ad cosderg the ssues of sustaablty ca be developed. Refereces Aggarwal, S. ad Jagg, C. (1995) 'Orderg polces of deteroratg tems uder permssble delay paymets', Joural of the operatoal research socety, Cheg, T. (1991) 'EPQ wth process capablty ad qualty assurace cosderatos', Joural of the Operatoal Research Socety, 42(8), Chug, K.-J. ad Huag, Y.-F. (2003) 'The optmal cycle tme for EPQ vetory model uder permssble delay paymets', Iteratoal Joural of Producto Ecoomcs, 84(3), Chug, K.-J., Lao, J.-J., Tg, P.-S., L, S.-D. ad Srvastava, H.M. (2015) 'The algorthm for the optmal cycle tme ad prcg decsos for a tegrated vetory system wth order-sze depedet trade credt supply cha maagemet', Appled Mathematcs ad Computato, 268, Das, K., Roy, T.K. ad Mat, M. (2004) 'Mult-tem stochastc ad fuzzy-stochastc vetory models uder two restrctos', Computers & Operatos Research, 31(11), De, S.K. ad Saa, S.S. (2015) 'Backloggg EOQ model for promotoal effort ad sellg prce sestve demad-a tutostc fuzzy approach', Aals of Operatos Research, 233(1), Goyal, S.K. (1985) 'Ecoomc order quatty uder codtos of permssble delay paymets', Joural of the operatoal research socety, He, Y., Zhao, X., Zhao, L. ad He, J. (2009) 'Coordatg a supply cha wth effort ad prce depedet stochastc demad', Appled Mathematcal Modellg, 33(6), Ho, C.-H., Ouyag, L.-Y. ad Su, C.-H. (2008) 'Optmal prcg, shpmet ad paymet polcy for a tegrated suppler buyer vetory model wth two-part trade credt', Europea Joural of Operatoal Research, 187(2), Huag, Y.-F. (2007) 'Ecoomc order quatty uder codtoally permssble delay paymets', Europea Joural of Operatoal Research, 176(2), Islam, S. ad Roy, T.K. (2006) 'A fuzzy EPQ model wth flexblty ad relablty cosderato ad demad depedet ut producto cost uder a space costrat: A fuzzy geometrc programmg approach', Appled Mathematcs ad computato, 176(2), Jamal, A., Sarker, B. ad Wag, S. (1997) 'A orderg polcy for deteroratg tems wth allowable shortage ad permssble delay paymet', Joural of the operatoal research socety, 48(8), Kauffma, A. ad Gupta, M.M. (1991) 'Itroducto to Fuzzy Arthmetc, Theory ad Applcato'. Lag, Y. ad Zhou, F. (2011) 'A two-warehouse vetory model for deteroratg tems uder codtoally permssble delay paymet', Appled Mathematcal Modellg, 35(5), Luhadjula, M.K. (1983) 'Lear programmg uder radomess ad fuzzess', Fuzzy Sets ad Systems, 10(1-3), Maham, R. ad Abad, I.N.K. (2012) 'Jot cotrol of vetory ad ts prcg for o-stataeously deteroratg tems uder permssble delay paymets ad partal backloggg', Mathematcal ad Computer Modellg, 55(5),

19 Maham, R. ad Karm, B. (2014) 'Optmzg the prcg ad repleshmet polcy for ostataeous deteroratg tems wth stochastc demad ad promotoal efforts', Computers & Operatos Research, 51, Maham, R., Karm, B. ad Ghom, S.M.T.F. (2017) 'Effect of two-echelo trade credt o prcgvetory polcy of o-stataeous deteroratg products wth probablstc demad ad deterorato fuctos', Aals of Operatos Research, 257(1-2), Madal, N.K., Roy, T.K. ad Mat, M. (2006) 'Ivetory model of deterorated tems wth a costrat: A geometrc programmg approach', Europea Joural of Operatoal Research, 173(1), Mutapcc, A., Koh, K., Km, S. ad Boyd, S. (2006) 'GGPLAB verso 1.00: a Matlab toolbox for geometrc programmg'. Pada, D., Kar, S. ad Mat, M. (2008) 'Mult-tem EOQ model wth hybrd cost parameters uder fuzzy/fuzzy-stochastc resource costrats: a geometrc programmg approach', Computers & Mathematcs wth Applcatos, 56(11), Pag, Q., Che, Y. ad Hu, Y. (2014) 'Coordatg three-level Supply Cha by reveue-sharg cotract wth sales effort depedet demad', Dscrete Dyamcs Nature ad Socety, Pramak, P., Mat, M.K. ad Mat, M. (2017) 'A supply cha wth varable demad uder three level trade credt polcy', Computers & Idustral Egeerg, 106, Sadjad, S.J., Hesarsorkh, A.H., Mohammad, M. ad Nae, A.B. (2015) 'Jot prcg ad producto maagemet: a geometrc programmg approach wth cosderato of cubc producto cost fucto', Joural of Idustral Egeerg Iteratoal, 11(2), Samad, F., Mrzazadeh, A. ad Pedram, M.M. (2013) 'Fuzzy prcg, marketg ad servce plag a fuzzy vetory model: a geometrc programmg approach', Appled Mathematcal Modellg, 37(10), Sarkar, B., Sare, S. ad Cárdeas-Barró, L.E. (2015) 'A vetory model wth trade-credt polcy ad varable deterorato for fxed lfetme products', Aals of Operatos Research, 229(1), So, H. ad Patel, K. (2012) 'Optmal prcg ad vetory polces for o-stataeous deteroratg tems wth permssble delay paymet: Fuzzy expected value model', Iteratoal joural of dustral egeerg computatos, 3(3), So, H.N. (2013) 'Optmal repleshmet polces for o-stataeous deteroratg tems wth prce ad stock sestve demad uder permssble delay paymet', Iteratoal joural of producto Ecoomcs, 146(1), Tabatabae, S.R.M., Sadjad, S.J. ad Maku, A. (2017) 'Optmal producto ad marketg plag wth geometrc programmg approach'. Talezadeh, A.A., Petco, D.W., Jabalamel, M.S. ad Aryaezhad, M. (2013) 'A EOQ model wth partal delayed paymet ad partal backorderg', Omega, 41(2), Twar, R., Dharmar, S. ad Rao, J. (1987) 'Fuzzy goal programmg a addtve model', Fuzzy sets ad systems, 24(1), Xu, G. (2014) 'Global optmzato of sgomal geometrc programmg problems', Europea Joural of Operatoal Research, 233(3),

20 Appedx. Trasformg SGP problems to a seres of stadard GP problems As meto earler, a global optmzato method s appled for solvg SGP problem proposed Steps 1, 2, ad 5. So ths secto, we frst preset a SGP problem, ad the expla ths approach detal for trasformg the SGP problem to a seres of stadard GP problem accordg to type of our problem. 1. SGP program A SGP problem s equal to a optmzato problem as follows: 0 M ψ 0 (y) = θ 0k c 0k y a 0k k=1 j m =1 m s.t ψ j (y) = θ jk c jk y a jk 1 k=1 =1 c 0k > 0, θ 0k = ±1 (1) c jk > 0, θ jk = ±1, a jk R, j = t (2) y > 0,, = m (3) j (j = t) show the umber of elemets of the objectve fucto ad costrats. ψ j (j = t) s a sgomal fucto. 2. Global optmzato approach Ths method defes all fuctos ψ j (j = t)as: ψ j (y) = ψ + j (y) ψ j (y) j = t (4) Where ψ + j (y) ad ψ j (y) are formulated as: j m ψ j + (y) = θ jk c jk y a jk k=1 j =1 m ψ j (y) = θ jk c jk y a jk k=1 =1 θ jk = +1, j = t (5) θ jk = 1, j = t (6) Next t defes a large umber, > 0, so that ψ + j (y) ψ j (y) + L > 0 ad rewrtes the model (1)-(3) as the followg problem: M ψ 0 (y) = ψ + 0 (y) ψ 0 (y) + L (7) s.t ψ + j (y) ψ j (y) + L 1 j = t (8) y > 0, = m (9) The model (7)-(9) coverts to the followg optmzato problem, by troducg a extra varable y 0 order to express costrats ad objectve fucto as quotet ad lear form, respectvely. M y 0 (10) s.t ψ + 0 (y) + L ψ 1 0 (y) y 0 (11) ψ + j (y) ψ j (y) j j 1, j = t (12) ψ + j (y) ψ j (y) j j 2, j = t (13) y > 0, = m (14) Where, j 1 = {j ψ j (y) + 1 are moomals} ad j 2 = {j j j 1 }. I the above model, the objectve fucto (10) s a posyomal fucto, costrat (12) s a posyomal equalty, ad costrat (14) s a 226

21 moomal equalty that all three equatos are allowable stadard GP problem, but costrats (11) ad (13) are ot permtted a stadard GP problem. So ths method used from arthmetc geometrc mea approxmato to approxmate every deomator of costrats (11) ad (13) wth moomal fuctos as follows: f(y) f (y) = ( v u (y) w u (x) w u (x) ) (15) u Where the parameters w u (x) ca be computed as: w u (x) = v u (x) f(x) u (16) Ad f(y) = u v u (y) s a posyomal fucto, v u (y) are moomal terms, ad x > 0 s a fxed pot. Usg the proposed moomal approxmato approach to every deomator of costrats (11) ad (13), fally we have: M y 0 (17) s.t ψ + 0 (y) + L ψ 0 (y. y 0 ) 1 (18) ψ + j (y) ψ j (y) j j 1, j = t (19) ψ + j (y) ψ 2j (y) 1 j j 2, j = t (20) y > 0, = m (21) Where ψ 0 (y. y 0 ) ad ψ 2j (y)are the correspodg moomal fuctos approxmated usg Equato (15). Now, the problem (17)-(21) s a stadard geometrc programmg that ca be optmzed effcetly usg GGPLAB solver MATLAB (Mutapcc et al. 2006). So, the proposed algorthm ca be summarzed as a teratve algorthm as follows: Algorthm Step 0: Select a tal soluto for decso varables y 0 ad y, y 0 (0) ad y (0) respectvely. Cosder a soluto accuracy ε > 0 ad put terato couter r = 0. Step1: I terato r, calculate the moomal compoets the deomator posyomals of Equatos (11) ad (13) by the determed y 0 (r 1) ad y (r 1). Calculate ther correspodg parameters w u (y 0 (r 1). y (r 1) ) usg equato (16). Step2: Do the codesato o the deomator posyomals of equatos(11) ad (13) Equato (15) by parameters w u (y 0 (r 1). y (r 1) ). Step3: Solve the stadard GP (17)-(21) to obta (y 0 (r). y (r) ). Step4: If y (r) y (r 1) ε, so stop. Else r = r + 1 ad retur to Step1. usg 227