An EPQ model with trapezoidal demand under volume flexibility

In this paper, we explored an economic production quantity model (EPQ) model for finite production rate and deteriorating items with time-dependent trapezoidal demand. The objective of the model under study is to determine the optimal production run-time as well as the number of production cycle in order to maximize the profit.

International Journal of Industrial Engineering Computations (2014) 127–138 Contents lists available at GrowingScience International Journal of Industrial Engineering Computations homepage: www.GrowingScience.com/ijiec An EPQ model with trapezoidal demand under volume flexibility Himani Dema*, S.R Singhb and Jitendra Kumarc a Department of Applied Science, Meerut Institute of Technology, Meerut-250103, India Department of Mathematics, Devanagri College, Chaudhary Charan Singh University, Meerut – 250001, India c Department of Mathematics and Statistics, Banasthali University, P.O Banasthali Vidyapith, Rajasthan – 304022, India b CHRONICLE ABSTRACT Article history: Received June 2013 Received in revised format August 15 2013 Accepted August 30 2013 Available online September 2013 Keywords: Production Trapezoidal demand Volume flexibility In this paper, we explored an economic production quantity model (EPQ) model for finite production rate and deteriorating items with time-dependent trapezoidal demand The objective of the model under study is to determine the optimal production run-time as well as the number of production cycle in order to maximize the profit Numerical example is also given to illustrate the model and sensitivity analyses regarding various parameters are performed to study their effects on the optimal policy © 2013 Growing Science Ltd All rights reserved Introduction Most of the researchers considered the time varying demand as an increasing or decreasing function of time, while in practice, this assumption is not suitable for all products The demand shows two-fold ramp type pattern for items like fashion apparel, particular kind of eatables and festival accessories have limited sales period and become obsolete at the end of period This kind of pattern has been termed as ‘‘trapezoidal ramp-type’’ In the beginning of the season, the demand increases up to a certain time point and stabilizes afterwards but starts declining towards end the of the season The economic order quantity (EOQ) model with ramp-type demand rate was initially proposed by Hill (1995) Since then several researchers and practitioners have paid significant consideration to study ramp-type demand Mandal and Pal (1998) developed the EOQ model with ramp-type demand for exponentially deteriorating items with shortages Wu and Ouyang (2000) investigated two inventory models assuming different replenishment policies: one started with shortage and another had shortage after inventory consumption After that, Wu (2001) developed a model for deteriorating items with * Corresponding author E-mail: himanidem@gmail.com (H Dem) © 2014 Growing Science Ltd All rights reserved doi: 10.5267/j.ijiec.2013.09.002 128 ramp-type demand and partial backlogging Giri et al (2003) extended ramp-type demand inventory model with more general Weibull distribution deterioration rate Manna and Chaudhuri (2006) studied an EPQ model with ramp-type two time periods categorized demand pattern assuming demand dependent production Deng et al (2007) focussed on the doubtful results found by Mandal and Pal (1998) and Wu and Ouyang (2000) and obtained a more consistent solution Panda et al (2008, 2009) extended Giri et al.’s (2003) one-fold demand model to two-fold demand Model studied by Hill (1995) was extended to trapezoidal-type demand rate by Cheng and Wang (2009) Panda et al (2009) worked on a single-item economic production quantity (EPQ) model with quadratic ramp-type demand function in order to determine the optimal production stopping time Model of Deng et al (2007) was extended to more general ramp-type demand rate, Weibull distribution deterioration rate, and general partial backlogging rate by Skouri et al (2009) Hung (2011) extended the model of Skouri et al (2009) by applying arbitrary component in ramp-type demand pattern Shah and Shah (2012) studied a joint vendor-buyer strategy for trapezoidal demand which is beneficial to both the players in the supply chain In most of the articles mentioned above, the constant rate of production is considered But constant production rate is not always realistic For example, when production model is based on time varying demand, the assumption of constant production rate is not suitable Such scenarios results into application of variable production rate The study of the model with changeable machine production rate was initiated by Schweitzer and Seidmann (1991) Khouja (1995) established a production model with unit production cost depending on used raw materials, engaged labor and tool wear and tear cost Bhandari and Sharma (1999) measured the marketing cost in addition to generating a generalized cost function The related studies done by Sana et al (2007) and Sana (2010) may be noted Dem and Singh (2012) worked on the EPQ model for damageable items with multivariate demand and volume flexibility Dem and Singh (2013) developed an EPQ model with volume flexibility under imperfect production process Goyal et al (2013) developed a production model with ramp type demand and volume flexibility In the present paper, we develop an EPQ model for deteriorating items trapezoidal type demand rate with volume flexibility We also assume that the inventory system includes several replenishments and all the ordering cycles are of fixed length Such type of demand pattern is generally seen in the case of any fad or seasonal goods coming to market The demand rate for such items increases with the time up to certain time and then stabilizes but in final phase, the demand rate decreases to a constant or zero, and then the next replenishment cycle starts We observed that such type of demand rate is very reasonable and proposed a practical inventory replenishment policy for such type of inventory model The remaining paper is structured as follows In Section 2, we explain the assumptions and notation used throughout this paper In Section 3, we formulate the mathematical model and the necessary conditions to find an optimal solution In Section 4, we provide numerical example for each case to illustrate the model Finally, the study is concluded in section Assumptions and Notations 2.1 Assumptions The inventory involves single item Demand rate is dependent on time given by (i  1)T  t  (i   u )T  a  b1t ,  D  t    a  b1 (i   u )T , (i   u )T  t  (i   v)T a  b t, (i   v)T  t  iT  where a, b1 , b2  , i=1,2,…, n , u and v are time parameters 129 H Dem et al / International Journal of Industrial Engineering Computations (2014) The function defined above is known as trapezoidal function Production rate is k times demand rate, where k >1 The unit production cost is dependent on production Time horizon is finite Deterioration rate is a constant The deterioration occurs when the item is effectively in stock 2.2 Notations D(t) Demand rate P(D(t)) Production rate, P(D(t))=kD(t) θ Deterioration rate Set up cost C1 C2 Holding cost per unit per unit time So Selling price per unit G kD  t  C(P) Production cost per unit given by C  P   R  R G I(t) T ti-1+u ti-1+v ti-1+r n H Material cost per unit Factor associated with costs like labor and energy costs Inventory level at any time t Constant scheduling period per cycle Time up to which demand stabilizes and equals to (i-1+u)T Time till the demand remains stable and equals to (i-1+v)T Production run time and equals to (i-1+r)T Total number of cycles length of planning horizon and equals to nT Model Formulation We have considered the following different cases based on the occurrence of time points of demand in different phases 3.1 Case (I) Inventory When (i   v )T  (i   r )T Q0 tu tv tr Q1 t1 t1+u Qn-1 t1+r t2 tn-1 tn-1+r tn Time Fig.1 Graphical representation of the system for Case I The differential equations governing the system are given as follows: dI11 (t )   k  1 ( a  b1t )   I11 (t ) , dt (i  1)T  t  (i   u )T , i  1, 2, , n (1) 130 (2) dI12 (t )   k  1 {a  b1 (i   u )T }   I12 (t ) , (i   u )T  t  (i   v )T dt dI13 (t ) (i   v )T  t  (i   r )T   k  1 ( a  b2 t )   I13 (t ) , dt dI (t ) (i   r )T  t  iT  ( a  b2 t )   I (t ) , dt Solving the Eq (1) to Eq (4) using the boundary conditions, I11  (i  1)T   , (3) (4) I12  (i   u )T   I11  (i   u )T  , I13  (i   v)T   I12  (i   v)T  , I  iT   1 1  I11 (t)   b1k  b1  2e (i 1)T eT   b1k  b1   (a  ak  Tb1(i 1) Tb1k(i 1))   a  ak  bt  b1kt        1 I12 (t )  Tb1  a  ak  Tb1k  Tb1i  Tb1u  eT e (i u 1)T b1  b1k   (b1k  b1  a  Tb1  ak  Tb1k  Tb1i  Tb1ki)e (i 1)T e (i u 1)T   Tb1ki  Tb1ku  I13 (t )   (b2 k  b2   2e (i  v 1)T e T (   b2 k  b2    (5) (6) ( a2  a2 k  Tb2 (k  1)(i  v  1)) 1  (e vT (k  1)(Tb1  a  Tb1e vT  ae vT  Tb1i  Tb1ie vT  Tb1ue vT ))  b1e vT (e uT  1)(k  1))  (a  ak  b2t  b2 kt ) (7) 1 I (t )   (a  b2t )  (b2  e T (b2e iT  a e iT  Tb2i e iT )) (8)      Holding cost for ith cycle is HCi1  C2  C2 ( a   b1   ( i 1 u ) T ( i 1) T  Tb1   I11  t dt   ( i 1 v )T I12  t dt   ( i 1 u ) T ak   b1k   ae  uT   I13  t dt   ( i 1 v ) T b1e uT  ( i 1 r ) T  iT ( i 1 r )T I  t dt  T 2b1u Tb1e uT ake uT   2    b1ke  uT T 2b1u Tb1k Tb1i Tau Tb1u Tb1ki Taku Tb1ku T 2b1ku                    2 T b1iu Tb1ke  uT Tb1ie  uT T 2b1ku Tb1kie  uT T b1kiu        2  2 2 2 b1 b1k T 2b1u T 2b1v T 2b1u Tau Tav b1  ( u  v)T Taku Takv T 2b1ku  3       3e               T b1kv T 2b1iu T 2b1iv T 2b1uv T 2b1ku T 2b1kuv T 2b1kiu T 2b1kiv                  a b Tb ak (b k  b ) Tb k Tb i Tb ki     13  21    12  21  12   e uT  e vT           b1 Tb1 Tb2 b2 k a  rT b  vT T 2b2 r T 2b2v Tb1 ak  rT  3    2e  3e    (  ) e  e  vT            T b2 r T 2b2 v Tb1k Tb2 k Tb1i Tb2 i Tar Tb2 r Tb1u Tav  e e             2 2         b1  (u r )T b2  ( v  r )T b1  ( u v )T Tb1ki Tb2 ki Takr Tb2 kr Tb1ku Takv Tb1  ( v  r )T  3e  3e  3e        e b1k  rT    vT  b1k 3 e ( u  r )T   b1k 3 e ( v  r )T   b1k 3 e ( u  v)T   T 2b2 kr    T 2b2ir      T 2b2 kv T b2iv Tb1k  rT Tb1i  rT    e  e     T 2b2 kr T b2 kv Tb1ki  rT Tb2 ki  vT Tb1k  ( v r )T Tb2 k  ( v  r )T  e vT  e vT    e  e  e  e   2 2     Tb1i  ( v r )T Tb2i  ( v  r )T Tb1u  ( v r ) T Tb2v  ( v r )T T 2b2ikr T 2b2 kiv Tb1ki  ( v r )T  e  e  e  e    e Tb2 k Tb1k   Tb1ku 2   ( v  r )T e   Tb2 kv 2  ( v  r )T e   Tb2 r 2  T b2 ir    Tb2i 2  (1 r )T e   T 2b2 ki   b2 3 T 2b2 a T 2b2 r T 2b2 r a  (1r ) T b2  (1 r )T Tb2i Tar       2e  3e   )  2   2     Tb2   Ta  (9) 131 H Dem et al / International Journal of Industrial Engineering Computations (2014) Production cost for ith cycle is PCi1   ( i 1u ) T ( i 1)T  ( i 1 v ) T     G G R k (a  b1t ) dt  (i 1u )T  R  k{a  b1 (i   u )T }dt k (a  b1t )  k{a  b1 (i   u )T }    ( i 1 r )T ( i 1 v )T   G R k (a2  b2t )dt k (a2  b2t )   (10) (Tu (2G  Rak  RTb1k  RTb1ki  RTb1ku ))  T (u  v )(G  Rak  RTb1k  RTb1ki  RTb1ku )  (T ( r  v )(2G  Rak  RTb2 k  RTb2 ki  RTb2 kr  RTb2 kv ))  Sales revenue for ith cycle is SRi1  S0  ( i 1 u ) T ( i 1)T (a  b1t )dt   ( i 1 v )T ( i 1 u )T (a  b1 (i   u)T )dt   iT ( i 1 v ) T (a  b2t )dt   (TS0u (2a  2Tb1  2Tb1i  Tb1u ))  TS0 (u  v)( a  Tb1  Tb1i  Tb1u)  (TS (v  1)(2a  Tb2  2Tb2i  Tb2v)) Total profit per unit time of the system is (11) (12)   SRi1  PCi1  HCi1  C1  H i 3.2 Case (II) When (i   r )T  (i   v )T  iT Inventory TP1  Q0 tu tr Q1 tv t1 t1+u t1+r Qn-1 t2 tn-1 tn-1+r tn Time Fig Graphical representation of the system for Case II The differential equations governing the system are given as follows: dI11 (t ) (i  1)T  t  (i   u )T , i  1, 2, , n   k  1 ( a  b1t )   I11 (t ) , dt dI12 (t )   k  1 {a  b1 (i   u )T }   I12 (t ) , (i   u )T  t  (i   r )T dt dI 21 (t ) (i   r )T  t  (i   v )T   a  b1 (i   u )T    I 21 (t ) , dt (13) (14) (15) 132 (16) dI 22 (t )  (a  b2t )   I 22 (t ) , (i   r )T  t  iT dt Solving the Eqs (13-16) using the boundary conditions, I11  (i  1)T   , I12  (i   u )T   I11  (i   u )T  , I 21  (i   r )T   I 22  (i   r )T  , I 22  iT    1  b k  b1   e (i 1)T e T   b1k  b1   ( a  ak  Tb1 (i  1)  Tb1k (i  1))       I11 (t )    (17)  a  ak  b1t  b1kt   1 Tb1  a  ak  Tb1k  Tb1i  Tb1u  eT e (i u 1)T b1  b1k    I12 (t )  (18) (b1k  b1  a  Tb1  ak  Tb1k  Tb1i  Tb1ki)e (i 1)T e (i u 1)T   Tb1ki  Tb1ku  1 I 21 (t )   (Tb1  a  Tb1i  Tb1u  (e T e (i r 1)T (b1e (i  r 1)T e (i u 1)T  ak  b1e (i 1)T e  (i r 1)T    ( i 1)T  ( i  r 1)T b1ke  ( i 1)T  ( i  r 1)T  a e e e (19)  b1ke (i u 1)T  (i  r 1)T  Tb1k  Tb1ki  Tb1k u Tb1 e (i 1)T  (i r 1)T  ak e (i 1)T e  (i r 1)T  Tb1k e (i 1)T  (i r 1)T  Tb1i e (i 1)T  (i  r 1)T Tb1ki e (i 1)T  (i  r 1)T ))) 1 I 22 (t )   (a  b2t )  (b2  eT (b2e iT  a e iT  Tb2i e iT ))  (20)  Holding cost for ith cycle is  HCi  C2 ( i 1 u ) T I11  t dt   ( i 1) T ( i 1 r ) T I12  t dt   ( i 1 u ) T ( i 1 v )T ( i 1 r )T b1   Tb1    ak  b1k      b1e uT iT ( i 1 v )T I 22  t dt  T 2b1u Tb1e uT ake uT b1ke uT T 2b1u             2 2 3 2  uT  uT 2 Tb k Tb i Tau Tb1u Tb1ki Taku Tb1ku T b1ku T b1iu Tb1ke Tb1ie T 2b1ku  12  21                    2 2 2 Tb1kie uT T 2b1kiu b1 b1k a b1 Tb1 ak b1k  rT  uT T 2b1u T 2b1r T 2b1u      (     )  e e    a  C2 ( ae uT I 21  t dt          2   Tau Tar b1  (u r )T Taku Takr b1k  (u r )T T b1ku T b1kr T b1iu T b1ir   3e    3e                T b1ru Tb1k  rT Tb1i  rT Tb1k  uT Tb1i  uT T b1ku Tb1ki  rT  uT  e  e  e  e   e e   T 2b1kir T 2b1kiu T 2b1kru ak a b Tb ak b k Tb k Tb i Tb ki     (  13  21    12  21  12 )  e  rT  e  vT        T b1r  Tb1k 2 ( a                T b1v Tb1k Tar Tav b1k  (u r )T b1k  (u v )T T b1ir T b1iv T 2b1ru T 2b1uv     3e  3e     b2  (21)  2   e (v r )T     Tb1ki Tb2i  2   e (vr )T  )e ( v1)T   Tb1ku 2   e (v r )T  T b2i   b2 3 Tb2i Tav Tb2 v Tb2iv    )         Ta Tb2 T b2 a T b2v T 2b2 v       2 2   2 133 H Dem et al / International Journal of Industrial Engineering Computations (2014) Production cost for ith cycle is PCi   ( i 1 u ) T ( i 1) T  ( i 1 r )T     G G R k (a  b1t )dt  (i 1u )T  R  k{a  b1 (i   u )T }dt k (a  b1t )  k{a  b1 (i   u )T }    (22) (Tu (2G  Rak  RTb1k  RTb1ki  RTb1ku ))+T ( r  u )(G  Rak - RTb1k  RTb1ki  RTb1ku ) Sales revenue for ith cycle is SRi  S0  ( i 1 u ) T ( i 1) T (a  b1t )dt   ( i 1 v ) T ( i 1 u ) T (a  b1 (i   u )T )dt   iT ( i 1 v ) T (a  b2t )dt   (TS0u (2a  2Tb1  2Tb1i  Tb1u ))  TS0 (u  v)( a  Tb1  Tb1i  Tb1u)  (TS0 (v  1)(2a  Tb2  2Tb2i  Tb2v)) Total profit per unit time of the system is (23) (24)   SRi  PCi  HCi  C1  H i 3.3 Case (III) When (i   r )T  (i   u )T  iT Inventory TP2  Q0 tr Q1 tu t1 tv Fig Qn-1 t1+r t2 tn-1 tn-1+r t1+u tn Time Graphical representation of the system for Case III The differential equations governing the system are given as follows: (25) dI11 (t ) (i  1)T  t  (i   r )T , i  1, 2, , n   k  1 ( a  b1t )   I11 (t ) , dt (26) dI 21 (t ) (i   r )T  t  (i   u )T   (a  b1t )   I 21 (t ) , dt (27) dI 22 (t ) (i   u )T  t  (i   v )T   a  b1 (i   u )T    I 22 (t ) , dt (28) dI 23 (t ) (i   v )T  t  iT  ( a  b2t )   I 23 (t ) , dt Solving the Eqs (25-28) using the boundary conditions, I11  (i  1)T   , I 21  (i   r )T   I1  (i   r )T  , I 21  (i   u )T   I 22  (i   u )T  , I 23  iT   I11 (t )    1  b k  b1   e (i 1)T eT   b1k  b1   (a  ak  Tb1 (i  1)  Tb1k (i  1))   a  ak  b1t  b1kt        (29) 134 I 21 (t )    1 b (b1   e T e (i  r 1)T ( (a  ak  Tb1 (i  r  1)  Tb1k (i  r  1))  (a  Tb1 (i  r  1))  2    (b1k  b1   e (i 1)T e (i  r 1)T (   1 ( b1k  b1 )  (a  ak  Tb1 (i  1)  Tb1k (i  1))))))  (a  b1t )   I 22 (t )   (a  Tb1  Tb1i  Tb1u  (e T e (i u 1)T (b1  (b1  b1k  a  Tb1  ak  Tb1k  Tb1i   ( i 1)T  ( i  u 1)T Tb1ki )e e (31)   (b1k  ak  Tb1k  Tb1kr  Tb1ki )e (i r 1)T e  (i u 1)T ))) (32) 1 I 23 (t )   (a  b2t )  (b2  eT (b2 e iT  a e iT  Tb2i e iT ))  (30)  Holding cost for ith cycle is HCi  C2  C2 ( a        I1  t dt   ( i 1) T b1 b1ke uT ( i 1 r )T  Tb1  ( i 1 u )T ( i 1 r )T  ak   b1k   aeuT  I 21  t dt   ( i 1 v ) T ( i 1 u ) T  b1euT   T 2b1u   I 22  t dt   iT ( i 1 v ) T Tb1euT  I 23  t dt  akeuT  2 T 2b1u Tb1k Tb1i Tau Tb1u Tb1ki Taku Tb1ku T 2b1ku         3 2         T 2b1iu Tb1ke uT Tb1ie uT T 2b1ku Tb1kie uT T 2b1kiu        2 2 2 2  T 2b1r T 2b1u T 2b1r T 2b1u ak b k a b Tb ak b k Tb ki Tb k Tb i   13  (  13  21   13  12  12  21 )  e rT  e uT                  2 2 2 Tb k Tar Tb1r Tau Tb1u Tb1ki Tb1kr ak  (u r )T b1k  (u  r )T T b1ir T b1iu  12        2e  3e     Tb1ki 2  e (u  r )T  2    Tb1k 2  e ( u r )T   b1  ( 3 a  b1 2 3   Tb1 2   ak 2  b1k 3   Tb1k 2   Tb1i Tb1ki uT  vT  )e e     T b1u T b1v T b1u Tau Tav b1  (u v)T ak  (u r )T ak  (v r )T b1k  (u r )T      3e  2e  2e  3e  b1k 3 (33)        T 2b1iu T 2b1iv T 2b1uv Tb1k  (u  r )T Tb1k  ( vr )T Tb1ki  ( u r )T Tb1ki  (v r )T     e  e  e  e e (v r )T    Tb1k T 2b2i     Ta Tb T 2b2 a T 2b2v T 2b2 v2 a  ( v1)T  e (u r )T  e (v r )T     22      2e 2  2         b Tb i Tav Tb2v T b2iv Tb2i  (v 1)T  23 e ( v1)T  22     e ) Tb1kr b2      Production cost for ith cycle is PCi   ( i 1 r )T ( i 1) T    G R k (a  b1t ) dt  (Tr (2G  Rak  RTbk  RTbki  RTbkr )) k ( a  b1t )   (34) Sales revenue for ith cycle is SRi3  S0  ( i 1 u )T ( i 1) T (a  b1t )dt   ( i 1 v )T ( i 1 u ) T (a  b1 (i   u)T )dt   iT ( i 1 v ) T (a  b2t )dt  (TS0u (2a  2Tb1  2Tb1i  Tb1u ))  TS0 (u  v)( a  Tb1  Tb1i  Tb1u)  (TS0 (v  1)(2a  Tb2  2Tb2i  Tb2v))  (35) 135 H Dem et al / International Journal of Industrial Engineering Computations (2014) Total profit per unit time of the system is   SRi  PCi3  HCi  C1  (36) H i Our objective is to find maximum total profit per unit time in each case, i.e., max TPm (n, r )    SRim  PCim  HCim  C1  , where n is a positive integer and 0