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International Journal of Industrial Engineering Computations 5 (2014) 151–168

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Optimal trade-credit policy for perishable items deeming imperfect production and stock dependent demand

S. R. Singh and Swati Sharma *

Department of Mathematics, D. N. College, Meerut 250001, India

CHRONICLE

ABSTRACT

Article history: Received June 1 2013 Received in revised format August 15 2013 Accepted August 15 2013 Available online August 17 2013 Keywords: Perishable items Trade credit Delay in payment

Trade credit is the most succeeding economic phenomenon which is used by the supplier for encouraging the retailers to buy more quantity. In this article, a mathematical model with stock dependent demand and deterioration is developed to investigate the retailer’s optimal inventory policy under the scheme of permissible delay in payment. It is assumed that defective items are produced during the production process and delay period is progressive. The objective is to minimize the total average cost of the system. To exemplify hypothesis of the proposed model numerical examples and sensitivity analysis are provided. Finally, the convexities of the cost functions and the effects of changing parameters are represented through the graphs.

© 2013 Growing Science Ltd. All rights reserved

1. Introduction In long-established inventory models, it is often assumed that the purchasing cost for the items is paid by the retailer to the supplier as soon as the items have been received. In practice, a delay period known as trade credit period is offered by the supplier to the retailer, in paying for purchasing cost. Up to the end of the trade credit of a cycle, the retailer is free of charge, but he/she is charged on an interest for those items not being sold before this end. During the trade credit period, the retailer can accumulate revenues by selling items and earning interests. Goyal (1985) is the first person who developed the EOQ model under conditions of permissible delay in payments. Shah et al. (1988) studied the same model, incorporating shortages. Later on, Aggarwal and Jaggi (1995) discussed the inventory model considering deterioration and permissible delay in payment. Other motivating mechanisms in this research area are those of Teng (2002), Ouyang et al. (2006), Khanra et al. (2011), Singh et al. (2011), Teng et al. (2012), Singhal & Singh (2013) and Singh and Sharma (2013). * Corresponding author. E-mail: [email protected] (S.

Sharma)

© 2014 Growing Science Ltd. All rights reserved. doi: 10.5267/j.ijiec.2013.08.002

152

Deterioration of goods plays an important role in inventory system since in real life situations most of the physical goods deteriorate over time. Foods, pharmaceuticals, drugs, radioactive substances are some examples of items in which sufficient deterioration can take place during the normal cargo period and thus it plays an important role in analyzing the system. Generally, deterioration is defined as decay, damage or spoilage and obsolescence, which result in decrease of value of the original one. Ghare and Schrader (1963) presented the first model for decaying items. Covert and Philip (1973) extended their model considering Weibull distribution deterioration. Raafat (1991) presented a survey of literature on deteriorating inventory models. Hariga and Benkherouf (1994) proposed an inventory model for deteriorating items and later on Goyal and Giri (2001) provided a detailed review of deteriorating inventory literatures. Some other models dealing with the same issue are Yang and Wee (2006) and Kumar et al. (2012). Many business practices reveal that the presence of a larger quantity of goods displayed attract customers to buy more quantity. This phenomenon implies that the demand may have a positive correlative with stock level. As Levin et al. (1972) observed that ‘‘large piles of consumer goods displayed in a supermarket will lead the customer to buy more. Yet, too much piled up in everyone’s way leaves a negative impression on buyer and employee alike”. Gupta and Vrat (1986) and Baker and Urban (1988) were the first to initiate a class of inventory models in which the demand rate is inventory dependent. Mandal and Phaujdar (1989) then developed a production inventory model for deteriorating items with uniform rate of production and linearly stock-dependent demand. Other papers related to this research area are by Zhou and Yang (2005), Lee and Dye (2012). Most of the existing production inventory models ignored the presence of the imperfect production process. However, in real life situation, it is often observed that some of the items may be imperfect in nature, which are reworked at a cost to make them perfect. The production of defective items may be due to machine breakdown, labor problem, etc. Rosenblatt and Lee (1986) presumed that the time between the beginnings of the production run until the process goes out of control is exponential and that defective items can be reworked instantly at a cost and kept in stock. Kim and Hong (1999) determined the optimal production run length in deteriorating production processes. Salameh and Jaber (2000) developed an economic production inventory model for items with imperfect quality items. Goyal and Barron (2002) extended the model presented by Salameh and Jaber's (2000). An inventory model is developed by Chung and Hou (2003) to obtain an optimal run time for a deteriorating production system with shortages. Yu et al. (2005) generalized the models of Salameh and Jaber (2000), considering deterioration and partial backordering. Later on, Kang (2010) presented an inventory model considering trade credit and items of imperfect quality. Recently, Sarkar and Moon (2011), Singh and Singh (2011), Sarkar (2012) and Singh et al. (2012) established some motivating inventory models with imperfect production processes. An enormous work has been done in the field of trade-credit. Many previous economic order quantity inventory models are developed with trade-credit, a very few production inventory models are developed under allowable delay in payment. In addition, the inventory models for perishable items with imperfect production, stock dependent demand under trade-credit in which production rate depends on demand factor are much rare. Therefore, the present model is developed with these unique features. This model is an extension of the model Sarkar (2012) by considering deterioration and demand dependent production. The most favorable solution of the proposed model not only exists but also is unique. To obtain the optimal solution some lemmas are provided and with the help of sensitivity analysis, the effect of change in the parameters on the optimal policy is also disclosed.

2. Assumptions and notations

S.R. Singh and S. Sharma / International Journal of Industrial Engineering Computations 5 (2014)

153

The following assumptions and notations are taken to discuss the model. Assumptions 1. The inventory organism deals with a single type of items. 2. The replenishment rate is finite. 3. The delay in payment is offered to the retailer. 4. The demand is stock-dependent. 5. There is no repair or replacement of the deteriorated units. 6. Shortages are not permitted. 7. The lead time is zero. 8. The production of imperfect items is considered. Notations I1(t) I2(t) p D

On-hand inventory at time t where 0 ≤ t ≤ t1 (units) On-hand inventory at time t where t1 ≤ t ≤ T (units) Selling price per unit ($/units) Stock-dependent demand i.e. D  a  mI (t ) , a > 0; m > 0 (units)

T P R S Ie Ic1 Ic2 CA C Cd Ch Cp Cr Zi

Length of inventory cycle (year) Production rate (units per year), defined as P = ka, and k>1 The 1st offered trade-credit period without any charge (years) The 2nd offered trade-credit period with charge (years) Rate of interest earned due to financing inventory (/year) Rate of interest charged due to the credit balance for [R, S] ($/year) Rate of interest charged due to the credit balance for [S, T] ($/year) Ordering cost per order ($/order) Production cost ($/unit) Deterioration cost ($/unit) Holding cost ($/unit item/unit time) Purchasing cost ($/unit) Rework cost for the defective cost ($/item) Total cost of the system for i={1,2,3} ($)

3. Formulation of the Model We consider an inventory model with stock-dependent demand model with different types of delay period. Depending on this policy, there may arise some cases: Case (1): If the retailer pays the purchasing cost within the time R (i.e., T ≤ R), then there is no interest charged. Case (2): If the retailer pays the purchasing cost after R and before S (i.e., R ≤ T ≤ S), then the supplier can charge a rate of interest Ic1 to the retailer.

154

Case (3): If the retailer pays the purchasing cost after S and before T (i.e., T ≥ S), the supplier can charge a rate of interest Ic2 on the unpaid balance (see Figs. 1–3). Inventory Level

I1(t)

I2(t)

t1

T

R

Time

Fig. 1. Inventory versus time (Case-1: T ≤ R) Inventory Level

I1(t)

I2(t)

t1

R

T

S

Time

Fig. 2. Inventory versus time (Case-2: R ≤ T ≤ S)

Inventory Level

I1(t)

I2(t)

t1

R S T

Time

Fig. 3. Inventory versus time (Case-3: T ≥ S) Now, the present state of the on-hand inventory is described by the following differential equations: I1 '(t )  P  D( I1 (t ))   I1 (t )  ka   a  mI1 (t )    I1 (t ), 0 ≤ t ≤ t1

and

(1)


I 2 '(t )   D ( I 2 (t ))   I 2 (t )    a  mI 2 (t )    I 2 (t ),

155

(2)

t1 ≤ t ≤ T

with boundary conditions I1 (0)  0 and I 2 (T )  0 . The solutions of the equations (1) and (2) are given as follows: I1 ( t ) 

 k  1 a   m  t 1  e    ,  m    

(3)

0 ≤ t ≤ t1

and I 2 (t ) 

a

 e  m   t T   1 , 

(4)

t1 ≤ t ≤ T

m    

Since I1 (t1 )  I 2 (t1 ) , we have

 k  1  a    m   t 1 e m    

1



m  T  1  e  1 a   m   t1  T      1  t1  ln 1   m    e k m     

  .

(5)

 

Now, different costs of the inventory system are as follows: Ordering cost is OC and is given by (6)

CA . T

OC 

Inventory holding cost is HC and is given by a  kt1  T   .  m    

(7)

Deterioration cost for deteriorating items is DC and is given by t T  C 1 a  kt1  T   C DC  d   I1 (t )dt   I 2 (t )dt   DC  d  a 2  k 1 e m t  e m t T   k  . T  0 T   m     m      t1 Production cost is PRC and is given by

(8)

HC 

Ch T

 t1   I 1 ( t ) dt   0

T

I t1

2

 C ( t ) dt   HC  h T 

 a  2   m   

 k 1 e 

 m   t1



PRC 

C T

t1

 Pdt  0



 e m   t1 T   k 

1

1



(9)

Ckat1 . T

Purchasing cost is PUC and is given by (10) C t1 C kat PUC  p  Pdt  p 1 . T 0 T Along with the trade credit, the paper considers the production of imperfect items. The lifetime of defective item follows a Weibull distribution defined as (t )   t  ,   1 , where α, β are two parameters and t is the time to failure. Hence, the total number of defective items is:  t  (11)     1 t1    ( ) d     t1      1  N   P (t )e  0 dt  ka  1  e    .   0   The rework cost is RC and is given by    (12)   t C N C ka  . RC  r  r  1  e   1   1

1

T

T

 

 

Now, for different delay periods: Case (1): T ≤ R In this case, interest earned is IE1 and is given by IE1 

t T T  t1   pI e  1    t1  t  D  I1  t   dt   T  t  D  I 2  t   dt   R  T    D  I1  t   dt   D  I 2  t   dt   . T 0 t1 t1  0   

156

IE1 

   m k 1 at1 m k 1 a  m t1 pIe  ma  m t T  R T  t1  at1   e 1   R a  T  t1   e   1  1 2 2 T   m    m    m        m t ma  T  t1    at12 m k 1 at12 m k 1 at1e   1 m k 1 a  m t1  e      1    2 3  m     2 2  m    m    m   











(13)



 a T 2  t12  ma T 2  t12   ma ma  m  t1 T   m  t1 T  te   e T  1   . 2 1 3 2 2  m     m    m    









In this case, interest charged is IC1 and is given by (14)

IC1  0 . Case (2): R ≤ T ≤ S In this case, interest earned is IE2 and is given by IE 2 

t pI e  1    t1  t  D  I 1  t   d t  T  0

T

 T t1

  t  D  I 2 t  d t  . 

  m k 1 at1 m k 1 a  m t1 ma T  t1   pI   ma  1   T a T  t1   IE2  e t1 at1  e e m t1T  1   2 2 T    m    m    m     m       m t1 2  m k 1 a  m t1 at12 m k 1 at1 m k 1 at1e e     1  2 3 2 m    m    m    2  a T 2  t12  ma T 2  t12   ma ma  m  t1 T   m  t1 T  t e T      1  e  . 3 2 1 2 2 m     m    m    















(15)







In this case, interest charged is IC2 and is given by IC 2 

I c1 C P T

T

 I 2 (t ) dt  R

(16)

I c1 C P a  1 e   m   R  T   .  R  T   T m     m    m    

Case (3): T ≥ S In this case, interest earned is IE3 and is given by IE 3  IE3 

t pI e  1    t1  t  D  I 1  t   dt  T  0



S

.

 T  t  D  I  t   dt  2



t1

m k 1 at1 m k 1 a  m t1 pIe   ma    m t T  m S T t1 at1  e e   1   e     1   T a  S  t1   2 2 T   m      m m         











(17)

 m t  ma  S  t1   at12 m k 1 at12 m k 1 at1e   1 m k 1 a  m t1 e   1     2 3  m     2 2 m    m    m   





2 2 ma  S 2  t12   ma ma  a  S  t1   m  t1 T   m  S T   m  t1 T   m  S T   Se  e  e  t e    . 1 2 3 2  m     2  m    m    









In this case, interest charged is IC3 and is given by IC 3 

Ic2CP T

T

 I 2 (t ) dt  S

Ic2CP a  1 e   m    S  T   .  S  T   T m     m    m    

Thus, the total average cost for case (1): is Z1(T) and is given by, Z1 (T )  OC  HC  DC  PRC  PUC  RC  IC1  IE1 . The total average cost for case (2) is Z2(T) and is given by

(18)

(19)


157

(20)

Z 2 (T )  OC  HC  DC  PRC  PUC  RC  IC2  IE2 . The total average cost for case (3) is Z3(T) and is given by

(21) Z3 (T )  OC  HC  DC  PRC  PUC  RC  IC3  IE3 . Our objective is to minimize the total cost of the inventory system. The necessary conditions for the existence of the optimal solutions are (22) dZ1 (T )  0, dT (23) dZ 2 (T )  0, dT (24) dZ 3 (T )  0. dT Using the software Mathematica-8.0, from eq. (22) to Eq. (24) we can determine the optimum values of T = Ti*, where i=1, 2, 3 and the optimal value Zi(Ti*), where i=1,2,3 of the total cost can be determined by (21) provided they satisfy the sufficiency conditions for minimizing Zi(Ti*), where i=1, 2, 3 given by d 2 Zi (T )  0, where i  1, 2,3 dT 2 T T * i

For the cost minimization we may formulate the three lemmas (motivated by Sarkar (2012)) as follows: Lemma

Z1(T*)

must have a minimum value at T* if it satisfies the equation  C  C    C ka    C   C    pI  and the inequality 16 T * 2  17 T *  18  0 where all the values of 1.

CA  2 p 4 r 1 h d 3 e T*   5  Ch   Cd   11  C p  C   10 Cr ka  6 pI e   

ϕi’s are given in Appendix 1. Proof. For minimization of the total cost Z1,

dZ1 d 2 Z1  0 and dT T T * dT 2

 0 must be satisfied. T T *

Now, a  kt1  T    Cp  C  kat1 CA  Ch  Cd   a k 1 em t1  e m t1T   k     2  T T T  m     m   



Z1 



   m k 1 at1 m k 1 a  m t1 pIe  ma  R  T  t1  at1  e e m t1T  1  1   R a T  t1   2 2 T  m      m    m        m t  ma T  t1    at12 m k 1 at12 m k 1 at1e   1 m k 1 a  m t1 e  1       2 3  m     2 2 m    m    m        1  a T 2  t12   ma T 2  t12   Cr ka  t1  ma ma  m  t1 T   m  t1 T    1    . t e T e e      1   1   2 1 3  2 2  m     T   m    m      























Differentiating the above expression with respect to T, we get dZ1 5  Ch   Cd  1  Ch   Cd  CA 2  Cp  C  11  Cp  C  3 pI e 6 pI e 4Cr ka 10Cr ka dT



T



T2



T2



T2



T



T2



T



T2

and again, differentiating the above expression with respect to T, we find d 2 Z1 2CA 21  Ch  Cd  25  Ch  Cd  12  Ch   Cd  22  Cp  C  211  Cp  C  dT 2



    T3 T2 T T3 13  Cp  C  23 pIe 26 pIe 14 pI e 24Cr ka 210Cr ka 15Cr ka        , T T3 T2 T T3 T2 T T3



T2



T

,

158

For minimum of Z1,

dZ1 d 2 Z1  0 and dT T T * dT 2

 0 imply that T T *

5  Ch Cd  11  Cp  C  pI  C ka  C   C C  2  Cp  C  pI  C ka     3 2 e  10 r    A2  1 h 2 d   6 e  4 r2  , T T T T  * T T T2 T T  *  T T  T T

and  2C 21  Ch Cd  12  Ch Cd  22  Cp  C 13  Cp  C  2 pI 2 C ka  C ka   3A      6 2 e  4 3r  15 r  T3 T T3 T T T T  *  T T T  2  C C  211  Cp  C 2 pI  pI 2 C ka   5 h2 d   3 3 e  14 e  10 2r  . 2 T T T T T  T T*

After some simplification, we get C A  2  C p  C   4 Cr ka  1  C h   C d   3 pI e    * T 

5  Ch   Cd   11  C p  C   10 C r ka  6 pI e   

2

16 T *   17 T *  18  0.

and

Hence the proof. 2. Z2(T*) must have a minimum value at T* if it satisfies the equation 2 CA  1  Ch   Cd   2  C p  C   4 Cr ka  21Cp Ic1  19 pIe   and the inequality 25 T*  26T* 27  0 where all the T*   5  Ch   Cd   11  C p  C   10 Cr ka  22 C p I c1  20 pIe    values of ϕi’s are given in Appendix 2. Lemma

Proof. For minimization of the total cost Z2,

dZ 2 dT

 0 and T T *

d 2 Z2 dT 2

 0 must be satisfied. T T *

Now, C A  C h   C d   a   2 T T   m   

Z2 

 k  1 e 

 m  t1

e

  m   t1 T 



k 

a  kt1  T    C p  C  kat1  T  m    



m  k  1 at1 m  k  1 a  m t1 pI e   ma   t1 at1    1   T a T  t1   e e m t1 T   1 2 2 T   m        m m         



ma T  t1    at12 m  k  1 at12 m  k  1 at1e  m  t1 m  k  1 a  m  t1     e 1    2 3 2 m    m      2 m   m   













 a T 2  t12  ma T 2  t12   ma ma  m  t1 T    m   t1 T    e     1 t e T  1 2 3 2 2  m     m   m    











    1  m  R T   t1 Cr ka  I C a  1 e     1  e   1    c1 P  R  T   .  T  m     T   m     m      

Differentiating the above expression with respect to T, we get dZ2 5  Ch  Cd  1  Ch  Cd  CA 2  Cp  C  11  Cp  C  19 pIe 20 pIe 4Cr ka 10Cr ka 21Cp Ic1 22Cp Ic1    2   2   2    , dT T T2 T T2 T T T T T T2 T

and again, differentiating the above expression with respect to T, we have d 2Z2 2CA 21  Ch   Cd  25  Ch   Cd  12  Ch   Cd  22  Cp  C  211  Cp  C  13  Cp  C   3       dT 2 T T3 T2 T T3 T2 T 219 pI e 220 pIe 23 pIe 24Cr ka 210Cr ka 15Cr ka 221Cp I c1 222Cp Ic1 24Cp Ic1          . T3 T2 T T3 T2 T T3 T2 T


For minimum of Z2,

dZ 2 dT

 0 and T T *

d 2Z2 dT 2

0 T T

159

imply that

*

  C C  11  Cp  C  pI  C ka 22Cp Ic1  C   C C  2  Cp  C  pI  C ka 21CpIc1  d  5 h   19 2 e  10 r     A2  1 h 2 d   20 e  4 r2   , T T T T T  * T T T2 T T T2  *  T T  T T

and  2C 21  Ch Cd  12  Ch Cd  22  Cp  C 13  Cp  C 2 pI 2 C ka  C ka 221Cp Ic1 24Cp Ic1   3A      202 e  4 3r  15 r    3 3 T T T T T T T T T3 T  *  T T  2  C C  211  Cp  C 2 pI  pI 2 C ka 222Cp Ic1   5 h 2 d   193 e  23 e  10 2r   . T T2 T T T T2  *  TT

After some simplification, we get CA  1  Ch   Cd   2  C p  C   4 Cr ka  21Cp I c1  19 pI e    * T 

5  Ch   Cd   11  C p  C   10 Cr ka  22C p Ic1  20 pIe   

2

and 25 T *   26T *  27  0

Hence the proof. 3. Z3(T*) must have a minimum value at T* if it satisfies the equation 2 CA  1  Ch   Cd   2  Cp  C   4Cr ka  30Cp Ic2  28 pIe   and the inequality 36 T *   37 T *  38  0 where all the T*   5  Ch   Cd   11  Cp  C   10Cr ka  31Cp Ic2  29 pIe    values of ϕi’s are given in Appendix 3.

Lemma

Proof. For minimization of the total cost Z3,

dZ 3 dT

T T *

d 2 Z3  0 and dT 2

 0 must be satisfied. T T

*

Now, Z3 

CA  Ch   Cd   a   2 T T   m   

 k 1 e 

 m t1

e

 m  t1 T 



k 

a  kt1  T    Cp  C  kat1  T  m    



  m  k 1 at1 m  k 1 a  m t1 pI e   ma  m t T  m S T t1 at1  e e   1   e      1   T a  S  t1   2 2 T   m    m    m     



ma  S  t1    at12 m  k  1 at12 m  k 1 at1e m t1 m  k 1 a  m t1  e 1    2 3 2 m     m      2  m    m  









 a S  t     2 



ma

 m  

t e 

 m  t1 T 

2

1



 Se m  S T  

ma

e   m  

 m  t1 T 

3



 e m  S T  

2

2 1

 

ma  S 2  t12    2  m     

    1  m S T  I C a   t1 1 Cr ka  e     1  e   1    c 2 P S  T    .  T  m     T   m    m     

Differentiating the above expression with respect to T, we get dZ3 5  Ch  Cd  1  Ch  Cd  CA 2  Cp  C 11  Cp  C 28 pIe 29 pIe 4Cr ka    2   2   dT T T2 T T2 T T T T2  C ka  C I  C I  10 r  30 2p c2  31 p c 2 . T T T

and again, differentiating the above expression with respect to T, we have d 2 Z3 2CA 21  Ch   Cd  25  Ch   Cd  12  Ch   Cd  22  Cp  C  211  Cp  C  13  Cp  C   3       dT 2 T T3 T2 T T3 T2 T 228 pIe 229 pI e 34 pIe 24Cr ka 210Cr ka 15Cr ka 230Cp Ic 2 231Cp I c1 35Cp Ic 2          T T T T3 T2 T3 T2 T3 T2



160

For minimum of Z3,

dZ 3 dT

 0 and T T *

d 2 Z3 dT 2

 0 imply that T T *

  C Cd  11  Cp C  pI  Cka 31CpIc2  C   C C  2  Cp C  pI  Cka 30CpIc2  5 h   28 2 e  10 r     A2  1 h 2 d   29 e  4 r2  2  , T T T T T  * T T T2 T T T  *  TT  TT

and 2C 2  C C    C C  22  Cp  C 13  Cp  C 2 pI 2 C ka  C ka 230Cp Ic2 35Cp Ic2  d    292 e  4 3r  15 r    3A  1 h 3 d  12 h  T T T3 T T T T T3 T  *  T T T  2  C C  211  Cp  C 2 pI  pI 2 C ka 231CpIc1   5 h 2 d   283 e  34 e  10 2r   . T T2 T T T T2  *  T T

After some simplification, we get CA  1  Ch   Cd   2  C p  C   4Cr ka  30C p I c 2  28 pI e   and  T * 2   T *    0. T*    37 38 36  5  Ch   Cd   11 C p  C   10Cr ka  31C p I c 2  29 pI e   

Algorithm Step 1: Determine T1* from equation (22), if T1 * ≤ R then evaluate Z1 (T1*) from (19). Otherwise go to step 2. Step 2: Determine T2* from equation (23), if R ≤ T1* ≤ S then evaluate Z2 (T2*) from (20). Otherwise go to step 3. Step 3: Determine T3* from equation (24), if T1 * ≥ S then evaluate Z3 (T3*) from (21). Otherwise go to step 4. Step 4: Find out TC = min{ Z1 (T1*), Z2 (T2*), Z3 (T3*)}.

4. Numerical Examples All calculations are executed with the help of the software Mathematica 8.0, from where we get the optimal value. To illustrate the proposed model two examples are presented here in which Z1 and Z3 are the optimal solution. Example 1. We consider the following parameter values on the basis of the previous study: CA=$ 180/order, p = $20/unit, a = 15, m = 0.5, k = 2, θ = 0.1, C = 2, Cd = 15/unit, R = 1.5 years, Ie = $0.15/year, Ic1 = $0.18/year, Ic2 = $0.20/year, S = 1.74 years, Ch = $14/unit/year, Cp = $10/unit, α = 0.010, β = 0.053, Cr = $1.5/item. Then the optimal solutions are: In, case (1): {T1*=1.2529, Z1 (T1*)=398.759}, case (2): {T2*=1.7157, Z2 (T2*)=401.442}, case (3): {T3*=1.76178, Z3 (T3*)=400.971}. Among the above optimal solutions, the better optimal solution TC=min{ Z1 (T1*), Z2 (T2*), Z3 (T3*)}=398.759, T*=1.2529. From the numerical example, Figs. 4–6 show the convexity of the cost function. Average cost

Average cost

406

Average cost

440

405

440 430

404

430 403

420 402

420

401

410

410

400 1.1

1.2

1.3

1.4

1.5

Time

Fig. 4. Case 1: average cost versus cycle length

1.5

2

2.5

3

1.5

Time


2

2.5

3

Time



161

Example 2. We consider the following parameter values on the basis of the previous study: CA=$ 350/order, p = $20/unit, a = 15, m = 0.5, k = 2, θ = 0.1, C = 2, Cd = 15/unit, R = 2.1 years, Ie = $0.15/year, Ic1 = $0.18/year, Ic2 = $0.20/year, S = 2.75 years, Ch = $14/unit/year, Cp = $10/unit, α = 0.10, β = 0.53, Cr = $1.5/item. Then the optimal solutions are: For, case (1): {T1*=1.77939, Z1 (T1*)=480.611}, case (2): {T2*=2.66707, Z2 (T2*)=479.241}, case (3): { T3*=2.86137, Z3 (T3*)=476.712}. Among the above optimal solutions, the better optimal solution TC=min{ Z1 (T1*), Z2 (T2*), Z3 (T3*)}=476.712, T*=2.86137. From the numerical example, Figs. 7–9 show the convexity of the cost function. Average cost

Average cost

Average cost

540

540

560

530

530

520

520

510

510

500

500

490

490

540 520

500

2

1.5

2

2.5

2

3

3

4

Time


3

4

5

5

Time


Time


5. Sensitivity Analysis Sensitivity analysis of the numerical example (1) and (2) are presented in Table 1 and 2 respectively as follows. Table 1 (For example (1)) The effect of changing the parameter (i) while keeping all other parameters unchanged Parameter (i) Cp

CA

p

a

m

k

θ

Cd

Cr

C

Ch

% Change -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20

T1* 1.27669 1.26464 1.24146 1.23031 1.11981 1.18809 1.31476 1.37410 1.29398 1.27285 1.23402 1.21611 1.40293 1.32147 1.19412 1.14299 1.26871 1.26044 1.24604 1.23984 1.46530 1.33769 1.19275 1.14799 1.26114 1.25699 1.24887 1.24490 1.25873 1.25580 1.25001 1.24714 1.25294 1.25292 1.25287 1.25285 1.25756 1.25522 1.25058 1.24829 1.31048 1.28078 1.22665 1.20191

Z1(T1* ) 363.199 380.991 416.502 434.221 368.411 384.010 412.780 426.168 404.494 401.663 395.786 392.750 346.120 372.867 423.923 448.458 394.754 396.787 400.671 402.524 364.548 384.015 410.296 419.561 396.444 397.605 399.907 401.049 397.378 398.069 399.447 400.133 398.709 398.734 398.784 398.808 391.655 395.207 402.309 405.859 385.626 392.263 405.122 411.36

T2* 1.78904 1.75041 1.68424 1.65554 1.53682 1.62765 1.80141 1.88513 1.66931 1.69182 1.74109 1.76823 1.92633 1.81080 1.63576 1.56744 1.71542 1.71438 1.71950 1.72598 2.39638 1.95273 1.57058 1.47257 1.73148 1.72349 1.70808 1.70063 1.72995 1.72278 1.70870 1.70179 1.71580 1.71575 1.71564 1.71559 1.72707 1.72136 1.71009 1.70454 1.86431 1.78541 1.65353 1.59766

Z2(T2*) 363.798 382.663 420.145 438.778 379.296 390.674 411.679 421.445 406.189 403.836 399.006 396.525 340.934 371.508 430.844 459.794 396.728 399.144 403.624 405.686 348.934 379.477 418.069 431.129 398.526 399.990 402.883 404.313 399.585 400.515 402.366 403.286 401.389 401.416 401.469 41.4960 393.958 397.701 405.181 408.918 383.516 392.649 410.563 418.145

T3 * 1.80790 1.78406 1.74091 1.72130 1.63198 1.69824 1.82282 1.88157 1.72479 1.74342 1.77985 1.79762 1.91014 1.82946 1.70413 1.65439 1.75925 1.75957 1.76595 1.77218 2.14613 1.92372 1.64509 1.55900 1.77314 1.76740 1.75627 1.75088 1.77243 1.76709 1.75651 1.75128 1.76186 1.76182 1.76174 1.76170 1.77029 1.76602 1.75756 1.75336 1.86733 1.81281 1.71402 1.66930

Z3(T3* ) 363.295 382.157 419.743 438.474 379.757 390.567 411.014 420.732 405.896 403.450 398.462 395.923 340.384 370.898 430.682 460.090 396.267 398.681 403.136 405.173 349.976 378.815 418.345 432.275 398.008 399.495 402.437 403.891 399.070 400.022 401.918 402.862 400.918 400.945 400.998 401.025 393.451 397.212 404.729 408.485 382.803 392.007 409.708 418.231

Optimal Solution Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z3 (T1 *) Z3 (T1 *) Z3 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z3 (T1 *) Z3 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z3 (T1 *) Z3 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z1 (T1 *) Z3 (T1 *) Z3 (T1 *) Z1 (T1 *) Z1 (T1 *)

TC 363.199 380.991 416.502 434.221 368.411 384.010 411.014 420.732 404.494 401.663 395.786 392.750 340.384 370.898 423.923 448.458 394.754 396.787 400.671 402.524 349.976 378.815 410.296 419.561 396.444 397.605 399.907 401.049 397.378 398.069 399.447 400.133 398.709 398.734 398.784 398.808 391.655 395.207 402.309 405.859 382.803 392.007 405.122 411.360

162

Table 2 (For example (2)) The effect of changing the parameter (i) while keeping all other parameters unchanged Parameter (i) Cp

CA

p

a

m

k

θ

Cd

Cr

C

Ch

% Change -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20 -20 -10 +10 +20

T1 * 1.81323 1.79610 1.76310 1.74721 1.58531 1.68462 1.87037 1.95811 1.84287 1.81007 1.75060 1.72349 2.00092 1.88027 1.69341 1.61897 1.79006 1.78397 1.77627 1.77455 2.12770 1.91876 1.68079 1.60773 1.78812 1.78371 1.77517 1.77103 1.78769 1.78353 1.77528 1.77120 1.78072 1.78005 1.77873 1.77807 1.78602 1.78270 1.77610 1.77283 1.86115 1.81904 1.74200 1.70667

Z1 (T1* ) 442.892 461.768 499.422 518.201 438.994 460.402 499.791 518.076 488.940 484.832 476.285 471.860 421.533 451.679 508.515 535.531 476.407 478.573 482.525 484.317 430.249 458.746 497.859 511.782 477.607 479.114 482.098 483.575 478.694 479.653 481.567 482.520 480.234 480.423 480.799 480.988 473.078 476.845 484.376 488.139 462.387 471.592 489.454 498.128

T2 * 2.85506 2.75217 2.59462 2.53165 2.33311 2.50028 2.83369 2.99996 2.53810 2.59860 2.74591 2.83880 3.08277 2.85219 2.51545 2.38891 2.56167 2.60677 2.74735 2.85462 4.06453 3.25081 2.34213 2.14123 2.67743 2.67205 2.66246 2.65824 2.69617 2.68150 2.65286 2.63887 2.67172 2.66939 2.66475 2.66244 2.69027 2.6786 2.65568 2.64444 2.98032 2.8114 2.54265 2.43429

Z2(T2* ) 437.557 458.509 499.761 520.188 451.201 465.689 491.970 503.972 487.481 483.429 474.893 470.358 407.781 444.004 513.654 547.372 476.096 477.862 480.188 480.644 393.610 442.920 506.344 527.412 475.658 477.460 481.001 482.742 476.502 477.874 480.601 481.955 478.732 478.987 479.494 479.748 471.052 475.148 483.329 487.414 452.648 466.244 491.709 503.71

T3* 2.95676 2.90722 2.81884 2.77928 2.63309 2.75028 2.96645 3.06568 2.75589 2.80883 2.91322 2.96407 3.11316 2.97776 2.76063 2.67284 2.75873 2.80456 2.92970 3.00880 3.54119 3.19904 2.58945 2.38914 2.86441 2.86273 2.86033 2.85960 2.88205 2.87169 2.85111 2.84091 2.8647 2.86304 2.85971 2.85805 2.87790 2.86962 2.85316 2.84498 3.06107 2.95946 2.76771 2.67903

Z3(T3*) 434.985 455.891 497.456 518.130 451.236 464.239 488.723 500.326 485.854 481.339 471.977 467.139 404.793 441.028 511.933 546.76 474.394 475.777 477.175 477.145 392.950 439.201 506.624 530.605 473.100 474.917 478.484 480.235 473.816 475.266 478.154 479.591 476.178 476.445 476.979 477.246 468.397 472.556 480.866 485.016 449.024 463.054 490.006 502.947

Optimal Solution Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z1(T1*) Z1(T1*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z1(T1*) Z1(T1*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z1(T1*) Z1(T1*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z3(T3*) Z1(T1*) Z3(T3*) Z1(T1*) Z1(T1*)

TC 434.985 455.891 497.456 518.130 438.994 460.402 488.723 500.326 485.854 481.339 471.977 467.139 404.793 441.028 508.515 535.531 474.394 475.777 477.175 477.145 392.950 439.201 497.859 511.782 473.100 474.917 478.484 480.235 473.816 475.266 478.154 479.591 476.178 476.445 476.979 477.246 468.397 472.556 480.866 485.016 462.387 463.054 489.454 498.128

The behavior of the parameters changed with respect to the total average cost is shown graphically in Fig. 10 (for example (1)) and Fig. 11 (for example (2)) and some interesting results drawn from sensitivity analysis are given as follows. (1) The total average cost increases as the purchasing cost (Cp) increases, which is true in practical situation. As the purchasing cost per item increases, it is obvious to increase the optimal cost of the system. (2) The total average cost of the system increases with an increase in ordering cost (CA), which is quite natural as the per order growth of ordering cost implies an increase in total average cost of the system. (3) When the selling price increases the total average cost of the inventory system decreases. The fact is that due to the higher selling price retailer accumulates more revenue and earns more interest during the delay period. (4) As the demand parameters (a, m) increases the total average cost of the system increases. The motive is that more demand means more production consequently the total average cost increases. (5) An increase in production parameter (k) shows that the retailer produces more items therefore the holding cost and deterioration cost, etc. increases as a result the optimal cost of the system increases. (6) The total average cost decreases as the deterioration rate (θ) and the deterioration cost (Cd) decreases which according to the real situation. (7) An increase in rework cost per unit item indicates the growth of the total rework cost. To reduce the cost, the production of imperfect items will have to be reduced. (8) When the production cost (C) and the holding cost (Ch) increases the total average cost of the system increases. The reason is that per unit increase in production and holding costs increases the total production and holding costs therefore the total average cost of the proposed model increases.


-30

-20

-10

0 10 change (%)

Cp

430 420 410 400 390 380 370 360

TC

TC

440 430 420 410 400 390 380 370 360 350

Ch

20

-30

30

C

ϴ

163

-20

-10

0 10 change (%)

Cd

CA

p

20

30

m

500 400 TC

300 200 100 0

-30

-20

-10

0 change (%) a

10

Cr

20

30

k

Fig. 10. (For example (1)) The effect of changing parameters on the optimal cost function

-30

-20

-10

0 10 change (%) Ch

20

C

ϴ

-30

-40

30

-20

0 change (%)

Cd

CA

20

30

20

p

40

m

600 500 400 300 200 100 0

TC

Cp

510 500 490 480 470 460 450 440 430

TC

TC

540 520 500 480 460 440 420

-20

-10

0 10 change (%) a

Cr

k

Fig. 11. (For example (2)) The effect of changing parameters on the optimal cost function

164

6. Conclusion In this research article, an inventory model for deteriorating items with stock dependent demand rate considering imperfect production and delay in payment scheme has been developed. In this model, two delay periods have been provided by the supplier to attract the retailer. During the delay period an interest was earned on accumulated revenue by the retailer selling his/her commodity. In most of the papers, the examiners have considered the production of the perfect items through different machinery systems. However, in practical situation, due to employment problems, machine breakdowns, the system produces imperfect quality items, which may rework at a cost to make it perfect. In this model, the production of the imperfect items follows Weibull distribution and the production rate depends on the demand factor. An algorithm to determine the optimal policy has also been presented. In addition, sensitivity analysis is performed to examine the effect of parameters. From sensitivity analysis it is observed that the model is enough stable with respect to the changes in system parameters. Further, the model may be generalized by considering shortages and n cycles in a finite planning horizon. Appendix 1. The values are given as follows: 1 

ae   m  t1

m   



2



 k  1  e 

3   RT t1 at1   



m   T

   m ak 

2



a  kt1  T 

2  kat1

m   

m k 1 at1 m k 1 a  mt1   ma T t1  ma e e mt1T 1   1  RaT t1   2 2  m  m  m   m  









aT2 m k 1 at12 m k 1 at1emt1 m k 1 a  mt maT2 t12   ma ma  m t1T   m t1T 1 1 1  e t e T e           2 3 2 1 3 2 m  2 m    m  m  m   m  2 





4  1  e

    1  t1   1 

















 m  T  a  k  1 1  e   d 1  , 5  ,  dT   m    1   k  1  e   m   T   k  1  7 d 7 d 8 d 9

d 3   R  T  t1   R  dT dT dT dT  e  m   T   k  1       m  k  1  at1 m  k  1  a   m  t1  7   at1    1 e  2 m    m     

6 





8   a  T  t1   

ma

e 

 m    t1  T 

m   

2





1 

ma  T  t1     m    

 aT 2 m  k  1 at12 m  k  1 at1 e  m  t1 m  k  1 a  m  t ma 1    e 1  t1 e  m  t1 T   T 2 3 2 2m    m    m    m     2 ma T 2  t12   ma  m   t1 T   e  1   3 2m     m    



9  



d 7 a    m   T dT  1   k  1 e









 m  k  1  1  e   m   t1  1   m      









a  k  1    me  m    t1  T  d 8  dT  m     e  m  T   k  1

 m  t T  e   1   ma  kt1   k  1 e   m  t1    m      a T d 9  am       dT m     m   2   m    1   k  1 e   m  T    


165

    1  t1

d  t1 e   1  ka 10  4  11    m  T dT 1   k  1 e  1   k  1 e m  T        m  T  m  T ka  k  1 e d 5 d 11 ka  k  1 m    e   12  13  2 dT 1   k  1 e m T  2 dT 1   k  1 e m T      2  k  1 e m T d7  k  1 m    em T d d 27 d 28 d 29 14  6     R  T  t  R    7 1 2 2 2 2  m  T dT

  m T dT 1   k  1 e  dT     1   k  1 e     a  k  1 m  k  1 d 2 7   m  t1   m  T    m   t1    me  e   e 1 1    2 dT 2 m     1   k  1 e   m  T       



a  k  1 d 28  2 2 dT e m  T   k  1   

dT

dT







 m  k  1 e m  t1 T   e m  T   me  m  t1 T    









  m T  k  1 e m t1  kt1 ma  k  1 e m T k   k  1 e m t1 d 29 ma  k  1 e a    2 2  m  T   m  T   dT 2 m  1   k  1 e  1   k  1 e     

 1

    1  t1   1 

d 10  t1 e  k  1 m    t1 e m  T     t1 1   2     m  T dT 1   k  1 e    16  12  Ch   Cd   13  C p  C   15Cr ka  14 pI e  17  2 6 pI e  5  Ch   Cd   11  C p  C   10 Cr ka  18  2 CA  1  Ch   Cd   2  C p  C   4 Cr ka  3 pI e  Appendix 2.

15 

The values are given as follows:       

m k  1 at1

19  t1 at1 

m  



m  k  1 a

 m  

2

e 

 m t1

  ma  T  t1   ma  m t T  1   T a T  t1   e   1   1   2  m      m    







 aT 2 m k  1 at12 m  k  1 at1e m t1 m  k  1 a m t ma  m t T 1     1  e t e   1   T 2 3 2 1 2 2  m     m   m    m   





ma 3

 m  





e m t1 T   1 







ma T 2  t12    2  m     

d 19 7 d d d   t1 7  8  T 8  9  m  T dT dT dT dT 1   k  1 e     m  R  T  1  e     a e m   R T   1 a   22  21  R T   m    m     m       d 20 d 7 d 27 d 8 d 28 d 29  k  1 e  m T 7 2    t  2  T  23  1 2 dT dT dT 2 dT 2 dT 2 1   k  1 e m  T  dT 1   k  1 e  m  T     

20 





166

d22  m  R T  ae    dT  25  12  Ch   Cd   13  C p  C   15Cr ka  24 C p I c1  23 pI e  26  2 20 pI e  5  Ch   Cd   11  C p  C   10 Cr ka  22 C p I c1  27  2 C A  1  Ch   Cd   2  C p  C   4 Cr ka  21C p I c1  19 pI e 

24 

Appendix 3. The values are given as follows:       

28  t1 at1 

m k 1 at1

 m  



m k 1 a

 m  

2

e 

 m  t1

  ma  S  t1   ma  m t T  m S T 1   T a  S  t1   e   1   e      2  m     m    







aS 2 m k 1 at12 m k 1 at1e m t1 m k 1 a  m t ma 1     1  e t e m t1T   Se m  ST  2 3 2 1 2 2 m      m    m    m    2 2 ma  S  t1    ma  m  t  T  m  S T  e   1   e      3 2 m     m    





29 

30









d 28 7 d d d   t1 7  32  T 32  33   m  T dT dT dT dT 1   k  1 e   



 1  e  m   S T  a   S  T   m     m    



32  a  S  t1  

ma

e 

 m   t1 T 

2

   

 m   S T   1 d30 a e 31   dT m   



 e m  S T  

ma  S  t1     m    

m     aS 2 m k 1 at12 m k 1 at1e m t1 m k 1 a  m t ma  m t T  m S T 1 33    e t e   1   Se      1  2 3 2 1  2 2 m       m  m  m         ma  S 2  t12   ma  m t1T   m  S T   e  e   3 2 m      m     m   t   1   m  S T   d 32  a m  k  1 e mae         m    1   k  1 e m  T  dT m            e m t1 T   k 1 e m t1  d33 1 ma  em  S T    m  S T       t Se   1 dT  m     1   k 1 e m T    m    m      m      















  m   t1 T    e  m  T  me m  t1  d 232 a  k  1  m  k  1 e   mae m   S T  2 2 m T      dT 1   k  1 e   

   m T  d 233 ma   k 1 e e m T   m  t1T   m t1  m  S T      k 1 e  t1  m     e   S  m    1 e  k 1  dT 2  m    1  k 1 e m T 2      




167

 m T

7 d d7 d 27 d32 d 232 d 233  k  1 e 2 34  29  t 2 T      1 2 dT dT 2 dT dT 2 dT 2 1   k  1 e m T  dT 1   k  1 em T      d31  m  S T 35   ae    dT 36  12  Ch   Cd   13  C p  C   15 Cr ka  35 C p I c 2  34 pI e 

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