Quality Deteriorating Items with Initial-Inspection and

65

Chapter 5

Ordering Policy for ImperfectQuality Deteriorating Items with Initial-Inspection and Allowable Shortage under the Condition of Permissible Delay in Payments Chandra K. Jaggi University of Delhi, India Mandeep Mittal Amity School of Engineering and Technology, India

ABSTRACT While developing the inventory model with shortages under permissible delay in payments, it has been observed in the literature that the researchers have not considered the fact that the retailer can earn interest on the revenue generated after fulfilling the outstanding demand as soon as he receives the new consignment at the start of the cycle. Owing to this fact, the present study investigates the impact of interest earned from revenue generated after fulfilling the stock out at the start of the cycle on a single commodity inventory model with shortages for deteriorating item, in which the whole lot goes through an inspection process on arrival before entering into inventory system, under the conditions of permissible delay in payments. After inspection, the non-defective items are retained to fulfill the demand and the defective items are returned to the supplier. The results have been demonstrated with the help of a numerical example using the tools of Matlab7.0.1.

DOI: 10.4018/978-1-4666-4506-6.ch005

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Ordering Policy for Imperfect-Quality Deteriorating Items with Initial-Inspection and Allowable Shortage

1. INTRODUCTION A very common assumption of the economic order quantity is that all the units produced or purchased are of good quality. But practically it is difficult to produce or purchase items with 100% good quality. Thus, the inspection of an ordered lot becomes indispensable in most of the organizations. The inspected items may be classified into two categories viz., non-defective and defective items. After inspection, the non-defective items are retained to fulfill the demand and the defective items are returned to the supplier. The fraction of defective items is constant and deterministic. However, when the nature of the item is deteriorating then inspection of the lot becomes more important. The deterioration is now a wellestablished fact, which may take place in the form of gradual decay, damage or perishability and the process is affected by extraneous and invisible factors. Moreover, the deterioration may also happen due to weather conditions, insufficient and unscientific storage structure etc. This situation occurs particularly in case of foodstuffs, which are damaged due to insects, spoilage, and rodents, where as in other commodities deterioration may occur during normal storage facilities. Ghare and Schrader (1963) were the first to propose an economic order quantity (EOQ) model for items with an exponentially decaying inventory. Covert (1973) introduced variable rate of deterioration in Ghare and Schrader (1963) model. A further improvement was introduced by Shah (1977) considering a model allowing complete backlogging of the unsatisfied demand. A good amount of work has been done by different researchers to explore the effect of deterioration on EOQ model under different circumstances (Dave, 1981; K. L. Mark, 1982; Hollier, 1983; Heng, 1991; Raafat, 1991; Widyadana et al., 2011). Moreover, while developing a mathematical model in inventory control, it is assumed that the payment will be made to the suppliers for the goods immediately after receiving the consignment.

66

However, in the day-to-day dealing, supplier does allow a certain fixed period to settle the account, during which the supplier charges no interest, but beyond this period interest is charged by the supplier under the terms and conditions agreed upon. Now, in case debt financing, it is often a short-term financing. Thus, interest paid here is nothing but the cost of capital or opportunity cost. Also, short-term loans can be thought of as having been taken from the suppliers on the expiry of the credit period. However, before the account has to be settled, the customer can sell the goods and continues to accumulate revenue and earn interest instead of paying the overdraft that is necessary if the supplier requires settlement of the account after replenishment. Therefore, it makes economic sense for the customer to delay the settlement of the replenishment account up to the last day of the credit period allowed by the supplier. Goyal (1985) presented the model by introducing permissible delay in payments for fixed time period and Aggarwal and Jaggi (1995) extended his work for deteriorating items. Further, Jamal et al. (1997) allowed shortages in the Aggarwal and Jaggi (1995) model, since then, many articles have been appeared under different situations (Raafat 1991; Shah et al. 2000; Goyal and Giri 2001). The primary benefit of taking trade credit is that one can have saving in purchase cost and opportunity cost, which becomes quite relevant for deteriorating items, because in such cases, one has to procure more units then required in the given cycle to account for the deteriorating effect. In particular, when the unit purchase cost is high and decay is continuous, the saving due to delayed payment appears to be more significant. Lot of work has been published by many authors for finding the economic order quantity with or without shortages for deteriorating items. However, in the literature, it has been found for the inventory models with shortages under permissible delay in payments that the researchers have not incorporated in the interest earned part by the retailer, which he could have also earned


on the revenue generated after fulfilling the back order at the start of the new cycle. In this study, an attempt has been made to formulate the retailer ordering policy for deteriorating items with initialinspection under the condition of permissible delay in payments. Shortages are allowed and are fully backlogged.

2. ASSUMPTIONS AND NOTATIONS The following assumptions are used in developing the model: 1. The lead time is negligible 2. Shortages are allowed 3. Demand is deterministic at a constant rate and known 4. Interest paid is greater than interest earned 5. Replenishment is instantaneous 6. A constant fraction (θ) of the on-hand inventory deteriorate per unit time. The notations adopted in this chapter are as below: A Order cost of inventory (dollars per order) N Number of items received before inspection R Demand rate (units per unit time) C Cost price of one item (dollars/unit) C1 Holding cost per unit per unit time C2 Shortage cost per unit per unit time S Selling price of one item I Inspection cost per item θ Constant fraction of the on-hand inventory deterioration per unit Np Number of non defective item (t = 0) p fraction of non-defective items (0 ≤ p ≤ 1) Ie Interest earned Ip Interest paid, Ip ≥ Ie M Permissible delay in settling the accounts T Inventory cycle length t1 Length of the period with positive stock of the item

S1 Quantity consumed during time t1 D (T) Deteriorating units per cycle TVC1 (S1, T) Total variable cost per cycle per unit time for case 1, when 0 ≤ M ≤ t1 ≤ T TVC2 (S1, T) Total variable cost per cycle per unit time for case 2, when 0 ≤ t1 ≤ M ≤ T

3. MATHEMATICAL FORMULATION AND SOLUTION Let N be the items received at the beginning of the period. Further, It is assumed that after inspection Np be the number of non-defective items at t = 0, which are retained to fulfill the demand. The fraction of non-defective items p (0 ≤ p ≤ 1) is known, as it can be easily estimated from the past data. At the beginning of the cycle, a batch of Np units enters the inventory system, after inspection (Np – S1) units are delivered towards back-order leaving a balance of S1 units as the initial-inventory. With the passage of time, the inventory level gradually decreases mainly due to demands and partly due to deterioration of items up to time t1 (0 ≤ t1 ≤ T). Further demand during (t1, T) are backlogged shown in Figure 1 and Figure 2. Figure 1. Inventory system for case 1, M < T1 with complete back ordering

67


Figure 2. Inventory system for case 2, M > T1 with complete back ordering

t1 =

1 log θ

  1 + S1θ    R  

(6)

Further, total number of non-defective items is Np = R ( T − t1 ) + S1

(7)

or N =

1 R ( T − t1 ) + S1   p 

(8)

The total demand during t1 is Let I(t) be the inventory level of the system at anytime t (0 ≤ t ≤ T). The instantaneous states of I(t) over the period (0, T) are given by dI (t ) + θ I (t ) = − R dt dI (t ) =− R dt

0 ≤ t ≤ t1 (1)

t1 ≤ t ≤ T (2)

The solution of the above differential equation (boundary condition at t =0, I (t) = S1 = initial inventory) is  −θ t R S e − [ 1 − e−θ t ] I (t ) =  1 θ R ( t − t ) 1 

 0 ≤ t ≤ t1  (3)  (4)  t1 ≤ t ≤ T    

Since, at t = t1, I(t1) = 0, from Equation (1) and Equation (2), we get S1 = i.e.

68

R θ t1 e −1 θ

(

)

(5)

R t1 =

R log θ

  1 + S 1 θ    R 

(9)

Therefore, number of units that deteriorated D(T), during one cycle is D(T ) =

R θ t1 e − 1 − R t1 θ

(

)

(10)

3.1 Inventory Scenarios Since, we are considering the permissible delay in payments in this model, where stock out are allowed. Under this situation there will be two cases: Case 1: 0 ≤ M ≤ t1 ≤ T Case 2: 0 ≤ t1 ≤ M ≤ T

Case 1: 0 ≤ M ≤ t1 ≤ T In this case, it is assumed that one can earn interest on revenue generated from the sales up to t1. Although, he has to settle the account at M, for that, he has to arrange money at some specified rate of interest in order to get his remaining stocks financed for the period M to t1.


t1

Now, the interest paid per cycle is

C I e ∫ R t dt 0

t1

= CI p ∫ I (t ) dt M  S −θ M −θ t = CI p  1 ( e −e 1 )  θ  R  1 −θ t −θ M  −  (t1 − M ) + ( e 1 − e )   θ  θ 

i.e. Jamal et al. (1997) have not taken into account the interest earned from revenue generated after fulfilling the back orders at the start of the cycle, which has got significant impact on their total cost reduction. t1

where, R θ t1 e −1 θ CI p R θ ( t1 − M ) CI p R = 2 (e − 1) − ( t1 − M ) θ θ

(

S1 =

Inventory Carrying Cost = C1 ∫ I (t ) dt 0  C1  R − θ t1 ( 1 − e )( S1 + ) − R t1  (15) =  θ  θ  C1 R  θ t1 = 2 ( e − θ t1 − 1)  θ 

)

(11)

Interest earned per cycle has got two parts. Part 1 In first part, one can earn interest till the time period (M), Interest earned = S I e ∫

M 0

R t dt

(12)

Second part includes the revenue generated after fulfilling the back orders (at the start of the cycle) for the time period (M). Interest earned = SI e R(T − t1 ) M

(13)

∴ Total interest earned in this case M 0

t1

= C2 R(T − t1 )2 / 2 Inspection Cost = I N

R t dt + S I e R (T − t1 )M

(14)

Whereas Jamal et al. (1997) has considered the interest earned part as

(16) (17)

The total variable cost per cycle per unit time (TVC1) is the sum of the order cost, cost of deteriorated units, inspection cost, carrying cost, shortage cost and interest paid, but the interest earned is subtracted from this sum, i.e. TVC1(S1, T ) =

Part 2

S Ie ∫

T

Shortage Cost = C2 ∫ R (t − t1 ) dt

A CD(T ) IN + + T T T

C1 R  θ t1  C2 R (T − t )2 ( e − θ t1 − 1) + 1 2 2T θT CI R CI R θ ( t − M ) + 2p ( e 1 − 1) − p ( t1 − M ) θT θT SI e R ( T − t1 ) M R M2 − − SI e T 2T (18) +

Now, we discuss some special cases. Special Case 1 Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle) the equation (18) reduces to,

69


TVC11(S1, T ) =

A CD(T ) C1 R  θ t1 + + 2 ( e − θ t1 − 1)  T T θT 

CI R θ ( t − M ) C2 R − 1) (T − t1 )2 + 2p ( e 1 2T θT R M2 CI R − p ( t1 − M ) − SI e θT 2T +

(19) When S=C, the above expression reduce to Jamal et al. (1997). Special Case 2 Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), the equation (18) reduces to, TVC12 (S1, T ) =


CI R θ ( t − M ) C2 R − 1) (T − t1 )2 + 2p ( e 1 2T θT R M 2 SI e R ( T − t1 ) M CI R − p (t1 − M ) − SI e − T θT 2T +

(20) Whereas Jamal et al. (1997) has not taken into account the interest earned from revenue generated after fulfilling the backorders at the start of the cycle on the selling price which will help retailer to reduce his total variable cost. Special Case 3

Case 2: 0 ≤ t1 ≤ M ≤ T In this case, one can earn interest on sales revenue up to the permissible period (M), and pay no interest for the units kept in the stock. Now, interest earned per cycle has got three parts. Part 1 In first part one can earn interest till the time period (t1), t1

Interest earned = S I e ∫ R t dt 0

(22)

Part 2 Second part is having interest earned for the time period (M – t1), Interest earned = S I e R t1 ( M − t1 )

(23)

Part 3 Third part includes the revenue generated after fulfilling the back orders (at the start of the cycle) for the time period (M),

Interest earned = SI e R (T − t1 ) M

(24)

∴ Total interest earned in this case

With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), the equation (18) reduces to,

S I e ∫ R t dt + SI e R t1 ( M - t1 ) + SI e R (T - t1 ) M

A CD(T ) IN + + T T T

Whereas Jamal et al. (1997) has considered the interest earned part as

TVC13 (S1, T ) =

C1 R  θ t1  C2 R (T − t )2 ( e − θ t1 − 1) + 1 2 2T θT CI R CI R θ ( t − M ) + 2 p (e 1 − 1) − p (t1 − M ) θT θT 2 RM −SI e 2T (21) +

70

t1

0

(25)

t1

C I e ∫ R t dt + CR t1 I e ( M − t1 ) 0

Similar to case 1, here also he has not taken into account the interest earned from revenue gener-


ated after fulfilling the back orders at the start of the cycle, which may lead to inaccurate analysis.

Special Case 2

t1

Carrying Cost = C1 ∫ I (t ) dt 0

=

When S=C, the above expression reduce to Jamal et al. (1997).

C1 R  θ t1  e t ( θ 1 ) − −   1 θ2 

(26)

Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), the Equation (29) reduces to,

(27)

TVC22 (S1, T ) =

T

Shortage Cost = C2 ∫ R ( t − t1 ) dt t1

= C2 R(T − t1 )2 / 2

Inspection Cost = I N

R t12 C2 R (T − t1 )2 / 2 − SI e 2T T S I e R (T-t1 ) M S I e R t1 (M-t1 ) − − T T +

(28)

The total variable cost per cycle per unit time (TVC2) is the sum of the order cost, cost of deteriorated units, inspection cost, carrying cost and shortage cost but the interest earned is subtracted from this sum, i.e. A IN CD(T ) + + T T T CR CR θt + 21 ( e 1 − θ t1 − 1) + 2 (T − t1 )2   2T θT R t12 S I R t (M-t ) 1 1 − SI e − e T 2T S I R (T-t1 ) M − e T (29)

TVC2 (S1, T ) =

Now, we discuss some special cases. Special Case 1 Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), the Equation (29) reduces to, TVC21 (S1, T ) =


A CD(T ) + T T

C1 R  θ t1  C2 R (T − t )2 / 2 ( e − θ t1 − 1) + 1 2 T θT 2 R t1 S I R t1 (M-t1 ) − SI e − e T 2T +

(30)

(31)

Whereas Jamal et al. (1997) has not taken into account the interest earned from revenue generated after fulfilling the backorders at the start of the cycle on the selling price which will help retailer to reduce his total variable cost. Special Case 3 With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), the Equation (29) reduces to, A CD(T ) IN + + T T T C1 R  θ t1 CR + 2 ( e − θ t1 − 1) + 2 (T − t1 )2 / 2  T θT  R t12 S I R t (M-t ) 1 1 − SI e − e T 2T

TVC2.3 (S1, T ) =

(32)

Total variable cost functions for both the cases are: TVC (S , T ) 0 ≤ M ≤ t ≤ T 1 1 1 TVC (S1, T ) =  TVC2 (S1, T ) 0 ≤ t1 ≤ M ≤ T  Our objective is to find such values of S1 and T (for case 1 and case 2) which minimize the 71


total variable cost function (TVC1 and TVC2), since this function is differentiable, therefore the general necessary condition for minimization of the function is,

∂TVC2 =0 ∂T

 S1 A CR   − + t  1 T 2 T 2  CR + θ S1   C1 I + 2 ( Rt1 − S1 ) − 2 (S1 − Rt1 ) T p T θ  SI R( M − t )t   SI RMt  1 1 − − e −  e 2 1  2  2T    T    ( M − 2t1 )  C2 R 2 2 1 (T − t1 ) = 0 − SI e R  + 2 T  ( R + θ S1 )  2T ⇒−

∂TVC(S1, T ) ∂TVC(S1, T ) = = 0 ∂T ∂S1 In Case 1, ∂TVC1 =0 ∂T

(35)

 S1 A CR   − + t  1 T 2 T 2  CR + θ S1   C I + 2 ( Rt1 − S1 ) − 21 (S1 − Rt1 ) T p T θ  SI RM 2   SI RMt  − − e 2  −  e 2 1  2T   T    S  CI  −θ M R  1 + R  − − 2p e  2  ( R + θ S1 ) θ θ  T    C R R − t1 − M  + 2 2 (T 2 − t12 ) = 0  θ 2T  ⇒−

(

⇒

C θ S1

T ( R + θ S1 )

+

I θ S1 Tp( R + θ S1 )

 SI e RM  C1 S1 − − T ( R + θ S1 )  T ( R + θ S1 )     e−θ M R  CI  t C2 R  ( 1 − 1) + p + T  θ ( R + θ S1 )  ( R + θ S1 ) T   SI e Rt1 + =0 T ( R + θ S1 )

+

(34) and in Case 2,

72

⇒

C θ S1

T ( R + θ S1 )

+

I θ S1 Tp( R + θ S1 )

 SI e RM  C1 S1 − − T ( R + θ S1 )  T ( R + θ S1 )    SI e R( M − 2t1 ) SI e Rt1 − − T ( R + θ S1 ) T ( R + θS1 )   C2 R  t1 − 1 = 0 + ( R + θ S1 )  T  +

)

(33) ∂TVC1 =0 ∂S1

∂TVC2 =0 ∂S1

(36)

Solving the above Equations (33), (34), (35) and (36), we get the optimum values of T* of T and S1* of S1 for TVC1 and TVC2. For sufficient condition, determine the following for both the cases (TVC1 and TVC2) ∂ 2TVC(S1, T ) ∂ 2TVC (S1, T ) ∂ 2TVC(S1, T ) , and ∂S1∂T ∂S12 ∂T 2

The sufficient condition for minimization, TVC (S1, T) is ∂ 2TVC(S1, T )

> 0,

∂ 2TVC(S1, T )

>0 ∂S1 ∂T 2 2  ∂ 2TVC(S , T )   ∂ 2TVC(S , T )   ∂ 2TVC(S , T )  1 1 1       and    −  ∂S ∂T  >0 ∂S12 ∂T 2       1 2


Since, these expressions (Appendix A) are complicated, and it is very difficult to prove these conditions mathematically. Alternatively, we have shown convexity by graph for both the cost functions TVC1 and TVC2 using plotting tool of Matlab7.0.1. (see Figure 3 and Figure 4).

4. NUMERICAL EXAMPLES Example 1 Given p = 0.85, θ = 0.1, Ip = 0.15 /year, Ie = 0.13 /year, R = 1000 units/year, A=200 dollar/order, i = 0.12 /year, C = 20 dollars/unit, I = 0.05, C1 = 2.4 dollars/unit/yr, C2 = 20 dollars/unit/yr, S = 24 dollars/unit and M = 15 days.

Figure 3. Optimal total variable cost with respect to S1 and T for Case 1

Figure 4. Optimal total variable cost with respect to S1 and T for Case 2

73


Matlab7.0.1 direct search tool is used to solve the Equation (18), we get following results, t1= 72 days T = 99 days N = 321 units S1 = 199 units TVC1 = $1407

Special Case 1 Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving Equation (19), we get t1 = 74 days, T = 99 days, N = 321 units, S1 = 204 units and TVC11 is $1382.

Special Case 2 Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), by solving equation (2.1.20), we get t1 = 72 days, T = 99 days,  N = 321 units, S1 = 199 units and TVC12 is $1348. While Jamal et al. (1997), T1= 96 days, T = 117 days, N = 323 units, S1 = 265 units and total cost function value is $ 1157.

Special Case 3 With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving Equation (21), we get t1 = 74 days, T = 99 days, N = 321 units, S1 = 204 units and TVC13 is $1441.

Example 2 Given p=0.85, θ=0.1 Ip = 0.15/year, Ie = 0.13/ year, R =1000 units/year, A = 200 dollar/order, i = 0.12/year, C = 120 dollars/unit, I = 0.90, C1 = 14.40 dollars/unit/yr, C2 = 10 dollars/unit/yr, S = 138 dollars/unit and M = 30 days.

74

Matlab7.0.1 direct search tool is used to solve the Equation (29), we get following results, t1 = 15 days T = 81 days N = 260 units S1 = 41 units TVC2 = $1391

Special Case 1 Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving Equation (30), we get t1 = 24 days, T = 77days, N = 247 units, S1 = 66 units and TVC21 is $1443.

Special Case 2 Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle), by solving Equation (31), we get t1 = 15 days, T = 81 days, N = 261 units, S1 = 41 units and TVC22 is $332. Because interest earned after fulfilling the back order (at the start of the cycle) is very high and reduces to $1203. While Jamal et al. (1997), t1 = 24 days, T = 78 days, N = 212 units, S1 = 66 units and total cost function value is $1481.

Special Case 3 With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), by solving Equation (32), we get t1 = 24 days, T = 77 days, N = 247 units, S1 = 66 units and TVC23 is $2503. Findings clearly indicate that in both the cases i.e. M < t1 and M > t1, it is beneficial for retailer to introduce inspection and interest earned on the revenue generated after fulfilling the back orders (at the start of the cycle) as he is able to reduce his total cost to that of Jamal et al. (1997).


5 SENSITIVITY ANALYSIS

24} dollars/unit/yr and M = {15, 30, 45} days and assuming rest of the data of example 2. No feasible solution marked as “—” in Table 3.

In order to gain more in sight of the above model the sensitivity analysis has been performed on certain parameters viz. purchase cost (C), Deterioration rate (θ), Shortage cost (C2) and permissible payment period (M). Now for fixed C2, taking θ = {0.1, 0.2}, C = {20, 40, 180, 200} dollars/unit, S = {24, 48, 216, 240} dollars/unit., C1 = {2.4, 4.8, 21.6, 24} dollars/unit/yr, I = {0.05, 0.1, 0.45, 0.50} and M = {15, 30, 45} days and assuming rest of the data of example 1. No feasible solution marked as “—” in Table 1. When purchase cost (C) is fixed and taking θ = {0.1, 0.2}, C2 = {20, 50, 100, 1000} dollars / unit/yr, S = 48 dollars/unit., C1 = 4.8 dollars/unit/ yr, I = 0.1 and M = {15, 30, 45} days and assuming rest of the data of example 1. The solution is given in Table 2. Further for case 2, keeping fixed C2 and taking θ = {0.1, 0.2}, C = {120, 150, 180,200} dollars/ unit, S = {138, 172.5, 207, 230} dollars/unit, I = {0.90, 1.13, 1.35, 1.50}, C1 = {14.40, 18, 21.6,

Observations •

•

It is evident from Table 1 that as M increases, t1 increases which suggests that the retailer should order more and total average variable cost decreases due to the revenue generated after fulfilling the back order (at the start of the cycle). However, when the deterioration rate (θ) increases, cycle length (T) reduces but the total variable cost increases. This fact encourages the retailer to order more frequently, which helps him to manage the loss due to deterioration effectively. However, Table 2 indicates that as C2 increases, t1 approaches to T that implies stock out tends to zero, which endorse the fact that if the inventory system is operated at all it is never optimum to have back order.

Table 1. Optimal cycle length and total variable cost with respect to C, θ and M For case 1: Results for C2 = 20 C

20

40

180

200

θ

0.1

0.2

M

t1

T

S1

N

TVC1

t1

T

S1

N

TVC1

15

72

99

199

321

1407

61

91

171

295

1539

30

72

99

199

321

1281

61

91

171

295

1412

45

72

99

199

321

1154

61

91

171

295

1285

15

45

79

124

255

1712

38

74

105

239

1837

30

45

79

124

255

1459

38

74

104

239

1583

45

—

—

—

—

—

—

—

—

—

—

15

—

—

—

—

—

—

—

—

—

—

30

—

—

—

—

—

—

—

—

—

—

45

—

—

—

—

—

—

—

—

—

—

15

—

—

—

—

—

—

—

—

—

—

30

—

—

—

—

—

—

—

—

—

—

45

—

—

—

—

—

—

—

—

—

—

75


Table 2. Optimal cycle length and total variable cost with respect to C2, θ and M For case 1: Results for C = 40 C2

20

50

100

θ

0.1

0.2

M

t1

T

S1

N

TVC1

t1

T

S1

N

TVC1

15

100

122

278

398

1137

79

105

221

343

1337

30

102

125

284

406

1034

80

107

225

350

1237

45

105

128

292

419

946

83

110

232

360

1155

15

73

86

202

279

1609

58

73

161

238

1907

30

76

89

209

289

1423

60

76

167

247

1735

45

80

94

221

305

1276

63

80

177

261

1605

15

76

82

210

268

1676

61

69

170

226

2024

30

79

85

218

278

1495

63

72

177

234

1854

45

83

90

230

294

1352

67

76

187

248

1731

N

TVC2

Table 3. Optimal cycle length and total variable cost with respect to C, θ and M For case 2: Results for C2 = 10 C

120

150

180

•

76

θ

0.1

M

t1

T

S1

15

—

—

—

30

15

81

41

45

16

80

42

15

—

—

30

12

45

13

15

0.2 N

TVC2

t1

T

S1

—

—

—

—

—

—

—

260

1391

12

79

33

255

1428

259

654

13

76

34

253

691

—

—

—

—

—

—

—

—

79

33

256

1321

9

78

27

250

1352

80

34

255

400

10

77

28

251

431

—

—

—

—

—

—

—

—

—

—

30

10

78

28

252

1242

8

77

22

249

1268

45

11

77

29

251

135

9

76

23

250

162

Further, Table 3 illustrates that as M increases then total variable cost decreases significantly because interest earned after fulfilling the back order increases significantly.

6 CONCLUSION The present paper investigates the effect of initialinspection, interest earned on selling price and interest earned after fulfilling the back orders (at the start of the cycle) on the retailer ordering policy. Findings have been very encouraging as retailer is not only able to increase his order quantity but also able to reduce his total variable cost in comparison to Jamal et al. (1997), where inspection cost and interest earned on revenue


generated after fulfilling the back order has not been incorporated in the Jamal et al. (1997). Further, three special cases viz. (i) Without inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle), (ii) Without inspection and including interest earned after fulfilling the back orders (at the start of the cycle) and (iii) With inspection and excluding interest earned after fulfilling the back orders (at the start of the cycle) have been discussed. Results have been validated with the help of numerical examples along with a comprehensive sensitive analysis using Matlab 7.0.1 tools.

7 FUTURE SCOPE Present paper investigates the impact of initialinspection on retailer’s ordering policy for deteriorating items under trade credit. In this study, we have considered initial-inspection process, in which the whole lot goes through an inspection process on arrival before entering in to inventory system. After inspection, the effective inventory is the numbers of non-defective items are retained to fulfill the demand and the defective items are returned to the supplier. The fraction of defective items is known and deterministic. Further, Jaggi and Mittal (2011) developed inventory model which for more realistic situation than the present study, where shortages are not allowed and permissible delay condition is relaxed. In this model, each item goes through in-process inspection, and screening rate is assumed to be more than the demand rate. This assumption helps one to meet his demand parallel to the screening process, out of the items which are of perfect quality. Further, the defective items are sold immediately after the screening process is over as a single lot at a discounted price. In this process, the fraction of defective items, α is a random variable with a known probability density function, f(α), which can be calculated from the past data and its expected value is given by

b

E(α ) = ∫ α f (α ) d α,

0 < a < b