An Inventory Model For Deteriorating Items With Three Parameter Weibull Deterioration And Price Dependent Demand

An Inventory Model For Deteriorating Items With Three Parameter Weibull Deterioration And Price Dependent Demand

1R. Amutha

Research Scholar

Department of Mathematics, Presidency College, Chennai-05

2Dr. E. Chandrasekaran

Associate Professor

Department of Mathematics, Presidency College, Chennai-05

Abstract

The paper presents an inventory model for deteriorating items with price dependent demand. Deterioration rate follows a three parameter Weibull distribution. Shortages are allowed and are completely backlogged. The results are illustrated with the help of numerical example. Sensitivity analyses are carried out to analyze the effect of changes in the optimal solution with respect to change in one parameter at a time.

Keywords: Deterioration items, holding cost, Inventory, Price dependent demand time, Shortages.

1. Introduction

   Inventory models are classified in to three categories (1) Deterioration (2) Obsolescence

   (3) no deterioration/no obsolescence. Deterioration makes the product value dull.

   Ajanta Roy [1] presented an inventory model for time proportional deterioration rate and demand is function of selling price. The Author discussed the model without shortage and also with shortages in which the shortages are completely backlogged. Mukesh Kumar, Anand Chauhan, Rajat Kumar [8] extended Ajanta Roy models with trade credit. Tripathy C.K and L.M. Pradhan [13] gave a model in which the demand of the product decreases with the increase of time and sale price and deterioration rate follows a three parameter Weibull distribution. Now Tripathy C.K and L.M. Pradhan [14] included salvage value and developed an EOQ model for three parameter Weilbull distribution deterioration rate under permissible delay in payments. Padmanabhan.G, Prem Vrat [9] formulated an EOQ model for perishable items under stock dependent selling rate. Sahoo.N.K., Sahoo .C.K. & Sahoo.S.K described an inventory model for price dependent demand and time varying holding cost. Vikas Sharma and Rekha Rani Chaudhary [16] explained and inventory model for two parameter Weibull deterioration rate.

   They found profit for their model. Sanjay JAIN and Mukesh KUMAR [12] explained an inventory model with ramp type demand and three parameter Weibull deterioration rate. The Authors also analyzed and summarized economic order quantity models done by few researchers. There are some products which start deteriorate only after some interval of time. This was explained by taking three parameter Weibull distribution deterioration rate. Anil Kumar Sharma, Manoj Kumar Sharma and Nisha Ramani [2] described an inventory model for two parameter Weibull distribution deterioration rate and demand rate is power pattern. Manoj Kumar Meher, Gobinda Chandra Panda, Sudhir Kumar Sahu [7] adopted a two parameter Weibull distribution deterioration to develop an inventory model under permissible delay in payments. Kun Shan Wu [6] made an attempt in his paper to obtain the optimal ordering quantity of deteriorating items for two parameter Weibull distribution deterioration under shortages and permissible delay in payments. P.K.Tripathy and S.Pradham [15] also define an inventory model with two parameter Weibull distribution as demand rate and deterioration rate increases with time.

   In this present paper, we have developed an inventory model for three-parameter Weibull deterioration rate and price dependent demand. Shortages are allowed and are completely backlogged. Holding cost is assumed to be constant. Our aim is to increase the profit.

2. Assumptions and notations

   1. The demand rate is a function of selling price.

   2. Shortages are allowed and are completely backlogged

   3. Lead time is zero.

   4. Replenishment is instantaneous

   5. A is the set up cost

   6. C is the unit cost of an item

   7. p is the selling price

   8. Demand D (t) = f (p) = a-p, where a > p.

   9. C2 is the shortage cost per unit time

   (x) (t) = (t-)-1, 0 < 1 , > 0 and

   T 2

   p) [ 1 +

   2

   ( ) 2

   ( 1)( 2)

   2 ( )2 2

   +

   +

   ]

   ]

   4(2 1)( 1)

   – < < is the deterioration rate. At time T1 the Inventory becomes Zero and shortages start occurring.

   (6)

   Stock loss due Deterioration

   1. h is the constant holding cost.

   2. T is the length of the cycle.

   T1

   D = (a-p) e (t ) 0

   T1

   dt (a-p) dt

   0

3. Mathematical formulation and solution

   (T

   ) 1

   2 (T

   )2 1

   = (a-p) [ 1 + 1 –

   Let I (t) be the inventory at time T ( 0 t T ) the differential equation for the instantaneous state over

   ( ) 1

   1

   –

   –

   2 ( )2 1

   2(2 1)

   (0,T) are given by

   dI (t)

   + (t )

   1

   I (t) = – (a-p), 0 t T1

   ]

   ]

   1 2(2 1)

   (7)

   dt

   (1)

   Order Quality

   dI (t)

   dt

   (2)

   = – (a-p), T1 t T

   T

   Q = [D + (a p)dt ]

   0

   With boundary condition I (T1) = 0

   (T

   ) 1

   2 (T

   )2 1

   = (a-p) [ 1 + 1 + T –

   Solving equation (1)

   ( ) 1

   1

   –

   –

   ]

   ]

   2 ( )2 1

   2(2 1)

   1

   1

   I (t) = – (a-p) [(t-T1) +

   2

   1

   ((t- ) 1 – (T – ) 1 ) +

   1 1

   1 1

   (8)

   1

   2(2 1)

   2(2 1)

   (3)

   ((t- )2 1 – (T – )2 1 ) ], 0 t T

   CQ

   Purchase cost =

   T

   Solving equation (2)

   C (T

   ) 1

   2 (T )2 1

   = – (a-p) (t-T ), T t T

   = (a-p)

   [ 1 + 1 + T –

   1 1

   (4)

   T 1

   2(2 1)

   Shortage Cost

   ( ) 1

   2 ( )2 1

   –

   ] (9)

   C T 1

   2(2 1)

   SC = 2 (a p) (t-T1) dt

   T T1

   Total Profit Per unit time is

   1

   C2

   = (a-p) (T-T1)2

   = p (a-p) –

   T

   [Ordering cost + purchase cost +

   2T

   (5)

   shortages cost + holding cost]

   Holding Cost

   1

   K(p,T, T1) = p (a-p) –

   T

   { A +

   C2

   (a-p) (T-T1)2 + C (a-p)

   2

   h T1

   (T

   ) 1

   ( ) 1

   –

   –

   HC =

   T

   I (t) dt

   0

   ( 1 –

   1

   1

   • h

   (T ) 2

   2 ( )2 1

   2 (T ) 1

   = (a-p) [

   T

   1

   ( 1 – T1 (T1- ) ) +

   1

   1

   1

   1

   +

   +

   2

   2(2 1)

   2(2 1)

   + T ) h (a-p)

   2 (T

   )2 2 [

   ( 1 – T (T – )2 1 )] + h (a-

   2(2 1) 2( 1) 1 1

   1

   (T ) 2

   2

   ( ) 2

   2 ( )2 2

   ( 1 -T (T – )

   1 ) + – ]

   2 1 1

   1 )

   1 )

   (T )2 2

   2(2 1)

   T 2

   ( 1)( 2)T 2

   4T 2 (2 1)( 1)

   (10)

   2 1 1

   ( – T1 (T1- )] + h (a-p) [ +

   K ( p,T )

   1 C (T Tv)2

   2( 1)

   ( ) 2

   2

   + ]}

   + ]}

   2 ( )2 2

   = (a-2p) –

   p

   { 2 – C

   T 2

   ( 2)( 1) 4(2 1)( 1)

   <>Let T1= vT, 0 < v < 1

   (Tv ) 1

   (

   (

   1

   ( ) 1

   –

   –

   –

   –

   1

   1

   K (p,T) = p (a-p) –

   T

   { A +

   C2

   (a-p) (T-vT)2 + C (a-p)

   2

   2 ( )2 1

   2(2 1)

   (Tv ) 2

   2 (Tv )2 1

   +

   +

   2(2 1)

   + T) + h [ (

   1

   2

   (

   (

   –

   –

   (vT ) 1 ( ) 1 – Tv (Tv- ) 1 ) + (

   1

   2 ( )2 1

   1

   +

   +

   2 (vT )

   –

   2 1

   2

   (Tv )2 2

   • Tv (Tv – )

     2 1

     2(2 1)

     (Tv)2

     )] h [ +

     2(2 1)

     2(2 1)

     + T ) h (a-p) [

     2( 1) 2

     (

     (

     1

     (Tv ) 2

     2

   • vT (vT – )

     1 ) +

     ( ) 2

     +

     +

     ( 2)( 1)

     2 ( )2 2

     4(2 1)( 1)

     ]} = 0. (11)

     2 For the maximization of profit we set,

     2(2 1)

     (

     (

     (vT )2 2

     + ].

     + ].

     2( 1)

   • vT (vT – )

   2 1

   )] + h(a-p) [

   K ( p,T )

   T = 0 &

   Provided

   K ( p,T )

   p = 0

   (vT)2

   ( ) 2

   2 ( )2 2

   2 K ( p,T )

   ( ) (

   ( ) (

   < 0,

   2 K ( p,T ) )

   2 ( 2)( 1) 4(2 1)( 1)

   T 2

   p2

   < 0 and

   K ( p,T )

   A C2 2

   ( 2 K ( p,T ) )( 2 K ( p,T ) ) ( 2 K ( p,T ) )

   –

   > 0

   –

   > 0

   = –

   T T 2 2

   (a-p) (1- v)

   – C(a-p) [

   T 2

   p2

   Tp

   +

   +

   T ( 1)v(vT ) (vT ) 1

   T 2 ( 1)

4. Numerical Example

   + +

   + +

   ( ) 1

   2 ( )2 1

   Let us consider C = 5, v = 0.2, C2

   = 5, = 0.4,

   T 2 ( 1) 2T 2 (2 1)

   ]

   ]

   2T (2 1)v(vT )2 2 (vT )2 1

   2T 2 (2 1)

   + h (a-p)

   [
   [

   –

   –

   	p	T	TP	Q
   0.5	32.4215	5.9714	141.5371	108.3591
   0.6	32.4534	5.9869	140.8515	109.2179
   0.7	32.4866	6.0031	140.1385	110.1126
   0.8	32.5210	6.0198	139.3986	111.0397

   	p	T	TP	Q
   0.5	32.4215	5.9714	141.5371	108.3591
   0.6	32.4534	5.9869	140.8515	109.2179
   0.7	32.4866	6.0031	140.1385	110.1126
   0.8	32.5210	6.0198	139.3986	111.0397

   T ( 2)v(vT ) 1 (vT ) 2

   = 2, = 1, h = 1, a = 50, A = 1000 in proper units we get p = 32.3909, T = 5.9564, TP = 142.1949, Q = 107.5330

5. Sensitivity Analysis

   ( 1)( 2)T 2

   v2 (vT ) +

   v 2T (2 2)(vT )2 1 2 (vT )2 2

   4(2 1)( 1)T 2

   2 v2 (vT )2

   –

   2

   v 2

   ]- h (a-p) [ –

   2

   a	p	T	TP	Q
   60	36.7900	5.2035	346.5236	124.2395
   70	41.3870	4.6959	605.7523	138.6358
   80	46.0894	4.3198	918.4367	151.5396

   H	p	T	TP	Q
   2	32.43.03	5.9277	139.9951	106.7867
   3	32.4698	5.8998	137.8099	106.0562
   4	32.5093	5.8125	135.6390	105.3385
   5	32.5488	5.8458	133.4822	104.6336

   H	p	T	TP	Q
   2	32.43.03	5.9277	139.9951	106.7867
   3	32.4698	5.8998	137.8099	106.0562
   4	32.5093	5.8125	135.6390	105.3385
   5	32.5488	5.8458	133.4822	104.6336

   From this table, it can be observed that

   C2	p	T	TP	Q
   4	31.8084	6.5334	177.8724	121.6404
   3	31.1694	7.3674	218.8579	141.8183
   2	30.4342	8.6899	267.7428	174.0637
   1	29.4950	11.0218	329.7243	234.6714

   C2	p	T	TP	Q
   4	31.8084	6.5334	177.8724	121.6404
   3	31.1694	7.3674	218.8579	141.8183
   2	30.4342	8.6899	267.7428	174.0637
   1	29.4950	11.0218	329.7243	234.6714

   1. p(t, p) is slightly sensitive to changes in , h and it is highly sensitive to changes in a, C2, C2 and v.

   2. p is slightly sensitive to changes in , h

      and moderately sensitivity to changes in a, C2, C2 and v.

   3. Q is slightly sensitive to changes in the values of , h and C and it is highly

      sensitive to changes in a, C2 and v.

   4. T is slightly sensitive to changes in the values of , a, h and C and moderately sensitive to changes in C2 and v

6. Conclusion

   C	p	T	TP	Q
   6	32.9898	6.0645	124.4512	105.7209
   7	33.5930	6.1782	107.3297	103.8447
   8	34.2008	6.2979	90.8342	101.8983
   9	34.8138	6.4246	74.9689	99.8804

   C	p	T	TP	Q
   6	32.9898	6.0645	124.4512	105.7209
   7	33.5930	6.1782	107.3297	103.8447
   8	34.2008	6.2979	90.8342	101.8983
   9	34.8138	6.4246	74.9689	99.8804

   A deterministic inventory model for deteriorating inventory model with three parameter Weibull distribution deterioration rate has been developed. Demand rate is function of selling price and holding cost is constant occurring shortages and completely backlogged. A numerical example is also given in support of the theory. A future research it may be consider to extend the model under permissible delay in payments.

7. References

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2. Anil Kumar Sharma, ManojKumar Sharma and Nisha Ramani An Inventory model with Weibull distribution deteriorating power pattern demand with shortages and time dependent holding cost. American Journal of Applied Mathematics and Mathematical Sciences (Open Access Journal Volume 1, Number 1-2 January

   December 2012, Pp. 17-22.

3. Biplab Karmakar and Karabi Dutta Choudhury, A Review on Inventory Models for Deteriorating Items with Shortages, Assam University Journal of Science& Technology, Physical Sciences and Technology, Vol.6,

   Number II, 51-59,2010

4. Ghosh S.K and K.S.Chaudhri, An order level inventory model for a deteriorating item with Weibull distribution deterioration time quadratic demand and shortages, AMO-

   Advanced Modeling and Optimization, volume 6, Number 1.

5. Goyal S.K, B.C.Giri Recent trends in modeling of deteriorating inventory European Journal of Operation Research, 134 (2001) 1-16

6. Kun Shan Wu An Ordering Policy for Items with Weibull distribution Deterioration under permissible Delay in Payments. Tamsui Oxford Journal of Mathematical Science 14 (1998) 39- 54.

7. Manoj Kumar Meher , Gobinda Chandra Panda, Sudhir Kumar Sahu, An Inventory Model with Weibull Deterioration Rate under the Delay in Payment in Demand Dec ling Market Applied Mathematical Sciences,Vol.6, 2012, no.23, 1121

   -1133.

8. Mukesh Kumar, Anand Chauhan, Rajat Kumar A Deterministic Inventory Model for Deteriorating Items with Price Dependent Demand and Time Varying Holding cost under Trade Credit International Journal of Soft computing and Engineering (IJSCE) ISSN:2231-2307, Volume-2, Issue-1, March 2012

9. Padmanabhan.G., Prem Vrat EOQ models for perishable items under stock dependent selling rate

10. Pattnaik.M, A Note on Non Linear Optimal Inventory Policy Involving Instant Deterioration of Perishable Items with Price Discounts, The Journal of Mathematics and Computer SCienceVol.3, No.2 (2011) 145-155.

11. Sahoo.N.K, Sahoo .C.K. & Sahoo.S.K. An Inventory Model for Constant Deteriorating Items with Price Dependent Demand and Time p Varying Holding Cost International Journal of computer cuience & communication Vol.1,

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12. Sanjay JAIN, Mukesh KUMAR An EOQ Inventory Model for Items with Ramp Type Demand, Three-Parameter Weibull Distribution Deterioration and Starting with Shortage, Yugoslav Journal Of Operations Research Volume 20(201), No. 2, 249-259

13. Tripathy C.K and L.M. Pradhan, Optimal Pricing and Ordering Policy for Three Parameter Weibull Deterioration .Int. Journal of math .Analysis, Vol.5,2011,no.6,275-284

14. Tripathy C.K and L.M. Pradhan An EOQ model for three parameter Weibull deterioration with permissible delay in payments and associated salvage value, International Journal of Industrial Computations3(2012)115-122

15. TripathyP.K, S.Pradhan, An Integrated Partial Backlogging Inventory Model having Weibull Demand and Variable Deterioration rate with the Effect of Trade Credit, International Journal of Scientific & Engineering Research Volume 2, Issue 4, April 2011

16. Vikas Sharma, and Rekha Rani Chaudhary, An inventory Model for deteriorating items with Weibull Deterioration with Time Dependent Demand and Shortages, Research journal of Management sciences, Vol. 2(3), 28-30, March (2013) ISSN 2319-1171.