Issue 1 volume 1 p2 (1)

International Journal of Innovative Research in Applied Sciences and Engineering (IJIRASE)

Volume 1, Issue 1, June 2016 An Inventory Model for Simple way of Profit finding with fuzzy sense N. Nalini, Research scholar, Mathematics Department, Presidency College/ University of Madras, India Email:nrkhome@yahoo.com Dr. E. Chandrasekaran, Professor of Mathematics, Vel Tech University, India.

Email:e_chandrasekaran@gmail.com

Abstract—to develop fuzzy set theory and the fuzziness in the inventory problem, the purpose is to find the optimal order quantity corresponds to the total costand the associated cost also fuzzy. This approach is to find the quantity which has the minimum cost with maximum profit. When the profit gained from selling one unit of the item without deterioration and other shortage cost must be avoided. Mathematical model has been developed in two steps. (i) The Fuzzy economic order quantity and with the fuzzy total cost (ii) the Fuzzy revenue and the related with Fuzzy Net profit. Our aim is to find total cost and Profit with fuzzy sense.

Numerical examples are given and sensitivity analysis is carried out to conclude the result.

KeywordsTrapezoidal fuzzy numbers, Economic order quantity, Total cost, Revenue, Fuzzy Profit and Gain Percentage 1.

Introduction

The word inventory refers to all kind of trades that has economic value and is maintained to fulfil the present and future needs of a business.A business or other organization whose first goal is making profit, as contradict to a non-profit organization which focuses a goal such as helping the community and is concerned with profit only as much as possible to keep the organization operating. Most companies considered to be businesses are for profit.If you have a for profit organization then you must try to make sure that you always have a good reputation with your customers and acquaintances. Profit or gain, there are only two ways to make money: increase sales and decrease costs. Firms utilize strategies such as perfect

planning and calculations to minimize cost, maximize revenue, and thereby optimize profits.

Many decision-making task are too complex to be understood quantitatively, Fuzzy logic resembles human reasoning in its use of imprecise techniques to reach decisions. Unlike classical logic which requires to know the system deeply, exact equations, and precise numeric values, fuzzy logic incorporates a right way of thinking, which originate complex systems into abstracted one. Fuzzy logic allows expressing this background knowledge with concepts such as very big and a long time which are matched into correct numeric values. This idea can be expressed in a more suitable by using fuzzy sets, many decision issues.Problems can be greatly simplified. Thedevelopment of fuzzy logic was motivated to calculate large measure by the need for a new developments in profit finding.

In past, so many authors stated the operations for subtraction, multiplication and inverse too, it is really helpful for this inventory model.D. Chakraborty 14  determined the optimal order quantity in maximizing the profit with fuzzy demand. Abbasbandy and Amirfakhrian 24 analyzed that the nearest trapezoidal fuzzy number related to any power of trapezoidal fuzzy number and found the LR fuzzy number related to trapezoidal one. Lee and Yao 5 investigate the production quantity model with fuzzy demand and fuzzy production quantities.

Sanhita Banerjee and Tapan Kumar Roy 19  discussed thedifference between arithmetic operations on generalized fuzzy numbers and solved elementary problems and calculations.Park and Vujosevic 15 16  developed the inventory models in fuzzy sense where the costs are fuzzy trapezoidal numbers, Vujosevic represented ordering cost by

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International Journal of Innovative Research in Applied Sciences and Engineering (IJIRASE)

Volume 1, Issue 1, June 2016 triangular fuzzy number and holding costs by trapezoidal fuzzy number. In this study an inventory model without shortages is developed by Trapezoidal membership function.We shall construct, the total cost and optimum order quantity, revenue and profit with fuzzy sense.

2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)

Selling cost is fixed.

Shortages and Deteriorations avoided.

Ordering and holding cost are constant.

H is the Holding cost per unit O is the Ordering cost per item L is the length of the Plan r is the Order quantity per cycle L r is the Length of a cycle D is the Demand quantity r* is the Economic order quantity Tc is the Total cost per unit

(xii)

R e is the Revenue of the product

(xiii)

S p is the Selling price Rs.3 per unit(fixed)

(xiv)

N p is the Net Profit of the transaction ~ T c is the Fuzzy total cost ~ H is the Fuzzy Holding cost ~ O is the Fuzzy Ordering cost ~ Re is the Fuzzy total Revenue ~ Dq is the Fuzzy demand quantity ~ Np is the Fuzzy Net Profit

(xvii) (xviii) (xix) (xx)



3. Mathematical formulation and solution



Economic order quantity, 2OD EOQ  r*  HL Total cost =ordering cost + Holding cost HLr OD TotalCost   2 r Now we are applying fuzzy variables for Total cost 2OD 1 r*  HL







We get, r  L*r ~ ~ HL r O D ~ Tc   2 r ~  ~ L   ~ D Tc   H  r    O   2  r 









~  ~ ~ ~ ~  L   ~ ~ ~ ~  D  T c   H1 , H 2 , H 3 , H 4   r     O1 , O2 , O3 , O4      r   2    ~  ~ L ~ L ~ L ~ L   ~ D ~ D ~ D ~ D T c   H1 r , H 2 r , H 3 r , H 4 r    O1 , O2 , O3 , O4  2 2 2  r r r r  2 ~  ~ L ~ D ~ L ~ D ~ L ~ D ~ L ~ D T c   H1 r  O1 , H 2 r  O2 , H 3 r  O3 , H 4 r  O4  r 2 r 2 r 2 r  2 ~ 4 Tc  1 , 2 , 3 , 4 (say) In the Left and Right form, AL    1   2  1  AR     4   4   3  By using signed distance method, 1 ~ 1 5 T c   AL    AR  d 20 ~  ~ ~ L ~ ~ D 1  ~ ~ ~ ~ L ~ ~ ~ ~ D Tc  (H1  H 4 ) r  (O1  O4 )   (H 2  H3  H1  H 4 ) r  (O2  O3  O1  O4 )  2 r  4 2 r  ~  ~ ~ ~ ~ ~ ~ L   ~ ~ ~ ~ ~ ~ D  Tc    2(H1  H 4 )  (H 2  H 3  H1  H 4 ) r    2(O1  O4 )  (O2  O3  O1  O4 )   8  4r    ~ ~ ~ ~ D ~ ~ ~ ~ ~ L T c  H 1  H 2  H 3  H 4 r  O1  O2  O3  O4 8 4r



To find total cost,







Assumption and Notations

(xv) (xvi)

~ ~ HLr OD ~ 2 Tc   2 r ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Let H  H1 , H 2 , H 3 , H 4 and O  O1 , O2 , O3 , O4 are trapezoidal fuzzy numbers, then we get? ~ ~ ~ ~ 2 D O1  O2  O3  O4 3 Lr *  ~ ~ ~ ~ H1  H 2  H 3  H 4 L







~ T c  Z r  say 

~ ~ (r )  H  H *

1

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   *



~ ~ ~ ~ D ~ ~ Lr 7  O1  O2  O3  O4 2  H3  H4 * 8 4 Lr

International Journal of Innovative Research in Applied Sciences and Engineering (IJIRASE)

Volume 1, Issue 1, June 2016 To find Profit, Sensitivity Analysis: Let Table (1): ~ ~ ~ ~ ~ ~ ~ ~ ~ Dq Dq1 , Dq2 , Dq3 , Dq4 and z(r)  Z r1 , Z r2 , Z r3 , Z r4  are the trapezoidal fuzzy numbers, then, D ~ ~ * O H Lr Revenue = (selling Price) ďƒ„ (Demand quantity) 100 15 9 18.2574 ~ ~ ~ ~ Revenue= đ?‘†đ?‘? ⊗ Dq1 , Dq2 , Dq3 , Dq4









~ Re = đ?‘†đ?‘? ⊗ đ??ˇđ?‘ž1 , đ?‘†đ?‘? ⊗ đ??ˇđ?‘ž2 , đ?‘†đ?‘? ⊗ đ??ˇđ?‘ž3 , đ?‘†đ?‘? ⊗ đ??ˇđ?‘ž4 1

~ Re = đ?‘†đ?‘? . đ??ˇđ?‘ž1 ,

đ?‘†đ?‘? . đ??ˇđ?‘ž2 , đ?‘†đ?‘? . đ??ˇđ?‘ž3 , đ?‘†đ?‘? . đ??ˇđ?‘ž4

~ Re

~ Np

G%

246.4751.

300

53.5249

21.72

200

20

12

25.8199

348.5685

600

251.4315

72.13

300

25

15

31.6228

426.9075

900

473.0925

110.82

400

30

18

36.5149

492.9503

1200

707.0497

143.43

1

~ Re =

5. Graphical Representation Z(r*)

đ?‘†đ?‘? đ??ˇđ?‘ž1 , 1

Profit=Revenue-Total Cost

100

= đ?‘†đ?‘? đ??ˇđ?‘ž1 , đ?‘†đ?‘? đ??ˇđ?‘ž2 , đ?‘†đ?‘? đ??ˇđ?‘ž3 , đ?‘†đ?‘? đ??ˇđ?‘ž4 ⊖

Z~r , Z~r , Z~r , Z~r  3

gain %

1

4

= đ?‘†đ?‘? đ??ˇđ?‘ž1 ⊖ đ?‘?(đ?‘&#x;1) , đ?‘†đ?‘? đ??ˇđ?‘ž2 ⊖ đ?‘?(đ?‘&#x;2) , đ?‘†đ?‘? đ??ˇđ?‘ž3 ⊖ đ?‘?(đ?‘&#x;3), đ?‘†đ?‘?đ??ˇđ?‘ž4⊖đ?‘?(đ?‘&#x;4) (9)

ЛиноКнаŃ? (Z(r*))

(8)

đ?‘ đ?‘? = đ?‘…đ?‘’ ⊖ đ?‘‡đ?‘? đ?‘ đ?‘? = đ?‘…đ?‘’ ⊖ Z r ∗

2

Re

1400 1200 1000 800 600 400 200 0

đ?‘†đ?‘? đ??ˇđ?‘ž2 , đ?‘†đ?‘? đ??ˇđ?‘ž3 , đ?‘†đ?‘? đ??ˇđ?‘ž4

1

Z r *

200

300

400

Example: 2 Let us consider an inventory system with the ~ following data O  15,20 ,25,30  ~ ~ H  9,12 ,15,18  and D  75,175 ,275 ,375  find the Gain percentage.

Sensitivity Analysis:

We can use the above Mathematical model for finding the profit through the total cost along with economic order quantity.

The following examples are derived from the equations (3), (7) (8)and (9).

4. Problem Calculations Example: 1 Let us consider an inventory system with the following data

Table (2):

D

~ O

~ H

L*r

Z r *

~ Re

~ Np

G%

75 175 275 375

15 20 25 30

9 12 15 18

15.81140 24.15232 28.86754 35.35537

213.4537 326.0559 409.19698 477.29707

225 525 825 1125

11.55 198.9 415.8 647.7

5.41 61.02 101.6 135.7

~ O  15,20 ,25,30 

~ ~ H  9,12 ,15,18  and D  100 ,200 ,300 ,400  And Find the Gain percentage.

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International Journal of Innovative Research in Applied Sciences and Engineering (IJIRASE)

Volume 1, Issue 1, June 2016 Graphical Representation:

1500 1000 500 0 75

175

275

375

Z(r*)

Re

gain %

Линейная (gain %)

From the tables (1) and (2), it can be observed that, 1.Gain percentage is increases when demand quantity increases 2.Revenue is increases when demand increase 3. Ordering and holding costs are same in both a tables but gain percentage is varies due to demand.

6. Conclusion A Fuzzy inventory model for demand with constant selling price, ordering cost and holding cost has been developed. Trapezoidal fuzzy models are found for profit and gain percentage. Without shortages and deteriorationonly profit has been found. A numerical example is also given in support of the theory. A future research it may be consider to extend the model under deterioration with shortages.

7. References [1] A Study on Fuzzy Inventory Models: Fuzzy Inventory Models Without Allowing Storage Constraint Fuzzy Inventory model without Shortages in Man Power Planning, K. Punniakrishnan, K. Kadambavanam, LAP LAMBERT Academic Publishing (2013).

[2] Inventory Models: Methods and Application of EOQ and Fuzzy, Monalisha.P, LAP LAMBERT Academic Publishing 2014.

[3] A.Roy,”An inventory model for deteriorating items with price dependent demand, time varying holding

cost”, Advanced modeling and optimization, volume 10,2008.

[4] S.K.Goyal, “An Economic order quantity for Deteriorating Items with Shortages and Linear trend in Demand”, Journal of the Operations Research Society, September 1992.

[5] Lee.H.M. and Yao.J.S., “EOQ in fuzzy sense for inventory without backorder model, Fuzzy systems”, Elsevier, volume.105 July 1999.

[6] T.K. Roy and M. Maiti, “A fuzzy EOQ model with demand dependent unit cost for limited storage capacity”, European Journal of Operations research, vol.99, June 1997.

[7] Kao.C. and Hsu.W.K., “A single period inventory with fuzzy demand”, Computers and Mathematics with Applications, Elsevier volume.43, April 2002.

[8] D. Dutta, Pavankumar,”Fuzzy inventory model without shortage using Trapezoidal Fuzzy number with sensitivity Analysis”, International organization for science and research journal of Mathematics, volume 4, December 2012.

[9] Mikael Bjork.k, ”An Analytical solution to a fuzzy economic order quantity Problem”, International Journal of Approximate Reasoning,Volume.50, Issue 3 March 2009.

[10] J. S. Yao and J. Chiang, “Inventory without backorder with fuzzy total cost and fuzzy storing cost by centroid and signed distance,” European Journal of Operational Research, volume.

148,(2003).

[11] A.R. Roy, P. Dutta, D. Chakraborty, “A single period inventory model with fuzzy random demand”, Mathematical and Computer Modelling, volume.41, Elsevier, May2005.

[12] L.Wang, Q.Liang Fu, Y.R. Zeng, “Continuous inventory models with a mixture of backorders and lost sales under fuzzy demand and decision situation”, Expert systems with Applications, under Elsevier, Volume.39, Issue 4,March 2012.

[13] Fuzzy logic with Engineering Applications, Timothy J. Ross, Third Edition, John Wiley & Sons Limited.

[14] P.Dutta, D.Chakra borty, A.R. Roy , “An inventory model for single period products with

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International Journal of Innovative Research in Applied Sciences and Engineering (IJIRASE)

Volume 1, Issue 1, June 2016 reordering opportunities under fuzzy demand�, An international journal of Computers and Mathematics with Applications, Science direct 53(10) May 2007.

[15] K.S. Park, “fuzzy set theoretic interpretation of economic order quantity�,Man and Cybernetics, 17(1987).

[16] M. Vujosevic et al.,�EOQ formula when inventory cost is fuzzy�,International journal of Production Economics, 45(1996),499-504.

[17] Debdulal Panda, Mahendra Rong, Manoranjan Maiti,�Fuzzy mixture two warehouse inventory model involving fuzzy random variable lead time demand and fuzzy demand�, under Springer, (2013).

[18] Liang-Yuh Ouyang, Hung-Chi Chang, “The variable lead time stochastic inventory model with a fuzzy backorder rate�, Journal of the Operations Research Society of Japan, volume.44 March 2001.

[19] Sanhitha Banerjee and Tapan Kumar Roy,�Arithmetic Operations on Generalized Trapezoidal Fuzzy number and its Applications�, Turkish Journal of Fuzzy systems,2012.

[20] R. S. Sachan , “On (T,đ?‘†đ?‘– ) inventory policy model for deteriorating items with time proportional demandâ€?,Journal Of Operational Research Society(1984).

[21] Mahata.G.C. and Goswami.A, “Economic order quantity fuzzy Models for Deteriorating Items with Stock Dependent Demand and Nonlinear Holding Costs� ,International Journal of Mathematical and Computer Science (2009).

[22] Kuo-Lung Hou , “An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting�, European Journal of Operational Research (2006).

[23] A. Nagoor Gani , “A New Operation on Triangular Fuzzy Number for Solving Fuzzy Linear Programming Problem�, Applied Mathematical Sciences, Vol. 6, (2012).

[24] Fuzzy sets and systems, Theory and Applications, Dubois, D., and H. Prade, H., Academic Press, New York, (1980).

[25] Dr.A.Kalaichelvi & K.Janofer , “Cuts of Triangular Fuzzy Number Matrices�, International Jr of Mathematics Sciences & Applications�, Vol. 2, No. 2, May 2012.

[26] S. Abbasbandy, M. Amirfakhrian, “The nearest trapezoidal form of a generalized left right fuzzy number�, International journal of approximate reasoning, Science Direct, under Elsevier, 43(2006)

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