an intuitionistic approach to an inventory model without...

Int. J. Pure Appl. Sci. Technol., 22(2) (2014), pp. 25-35

International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in

Research Paper

An Intuitionistic Approach to an Inventory Model without Shortages

Prabjot Kaur1,* and Mahuya Deb2

1 Department of Mathematics, Birla Institute of Technology, Mesra, Jharkhand, India 2 Usha Martin Academy, Ranchi, Jharkhand, India

* Corresponding author, e-mail: ([email protected])

(Received: 1-4-14; Accepted: 6-6-14)

Abstract: Inventory management policies are crucial for the successful operations of firms involving inflow, storage and outflow of physical goods. However, the parameters associated with inventory problems often deal with uncertainties, and as such it is justifiable to consider these factors in elastic form as deterministic values may fail to give the correct approximations. This paper deals with a basic Economic Order Quantity(EOQ)inventory model where the objective is to determine the optimal cost and an optimum order quantity of inventory by taking certain non-deterministic parameters as triangular intuitionistic fuzzy numbers. Two case studies of the mathematical model have been given in order to show the applicability and robustness of the proposed model. Sensitivity analysis has been carried out which shows the linear relation between holding cost, EOQ and total cost .The advantage of the proposed intuitionistic approach is that it is a robust model which deals with the varying parameters in a general business inventory consistent with human behaviour by reflecting and modelling the hesitancy present in real life situations. Keywords: Inventory management, Triangular intuitionistic fuzzy number (TIFN), Economic order quantity (EOQ).

1. Introduction: In any sort of business, a certain extent of inventory of resources is held to provide desirable services to the customers and to achieve the sales turnover target. Inventory control is the art of controlling an optimized amount of stock held in various forms within a business to economically meet the demands

Int. J. Pure Appl. Sci. Technol., 22(2) (2014), 25-35 26

placed upon that business. Inventory management policies are crucial to the successful operations of the firms. Any disturbance in the inventory can lead to an overall misbalance in the management policies. The most important task of inventory management is making a trade-off between the minimization of the total cost and maximization of the customer satisfaction. Two fundamental questions that must be answered in controlling the inventory of physical goods are when to replenish the inventory and how much to order for replenishment. The earliest derivation of what is often called the simple lot size formula was obtained by Ford Harris of the Westinghouse Corporation in 1915 [4] in which the objective is to obtain an optimal and economic order quantity in such a way that the total yearly inventory cost is minimized, using mathematical relations (given afterwards). The EOQ model not only takes into consideration the cost price of the inventory, but also the other expenses involved in maintaining the inventory, like holding costs (which may include various aspects such as electricity, storage facility etc.). It is also referred to as Wilson formula since it was derived by R.H Wilson[15] as an integral part of the inventory control scheme thereby arising interest in the EOQ model in academics and industries [8]. Shortly after the World War II, a stochastic version of the simple lot size model was developed by Whitin [19]. Later, Hadley et al [5] analysed many inventory systems. Research work on static lot size model has been reported in the literature by Arrow [7]. The parameters related to classical inventory model are all crisp. But, in real life situations, these parameters may have slight deviations from the exact value which may not follow any probability distribution. In such situations, if they are treated as fuzzy parameters, then such a model becomes more realistic. The theory of fuzzy set introduced by Zadeh [13] in 1965 has achieved successful applications in various fields including inventory control. Recently, the concept of fuzzy parameters has been introduced in the inventory problems by several researchers. Park [10], Kacpryzk and Staniew [9] introduced fuzzy sets in the inventory problem. Consequently researchers began to make use of fuzzy numbers to consider these uncertain factors in elastic form. Hsieh [2] discussed an inventory model where demand and lead time are assumed to be fuzzy trapezoidal numbers. De and Rawat [12], proposed an EOQ model without shortage cost by using triangular fuzzy number. Bai and Li. [16], Dutta. et.al. [3] have also used Triangular and Trapezoidal Fuzzy number in building inventory models for determining the optimal order quantity and the optimal cost. However the fuzzy set theory was extended to the intuitionistic fuzzy sets by Atanassov [11] by adding an additional non-membership degree. Among various extensions of fuzzy sets, IFSs have captured the attention of many researchers in the last few decades. This is mainly due to the fact that IFSs are consistent with human behaviour, by reflecting and modelling the hesitancy present in real life situations. Therefore in practice, it is realized that human expressions like perception, knowledge, and behaviour are better represented by IFSs rather than fuzzy sets The concept of an IFS can be seen as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of conventional fuzzy sets. The IF-set may express information more abundant and flexible than the fuzzy set when the uncertain information is involved. Banarjee and Roy [17] generalized the application of the intuitionistic fuzzy optimization in the constrained multi objective stochastic inventory model. Susovan Chakraborty et al. [18] gave the solution for the basic EOQ model using intuitionistic fuzzy optimization technique. Mahapatra [6] gave a multi-objective inventory model of deteriorating items with some constraints in an intuitionistic fuzzy environment. This paper is a novel approach in building up the intuitionistic inventory model considering the demand and ordering cost as TIFN and for defuzzification an accuracy function defined by [1] has been considered. The total variable cost is expressed as a function of these two decision variables and then the reorder point is obtained which minimizes the total cost. As the model parameters are generally imprecise or may be unable to exhibit the variability in such situations IFS theory can come as a rescue. The organisation of the paper is as follows: In section 2 the preliminaries on Intuitionistic fuzzy sets is detailed. Section 3 deals in establishment of the Inventory Model in the Intuitionistic fuzzy environment. Section 4 deals in optimising and finding a solution for the Inventory problem using TIFN. Section 5 deals with a numerical and a sensitive analysis to illustrate the result. Lastly the concluding remark is stated in Section 6.


2. Preliminaries on Intuitionistic Fuzzy Sets: Intuitionistic fuzzy set was introduced by Atanassov (1986) and it has found applications in various areas of research. In this section some basics and notations on intuitionistic fuzzy sets are reviewed. 2.1 Definition 1: A fuzzy set A/in X = {x} is given by (Zadeh [13]):

A/= {< x, µA(x) >|x ∈X} Where µA: X → [0, 1] is the membership function of the fuzzy set A; µ A ∈ [0, 1] 2.2 Definition 2: An intuitionistic fuzzy set A in X is given by (Atanassov [11]): A = {< x, µA(x), νA(x) >|x ∈X} Where µA: X → [0, 1] νA: X → [0, 1] With the condition 0<µA(x) + νA(x)<1 ∀ x ∈X The numbers µA(x), νA(x) ∈ [0, 1] denote the degree of membership and non-membership of x to A, respectively. 2.3 Definition 3: A TIFN [1] � � is an intuitionistic fuzzy set in R with the following membership function ��̿(x) and non- membership �̿ (x) (Figure 1).

��̿(x) =

�� , �� ≤ � ≤ ��

xa −3�� , �� ≤ � ≤ ��0, ��ℎ� !"#� $ and �̿(�) = (

�� ) *�� ′ , ��′ ≤ � ≤ ��′�� , �� ≤ � ≤ ��′1, ��ℎ� !"#�$

Where ��′ ≤ �� ≤ �� ≤ �� ≤ ��′ and ��̿ + �̿ ≤ 1

Figure 1: A Triangular Intuitionistic Fuzzy Number


2.4 Arithmetic Operations of Triangular Intuitionistic Fuzzy Number [1] is Given by: Various arithmetic operations are carried out on intuitionistic fuzzy sets.

If �̿ { }/ /1, 2 3 1 2 3( , );( , , )a a a a a a= and -. = {(0�, 0�, 0�)(0�′, 0�, 0�′)are two TIFNs, then

1. Addition of two TIFN is �̿ + -. = {(�� + 0�,�� + 0�, �� + 0�)(��′ + 0�′, �� + 0�, ��′ + 0�′)} is also a TIFN. (1) 2. Subtraction of two TIFN is �̿ − -. = {(�� − 0�,�� − 0�, �� − 0�)(��′ − 0�′, �� − 0�, ��′ − 0�′)} is also a TIFN. (2) 3. Multiplication of two TIFN is �̿ × - � = {(��0�, ��0�, ��0�)(��′0�′, ��0�, ��′0�′} is also a TIFN. (3) 4. If TIFN �̿ = (��, ��, ��)(��′, ��, ��′) and y=ka (with k>0) 4. ′ = 5�̿′is a TIFN {(ka1,ka2,ka3)(ka1

',ka2,ka3' )} (4)

5. Division of two TIFN is

B

A = {6��7� , ��7� , ��7�8 9 ′

′

3

1

b

a , ��7� , ′

′

1

3

b

a : } is also a TIFN (5)

2.5 Accuracy Function for Defuzzification:

Let �̿ { }/ /1, 2 3 1 2 3( , );( , , )a a a a a a= be a TIFN, then accuracy function [1] for defuzzification is defined

as

�̿′ = (��;��;��); )2( 321′++′ aaa< (6)

3. Notations and Assumptions 3.1 We define the following symbols: • Ch– Holding cost per unit quantity • Co– Ordering cost per order • D– Total demand over the planning period • TC– Total variable cost for the period • TC ....– Intutionistic Fuzzy total variable cost • *Q – Optimal order quantity

• CQ...– Triangular Intutionistic Fuzzy unit cost of placing an order • D � – Triangular Intutionistic Fuzzy annual demand over the planning period • P(Q)– Defuzzified total cost


• P(Q)* – Optimal total cost 3.2 Assumptions: In this paper the following assumptions are considered: • Shortages are ignored – Any sort of shortages in the stock/ inventory are assumed to be

nonexistent. That means the stock is always available and there is never a shortage of availability of stock.

• It is a general practice to give discounted prices (reduced cost for goods) on large scale purchases. This is generally done in good faith towards the customers and to maintain long term relations with them. In our model, such discounts are not allowed.

• Ordering costs are the costs incurred each time an order is placed for procuring items from outside suppliers. Ordering or set up cost and demand are considered as Triangular Intuitionistic Fuzzy Numbers.

4. Deterministic Single Item Inventory Model in a Crisp Environment First we deal with an inventory model by Harris [4] without shortages in a crisp environment. In this model, the economic order quantity is obtained by the following model equation:

02

h

DCQ

C= (7)

The cycle time i.e. the optimal interval between the successive orders is given by t*= Q*/ D (7.1) Optimal number of orders (N*) to be placed in the given time period N*=D/Q* (7.2) Thus the total variable inventory cost (Figure 1) is equal to the ordering cost plus the carrying cost and is given by the following equation:

0

2hDC QC

TCQ

= + (7.3)

The optimum *Q and *TC can be obtained by equating the first partial derivative of the total inventory cost to zero , and solving gives the resulting equations:

Optimal order quantity *Q = 02

h

DC

C (7.4)

Minimum total variable inventory cost *02 hTC DC C= (7.5)


As shown in the figure, the two cost components and their total have been plotted. It is seen that the replenishment or the ordering cost per unit time decreases as Q increases whereas the carrying charges tends to increase with the increase in the size of order. Hence the two opposing costs, one encourages increase in the order and the other discourages need a trade off in inventory management. The sum of the two costs is a u-shaped function whose minimum is at a value of Q*, which appears to be at the point where inventory carrying and ordering costs are equal. This is the Economic Order Quantity also known as the Wilson Formula. The total cost curve is quite shallow in the neighbourhood of EOQ which indicates that reasonable sized deviations from the EOQ will have little impact on the total relevant costs incurred. [14] 4.1 Formulation of Single Item Inventory Model under Fuzzy Environment Let the annual demand be TFN shown as follows:

D = (d1, d2, d3) Unit cost of placing an order per year is also TFN as:

0C = (c01, c02, c03)

The fuzzy economic order quantity using the signed distance [16] for defuzzification is:

01 1 02 2 03 32

2 h

c d c d c dQ

C

+ += (7.6)

The fuzzy total annual cost after further calculations is given by:


01 1 02 2 03 312

4 2 2 2h h hc d Qc c d Qc c d Qc

TCQ Q Q

= + + + + +

(7.7)

4.2 Formulation of Single Item Inventory Model under Intuitionistic Fuzzy Environment In this section, we propose the intuitionistic fuzzy inventory model by changing the crisp parameters such as ordering cost and demand from the previous model in intuitionistic sets using triangular intuitionistic fuzzy numbers (TIFN's). Let the annual demand be TIFN shown as follows: Y�=(d1,d2,d3)(d1',d2,d3') Unit cost of placing an order per year is also TIFN as: ZQ...=(c01,c02,c03) (c01',c02,c03') Annual holding cost of inventory per unit is constant which is denoted as Ch Thus the Total Inventory Cost in the Intuitionistic environment is given by: [Z.... = ZQ... ⊗ ]�̂ ⨁ Z` ⨂ �̂ (7.8)

Optimal number of orders (N*) to be placed in the given time period N*=Y�/Q* (7.9) The cycle time i.e. the optimal interval between the successive orders is given by t*= Q*/Y� (7.10)

4.3 Optimization and Solution With the above symbols and operations being defined we derive the total cost as: [Z.... = [cd�e�^ + ^cf� , cd�g�^ +

^cf� ,cd� e�^ + ^cf� ][cd�′g�′^ + ^cf� , cd�g�^ +

^cf� , id�′e�′^ + ^cf� ] (7.11)

Defuzzifying the total cost in equation (2.6) based on accuracy function defined by [14] we have:

01 1 02 2 03 3 01 1 02 2 03 31( ) ( ) 2( ) ( ) ( ) 2( ) ( )

8 2 2 2 2 2 2h h h h h hC d QC C d QC C d QC C d QC C d QC C d QC

P QQ Q Q Q Q Q

/ / / / = + + + + + + + + + + +

(7.12)

Computation of *Q at which ( )P Q is minimum:

( )P Q is minimum when( )

0P Q

Q

∂ =∂


And where

2

2

( )0,

P Q

Q

δδ

⟩

Now ( )

0,P Q

givesQ

δδ

= the economic order quantity Q* as

* 01 1 02 2 03 3 01 1 02 2 03 3( 2 ) ( 2 )

4 h

c d c d c d c d c d c dQ

C

/ / / /+ + + + += (7.13)

Also at *Q Q= ɶ we have 2

2

( )0,

P Q

Q

δδ

⟩

This shows that ( )P Q is minimum at Q=Q*. And from (2.7) we have the minimum total cost as:

* * * * * ** 01 1 02 2 03 3 01 1 02 2 03 3

* * * * * *

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

8 2 2 2 2 2 2h h h h h hC d Q C C d Q C C d Q C C d Q C C d Q C C d Q C

P QQ Q Q Q Q Q

/ / / / = + + + + + + + + + + +

(7.14)

4.4 Algorithm for Finding Intuitionistic Fuzzy Total Cost (IFTC) and Intuitionistic Fuzzy Optimal Order Quantity (IFOOQ) Step 1: Calculate first the order quantity and then the total cost for the crisp model as given in (2) for the crisp values of D, C0 and Ch. Step 2: Determine the optimal order quantity in intuitionistic sense using arithmetic operations on intuitionistic demand and ordering cost using triangular intuitionistic fuzzy numbers. Step 3: Use the Accuracy function for defuzzification of the total cost. Then the Intuitionistic fuzzy order quantity can be obtained by putting the first derivative of P(Q)equal to zero and where second derivative of P(Q) is greater than zero.

5. Case Studies Two case studies have been presented in this paper, which have been taken from [8]. These cases are first solved by the Crisp EOQ Model and then the same problem has been solved using the fuzzy and intuitionistic fuzzy approach (proposed in this paper). Thereafter, the results have been compared and the observations are given. Further, a sensitivity analysis has been carried out and accordingly, the variation in the results has been observed, in order to test the robustness of the fuzzy intuitionistic approach. 1. The Production department of a company requires around 3,600 kg of raw material for

manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs 36 and the cost of carrying inventory is 25 per cent of the investment in the inventories. The price is Rs 10 per kg. Help the purchase manager to determine an ordering policy for raw material and the minimum yearly variable inventory cost.

Int. J. Pure Appl. Sci. Technol

Solution: CRISP MODEL: Given that, D = 3600 kg per year Co=Rs 36 per order and Ch= 25 per cent of the investment=Rs 10*0.25= Rs 2.50 per kg

Comparison

Parameters CrispEOQ(Q) 321Variable total cost(TC) 804.98Cycle Time (t) 0.08Number of orders(N) 11.180

Table 2: Sensitivity Analysis

Sl No

Carrying Cost(Ch

)

For Fuzzy Set D=(3200,3600,4000)

C= (30,36,41)

CH Q TC

1 1.75 385.15 6732 2.00 360.27 660.593 2.25 339.62 700.664 2.50 322.24 738.475 2.75 307.24 774.666 3.00 294.16 809.067 3.25 282.64 841.09

Figure 2: Sensitivity Analysis

0

200

400

600

800

1000

Q

hnol., 22(2) (2014), 25-35

of the investment=Rs 10*0.25= Rs 2.50 per kg per year

Comparison among Crisp, Fuzzy and Intuitionistic model

Crisp Fuzzy Intuitionistic321 322 323.60804.98 805.60 809.0060.08 0.089 0.09 11.180 11.17 11.027

Sensitivity Analysis for Fuzzy and Intuitionistic Fuzzy Set

=(3200,3600,4000) For Intuitionistic Fuzzy Set Dɶ =(3200,3600,4000)(3000,3600,4200)

0Cɶ = (30,36,41)(28,36,44)

TC *Q 673 386.7 660.59 361.80 700.66 341.10 738.47 323.60 774.66 308.54 809.06 295.40 841.09 283.82

Sensitivity Analysis for Fuzzy and Intuitionistic Fuzzy Set

TCQ*

P(Q*)

33

model

Intuitionistic 323.60 809.006

11.027

Intuitionistic Fuzzy Set

Intuitionistic Fuzzy Set

=(3200,3600,4000)(3000,3600,4200)

*( )P Q 624.8 678.37 767.47 809.006 735.84 768.18 924.76

for Fuzzy and Intuitionistic Fuzzy Set

Series1

Series2

Series3

Series4

Series5

Series6

Series7


Observations from the sensitivity analysis reveal that 1. As carrying cost increases the economic order quantity decreases i.e. economic order quantity

is inversely proportional to the holding cost. 2. However the optimal total cost increases with increase in carrying cost, even though the

economic order quantity decreases. 2. A company works 50 weeks in a year for a certain part, included in the assembly of several

parts, there is an annual demand of 10,000 units. This part can be obtained from either an outside supplier or a subsidiary company. The following data relating to the part are given:

From outside supplier(Rs) Subsidiary company(Rs.) Purchase price/unit 12 13 Cost of placing an order 10 10 Cost of receiving an order 20 15 Storage and all carrying costs including capital per unit per annum

2 2

Which purchasing quantity and from which source would you recommend the company to buy the required product? What would be the minimum total cost? Solution: Comparison among crisp, fuzzy and Intuitionistic Model for outside suppliers and subsidiary company (ignoring the fixed cost)

Outside Suppliers

Parameters Crisp Fuzzy Intuitionistic Fuzzy

C0 30 (24,30,35) (24,30,35)(22,30,38) D 10,000 (9600,10000,10400) (9600,10000,10400)(9400,10000,10600)

Ch 2 2 2 EOQ(Q) 547.72 546.44 548.18 TC Rs 774.5 Rs 845 Rs 1096.36

Subsidiary Company

Parameters Crisp Fuzzy Intuitionistic Fuzzy

C0 25 (19,25,30) (19,25,30)(17,25,33) D 10,000 (9600,10000,10400) (9600,10000,10400)(9400,10000,10600) Ch 2 2 2 EOQ(Q) 500 498.59 459.08 TC Rs 1000 Rs 997.2 Rs 918

Solving the problem in crisp, fuzzy and intuitionistic environment we see that a linear relation exist between the economic order quantity and the total variable cost. The above analysis reveals that the company should order for 459 units incurring a cost of Rs 918 from the subsidiary company.


6. Conclusion In this paper we study an inventory problem in intuitionistic environment. Analysis of the problem under crisp and intuitionistic environment has shown that the optimal order quantity obtained under the intuitionistic environment is closer to the crisp and fuzzy Economic Order Quantity. Therefore, when membership function is not always accurately defined due to the lack of personal error, an intuitionistic fuzzy set may help in solving the problem. A comprehensive sensitivity analysis has been performed to illustrate the impact of carrying cost on economic ordering policy .It also establishes the fact that reasonable sized deviations from EOQ has little impact on the relevant cost incurred. It is therefore not worth making accurate estimates of the input parameters. Hence Intuitionistic fuzzy approaches are suitable to use when crude estimates should suffice.

References [1] A. Nagoorgani and K. Ponnalagu, A new approach on solving intuitionistic fuzzy linear

programming problem, Applied Mathematical Sciences, 6(70) (2012), 3467-3474. [2] C.H. Hsieh, Optimization of fuzzy inventory models under fuzzy demand & fuzzy lead time,

Tamsui Oxford Journal of Management Sciences, 20(2) (2004), 21-36. [3] D. Dutta and P. Kumar, Fuzzy inventory model without shortages using trapezoidal fuzzy

number with sensitivity analysis, IOSR Journal of Mathematics (IOSR-JM), 4(3) (2012), 32-37.

[4] F. Harris, Operations and Cost (Factory Management Series), Chicago: A.W. Shaw Co., (1915), 48-52.

[5] G. Hadley and T.M. Whitin, Analysis of Inventory Systems, Prentice–Hall, Englewoodclipps, NJ, 1963.

[6] G.S. Mahapatra and T.K. Roy, Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations, World Academy of Science, Engineering and Technology, 50(2009), 574-581.

[7] J.K. Arrow, F. Harris and J. Marschak, Optimal inventory policy, Econometrica, XIX(1951), 250-272.

[8] J.K. Sharma, Operations Research (Fourth Edition), Macmillan Publishers, 2009. [9] J. Kacpryzk and P. Staniewski, Long term inventory policy-making through fuzzy decision-

making models, Fuzzy Sets and Systems, 8(June) (1982), 117-132. [10] K.S. Park, Fuzzy set theoretical interpretation of economic order quantity, IEEE Trans.

Systems Man. Cybernet SMC, 17(1987), 1082-1084. [11] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), 87-96. [12] P.K. De and A. Rawat, A fuzzy inventory model without shortages using triangular fuzzy

number, Fuzzy Information & Engineering, 1(2011), 59-68. [13] L.A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-356. [14] R. Peterson and E.A. Silver, Decision Systems for Inventory Management and Production

Planning, New York: John Wiley, 1985. [15] R. Wilson, A scientific routine for stock control, Harvard Business Review, 13(1934), 116-

128. [16] S. Bai and Y. Li, Study of inventory management based on triangular fuzzy numbers theory,

IEEE, 978-1-4244-2013-1/08/ 2008. [17] S. Banerjee and T.K. Roy, Solution of single and multi-objective stochastic inventory models

with fuzzy cost components by intuitionistic fuzzy optimization technique, Advances in Operations Research, Article ID 765278(2010), 19 pages.

[18] S. Chakraborty, M. Pal and P.K. Nayak, Intuitionistic fuzzy optimization technique for the solution of an EOQ model, Fifteenth International Conference on IFS, Burgas, NIFS, 17(2) (2011), 52-64.

[19] T.M. Whitin, The Theory of Inventory Management, Princeton, N.J.: Princeton University Press, 1953.

an intuitionistic approach to an inventory model without...

Documents