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15
A FUZZY INNOVATIVE ORDERING PLAN USING STOCK DEPENDENT HOLDING COST OF INSPECTION WITH SHORTAGES IN TIME RELIABILITY DEMAND USING TFN
Sivan V1*, Thirugnanasambandam K2, Sivasankar N3, Sanidari4
department of Mathematics, Saveetha Engineering College Chennai- 602105, India.
*[email protected] 2Department of Mathematics,Muthurangam Govt Arts College Vellore- 632002,India. [email protected] 3Department of Economics, Pondicherry University, Puducherry 605014- India.
[email protected] 4Department of Mathematical Sciences, Kaduna State University, Kaduna, Nigeria.
Abstract
The considerations in this paper are, the demand is consistent with time deterioration, the holding cost is dependent based on the quantity of stock available in the system, and the ordering cost is linear and time-dependent. This system should be considered in terms offuzziness. It is assumed that the shortages are permitted partially, the order is inspected, defective items are identified, by using penalty cost, the defective items should be minimized. Under the classical model and fuzzy environment, the mathematical equation is arrived at to find the optimal solution of total relevant cost with optimal order quantity and time using triangular fuzzy numbers. Defuzzification has been accomplished through the use of the signed distance method of integration. The solutions have been arrived and the model numerical problem of three levels of values (lower, medium and upper) in parametric changes has been verified. Using Sensitivity analysis, the solution is used to validate the changes in different parameter values of the system. To demonstrate the convexity of the TRC function over time, it has used a three-dimensional mesh graph.
Keywords: Ordering plan, Triangular Fuzzy Numbers, Stock depending holding cost, Varying order Cost.
1. Introduction
In any type of business, maintenance of the stock plays crucial role. The stock should be with effective quality (with freshness); it means deterioration should be very less. In this paper, the demand is estimated so that the deterioration is reliable with time period. If the demand increases automatically the deterioration is to be minimized with time. The order placement depends on the demand of the system. Suppose the quantity supplied as part of order quantity is less than the demand, it will lead to shortage. The ordered stock should be in good condition if there is any defective product or service supplied, the firm will incur loss and also the goodwill of the customer. In order to meet the shortages, the lost sale cost is added, the items should be inspected. Sometimes when items are supplied by delay, then the penalty cost will also be added in the
process. Here the ordering cost is not fixed as it is linearly time-dependent. Also, the holding cost is not fixed for the entire time period, whenever the stock reduces, the holding cost also reduces.
Abhishek et al., [1] developed a paper in a fuzzy economic production quantity model deteriorating production depends proportional to population, selling price and advertisement, in this paper he explain clearly how demand is work with population and selling price with advertisement. Dutta .D. et al., [2] presented a article in Optimal Inventory Shortages Fuzziness in Demand,. the model is developed in crisp environment. After that it is convert in to fuzzy environment. All the functions convert in to (TFN) and the fuzzy trapezoidal number. In order to Defuzzification, the SDM used. The EOQ, optimal total cost derived and in both environment. Magfura Pervin et al., [10] explained the combined vendor buyer of quadratic demand inspection preservation technology applied the vendor applied the (PT) to reduce deterioration cost using this technique to reduce the total cost and fount the optimal total cost. Pavan Kumar [3] deals with Optimal inventory model with shortages applied fuzzy environment. The shortages were allowed partially backlogged method were applied for manage the stock for genuine customer is was very useful for manufacturer. Sankar Kumar Roy et al., [4] established a model of Imperfection and inspection and varying demand in trade credit using inspection policy easily found the defect of the items and supply to the customer and get the goodwill of the customer. This model is very useful for improve the relationship of customer and supplier. Sivan.V et al., [5] formulated a model of retailer supplier of price dependent demand; To demand will improve automatically when we reduce the cost of the item. Srabani Shee et al.,[6] proposed a model in Fuzzy Supply Chain Varying Holding Cost of supplier and retailer , supplier will more benefit then the retailer since retailer will spend more amount for holding the items so the holding cost of retailer is more than the supplier all the calculation doing by fuzzy and crisp environment. Thirugnanasambandam. et al.,[7] developed model of estimation of EOQ Model negative exponential Demand of linear term. The drugs are maintained stock and problem formed using negative exponential demand so day by day the demand is diminishing. Two types of demand functions formulated and calculated the optimal total cost and more quantity. Tripathi [8] investigated the innovative stock sensitive demand of EOQ for deterioration by means of inconsistent here the newly found stock dependent holding cost using the model holding cost to minimized and linear and nonlinear holding cost considered and verified with parametric changes. Sudip Adak et al., [9] established inventory model reliability dependent partial backordering in fuzziness. Here the deterioration is minimized using the demand if demand is increases automatically the deterioration is reduced and partial shortages are balanced with backlogging here also the results were found and compared with crisp and fuzzy environment.
The present paper has eleven sections. Basic definitions and fuzzy preliminaries are followed by introduction provided. In Section 3, notations and assumptions are introduced. The problems are described and formulated in the fourth section. In Section 5, comes out with numerical solutions and sample problems. The Sensitivity Analysis, Graphical representation and the impact of parametric changes are portrayed in the sixth section. In the section seven represents detailed observation. In Section 8, the Inventory model in fuzzy environment is formulated. Some numerical problems are using Triangular Fuzzy Numbers with different data sets are solved. Illustrative examples are given in the Section nine. In Section 10, comparative studies of crisp and fuzzy optimal values are explored. Conclusions and further developments are distinguished in the final section.
2. Definitions and Fuzzy Preliminaries
Definition 1. Membership value : A fuzzy set U is a universe of discourse. The following set of pairs is defined as X. U = {(x, ^q(x))/x € R}, where ^q(x) : X —> [0,1] is a mapping called membership value or degree of membership of x € R in the fuzzy set U.
Definition 2. Convex : A fuzzy set U of the universe if and only if the discourse X is Convex,
Vxi, x2, G R The following set of pairs is defined as X.
^g(px1 + (1 — p)x2)) > min [^g(x1), ^g(x2)], when 0 < p < 1.
Definition 3. Normal Fuzzy Set: A fuzzy set g of the universe X is referred to as a Normal Fuzzy Set, meaning that at least one exists x G R such that ^g(x) = 1.
Definition 4. Triangular Fuzzy Number (TFN): The Triangular Fuzzy Number g = [aF1, aF2, aF3] and is formed its continuous membership function ^g (x) : X —> [0,1] is,
m (x) = f (x)
x — aFi
aF2 — aFl'
aF3 — x
aF3 — aF2' 0,
for aFl < x < aF2;
for aF2 < x < aF3; Otherwise;
ftl(ji)
K2 ..........^Pk
K\ / j K
0
Figure 1: Triangular Fuzzy Number
Figure 2: Fuzzy number with cuts
Definition 5. Signed Distance Method : Signed Distance Method: Defuzzification of g can be
discovered using the Signed Distance Method. If g is a TFN then Sign distance from g to 0 is
described as:
1 r 1 ~
d(g,0) = 1 [3L(k), 3R(k),0] dk.
2 J0
3. Notations and Assumption
3.1. Notations
l1(t) The inventory level in the time period 0 < t < t1 I2(t) The inventory level in the time period t1 < t < T t1 The time stock reached to zero T The total cycle time
cFO Ordering cost depend of time dependent cpO Fuzzy Ordering cost depends of time dependent d(t) = ax Deterioration period a, A > 1 cF2 : Deterioration cost per unit time
9. Cf2 : Fuzzy deterioration cost per unit time
10. Cf3 : Holding cost per unit time
11. c F3 : Fuzzy holding cost per unit time
12. cF4 : Shortage cost per unit time
13. cF4 : Fuzzy shortage cost per unit time
14. cF5 : Inspection cost per unit time
15. c F5 : Fuzzy inspection cost per unit time
16. cF6 : Penalty cost per unit time
17. cF6 : Fuzzy penalty cost per unit time
18. Qf : The maximum order level in the time period (0 < t < t\)
19. TRC(t1, T) : The total relevant cost per cycle
20. TRC(t1, T) : The Fuzzified total relevant cost per cycle
21. pF : Unit of shortage cost (0 < pF < 1)
22. : Unit of lost sale cost (^F > 0)
23. 5f : Unit of penalty cost 5F > 0
24. A : Shape parameter A > 0
25. Dj : The total items deteriorated
3.2. Assumption
1. The demand function is written D (t) = pita + p2 for 0 < t < ti, ti < t < T for pi, p2 > 0
2. Deterioration per cycle DCF = ^x, a > 1, A > 1
3. Ordering cost cFO = r1t1 + r2, r1 > 0, r2 > 0
4. T is the complete cycle periods time horizon
5. t1 is the period of time when inventory level reduces to finish
6. Lead time is negligible.
7. Shortages partially allowed is pF, 0 < pF < 1 is backordered.
8. In some situation the demand may be considered high, in that period to maintain good relationship with customer and in to consider the inspection policy.
9. In the given cycle time, in the beginning of the process full inventory level is considered. 10. During the shortage period same demand is to be considered.
3.3. Decision Variables
• t1 : The time Period first level 0 < t < t1
• T : The time Period second level t1 < t < T
3.4. Objective Functions
1. TRC(ti, T) : Total relevant cost per cycle
2. ßF : Cost of preservation technology investment per unit time
3. Qf : The maximum order in the time period 0 < t < T
4. t1, T : Optimal time periods in 0 < t < t1, t1 < t < T
4. Problem Description and Mathematical Equation
Figure 3: Inventory level Vs Time.
4.1. Problem description.
Initially the inventory level is Qf, because of demand and deterioration the level of inventory is gradually reduced in time period t = t\, so the shortages in this inventory is taken partially.
4.2. Mathematical Equation.
The following formula were used to create the Mathematical model for this paper using differential equations. In shortage period also, the same demand function is maintained.
dJdr + ax 11 (t) = - (V1ta+P2), for o < t < t1,
(1)
dl1 (t) dt
- ß (p1ta + p2), for t1 < t < T,
(2)
Using the initial and the boundary conditions, let us find I1 (t), and I2(t). In I1(t) and I2(t) put
t = t1, t = T and I1 (t1) = 0, I2(T) = 0 and get the solution. In I1 (t) put t = 0 get order quantity Qf therefore Ii(0) = Qf.
Solution of 1 and 2 h (t) =
{P2(t1 — t)} + (t3 — t3) + ^ (ta+1 — ta+1) +
P1
a +1
2aA+1 + 6aA
X(ta+3—ta+3) + 2aA (t3—1112) + 12ak (t5—t211)
P2
+
P1
(a + 1)2aA
(ta+3 — ta+112) +
P1
(2a A+1 + 6aA )(2aA)
x(ta+5 — ta+312)
h (0)
Qf
(3)
{P2 (t1)} + 61 (t1) + -arh (ta+1)+ (^+3)
l2(t) = ^ ^ [(Ta+1 — ta+) + p2(T — t)]}
a +1
The ordering cost is given by,
DC = Cfo = r111 + T2, T1, T2 > 0
(4)
(5)
(6)
The total number of pieces that becomes deteriorated throughout that period of interval 0 < t < t1 is formed by,
D1 = Q — fh D(t) dt 0
(t1) + 6? ('1) + arr (';+1) + (t1+3)—f(P1''+P2)
dt
£ (t3) +
P1
-t'
.a+3
6aA' 2aAr1+6aA 1 Therefore the deteriorating cost is formed by,
CF2{62 (f)+
P1
.a+3
2aA+1 + 6aA 1
The Holding cost (HC) during the interval [0, t1 ] is formed by,
ch
HC
C3/q 1 e + f [ 11 (t)]dt
CF3 e t1 + f
i.4
P2^ t2 + ^^ ^ + -P1-ta+2
2 Jt1 + i2 + an^
-ta+2 +
P1
2a +1 + 6a
(a + 3 A ta+4 ( P2 \ t6 (
P1
a +1
3(2aA+1 + 2aAJ \a + 4
a+4
( P1 \ (a+3 ) ta+6 \6aA(2aA+1 + 6aAJ \a + 6 ' 1
Shortage Cost is formed by, Sh.C =
CF4 Pf fT (P1ta + P2) dt ■'h
CF4 -a+Y) [Ta+1 — t1 a+1 ] + P2(T — t1)
(7)
(8)
(9)
x
The lost sale cost is formed by,
Lo.SC = (1 - Pf)iT (pita + p2) dt
Jti
= ^ (1 - ) { (TH) [r+1 - tl8+1 ] + P2(T - tl) }
+
The Inspection cost in the interval [0, t1 ] is given by, In.C = cF5 Qf
(10)
c^r J W + p2 t3 + p1 t8+1 + cF5< p2h + TTTt1 + . -n t1 +
P1
.8 + 3
6aA 1 (a + 1) 1 T2«A+1 + 6aA 1 The penalty cost (PC) during the interval [0, t1 ] is formed by,
r t1
Pn.C = Cf6&f (P1 ta + P2) dt
= CF6 'F ( JaTT) '1+' + P2 '1)
The Total Average Relevant Cost,
TRC(t1, T) = TI Ordering Cost + Deteriorating Cost + Holding Cost +
Shortage Cost + Lost Cost + Inspection cost + Penalty cost
(11)
(12)
1 T
n'1 + T2 + CF2
(P2_ [6aA
('1) +
P1
.8 + 3
2aA+1 + 6aA 1
+CF3 ^'1 + f P1
P2 \ 2 + ( P2 \ 'l + P1 '8+2
YJ '1 +{8^) 2 + 8+2 '1
+
8 + 3 ^ ±8+1
'1 _
28A+1 + 68V V 8 + l) 1
P1 \(8 + ^ »8+1
P2 7282A
3(28A+1 + 28A)J \8 + 1) 1 P1
8 + 3 ^ '8 + 6
68A(28A+1 + 68A)J \ 8 + 6) 1 +CF1P f{ 10+Y)[Ta+i - '18+1 ] + P2(T - '1)}
+ ^(1 - PF) { ja+j) [T8+1 - '18+1] + P2(T - '1) }
+c^r!P„' + P2 '3 + V1 '8+1 + V1 '8+3! +CF5{ P2'1 + 6TA '1 +(8+1) 'a + 2aa+1 + 6aA 'a ]
+CF6'f{ (8+1) '8+1 + P2'1
For the convenience let us do this substitution,
'
91 =
94
P2 6aA
P2 8aA
92 =
95
P1 9 = P2
2aA +1 + 6aA 93 2
P2
97 = 24aA
910
P1
6aA (2aA +1 + 6aA) \a + 6
a + 3
911
P1 a + 2
P2 72a2A
P1
96
P1
a + 3
2aA+1 + 6aA V a + 4
P1 / a + 1
3 (2aA +1 + 6aA) Va + 4
(a +1)
Substitute the above 91, 92,94,95,96,97,98,99,910,911 in equation (13) The Total Relevant Cost will become
TRC(t1, T)
r*111 + r2 + CF2 ( 9111 + 92t1+3 )
+CF3 ( et1 + f 9312 + 91 -2 + 95t1+2 + 96ta+4 — 98t6 — 99t1+4
— 910ta+^ + CF4PF { 911 [Ta+1 — t1 a+1 ] + P2(T — t1)} +Uf (1 — P F )[ 911 [Ta+1 — t1 a+1 ] + P2 (T — t1)
+cf^ P2t1+9111+911 ta+1+92ta+3}
+CF6Sf (911 ta+1 + P2t^
(14)
TRC(t1, T)
T
r111 + r2 + CF2 ( 9111 + 92t1+3 )
+CF3 et1 + f
9311 + 914 + 95t1+2 + 96 ta+4 — 98t1 — 99t1+4
910ta+^ + [CF4PF + ^(1 — P)] {911 [Ta+1 — t1 a+1]
+P2(T—11)} +CF^ P2t1+9113+9nta+1+92 ta+3}
+CF6Sf (911 ta+1 + P21^ (15)
5. Numerical Solutions and Sample Problems
5.1. Numerical Solutions of Fuzzy Innovative Ordering Plan
For the solution purpose, MATLAB R2018b and Excel solver are used to find all the optimal solutions, all the graphs and convex mesh using MATLAB R2018b software.
To find the solution of the equation (15) using the below necessary and sufficient condition.
The necessary condition for the least value of TRC (t1, T) are,
d (TRC (t1, T)) dt1
1
and
д (TRC (t1, T))
дТ
0
The Sufficient condition for optimal TRC (t1, T), t1 > 0, T > 0.
aw > о dt2
> 0
and
d2 (TRC) дТ2
>0
д2 (TRC)
d2(TRC) д^дТ
a^çrRC)
дТ2
>0
дТдЬ
Therefore the optimal solutions of t*, T*, QF and TRC* are found and given in the table.
Example 1. Let's take the input: p1= 4.2, p2 = 9.5 , a =1.2, e = 1.8,f= 0.2, вF = 0.004, 5F = 3.85, =11, л = 3.99, cF2 =Rs.15, cF3 = Rs.10, cF4 =Rs.5 , cF5 =Rs.0.5, cF6 =Rs.1.03, rF1 = Rs.9.8, rF2 = Rs.40.
The Optimal Solutions are QF = 9.58728, t* = 0.82247,T* =0.90941 & TRC* = Rs.145.406.
Example 2. Let's take the input: p1= 4.2, p2 = 9.99, a =1.2 , e = 1.8, f = 0.2, = 0.004, 5F = 3.85, pF=11,л = 3.99, cF2 =Rs.15, cF3 = Rs.10, cF4 =Rs.5, cF5 =Rs.0.5, cF6 =Rs.1.03, rF1 =Rs. 9.8, rF2 = Rs.40.
The Optimal Solutions are QF = 10.1671, t* = 0.83244, T* = 0.86903 & TRC* = Rs.148.603
Example 3. Let's take the input: p1 = 4.2, p2 = 10.25, a = 1.2, e = 1.8, f = 0.2, eF = 0.004, 5F = 3.85, pF = 11, Л = 3.99, cF2 =Rs.15, cF3 = Rs.10, cF4 =Rs.5, cF5 =Rs. 0.5, cF6 =Rs.1.03, rF1 = Rs.9.8, rF2 = Rs.40.
The Optimal Solutions are QF = 10.4737, t* = 0.83734, T* = 0.84657 & TRC* = Rs.150.252.
5.3. Convexity of the optimal function
Convexity of Optimal total cost TRC(t1, T) versus t1 and T using Matlab R2018b are shown graphically Figure (4) and Figure (6).
Table 1: Тке optimal solution using crisp
i*__Т* QF TRC*(t1, ^(Rs.)
0.83244 0.86903 10.1671 148.603
5.2. Sample Problems
Figure 4: Convexity GraPh of t1, T with TRC(t1, T) Figure 5: Convexity GraPh of t1, T with TRC(t1, T)
6. Sensitivity Analysis and Graphical representation 6.1. Sensitivity analysis of Fuzzy Innovative Economic Order Quantity
Table 2: Sensitivity analysis of time reliability demand (Parameters P1, P2, a, e and f)
Parameter Changed values t* t1 t* QF TRC* (Rs.)
P1 4.1580 0.8318 0.8698 10.1431 148.2532
4.2000 0.8324 0.8647 10.1671 148.3721
4.2420 0.8331 0.8598 10.1912 148.4893
4.2840 0.8337 0.8548 10.2152 148.6049
P2 10.0879 0.8343 0.8563 10.2826 148.9947
10.1898 0.8362 0.8474 10.4028 149.6376
10.2917 0.8381 0.8384 10.5228 150.2749
10.4975 0.8418 0.8199 10.765 151.5447
a 1.2120 0.8459 0.8566 10.3447 147.8646
1.2240 0.8595 0.8485 10.5236 147.3534
1.2362 0.8733 0.8403 10.7076 146.8281
1.2485 0.8872 0.8321 10.893 146.2988
e 1.6940 0.8422 0.8454 10.32 147.3337
1.7464 0.8324 0.8647 10.1671 148.3721
1.7820 0.8324 0.8647 10.1671 147.8557
1.8000 0.8341 0.8615 10.1932 148.1983
f 0.18822 0.846 0.8532 10.3799 147.7543
0.19404 0.8393 0.859 10.2739 148.0635
0.19800 0.8347 0.8628 10.2028 148.2694
0.20000 0.8324 0.8647 10.1671 148.3721
Table 3: Sensitivity analysis of time reliability demand (Parameters Pp, Sp, pp, \ and cp2)
Parameter Changed values t* t1 t* Qp TRC* (Rs.)
Pp 0.00202 0.8339 0.8619 10.1902 148.3776
0.00288 0.8333 0.8631 10.1801 148.3753
0.00360 0.8327 0.8642 10.1718 148.3732
0.00400 0.8324 0.8647 10.1671 148.3721
Pp 3.7357 0.8469 0.8397 10.3946 147.0331
3.7734 0.8422 0.8481 10.3197 147.4823
3.8115 0.8373 0.8565 10.2438 147.9286
3.8500 0.8324 0.8647 10.1671 148.3721
pp 11.1100 0.8459 0.8377 10.3787 148.3928
11.3322 0.8592 0.8098 10.5892 148.3414
11.3333 0.8724 0.7807 10.7987 148.2116
11.4433 0.8853 0.7504 11.0072 147.9958
A 4.02990 0.8349 0.8633 10.2017 148.2932
4.07020 0.8374 0.8618 10.2368 148.2132
4.11090 0.84 0.8603 10.2723 148.1320
4.15201 0.8426 0.8587 10.3082 148.0497
cp2 14.5545 0.8419 0.8591 10.3146 148.0689
14.7015 0.8387 0.861 10.2652 148.1704
14.8500 0.8356 0.8629 10.2161 148.2714
15.0000 0.8324 0.8647 10.1671 148.3721
Table 4: Sensitivity analysis of time reliability demand (Parameters cp3, cp4 cp5, cp^, rpi and rp2)
Parameter Changed values t* t1 t* Qp TRC* (Rs.)
cp3 9.7030 0.8442 0.8492 10.3512 147.5376
9.8010 0.8403 0.8544 10.2902 147.8166
9.9000 0.8364 0.8596 10.2289 148.0947
10.0000 0.8324 0.8647 10.1671 148.3721
cp4 5.0500 0.8325 0.8647 10.1675 148.3722
5.1005 0.8325 0.8646 10.1679 148.3723
5.1515 0.8325 0.8646 10.1683 148.3724
5.2030 0.8325 0.8645 10.1687 148.3725
cp5 0.4851 0.8346 0.8615 10.2007 148.1968
0.4901 0.8339 0.8626 10.1896 148.2548
0.4950 0.8332 0.8636 10.1784 148.3132
0.5000 0.8324 0.8647 10.1671 148.3721
cp6 0.9994 0.8498 0.8352 10.4391 146.7880
1.0095 0.845 0.8436 10.3642 147.2414
1.0197 0.8373 0.8565 10.2438 147.9286
1.0300 0.8353 0.8604 10.2117 148.1393
rp1 9.5089 0.8351 0.8595 10.2093 148.0905
9.6050 0.8342 0.8612 10.1954 148.1837
9.7020 0.8333 0.863 10.1813 148.2776
9.8000 0.8324 0.8647 10.1671 148.3721
rp2 39.4020 0.8324 0.8561 10.1671 147.9095
39.8000 0.8324 0.8647 10.1671 148.3721
40.1980 0.8324 0.8733 10.1671 148.8301
40.6000 0.8324 0.8818 10.1671 149.2881
6.2. The graphical representation using Matlab 2018b
Figure 6: The imPact of a is comPared with Figure 7: The imPact of A is comPared with TRC(t1, T) TRC(t1, T)
Figure 8: The imPact of ^f (LSC) is comPared with Figure 9: The imPact of Deteriorating cost is comPared TRC(t1, T) with TRC(t1, T)
Figure 10: The impact of Holding cost is compared Figure 11: The impact of Shortage cost is compared
with TRC(ti, T)
with TRC(t1, T)
Figure 12: The impact of Inspection cost is compared Figure 13: The impact ofPanelty cost is compared with with TRC(t1, T) TRC(t1, T)
7. Observations using table values
Here the investigations are done by using tabular values, let us observe the following progress.
1. While p2 is raising, the following values ti, T, Qf and TRC are oscillating.
2. While the values of a, & A are raising, the value of t1 is raising, T is reducing, QF is mounting and TRC is gradually turning down.
3. During the augmentation of the following values, e, f, SF, cF2, cF3, cF5, cF6 and rF1, t1 is diminishing, T is growing, Qf is turning down and TRC is gradually raising.
4. During the mounting of the Cf4, the following value of ^is raising , T is growing, Qf turns up and TRC is gradually leading.
5. While the value of p1is raising, the following values ¿1 is raising, T is reducing, Qf is mounting and TRC is gradually raising
6. While the value of Pf is raising, ¿1 is reducing, T is raising, Qf and TRC are turning down
7. While the value of p1is raising, ¿1 is raising, T is reducing, Qf is mounting and TRC is gradually raising
8. While the value of rF2 is raising, the same values of t1 are repeated, T is raising, the same values of Qf are repeated and TRC is gradually raising.
8. The Proposed Inventory Model Produced in a Fuzzy Environment
Due to the decision making problem, sometimes the output will be uncertaint and vague, and so some new ideas can be applied to meet the difficulties in characterizing the vagueness and uncertainty. Let us apply the fuzzy environment using Triangular Fuzzy Numbers,
TRC(t1, T)
T
r1t1 + r2 + Cf2($111 + $2 ta+3)
+ CF3 i et1 + f
' t1 + $1 2 + $5 f?+2 + $6 f?+4 - $8 t1
>f?+4 - $1011+6
J + CF4PF { $11 [Ta+1 - t18+1 ] + P2 (T - t1 + № (1 - PF ) $11 [Ta+1 - t1 a+1 ] + P2 (T - t1) + CF5 j P2 t1 + $111
+ $11 ¿1+1 + $2 t1+3 j + CF6Sf ( $11 ¿1+1 + P211 For the convenience, let us do the following suitable substitution.
A1
t1 + !
a+3
(16)
A 2 = et1 + f
, t2 + $12 +.
t1+2 +
,t1+4 -
.<? -
ta+4 ,f1 -
a+6
A3 A4 A5
A6
Pf{$11 [Ta+1 - t1 a+1 ]+ P2 (T - i1)} P(1 - PF) [$11 [Ta+1 - t1 a+1 ] + P2 (T - t1)]
P211+$111+$11 ¿r1+$2 ta+3
Sf ($11 ta+1 + P2 ¿1)
1
In equation (16), substitute the above A1, A2, A3, A4, A5, A6
TRC (t1, T)
1 T
r111 + r2 + cp2 A1 +cp3 A2 + cp4 A3 + A4 + cp5 A5 + cp6 A6 The parameters and costs should be fuzzified using Triangular Fuzzy Number (TFN). rf1 = (rn, r12, r13) , rF2 = (r21, r22, r23),
cp2 = (cP2^ cP22, cP23, ) cp3 = (cP31, cP32, cp33 ^ cp4 = (cP41, cP42, cp43 ), cp5 = (cp51, cp52, cp53 ), cp6 = (cp61, cp62, cp63 )
TRC(t1, T)
( r P111 + r P2 ) + cp2 (01 t31 + 02 t\
j.a+3
+ cp3 (e t1 + f
.a+4 p
U — 010 t
■A+01-2+05 ta+2+06 ta+4 - 08 tf
TRC(t1, T)
} + cp4Pp { 011 [Ta+1 - t1 a+1] + p2 (T - t1 + pp (1 - Pp ) 011 [Ta+1 - t1 a+1] + p2 (T - t1) + cp51 p2 t1 + 011?
+ 011 ta1+1 + 02 ta+3} + cp6Sp ( 011 ta1+1 + p2t1
= T ((r11, r12, r13)t1 + (r21, r22, r23)) + j (cp21, cp22, cp23 ) X(01 t3 + 02 ta+3) +(cp31, cp32, cp33 )( et1 + f
. . . . . t1 + 01^ +05 ta+2+06 ta+4 - 0811 - 09 ta+4 - 010 ta+6}
+ (cp41, cp42, cp43 )Pp{ 011 [Ta+1 - t1 a+1] + p2 (T - t1)} + p(1 - p) 011 [Ta+1 - t1 a+1] + p2 (T - t1) + (cp51, cp52,cp53 )
X j p2 t1 + 0111 + 011 ta+1 + 02 ta+^ + (cp61, cp62, cp63 ) Sp
x( 011 ta+1+p211
( Up, Vp, Wp)
where,
Up
(rn) t1 + r21 + cp2^ 0113 + 02 ta+3} + cp31 e t1 + f
7 M
■ t1 + 01 ^
+05 ta+2+06 ta+4 - 0811 - 0911+4 - 010 ta+6
+
cp41 Pp + pp (1 - Pp ) X j 011 [Ta+1 - t1 a+1 ] + p2 (T - t1 ) } + cp51 j p2 t1 + 0113 + 011 ta+1
(17)
(18)
(19)
+$2 ta+3}+cF61 Sf ($11 ta+1+P2 ¿1)
Vf
(r12 )t1 + r22 + CF22 { $1 if + $2 ¿1+^ + CF32 e ¿1 + f
+ $1"2
+ $5 ¿1+2 + $6 ¿1+4 - $8 ¿1 - $9 f?+4 - $10 f?+6
+
CF42 PF + ^F (1 - PF ) x | $11 [Ta+1 - ¿1 a+1] + P2 (T - ¿1)} + CF5^ P2 ¿1 + $1 ¿3 + $11 f1+1 + $2 fa+3} + CF62SF($11 ¿r1 + P2f1)
WF
(r13)f1 + ^23 + {CF23($1 f? + $2 ¿a+3^ + CF33 e ¿1 + f
■¿1 + $1-2
+$5 i1+2+$6 ¿a+4 - $8 ¿1 - $9 ¿a+4 - $10 f1+6
+
CF43 Pf + ^F (1 - PF ) x{ $11 [Ta+1 - ¿1 a+1 ]+ p2 (T - ¿1)} + CFKjj P2 ¿1 + $1f3 + $11 fa+1
+$2 f1+M + CF63 Sf ($11 ¿a+1 + P2f1)
a+1
The K-cuts UL(k) & UR(k) of Triangular Fuzzy Numbers. TRC(i1, T) are given by
Ul (K )
uf + (vf - uf)k
(r11 )i1 + r21 + CF21 A1 + CF31 A2 + CF41 A3 + A4
+CF51 A5 + CF61 A6 + j (f21 - rn)i1 + (f22 - ^1) + (Cf22 - CF21 ) A1 + (CF32 - CF31) A2 + (CF42 - CF41 ) A3 + A4 + (CF52 - CF51 ) A5
ur (k)
+ (CF62 - CF61 ) A6 W
wf - (wf - vf)k
(r13)*1 + r23 + CF23 A1 + CF33 A2 + Cf43 A3 + A4
+CF53 A5 + CF63 A6 + j (r13 - f12)i1 + (f23 - r23) + (CF23 - CF23 ) A1 + (CF33 - CF33) A2 + (CF43 - ^F43 ) A3 + A4 + (CF53 - CF53 )A5
+ (CF63 - CF62 )A6 \k
(20)
(21)
(22)
(23)
By apply the Signed Distance Method, the defuzzified value of average TRC, using the fuzzy number
TRC(ti, T)
{ulk + UrK} dK
1
4T
(rii + 2ri2 + ri3)tl + (r21 + 2r22 + ^23) + (CF2i + 2CF22 +CF23)Ai + (cF3i + 2CF32 + CF33)A2 + [cf4i + 2Cf42 + CF43] A3 +4A4 + (CF5i + 2CF52 + CF53 ) A5 + (CF6i + 2CF62 +CF63 ) A6
(25)
9. Solutions and numerical problems using triangular fuzzy numbers of
DIFFERENT DaTa
9.i. Solutions using triangular fuzzy numbers
For the solution purpose of equation (i9), MATLAB R20i8b and Excel 20i0 solver are used to find all the optimal solutions. For optimization let us do the following:
The necessary condition for the least value of TRC (ti, T) are,
d(TRC (ti,T) = 0 and d(TRCiti,T) = 0 dti dT
The sufficient condition for optimal TRC (ti, T), ti > 0, T > 0.
d2(TRC) d2 (TRC) dti dti dT
d2(TRC) d2 (TRC)
>0
dT dti dT2
Therefore the optimal fuzzy solutions of t*, T*, QF and TRC* are found and given in the table.
Table 5: Optimal solution using fuzzy Numbers
i
ti * t* QF TRC(ti, T)* (Rs.)
0.5452652i 0.6455852 20.4035902 i35.998429
9.2. Sample problems using triangular fuzzy numbers
Example 4. Let's take the input: px = 8.25, p2 = 31.75, a = 1.25, e = 1.825, f = 0.385, £F = 0.0044, SF = 4.2, pF = 13.75, A = 4.25, cF2 = (8,11.5,15), cF3 = (2,2.5,3), c^ = (1.25,2.125,3), cF5 = (0.2,0.3,0.4), cf6 = (0.22,0.33,0.44), rF1 = (8.55,10.525,12.5), rF1 = (20.5,21.5,22.5). The optimal solutions are Q*F = 19.6661, t** =0.57313, T* =0.70023, & TRC* = Rs.127.092.
Example 5. Let's take the input: px = 8.25, p2 = 34.75, a = 1.2, e = 1.825, f = 0.385, = 0.0044, S* = 4.2, p* = 13.75, A = 4.25, cF2 = (8,11.5,15), cF3 = (2,2.5,3), c^ = (1.25,2.125,3),
cF5 = (0.2,0.3,0.4), cF6 = (0.22,0.33,0.44), rF1 =(8.55,10.525,12.5 ),rF2 = (20.5,21.5,22.5 ).
The optimal solutions are QF = 20.4035902, t* =0.54526521, T* = 0.6455852, & TRC* =
Rs.135.998429.
Example 6. Let's take the input: p1 = 8.25, p2 = 36, a = 1.2, e = 1.825 , f = 0.385, ßF = 0.0044, SF = 4.2, Pf = 13.75, A = 4.25, cF2 = (8,11.5,15), cF3 = (2,2.5,3), cF4 = (1.25,2.125,3), cF5 = (0.2,0.3,0.4), cF6 = (0.22,0.33,0.44), rF1 = (8.55,10.525,12.5), rF2 = (20.5,21.5,22.5). The optimal solutions are QF = 21.2316, t* =0.54833, T* =0.60289, & TRC*= Rs.138.968.
10. Comparison of Crisp and Fuzzy Optimal Solutions
Table 6: Comparison of crisp and fuzzy solutions
t1 * t* QF TRC(t1, T)* (Rs.)
Crisp 0.83244 0.86903 10.1671 148.603
Fuzzy 0.54526521 0.6455852 20.4035902 135.998429
11. Conclusion & Extending investigation scope
In this study, an attempt is made to formulate an inventory model of innovative economic order with the quantity of items. The considerations in this paper are (i) the demand is consistent with time deterioration,(ii) the holding cost has been used as dependent on the amount of stock available in the system, and (iii) the ordering cost is linear and time-dependent. This system should be considered in terms of crisp and fuzziness. It is assumed that the shortages are permitted partially and the quantity ordered is inspected to reduce defective items. To use the penalty cost delay of supplying items should be minimized. Under the classical model and fuzzy environment, a mathematical equation is arrived. The optimal solution of total relevant cost with optimal order quantity and time using triangular fuzzy numbers has been found. Defuzzification has been accomplished through the use of the signed distance method of integration. The solutions have been arrived at and verified by using model with a few numerical problems of three levels of values (lower, medium, and upper) in parametric changes. Sensitivity analysis is used to validate the changes in different values of the system's parameters. To demonstrate the convexity of the total relevant cost function over time, a three-dimensional mesh graph has been used. This model can be modified and developed further by changing the demand into probabilistic, price, advertisement dependent etc.
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