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34
INVENTORY MODEL WITH EXPONENTIAL DETERIORATION AND SHORTAGES FOR SEVERAL LEVELS OF PRODUCTION
A. Lakshmana Rao1, S. Arun Kumar2 and K. P. S. Suryanarayana3
Department of Mathematics123, Aditya Institute of Technology and Management, Tekkali, India agatamudi111@gmail.com1, arunkumarsaripalli9788@gmail.com2, suryanarayana.kornu@gmail.com3
Abstract
The EPQ models are mathematical models which represent the inventory situation in a production or manufacturing system. In production and manufacturing units, the EPQ model is extremely significant and also be utilized for scheduling the optimal operating policies of market yards, warehouses, godowns, etc. In this research study, we provide inventory model of economic production for deteriorating commodities at multiple levels, in which various production stages are mentioned as well as deterioration rates follow exponential distributions. After a specific period of time, it is feasible to swap production rates from one to another, which is advantageous by starting with a production of low rate, an enormous amount of manufacturing articles is avoided at the outset, resulting in lower holding costs. Variation in output level allows for customer happiness as well as potential profit. The goal of this study is to determine the best production time solution so as to reduce total cost of the entire cycle. Finally, numerical illustrations and parameter sensitivity analyses have been used to validate proposed inventory system's results.
Keywords: EPQ, optimal operating policies, Exponential distribution, multiple levels of production, cycle time.
1. Introduction
Inventory system plays a leading role in many real applications at certain places such as production processes, manufacturing units, transportation, market yards, ware houses, assembly lines etc. One of the vital sections of operations research is inventory management, which is used for determining the optimal operating policies for inventory management and control. Inventory models provide the basic frame work for analyzing several production systems. The inventory models are broadly categorized into two groups namely, (i) Economic Order Quantity models (EOQ models) and (ii) Economic Production Models (EPQ models). The EPQ models are more common in production and manufacturing processes, warehouses, etc. Recently much emphasis was given for analyzing EPQ models for deterioration items. Deterioration is a natural phenomenon of several commodities over time. For commodities like glassware, hardware, and steel, deterioration can be quite minor at times, causing deterioration to be taken into account when determining economic lot sizes. In general, some commodities decay at a faster pace than others, such as medicine, gasoline, strawberries, fish, blood, and food grains, which must be taken into account when determining the size of a production lot.
Damage, decay, spoiling, evaporation, and obsolescence are all examples of deterioration. In modern years, the issue of decaying inventory has gotten a lot of attention. The majority of studies on deteriorating inventory assumed a constant rate of deterioration. In general the exponential delivery is commonly used to describe a product in stock that deteriorates over interval. The rate of deterioration rises with age, therefore the longer an object is left unused, the faster it will fail. Many commodities decay in real life due to their inherent nature, such as fruits, vegetables, food items, seafood, agricultural products, textiles, chemicals, medicines, electronic components, cement, fertilisers, oils, gas, and so on, which are held in inventory at various locations.
The first economic quantity model was developed by Harris [1]. The inventory model of a decaying item at the end of a scarcity period was studied by Wagner and Whitin [2]. As a result, deterioration functions come in a variety of shapes and sizes, including constant and time-dependent functions. We used the Exponential as the function of deterioration in our proposed model. Berrotoni [3] explored that the leakage failure of both dry batteries and ethical drugs life expectancy may be described as an exponential distribution. In some circumstances, the deterioration rate rises with time. The longer an object is left unused, the faster it deteriorates. This study prompted Covert and Philip [4] to create an inventory model for deteriorating items with varying rates. It was made use of two variables the Weibull distribution will deteriorate as a time distribution.
Balki and Benkherouf [5] also proposed a model as such but with a stock-dependent and time-varying demand rate across a finite time horizon. Chang [6] improved previous model by accounting for profit in the inventory system. Additionally, Begam et al [7] devised an instantaneous replacement policy. They used a three-parameter Weibull distribution to time based model inventory deterioration rate. Begam et al [8] re-examined the previous model, ignoring scarcity and assuming demand to be a linear function of price. Rubbani et al [9] proposed an integrated methodology for deteriorating item pricing and inventory control. For decaying products, Sivasankari and Panayappan [10] suggested a production inventory model that considers two different degrees of output. Cardenas-Barron et al [11] projected substitute heuristic algorithm for a multi-product EPQ (Economic Production Quantity) a vendor-buyer cohesive model with JIT view point and a budget constraint. Sarkar et al. [12], in his investigation on EPQ model with rework in a manufacturing system of single-stage with scheduled backorders, and produced 3 different inventory models for 3 different density functions of distribution like Triangular , Uniform, and Beta. Cardenas-Barron et al [13] determined the ideal replenishment lot size and dispatch strategy for an EPQ inventory model with multiple deliveries and rework. When using a multi-shipment policy, Taleizadeh et al [14] presented study work addresses the problem of determining price for sale, lot size of replenishment, and shipments quantity for a model of economic quantity with rework for defective goods. Karthikeyan and Viji [15] modified this model by using the Exponential distribution for deterioration.
Determining the ideal quantity of production boxes for different periods as an aim in order to reduce total inventory costs. Lately, Viji and Karthikeyan [16] have established an inventory model of economic production quantity for constantly deteriorating products that takes into account three levels of manufacturing. Researchers have established an economic production inventory model for many levels of production with exponential distribution deterioration, demand is time dependent and continuous, and multiple rates of production are examined in this research. The following is a breakdown of the paper's structure. The assumptions as well as notations are presented in Section 2. The third section is dedicated to mathematical modelling. Section 4 includes a numerical example as well as a sensitivity analysis. The paper comes to a conclude with Section 5.
A. Lakshmana Rao, S. Arun Kumar, K. P. S. Suryanarayana RT&A, No 3 (69)
INVENTORY MODEL FOR SEVERAL LEVELS OF PRODUCTION Volume 17, September 2022
2. Assumptions
For developing the model the following assumptions are made:
• Multiple production rates are taken into account.
• The demand rate is continuous and linear which is D(t) = a + pt (1)
• The production system has a limited time horizon.
• Shortages are permitted, as are entire backlogs.
• The exponential distribution governs the time it takes for an item to deteriorate, which is
f(t) = 0e-te, Q>0,t>0 (2)
• Consequently, the instantaneous rate of production is
h(t)=-££)- = 8,8>0 (3)
V ' l-F(t) V '
• Production rate (K), which is greater than demand rate(R). The following notions are used to developing this model.
K is the production rate in units @ unit time. R is the demand rate in units@ unit time. The holding cost @ unit of time is denoted by the C1 The shortages cost @ unit of time is denoted by C2 The ordering cost @ unit of time is denoted by C3 The production cost @unit of time is denoted by Cp S is the shortage level. Q is the optimum production quantity.
Qi, Q2, Q3 and Q4 are the maximum possible inventory level at time ti, t2, t3 and t4. T is the total cycle length.
3. Mathematical formulation of the model
The following is a description of Figure 1.
Let's the production be assumed to begin at t = O and finishes at t = T. Let the rate of production be 'K' and the rate of demand be 'R' during the time interval [0, ti], where R is less than K. At time t = ti, the stock reaches a level Q1. Through the gaps of time [ti, fe], [t2, t3] and [t3, t4]. Let's call the rate of growth ai(K-R), a2(K-R), and a3(K-R), where ai, a2, and a3 are constants. At times t2, t3, and t4, the inventory level reaches levels Q2, Q3, and Q4, respectively. The product turns into technically superseded or else, customer taste changes during the decline time T. It is important to keep an eye on the product's stock levels. Due to demand, inventory levels begin to decline at a rate of R. To consume all units Q at the demand rate, it will take time T.
Figure!: Schematic diagram of the inventory level
The model governs distinguished equations as follows. d
— I(t) + 9I(t) = (K-R)-(a + Bt), 0 <t <ti (4) at
d
— I(t) + dl(t) = ai(K-R)-(a + Bt), t1<t<t2 (5) dt
d
-Tzl(t) + ei(t) = U2(K-R)-(a + (it), t2<t<t3 (6)
dt
d
— I(t) + 9I(t) = a3(K-R)-(a + (3t), t3<t<t4 (7) d
— I(t) + 9I(t) = -R, t4<t <t5 (8) dt
d , .
— I(t) = -R,ts<t<t6 (9) d
— I(t) = (K-R),t6<t<T (10)
I(0) = 0,1(t1) = Qi,1(t2) = Q2,1(t3) = Q3,1(t4) = Q4,1(t5) = 0,1(t6) = S, and I(T) = 0 are the initial conditions.
The solutions of equations (4) - (10), using the initial conditions, the on-hand inventory at time't' is calculated as follows:
K-R a Bt B ,
Kt)= ---§-PY + f2(l-e-s^),0<t<t1 (11)
ai(K-R) a Pt B
I(t)= -a--+ ),ti<t<t2 (12)
e 8
ai(K- R)
8
d2(K - R)
8
d3(K- R)
Kt)= \ -~-Y + ^(l-e-0t),t2<t<t3 (13)
I(t)= U3 (1\- liJ-^-Y + y2(1- e-0t), t3<t<t4 (14)
I(t)= -l(1-e9(t5-t)),t4<t<t5 (15)
I(t)= -R(t-t5),t5<t<t6 (16)
I(t) = (K - R)(t -T),t6<t<T (17) Maximum inventories Qi, Q2, Q3 and Q4:
The maximum inventories are estimated using I(ti) = Qi, I(ti) = Q2, I(t3) = Q3, and I(t4) = Q4 during the times ti, t2, t3, and t4 and equations (11) - (14).
We have omitted the second and higher powers of in the e-0t for 6 values
Therefore
/■K-R a Bt B\
Qi=9ti{—-ë-lï + ë2) (18)
a1(K-R) a pt p — _ — __ + _
e e e e
Q2=et2(—---ë-'T + -k) (19)
o = et (a2(K-R) a P'+P) (20)
Q3= ^--e--ë-T + d2) (20)
Q*= ^(^-l-^ + i;) (21)
Shortage level S:
From equations (i6) and (i7) and using I(t6) = S, we get, I(t6) = S=> I(t6) = - R(t6 - t5) = S and I(t6) = S => (K - R)(t6 - T) = S Therefore - R(t6 - t5) = (K - R)(t6 - T) On simplification
Therefore R
t6=-(t5-T)+T
(22)
As a result, total cost equals to the entirety of production, ordering, holding, deteriorating and shortage costs.
The overhead costs independently
(i) Production cost per unit time = R Cp
(ii) Ordering cost per unit time = C3/T
(iii) Holding cost per unit time
= y(j I(t)dt+ j I(t)dt + j I(t)dt+ j I(t)dt+ j I(t)dt
\0 ti t2 t3 t* j
t*
ts
c1
T
C K-R a pt p , _ f
a1(K-R) a pt p
9
—e-T+h(1-e-et)dt
+
** 3 j
a2(K-R) a pt p
ti 14
e
a3(K-R) a pt p
9
—9-T + T2(l-e-et)dt
ts
+ j -l(l-e8(ts-t))dt
14
We have simplified the expansion of e-0t for small values of 6 by ignoring the second and higher powers of in the expansion of e'6t.
Holding cost per unit time
Ci T
(K-R)l(1-ai)t-^+(ai-a2)t-^+(a2-a3)t-2 + a3t-4
-{a-li + R)ti-R{Y-t4t5
3
- B — P 3
Deteriorating cost per unit time
i
2
3
i
2
0
2
3
t4
ts
= y(j h(t)I(t)dt+ j h(t)I(t)dt+ j h(t)/(t)dt+ j h(t)I(t)dt + j h(t)I(t)dtj
t4
, where h(t) = 6
cL
T
tl t2
+
l t4
ts
+ j e(ll(1-e9(ts-t)))dt
4
We have simplified the expansion of e-6t for small values of 6 by ignoring the second and higher powers of in the expansion of e-6
Deteriorating cost per unit time
Cl
T
e(K-R)l(1-a1)t-^+(a1-a2)t-^+(a2-a3)t-2+a3t-4
l
2
3
o
l
2
3
u
3
2
3
2 2 3
+ (ß-8a-R8)^-R8l-5-t4t5\-ß8^
Shortage cost per unit time
Cl
T
"6 l j I(t)dt+ j I(t)dt
ts t6
c
" 6 I
j -R(t - t5)dt + j(K- R)(t - T)dt
s
On simplification
c
T2
R[T-Y + t5t6)-K(Tt6-T-^
T2
Substitute t6 value in (25) from (22) and on simplification, we have Shortage cost
Cl
T
R2t2
Optimum quantity of the model: Q
R2 T2
, (2 + K)RT2 KT2 R2TU Rt2
- + (k-r)t2+—2Ï---2 2 + RTts -~2 22K
rti r t 2 rt3 r t 4 rT
= j h(t)dt + j h(t)dt + j h(t)dt + j h(t)dt + j h(t)dt
'o Jt, J t2 't3 ->t6
(24)
(25)
6
2
2
Q = Qi + Q2+Q3 + Q4 + 0(T- te)
On simplification
R
Q = Qi + Q2 + Q3 + Q4 + ^(t5-T)
(27)
Therefore, total cost is the sum of the costs pertaining to Production, Setup, Holding, Deteriorating and Shortage.
^ -, ^3 Ci
TC(ti, t2, t3, t4, ts, t) = rcp+y + y
I L-i Ln
(K-R)Ul-ai)±+(ai-a2)^
+ (a
t2 t2 . 3 . L4
2 - ^3)-+a3-
t'i t'i ( ß \t'i ft'2 \ t4
T
e(K-R)l(l-a1)t-±+(ai-a2)t-^+(a2-a3)t-^ + a3t-4
+ (ß-8a-R8)-4-R8(-f-t4t5)-ß8f
+ 1 T
R2t2
, (2+K)RT2 KT2 R2Tts Rt2 R2T2
1T + (k-r)t2+—2Ï---2 2 + RTts -~2 2K~
Let us consider that t1 = u t5, t2 = v t5, t3 = w t5 and t4 = x t5
Therefore total cost becomes from (28)
(28)
Tc(t5, t) = rcp+y + y
(K-R)l(l- (ai - U2)
+ (a2 -
' L5 , * 1-5
2 3 2
+ '£
T
I, ,u2ts , v2t 5
2
2
+ (a2 - a3)
■+(ß -da- Rd)
№ Rdl-5-xt
ßß-
T
K
+ (K- R)T2 +
(2+K)RT2 KT2 R2Tt5
2K
2
2
+ RTt5 -
52
2K
(29)
Since TC(t5,T) is minimum so that differentiating (29) with regard to t5 and T likening to zero that is = 0 and = 0 also satisfy the condition {(^¡^F11) -
(d2TC(t
—gt gsT—)} > 0 and then solving to get t5 and T.
MATHCAD is used to find the optimum solution to equation (29).
2
2
2
v2t2
2
2
x3tê
t
5
5
2
3
w2té
x2t2
x2t2
5
2
2
2
2
3
R2t2
Rté R2T2
4. Numerical Illustration
The following numerical illustration analyses the above stated model by considering the values of the following. K= 1000, R = 500, Ci = 5, C2 = 0.5, C3 = 50, ai = a2 = a3 = 10, 6 = 2, a = 0.2, p = 2, u = 0.2, v = 0.4, w=0.6 and x= 0.8. The optimum values are found as T5 = 0.9 and T = 6.049, production cost = 25000, Holding cost = 198, setup cost = 8.333, deteriorating cost = 4.08, shortage cost = 8.825 and total cost = 364500.
Table 1 and 2 shows when there is an increase in rate of deterioration, the Cycle Time (T), Order Quantity (Q), and Total Cost (TC) increases. All increase as the rate of demand parameter increases.
Table 1: Parameter variations on optimal values
Parameters Optimum values
ti t2 t3 t4 t5 T
Ci 5.0 0.18 0.37 0.56 0.74 0.893 6.049
5.25 0.176 0.352 0.53 0.704 0.88 6.293
5.50 0.17 0.339 0.51 0.679 0.848 6.682
5.75 0.166 0.332 0.49 0.664 0.831 6.922
C2 0.5 0.18 0.37 0.56 0.74 0.893 6.049
0.525 0.517 1.033 1.55 2.066 1.051 8.27
0.550 0.519 1.037 1.56 2.074 1.327 10.21
0.575 0.526 1.053 1.58 2.105 1.501 11.1
C3 50 0.18 0.37 0.56 0.74 0.893 6.049
55 0.222 0.445 0.67 0.89 1.591 7.272
60 0.516 1.033 1.55 2.066 2.282 8.52
65 0.518 1.036 1.55 2.071 2.609 10.17
Cp 50 0.18 0.37 0.56 0.74 0.893 6.049
55 0.203 0.407 0.61 0.813 1.017 6.476
60 0.204 0.409 0.61 0.818 1.022 6.993
65 0.205 0.41 0.62 0.821 1.026 7.462
6 2 0.18 0.37 0.56 0.74 0.893 6.049
2.1 0.22 0.44 0.66 0.879 1.098 7.621
2.2 0.265 0.531 0.79 1.062 1.327 9.2
2.3 0265 0531 0.80 1.062 1.327 9.2
a 0.2 0.18 0.37 0.56 0.74 0.893 6.049
0.21 0.225 0.451 0.68 0.901 1.127 7.574
0.22 0.248 0.496 0.74 0.992 1.440 10.59
0.23 0.367 0.734 1.10 1.468 1.836 11.75
ß 2 0.18 0.37 0.56 0.74 0.893 6.049
2.1 0.308 0.617 0.93 1.233 1.241 8.768
2.2 0.318 0.637 0.96 1.274 1.592 9.071
2.3 0.354 0.708 1.06 1.416 1.770 8.48
Table 2: Parameter variations on optimal values
Optimum values
Parameters Q1 Q2 Q3 Q4 Q TC
5.0 629.98 12610 18910 25210 57340 127000
5.25 615.63 12320 18470 24630 56040 116300
5.50 593.62 11880 17810 23750 54030 101400
Optimum values
Parameters Qi Q2 Q3 Q4 Q TC
Ci 5.75 581.18 11630 17440 23250 52910 98420
C2 0.5 629.98 12610 18910 25210 57340 127100
0.525 1807 36160 54240 72310 61500 156300
0.550 1814 36300 54440 72590 65100 170100
0.575 1841 36840 55260 73680 67600 200700
C3 50 629.98 12610 18910 25210 57340 127100
55 778.32 15570 23360 31140 60850 156100
60 1807 36150 54220 72290 63500 182400
65 1812 36240 54360 72480 67900 215500
Cp 50 629.98 12610 18910 25210 57340 127100
55 711.45 14230 21350 28470 60760 189100
60 715.25 14310 21460 28620 62110 234700
65 717.88 14360 21540 28720 65350 272700
6 2 629.98 12610 18910 25210 57340 127100
2.1 768.4 15370 23060 30750 64950 158100
2.2 928.59 18580 27870 37160 72530 187100
2.3 932.58 18780 27970 37260 78730 198100
a 0.2 629.98 12610 18910 25210 57340 127100
0.21 788.49 15770 23660 31550 69770 152600
0.22 867.58 17360 26040 34710 78970 172400
0.23 1284 25700 38540 51390 84900 213100
P 2 629.98 12610 18910 25210 57340 127100
2.1 1079 21580 32370 43160 58180 145700
2.2 1114 22290 33430 44580 60020 152500
2.3 1239 24790 37180 49570 62200 183700
5. Sensitivity analysis
The sensitivity analysis is carried out to effect the change in parameters -15% to 15%, the optimum values are varied to identify the relation between the parameters and optimum values of the production schedule are shown in Figure 2. The true solution with model parameters which are considered to be stationary at its value, in turn is the total cost function. It makes sense to investigate the sensitivity, or the effect of changing model parameters over a given optimum solution.
I. The optimal quantity (Q), production times ti, t2, t3, and t4, cycle time (T), maximum inventories Qi, Q2, Q3, and Q4, and total cost (TC) all increase when the value of the deteriorating parameter 6 increases.
II. The optimal quantity (Q), production times t1, t2, t3, and t4, cycle time (T), maximum inventories Q1, Q2, Q3, and Q4, and total cost (TC) all increases when the value of ordering cost per unit (C3) increases.
III. The optimal quantity (Q), production times ti, t2, t3, and t4, maximum inventories Q1, Q2, Q3, and Q4, and total cost (TC) fall when the value of holding cost per unit (C1) decreases, but cycle time (T) increases.
IV. The optimal quantity (Q), production times t1, t2, t3, and t4, cycle time (T), maximum inventory Q1, Q2, Q3, and Q4, and total cost (TC) all increases when the value of shortage cost per unit (C2) increases.
A. Lakshmana Rao, S. Arun Kumar, K. P. S. Suryanarayana RT&A, N0 3 (69)
INVENTORY MODEL FOR SEVERAL LEVELS OF PRODUCTION Volume 17, September 2022
V. The optimal quantity (Q), production times ti, t2, t3, and t4, cycle time (T), maximum inventory Q1, Q2, Q3, and Q4, and total cost (TC) increases as the value of demand parameters (a, p) increases.
85000 80000 75000 70000 65000
C1 C2 C3 Cp e
a
P
-15 -10 -5 0 5 10 15 Parameters change in percentage
290000 270000 250000 230000 210000 190000 170000 0000
C1 C2 C3 Cp
e
a
P
-15 -10 -5 0 5 10 15 Parameters change in percentage
Figure 2: Relationship between parameters and optimal values (t5, T, Q, TC)
6. Conclusion
The study explored an inventory model intended to deteriorating items that takes into account many levels of manufacturing. Researchers supposed that the rate of demand is reliant on time and which is linear and rate of deterioration is follows exponential distribution. The projected model is appropriate for the products introduced newly which have a consistent harmony up to a certain point in time. Such circumstances are beneficial, because, by starting at a modest production rates, a significant quantity of manufacturing goods will be avoided at the outset, resulting in a reduction in holding costs. As a result, we will receive customer happiness as well as possible profit. We developed a solution through mathematical model for this problem. Numerical illustration and sensitivity analysis are contributed to demonstrate the model. The suggested inventory model shall help manufacturers as well as retailers to indeed calculate the best order quantity, cycle time, and total inventory cost. This model is extended in a variety of ways for additional research, including demand with selling price, power demand, on-hand inventory demand etc.
Funding
No funding was provided for the research.
Declaration of Conflicting Interests The Authors declare that there is no conflict of interest.
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