Inventory Model for the Deteriorating Items With Price and Time Dependent Trapezoidal Type Demand Rate
Jitendra Kaushik1, Ashish Sharma*2
1,2Department of Mathematics, Institute of Applied sciences and Humanities, GLA University, NH-2, Mathura-281406, INDIA.
Email: 1[email protected] , 2[email protected]
Abstract
In real life, many products (e.g. Fruits, vegetables, and fashionable items) degrade naturally following a trapezoidal type demand rate. Inventory, Procurement, pricing and replenishment decision are imperative in case of deteriorating items. In the light of these aspects, we developed a price and time-dependent inventory model with trapezoidal demand rate for deteriorating items. In addition, optimal procurement and pricing policies are determined. The concavity of profit functions with respect to decision variables is discussed analytically. The numerical solution of the model is demonstrated and examined. In order to highlight the effect of changes in the parameters on the optimum solution sensitivity analysis is performed.
Keywords: Trapezoidal type demand, Deteriorating items and Inventory Model.
1. Introduction
The current lifestyle and arrival of new technology within the split time period have reduced the lifespan of products. Therefore, the management of products with a short life span is outmost importance for the profitability of an organization. The lifecycle of such a product can be depicted in trapezoidal demand function which depicts a steady increase in demand, reach on maturity stage then decline in the final stage. In practice, it is observed that a trapezoidal demand pattern quite often exists in deteriorating items. In general, deterioration is considered as a reduction in the value and usefulness of original products, (i.e. Decline or physical depletion). Such products include fruits, vegetables, milk products, and foodstuffs which deteriorate with time while kept in store.Moreover, electronic and radioactive substances also lose its utility over the period of time (Zhao, 2014). However, the life cycle of the products may vary from product to product, for example, in the case of milk products it may be a day; in case of fruits it may be a week, etc. Generally retailers of such products store only limited quantity so that they end with shortages in the demand season and in the next replenishment they partially backlogged the shortages. During shortages, only the loyal customers will wait up to the next replenishment and the proportion of such customers will decrease as waiting time increases.
Many researchers have studied inventory models for deteriorating items.
Harris(1913) developed the Economic Order Quantity (EOQ) method with an assumption of constant demand rate, which was later extended by Rest et al. (1976)considering an increasing demand linearly. The demand rate of the trapezoid form is an extension of the demand rate of the ramp type. Ritchi (1980) developed a ramp type demand trend which
was further developed by Hill (1995). Mandal and Pal (1998) demonstrated Economic Order Quantity (EOQ) inventory model for deteriorating items using a ramp type demand function with an assumption of constant deterioration rate. Further, the study discussed probabilistic and deterministic cases of demands. A study by Wu and Ouyang (2000) considered a demand rate a ramp time function of time for the development of order-level inventory system. This study extends inventory model started with a shortage and without shortage. A similar study is reported by Deng et al. (2007) firstly raised some questions on results derived by Wu and Ouyang (2000) and Mandal and Pal (1998) and further related problem was solved by an robust method for achieving the optimum solution.
Furthermore, several development shape been reported by Giri et al. (2003). This study presented a single item EOQ model for deteriorating items using ramp-type demand and weibull distribution over an infinite planning period. A study by Cheng and Wang (2009) put forward the replenishment policy for the inventory model of deteriorating items using trapezoid-type demand-rate. Cheng et al. (2011) demonstrated an inventory model for time-dependent deteriorating items using the demand rate as trapezoidal function and partial backlogging. This study further advances a useful replenishment policy for such inventory model. Similarly Singh and Pattnayak (2013) considered work of Cheng and Wang (2009) and analyze EOQ model for deteriorating items using trapezoidal demand rate and two-parameter weibull distribution. Ahmed et al. (2013) considered a ramp type demand function with partial backlogging and common deterioration rate for finding EOQ for an inventory system. Taleizadeh and Nematollahi (2014) proposed an optimal EOQ model for inventory system of perishable item considering constants deteriorating rate with an assumption of demand in the back order and delay payment over the finite horizon planning. Similarly Lin et al. (2014) examined the replenishment policy reported by Cheng et al. (2011) and provide detailed analysis of some questionable results and model from the above-mentioned study. This study explicitly stated that only one of the four possible scenarios was addressed by Cheng et al. (2011).Further provides a complete inventory model solution with continuous trapezoidal demand rate function of time with partial backlogging, exponential backordered rate and linear deterioration. Mishra (2015) takes into consideration the role of preservation technology for reducing the deterioration rate. Inventory model presented in this study assumes trapezoidal demand function by allowing preservation technology cost as decision variable. Firstly, no shortage scenario was considered which was further extended to a scenario where shortage was allowed. In order to minimize total average cost via optimization of procurement time Singh et al.
(2018) demonstrated EOQ model considering ram-type demand, deterioration rate proportional to time allowing shortages and completely backlogged. Recently Sharma et.al (2019) developed a deterministic inventory model considering the constant deterioration with items time and price dependent seasonal demand. With a view of profit maximization Kaushik and Sharma (2019) developed a price and time-dependent inventory model with trapezoidal type demand with time dependant deterioration rate for deteriorating items.
In the presented study, we have considered trapezoidal demand rate with an assumption, the deterioration rate considers constant and shortage allowed with partial backlogged. Three phases of the trapezoidal type demand function are represented by three different prices depending on the time and Iso-elastic demand functions. We assume that during shortages, the proportion of customer’s decreases as enhance the waiting time.
The selling price and the optimum order quantity are determined simultaneously.
Analytical results for concavity of the profit function regarding the variables of the decision are discussed. Numerical examples and Solution procedure are provided. Further paper discussed as following: section 2 highlights notations and assumptions used in the model. In section 3 mathematical models are established and analyzed. Numerical examples are presented in section 4 in order to describe the solution procedure. Section 5 shows sensitivity analysis to observe profit maximization point. Finally concluding remarks are given in section 6.
2. NOTATIONS AND ASSUMPTIONS
In the presented model we consider constant deterioration rate and take the price and time to be dependent on demand function. The linear demand function took in this model with respect to time and power function of price. Once the Inventory is restored, it will be used in a total demand season. Inventory will be depleted in either phase 1, 2 or 3, there are three possibilities so condition of all three cases are discussed. We aim to identify the point of maximum profit for the organization via optimal profit situations. This will provide the profit function's optimum overall value. In case of shortage and partial backlogged assumption has been made that more waiting time decrease the probability of the customer buying the items. We provide loyal customers with a discounted price for those were waiting for the next replenishment.
The stated below notations are used in this manuscript throughout:
2.1 Notations
T - Denoted for Interval length
T1 - Denoted for Time up to which demand increases.
T2 - Time to which demand remains constant and declines afterwards.
TLF - Time epoch when Inventory are depleted and shortages begin. F = 1, 2 and 3 (variable decision)
I - Initial level of Inventory Ib - Backlogged shortage C0 - Ordering cost
C1 - Holding cost, per unit per unit time C - Purchase price per unit
- Constant deterioration rate
P - Per unit selling price of item. (Decision variable) λP - Backlogged price ; (0 < λ < 1)
2.2 Demand function
We are considering a linear demand function which is depending on price and time with constant deterioration rate in the trapezoidal type demand model, where demand moves in three stages. The linear demand function is described as time and power function of price as stated below three different phases.
For Phase I: f1
P t, a jbtP
; 0 ≤ t ≤ T1
For phase II: f2
P t, a bTj 1P
; T1 ≤ t ≤ T2
For phase III: f3
P t, a jbt rP
; T2 ≤ t ≤ T
Where a, b and j are parameters of constant demand. Parameter ' r ' it gives versatility that a jump will start with the demand carried in III Phase.
2.3 Assumptions
The following are the model's key assumptions.
(1) Customers will sustain with us it is depending on waiting time during the shortage period. If waiting time increases customer probability of waiting for next replenishment is
( ) 1 ( / ) ; 0 T T
(2) Price of Backorder is P such that CP < P for this C 1 P (3) Positive Demand function; fF(P, T)0
(4) The sale price per unit would surpass the purchase price and the total holding cost C + TC1
P The RHS of the expression is known as a price floor.
(5) Replenishment rate is infinite.
(6) Demand decrease as a price increase
(7) Backlogs may clear at the arrival of next replenishment and shortage time cannot be exceeded the length T
3. ANALYSIS
We are describing three cases of Inventory Model into three phase, growth in the first stage, constant in second stage and decline in third stage.
Case 1: Inventory depletes in first stage known growing stage (0 ≤ TL1 ≤ T1) Holding cost is given by
1
1 1 L1 1
0
[ {(T 0) f (P, t) }dt]
TL
H C
Total revenue earned into first phase
1
1 1 1
0
( , )
TL
R Pf P t dtPIb
Here Ib1 is denoting for backlog amount
1 2
1 1 2
1 1(P, t) * (T1 t) dt + 2(P, t) * (T2 t) dt + ( 3(P, t) r) * (T t) dt
L
T T T
b
T T T
I
f
f
f The initial inventory level is
1
1 1
0
{ ( , ) }
TL
I
f p t dtProfit per unit time for the first stage
Net1 = (R1 – H1 - C0 – C (I1 + Ib1)) / T (1)
Case 2: Inventory depletes in second stage known constant phase (T1 ≤ TL2 ≤ T2) Holding cost is given by
1 2
1
2 1 1 1 L 2 2
0
[ {(T 0) f (P, t) }dt + {(T 0) f (P, t) }dt ]
T TL
T
H C
R2 the revenue earned
1 2
1
2 1 2 b 2
0
f (P, t) dt f (P, t) dt PI
T TL
T
R
P
P Backlogged amount is
2
2 2
2 2(P, t) * (T2 t) dt + ( 3(P, t) r) * (T t) dt
L
T T
b
T T
I
f
f and the first level of inventory is
1 2
1
2 1 2
0
{ ( , ) } { ( , ) }
T TL
T
I
f p t dt
f p t dt Profit for the second stage per unit timeNet2 = (R2 – H2 - C0 – C (I2 + Ib2)) / T (2) Case 3: Inventory depletes in decreasing phase (T2 ≤ TL3 ≤ T)
Holding cost is given by
3
1 2
1 2
3 1 1 1 2 2 3 3
0
[ {(T 0) f (P, t) }dt + {(T 0) f (P, t) }dt + {(T 0) f (P, t) }dt]
TL
T T
L
T T
H C
The revenue earned is
3
1 2
1 2
3 1 2 3 b 3
0
f (P, t) dt f (P, t) dt (f (P, t) r) dt + PI
TL
T T
T T
R
P
P
P Here backlogged amount is
3
3 ( 3(P, t) r) * (T t) dt
L
T b
T
I
f For the third stage, profit per unit time
Net3 = (R3 - H3 - C0 - C (I + Ib3)) / T (3) Let us discuss the concavity of function NetF regarding the decision variables TLF.
Theorem
1: (a) Net1 is concave function in TL1. (b) Net2 is concave function in TL2.(c) Net3 is concave function in TL3 if
2
1 1 3
3
3 1 2 1 1
[3{3(1 ) ( 4 )} 6 ]
3 [ 2 ] [4 1] (2 Pr )
3
j
j j
P j C P r bC T a C PbT
T
bP T P CT bT C P T b
(4)
Proof: (a) On partially differentiating Net1from expression (1) with respect to TL1, we get
3
2
1 1
1 1
1 1
2
2 1 1
1 1 1 1
1 1
1 1
1 1 ( )
( ) ( )
L
j j
aP T bP T
j j j j
aP bP T C aP bP T
T T
j j
aP T bP T
j j
C aP T bP T T P
T T
Net
T T
(5)
1 1 1 1 1 1 1 1
1 3 2
1 2
1
C 2 2
j j
L
P T P a C T C P b T C P C T T C P
T Net
T T
(6)
Now the rhs of the above expression is
1 1 1 1 1 1 1 1 1 1 1
1 3
2 2 2
0
j j j
T C P aC aT C P aPP bT C bT P C T bT T C T P T T
Thus on simplification
1 1 1 1
1 3
[ 2 ] [ ] [ ] 2 [ ]
0
j j
T C aP bT aP C P bT C P T C P
T T
Since C < Pλ Thus, the stated above expression is negative. In this way Net1is a concave function for TL1
(b) On partially differentiating Net2 from expression (2) with respect to TL2, we get
1 1 1 1
1 1
2 2
1 1 1
1 1 1 1
2 2
2 2 3
( ) P (( ) )
( )( ) ( )
j
j j
j j
L
aP T bT P jT
a bT C a bT P
T T
aP T bT P jT C P j T T a bT P P
T T
Net T T
(7)
and
1
1 1
2 2 2 2
3 2
2 2
2
T T1
C T
T T T T
T
j j j
L
j
j j
aP b P aP b P
C P a
Net T
b P P
(8)
The Right Hand Side of the expression is
1 1
2 2 2 2
3
T1 T1
T T T T T
0
j j j j
j j
aP b P aP b P
C C P a b P P
T
1 2 1 1 1
2 3
( ) ( ) ( )
0
j j j j
CP a bT T C P a bT P PP a bT
T T
1 1 1
2 3
( ) ( )
0
j j j
a bT P C P C P a bT P T T
Since CPthere is a negative expression above
(c) On partially differentiating Net3 from expression (3) with respect to TL3, we get
3
1 1
3 3
2
1 1 1
1 2 1 2 1 2 1
3 3 3
2
1 1 1
3 3 3
1 1
1 ( Pr bP (
1 2 1
) ( ( ) ( )(3 ( 2 )
3 3
3 ( ))) ( ) )
L
Net aP j jT P C aP j r bP jT
T T
j j
aP T rT bP T C bP j T T P j T T a b T T
T T T
j j
aP T rT bP T
Pj r P
T T T
(9) And
1
1 1
1 2 1
3 3 3
2 1 1
3 3 3
3
( 2 4
( ) C ( ( )
3 1 2
2 (3 ( 2 ) 3 ( ))) ( )
3 3 2
3
j j j j
j
j j
j j
b P C bP aP T r bP T bP T T
T T T
aP T r bP T
P a b T T P r P
Net T T T
TL T
(10) The Right hand side of the stated above expression (9) is negative iff
1 1
1 2 1 2 1
3 3 3
3
1 1
3 3 3
1 2 4 1
( )[ ( ) ( ( ) (3 ( 2 )
3 3
3 ( ) ( 2 ) ]< 0
j j
j j j j
j j
j
aP r bP T
bP C bP C bP T T P a b T T
T T T
T
aP T r bP T
P r P
T T T
That is
3 3 1
3 1 2 1 1
3
2 1 1
3
2 4 4
(1 / T )[ 3
3 3
2 2
] < 0
3 3 3 3 3
j j j j
T bPP CT bP CaP rC bCP T bC P jT bC P jT P j T
j j
bT T b P rj Pj PP a Pr PbP T T
On simplification
3
3 1 1
3
1 2 2 1
3 1
3
(1 / T )[ 4 3
3 3
2 1 2 4 2 Pr ] < 0
3 3 3 3
j j j
CT bP CaP rC bC P jT P j P r Pj T
j j
PbP T T bPP j bCP jT bC P T bT T b T
Implies that
3 3 3 1 1 1
3 1 2 1 1
3 [ ] 3 [ ] [ 4 ] 6
3
3 [ 2 ] [4 1] 2 Pr
3 3
j j j j j
CP T a T P P T P r bC T PbP T
T
j j
bP T P CT bT C P T b
Thus
2
1 1 3
3 1 2 1 1
[3{3(1 ) ( 4 )} 3 6 ]
3 [ 2 ] [4 1] (2 Pr )
3
j
j j
P j C P r bC T aC PbT
T
bP T P CT bT C P T b
Therefore, if all these stated above condition satisfied, then we can say the profit function is concave
with respect to TL3.
Now, we're testing the profit function's concavity relative to P.
2 2 3
1 3 2 3 1 3
2 1 2 3
2 2
1 2 3 2 3 1
2
1 2 2 2 3 3 1 1 1
4
3 1 3
1 1
1 1 ( 3 ( ) 3 ( ( ( 1 ) )
6
( ( 1 ) ) T ( ( 1 ) )
( ( ( 1 ) ) ( ( 1 ) ) ( (2 2 2 )
( )))) b( 3T (
j
j j
j j j
j
Net j j
P T T T T P r a T T C j P
P T T T
T T T C j P T T C j P
T T T C j P T T C j P T T C P jP C jT
T C P jP T
13 2 3
3 2 3 3
1 2 3 2 3 1 1 2 2
2 2
3 1 1 1 3
( 1 ) ) 2 ( ( 1 ) )
3 ( ( 1 ) ) 2 T ( ( 1 ) ) (2 ( ( 1 ) )
( (3 3 3 2 ) 2 ( )))))
j j
j j j
j j
C j P T T T C j P
T T T C j P T T C j P T T T C j P
T T C P jP C jT T C P jP
(11)
and
2
3 2
1
1 3 1 2 3
2 2
1 2 3
2 2
2 3 1 1 2 2 2 3
3 1 1 1 3
4 1 3
1 2 (3 (C ) (C C )
6
(C ) ( (C ) ( j )
( (2 C(1 j) 2( 1 j) P (1 ) ) ( j )))
(3 ( j
j j
j j
Net jP j aT T C P jP T T T P jP
P T T T
T T T C P jP T T T C P jP T T C C P jP
T T C j T T C C P jP
b T T C C P jP
13 2 3
2 2 3
1 2 3 2 3 1
3 2
1 2 2 3 1
2
1 1 3
) 2 ( j )
3 ( j ) 2 ( j )
(2 ( j ) ( (3 (1 ) 3( 1 )
2 (1 ) ) 2 ( j )))
T T T C C P jP T T T C C P jP T T T C C P jP
T T T C C P jP T T C j j P
C j T T C C P jP
Furthermore, phase II and phase III expressions are also complicated; thus, analytically, It is difficult to check the profit function about the concavity with respect to P. Therefore we will test numerically the concavity of the profit function in relation to P as well as the joint concavity of function NetF in relation to TLF and P.
Let
2 2
F LF
U Net T
,
2 2
F LF
V Net P
and
2 2 2
2
2 2
( F* F) ( F)
LF LF
Net Net Net
W T P T P
4. SOLUTION PROCEDURE
For F =1
Step no.1: Solve from expression (4) and 1 0
P
Net Value obtain from the expression
(11) for TL*1 and P*
Step no. 2. Check 0 <TL*1< T1 and P*> Price floor. If yes, test the initial values of the parameters for the next step.
Step no.3. For this set of (TL*1,P*) find the value of 2 1
2
P Net
as an expression (12). Go
to the next step, if this value is negative.
Step no.4. For this set of (TL*1,P*) find out the value of Net1 from expression (1).
Step no.5. Repeat step 1 to 4 for F = 2 and 3. From Net1 , Net2 and Net3 select the maximum one.
Now we illustrate this procedure in the Numerical example given below.
Example. Now for the T1 = 40, T2 = 75, T = 100, r = 1, a = 20, b = 3.3, j = 1.5, C0 = 100, λ = 0.99, C1 = 0.001, C = 0.5 by the findings of the solution procedure are shown.
Result: In the table 1, the optimal value of,TL*, P*, Net*, for the phases I and II are given. Since the concavity of Net1 and Net2 with respect to TL is already proven in Theorem 1 (a) and 1 (b). We therefore test numerically the Net concavity in column ' X ' and the joint concavity in column ' Y ' with respect to P. In Table 2 the numerical results for case III are given. For Net3 condition for Net3 concavity in relation to TL3 is shown under column ' LHS ' and ‘RHS’. Joint Net concavity is shown in column ' Y '
Table 1: Test Results of the Phase 1 and 2 of numerical example
θ *
TL P* Net* I* I b* X Y 0.2 39.235 1.744 7.622 722.57 728.856 -1.6975 .03454 0.4 39.182 1.74342 7.581 730.66 725.839 -1.6987 .03449 0.6 39.129 1.74286 7.541 738.72 722.82 -1.6987 .03444 0.8 39.076 1.74231 7.499 746.76 719.798 -1.7011 .03439 1.0 39.022 1.745938 7.459 754.77 716.774 -1.7022 .034334
Phase II , T1< TL2< T2
0.2 73.884 1.70697 14.199 1308.64 2555 -3.4842 .03979 0.4 73.751 1.70654 14.122 1323.75 2550 -3.4861 .03982 0.6 73.657 1.7061 14.046 1338.82 2544.71 -3.4879 .03986 0.8 73.564 1.70567 13.969 1353.84 2539.43 -3.4897 .03988 1.0 73.470 1.70525 13.892 1368.81 2534.15 -3.4914 .03992 From Table 1 we can see that the maximum profit = 14.199 attain in Phase II at θ = 0.2 where TL* = 73.884,
P* = 1.70697, I* = 1308.64, I = 2555, X = -3.4842 and Y = .03979 b*
Table 2: Test Results of Phase III of the numerical example
θ *
TL P* Net* I* I b* LHS RHS Y 0.2 99.308 1.7601 9.917 1018.65 90.514 14.92 476.50 .2474 0.4 99.189 1.7855 9.900 1000.05 103.703 14.86 474.16 .2553 0.6 99.063 1.8126 9.884 980.89 116.99 14.76 470.73 .2631 0.8 98.928 1.8417 9.869 961.13 130.379 14.67 467.41 .2708 1.0 98.7851 1.8731 9.856 940.69 143.873 14.58 464.19 .2785 The lhs and rhs are the expression (4) is left hand side and the right hand side.
From the table 2 we can observe that the maximum profit = 9.917 at θ = 0.2 where TL* = 99.308, P = 1.7601, I* = 1018.65, I = 90.514 and Y = .0397 b*
Thus, We can find maximum profit, from overall profit functions which is 14.199 in phase II for θ = 0.2. The corresponding values of TL* = 73.884, P = 1.70697,
I* = 1308.64, I = 2555, X = -3.4842 and Y = .03979. b*
Figure 1, 2 and 3 shows the joint profit function concavity Net1, Net2 and Net3
respectively with respect to TL1 and P.
Fig 1: Shows Joint concavity of Net1 with Respect to TL1 and P.
Fig 2: Shows Joint concavity of Net2 with Respect to TL2 and P
Fig. 3: Joint concavity of Net3 with respect to TL3 and P
Therefore, we apply sensitivity analysis on profit function which carried highest profit as per the result of Phase I, Phase II and Phase III with respect to decision variables and demand parameters.
5. SENSITIVITY ANALYSIS
We have to apply sensitivity analysis due to high profit procurement in phase II for θ = 0.2. We can check how parameter variables (a, b and j) effected the profit outcome in decision variable (T1, P).
Table 3: Test Results of the Sensitivity analysis of profit function
-100% -75% -50% -25%
0%
25% 50% 75% 100%
a 0.7273 0.5455 0.3636 0.1818 0 -0.1818 -0.3636 -0.5455 -0.7273 b 0.3302 0.2476 0.1651 0.0826 0 -0.0826 -0.1651 -0.2476 -0.3302 j -1.3009 -0.8724 -0.5217 -
0.2348
0 0.1921 0.3494 0.4780 0.5833 T1 2.2077* 1.2418* 0.5519* 0.1379 0 0.1379* 0.5519* 1.2418* 2.2077*
P Not valid
1.9783 0.2859 0.0364 0 0.0152 0.0442 0.0757 0.1059
From stated above table no 3 we can observe that there is same degree of percentage change came into existence in both of side of sensitivity analysis for parameter ‘a’. When we increase values as 25% to up to 100% we procure same values as we decrease by - 25% to -100%. This same result is occurring for parameter ‘b’ as same degree of change in percentage has to seen in both of cases as parameter ‘a’. But there is change in results in case of ‘j’ parameter. When we increase value by 25% to 100% then profit is also increases, but when we decrease the value by -25% to -100% we getting more loss as compare to percentage change in ratio. Yet results are same in case of parameter ‘a’ and
‘b’ but emerging more loss in case of ‘j’ parameter. When we increase 25% in ‘j’ we get 0.1921 but when we decrease by -25% we get -0.2348. Which shows more changes in percentage when we reduce value by -25% to -100%.
6. CONCLUSION
Products like fruits and vegetables are deteriorating in nature and pusses the trapezoidal type demand pattern. We developed a Mathematical model for such products by considering Iso-elastic demand function with price and
Time-dependent demand function and constant rate of deterioration allowing Shortages are partial backlogging. The trapezoidal demand pattern contains three phases first is increasing, second is maturity and the third is declining. We put numerical examples to show the result of profit function and sensitivity analysis applied with different parameters which show how the change has arrived due to changes in parameters and optimal profit. We find the highest profit in the second phase.
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Authors
Jitendra Kaushik is Ph.D. Research scholar in Department of Mathematics, GLA University Mathura. He received degree of M.Phil in Mathematics from Bundelkhand University as merit holder. His research interest area in Inventory modeling.
Dr. Ashish Sharma is Associate Professor in the Department of Mathematics at GLA University, Mathura, India.
He received the Ph.D. degree in Statistics from Devi Ahilya University, Indore, India. His research interests are primarily in inventory modeling, pricing and facility allocation. His articles have appeared in journals such as International journal of production economics, Computer and Industrial Engineering, Mathematical and Computer Modeling.
