Multi item inventory model include lead time with demand dependent production cost and set-up-cost in fuzzy environment

Corresponding Author: satyakrdasmath75@gmail.com
10.22105/jfea.2020.254081.1025
E-ISSN: 2717-3453 | P-ISSN: 2783-1442
|
Abstract
1 | Introduction
Inventory models deal with decisions that minimize the total average cost or maximize the total
average profit. In that way to construct a real life mathematical inventory model on based on various
assumptions and notations and approximations. Multi-item is also an important factor in the
inventory control system. The basic well known Economic Order Quantity (EOQ) model was first
introduced by Harris in 1913; Abou-el-ata and Kotb studied a multi-item EOQ inventory model with
varying holding costs under two restrictions with a geometric programming approach [1]. Chen [7]
presented an optimal determination of quality level, selling quantity and purchasing price for
intermediate ﬁrms. Liang and Zhou [11] discussted two warehouse inventory model for deteriorating
items and stock dependent demand under conditionally permissible delay in payment.
Journal of Fuzzy Extension and Applications
www.journal-fea.com
J. Fuzzy. Ext. Appl. Vol. 1, No. 3 (2020) 227–243.
Paper Type: Research Paper
Multi-Item Inventory Model Include Lead Time with
Demand Dependent Production Cost and Set-Up-Cost in Fuzzy
Environment
Satya Kumar Das*
Department of Mathematics, General Degree College at Gopiballavpur-II, Jhargram, West Bengal, India; satyakrdasmath75@gmail.com.
Citation:
Das, S. K. (2020). Multi item inventory model include lead time with demand dependent
production cost and set-up-cost in fuzzy environment. Journal of fuzzy extension and application, 1 (3),
227-243.
Accept: 12/08/2020
Revised: 10/05/2020
Reviewed: 02/05/2020
Received: 23/03/2020
In this paper, we have developed the multi-item inventory model in the fuzzy environment. Here we considered the
demand rate is constant and production cost is dependent on the demand rate. Set-up- cost is dependent on average
inventory level as well as demand. Lead time crashing cost is considered the continuous function of leading time.
Limitation is considered on storage of space. Due to uncertainty all cost parameters of the proposed model are taken as
generalized trapezoidal fuzzy numbers. Therefore this model is very real. The formulated multi objective inventory
problem has been solved by various techniques like as Geometric Programming (GP) approach, Fuzzy Programming
Technique with Hyperbolic Membership Function (FPTHMF), Fuzzy Nonlinear Programming (FNLP) technique and
Fuzzy Additive Goal Programming (FAGP) technique. An example is given to illustrate the model. Sensitivity analysis
and graphical representation have been shown to test the parameters of the model.
Keywords: Inventory, Multi-Item, Leading time; Generalized trapezoidal fuzzy number, Fuzzy techniques; GP technique.
Licensee Journal
of Fuzzy Extension and
Applications. This rticle
is an open access article
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the Creative Commons
Attribution (CC BY)
license
(http://creativecommons.
org/licenses/by/4.0).

228
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|J.
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Das et al. [12] developed a multi-item inventory model with quantity dependent inventory costs and
demand-dependent unit cost under imprecise objectives and restrictions with a geometric programming
approach. Das and Islam [13] considered a multi-objective two echelon supply chain inventory model
with customer demand dependent purchase cost and production rate dependent production cost. Shaikh
et al. [30] discussed an inventory model for deteriorating items with preservation facility of ramp type
demand and trade credit.
The concept of fuzzy set theory was first introduced by Zadeh [27]. Afterward, Zimmermann [28]
applied the fuzzy set theory concept with some useful membership functions to solve the linear
programming problem with some objective functions. Bit [2] applied fuzzy programming with
hyperbolic membership functions for multi objective capacitated transportation problems. Bortolan and
Degani [4] discussed a review of some methods for ranking fuzzy subsets. Maiti [19] developed a fuzzy
inventory model with two warehouses under possibility measure in fuzzy goals. Mandal et al. [22]
presented a multi-objective fuzzy inventory model with three constraints with a geometric programming
approach. Shaikh et al. [26] developed a fuzzy inventory model for a deteriorating Item with variable
demand, permissible delay in payments and partial backlogging with Shortage Following Inventory (SFI)
policy. Garai et al. [29] discussed multi-objective inventory model with both stock-dependent demand
rate and holding cost rate under fuzzy random environment.
In the global market system lead time is an important matter. Ben-Daya and Rauf [3] considered an
inventory model involving lead-time as a decision variable. Chuang et al. [8] presented a note on periodic
review inventory model with controllable setup cost and lead time. Hariga and Ben-Daya [14] discussed
some stochastic inventory models with deterministic variable lead time. Ouyang et al. [20] studied
mixture inventory models with backorders and lost sales for variable lead time. Ouyang and Wu [21]
established a min-max distribution free procedure for mixed inventory models with variable lead time.
Sarkar et al. [24] developed an integrated inventory model with variable lead time and defective units
and delay in payments. Sarkar et al. [25] studied quality improvement and backorder price discount under
controllable lead time in an inventory model.
Geometric Programming (GP) is a powerful optimization technique developed to solve a class of non-
linear optimization programming problems especially found in engineering design and manufacturing.
Multi objective geometric programming techniques are also interesting in the EOQ model. GP was
introduced by Duffin et al. in 1966 [10] and published a famous book in 1967 [9]. Beightler et al. [5]
applied GP. Biswal [6] considered fuzzy programming techniques to solve multi-objective geometric
programming problems. Islam [16] discussed multi-objective geometric-programming problem and its
application. Mandal et al. [22] developed a multi-objective fuzzy inventory model with three constraints
with a geometric programming approach. Mandal et al. [23] discussed an inventory model of
deteriorating items with a constraint with a geometric Programming approach. Islam [17] studied a
multi-objective marketing planning inventory model with a geometric programming approach. Kotb et
al. [18] presented a multi-item EOQ model with both demand dependent on unit cost and varying lead
time via geometric programming.
In this paper, we have developed an inventory model of multi-item with space constraint in a fuzzy
environment. Here we considered the constant demand rate and production cost is dependent on the
demand rate. Set-up- cost is dependent on average inventory level as well as demand. Lead time crashing
cost is considered the continuous function of leading time. Due to uncertainty all cost parameters are
taken as generalized trapezoidal fuzzy numbers. The proposal has been solved by various techniques
like GP approach, FPTHMF, FNLP, and FAGP. Numerical example is given to illustrate the model.
Finally sensitivity analysis and graphical representation have been shown to test the parameters of the
model.

229
Multi
item
inventory
model
include
lead
time
with
demand
dependent
production
cost
and
set-up-cost
in
fuzzy
environment
2| Mathematical Model
2.1| Notations
ℎ𝑖: Holding cost per unit per unit time for ith item.
𝑇𝑖: The length of cycle time for 𝑖thitem, 𝑇𝑖 > 0.
𝐷𝑖 ∶ Demand rate per unit time for the ith item.
𝐿𝑖 ∶ Rate of leading time for the ith item.
SS: Safety stock.
𝑘: Safety factor.
𝐼𝑖(𝑡): Inventory level of the ith item at time t.
𝐶𝑝
𝑖
: Unit production cost of ith item.
𝑆𝑐
𝑖
(𝑄𝑖, 𝐷𝑖): Set up cost for ith item.
𝑅𝑖( 𝐿𝑖): Lead time crashing cost for the ith item.
𝑄𝑖: The order quantity for the duration of a cycle of length 𝑇𝑖for ith item.
𝑇𝐴𝐶𝑖(𝐷𝑖, 𝑄𝑖, 𝐿𝑖): Total average profit per unit for the ith item.
𝑤𝑖: Storage space per unit time for the ith item.
𝑊: Total area of space.
𝑤𝑖
̃: Fuzzy storage space per unit time for the ith item.
ℎ𝑖
̃: Fuzzy holding cost per unit per unit time for the ith item.
𝑇𝐴𝐶𝑖
̃ (𝐷𝑖, 𝑄𝑖, 𝐿𝑖): Fuzzy total average cost per unit for the ith item.
𝑤𝑖
̂ : Defuzzyfication of the fuzzy number 𝑤𝑖
̃.
ℎ𝑖
̂: Defuzzyfication of the fuzzy number ℎ𝑖
̃.
𝑇𝐴𝐶𝑖
̂(𝐷𝑖, 𝑄𝑖, 𝐿𝑖): Defuzzyfication of the fuzzy number 𝑇𝐴𝐶𝑖
̃ (𝐷𝑖, 𝑄𝑖, 𝐿𝑖).
2.2| Assumptions
 Multi-item is considered.
 The replenishment occurs instantaneously at infinite rate.
 The lead time is considered.
 Shortages are not allowed.
 Production cost is inversely related to the demand. Here considered 𝐶𝑝
𝑖 ( 𝐷𝑖) = 𝛼𝑖𝐷𝑖
−𝛽𝑖
, where 𝛼𝑖 > 0 and 𝛽𝑖 > 1 are
constant real numbers.

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 The set up cost is dependent on the demand as well as average inventory level. Here considered 𝑆𝑐
𝑖
(𝑄𝑖, 𝐷𝑖) =
𝛾𝑖 (
𝑄𝑖
2
)
𝛿𝑖
𝐷𝑖
𝜎𝑖
where 0 < 𝛾𝑖, 0 < 𝛿𝑖 ≪ 1 and 0 < 𝜎𝑖 ≪ 1 are constant real numbers.
 Lead time crashing cost is dependent on the lead time by a function of the form 𝑅𝑖( 𝐿𝑖) = 𝜌𝑖𝐿𝑖
−𝜏𝑖, where 𝜌𝑖 > 0
and 0 < 𝜏𝑖 ≤ 0.5 are constant real numbers.
𝐿𝑖.
 Deterioration is not allowed.
2.3| Formulation of the Model
The inventory level for ith item is illustrated in Fig. 1. During the period [0, 𝑇𝑖] the inventory level reduces
due to demand rate. In this time period, the governing differential equation is
With boundary condition, 𝐼𝑖(0) = 𝑄𝑖
, 𝐼𝑖(𝑇𝑖) = 0.
Solving Eq. (1) we have,
Fig. 1. Inventory level for the ith item.
Now calculating various averages cost for ith item,
Average production cost (𝑃𝐶𝑖) =
𝑄𝑖𝐶𝑝
𝑖 ( 𝐷𝑖)
𝑇𝑖
= 𝛼𝑖𝐷𝑖
(1−𝛽𝑖)
;
Average holding cost(𝐻𝐶𝑖) =
1
𝑇𝑖
∫ ℎ𝑖𝐼𝑖(𝑡)𝑑𝑡
𝑇𝑖
0
+ ℎ𝑖𝑘𝜔√𝐿𝑖 = ℎ𝑖 (
𝑄𝑖
2
+ 𝑘𝜔√𝐿𝑖) ;
Average set-up-cost (𝑆𝐶𝑖) =
1
𝑇𝑖
[𝛾𝑖 (
𝑄𝑖
2
)
𝛿𝑖
𝐷𝑖
𝜎𝑖
] =
𝛾𝑖𝑄𝑖
𝛿𝑖−1
𝐷𝑖
𝜎𝑖+1
2𝛿𝑖
;
Average lead time crashing cost (𝐶𝐶𝑖) =
𝜌𝑖𝐿𝑖
−𝜏𝑖
𝑇𝑖
=
𝐷𝑖𝜌𝑖𝐿𝑖
−𝜏𝑖
𝑄𝑖
.
Total average cost for ith item is
dIi(t)
dt
= −Di,0 ≤ t ≤ Ti. (1)
Ii(t) = Qi − Dit, 0 ≤ t ≤ Ti . (2)
Ti =
Qi
Di
. (3)

231
Multi
item
inventory
model
include
lead
time
with
demand
dependent
production
cost
and
set-up-cost
in
fuzzy
environment
A Multi-Objective Inventory Model (MOIM) can be written as:
2.4| Fuzzy Model
Due to uncertainty, we consider all the parameters (𝛼𝑖, 𝛽𝑖, ℎ𝑖, 𝜌𝑖, 𝛾𝑖, 𝛿𝑖, 𝜎𝑖, 𝜏𝑖)of the model and storage space
𝑤𝑖 as Generalized Trapezoidal Fuzzy Number (GTrFN)(𝛼𝑖
̃, 𝛽𝑖
̃, ℎ𝑖
̃, 𝜌𝑖
̃, 𝛾𝑖
̃, 𝛿𝑖
̃, 𝜎𝑖,
̃ 𝑤𝑖
̃, 𝜏𝑖
̃). Here
𝛼𝑖
̃ = (𝛼𝑖
1
, 𝛼𝑖
2
, 𝛼𝑖
3
, 𝛼𝑖
4
; 𝜑𝛼𝑖
), 0 < 𝜑𝛼𝑖
≤ 1; ℎ𝑖
̃ = (ℎ𝑖
1
, ℎ𝑖
2
, ℎ𝑖
3
, ℎ𝑖
4
; 𝜑ℎ𝑖
), 0 < 𝜑ℎ𝑖
≤ 1;
𝛽𝑖
̃ = (𝛽𝑖
1
, 𝛽𝑖
2
, 𝛽𝑖
3
, 𝛽𝑖
4
; 𝜑𝛽𝑖
), 0 < 𝜑𝛽𝑖
≤ 1; 𝜌𝑖
̃ = (𝜌𝑖
1
, 𝜌𝑖
1
, 𝜌𝑖
1
, 𝜌𝑖
1
; 𝜑𝜌𝑖
), 0 < 𝜑𝜌𝑖
≤ 1;
𝛾𝑖
̃ = (𝛾𝑖
1
, 𝛾𝑖
2
, 𝛾𝑖
3
, 𝛾𝑖
4
; 𝜑𝛾𝑖
), 0 < 𝜑𝛾𝑖
≤ 1; 𝑤𝑖
̃ = (𝑤𝑖
1
, 𝑤𝑖
2
, 𝑤𝑖
3
, 𝑤𝑖
4
; 𝜑𝑤𝑖
), 0 < 𝜑𝑤𝑖
≤ 1;
𝛿𝑖
̃ = (𝛿𝑖
1
, 𝛿𝑖
2
, 𝛿𝑖
3
, 𝛿𝑖
4
; 𝜑𝛿𝑖
), 0 < 𝜑𝛿𝑖
≤ 1; 𝜎𝑖
̃ = (𝜎𝑖
1
, 𝜎𝑖
2
, 𝜎𝑖
3
, 𝜎𝑖
4
; 𝜑𝜎𝑖
), 0 < 𝜑𝜎𝑖
≤ 1;
𝜏𝑖
̃ = (𝜏𝑖
1
, 𝜏𝑖
2
, 𝜏𝑖
3
, 𝜏𝑖
4
; 𝜑𝜏𝑖
), 0 < 𝜑𝜏𝑖
≤ 1; ( 𝑖 = 1,2, … … … , 𝑛).
Then the above inventory Model (5) becomes the fuzzy inventory model as
𝜆 −Integer method is used to defuzzify the fuzzy number. In this method the defuzzify value of the fuzzy
number 𝐴̃ = (𝑎, 𝑏, 𝑐, 𝑑; 𝜑) is 𝜑 (
𝑎+𝑏+𝑐+𝑑
4
). So using the defuzzified values (𝛼𝑖
̂, 𝛽𝑖
̂, ℎ𝑖
̂, 𝜌𝑖
̂, 𝛾𝑖
̂, 𝛿𝑖
̂, 𝜎𝑖
̂, 𝑤𝑖
̂ , 𝜏𝑖
̂) of the
GTrFN (𝛼𝑖
̃, 𝛽𝑖
̃, ℎ𝑖
̃, 𝜌𝑖
̃, 𝛾𝑖
̃, 𝛿𝑖
̃, 𝜎𝑖,
̃ 𝑤𝑖
̃, 𝜏𝑖
̃), the above fuzzy inventory Model (6) reduces to
TACi(Di, Qi, Li) = (PCi + HCi + SCi + CCi) = αiDi
(1−βi)
+ hi (
Qi
2
+ kω√Li) +
γiQi
δi−1
Di
σi+1
2δi
+
DiρiLi
−τi
Qi
.
(4)
Min {TAC1, TAC2, TAC3, … … … … … , TACn},
TACi(Di, Qi, Li) = αiDi
(1−βi)
+ hi (
Qi
2
+ kω√Li) +
γiQi
δi−1
Di
σi+1
2δi
+
DiρiLi
−τi
Qi
,
Subject to
∑ wiQi ≤
n
i=1 W, Di > 0, Qi > 0, Li > 0, for i = 1,2, … … … . . n.
(5)
Min {TAC1
̃ , TAC2
̃ , TAC3
̃ , … … … … … … … . , TACn
̃ },
Subject to
∑ wi
̃Qi ≤
n
i=1 W, for i = 1,2, … … … . . n.
Where
TACi(Di, Qi, Li)
̃ = αi
̃ Di
(1−βi
̃)
+ hi
̃ (
Qi
2
+ kω√Li) +
γi
̃ Qi
δi
̃−1
Di
σi
̃+1
2δi
̃ +
Diρi
̃ Li
−τi
̃
Qi
.
(6)

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|J.
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(2020)
227-243
3| Fuzzy Programming Techniques to Solve MOIM
Solve the MOIM as a single objective NLP using only one objective at a time and ignoring the others.
So we get the ideal solutions. Using the ideal solutions the pay-off matrix as follows:
3.1|Fuzzy Programming Technique Using Hyperbolic Membership Function
(FPTHMF)
Now fuzzy non-linear hyperbolic membership functions 𝜇𝑇𝐴𝐶𝑘
𝐻
(𝑇𝐴𝐶𝑘(𝐷𝑘, 𝑄𝑘, 𝐿𝑘)) for the kth objective
functions 𝑇𝐴𝐶𝑘(𝐷𝑘, 𝑄𝑘, 𝐿𝑘) respectively for 𝑘 = 1,2, … . , 𝑛 are defined as follows:
Where 𝛼𝑘 is a parameter, 𝜎𝑘 =
3
(𝑈𝑘−𝐿𝑘)
2
⁄
=
6
𝑈𝑘−𝐿𝑘
.
Min {TAC1
̂ , TAC2
̂ , TAC3
̂ , … … … … … … … … … … . , TACn
̂ },
Subject to
∑ wi
̂Qi ≤
n
i=1 W,
Where
TACi(Di, Qi, Li)
̂ = αi
̂ Di
(1−βi
̂)
+ hi
̂ (
Qi
2
+ kω√Li) +
γi
̂ Qi
δi
̂−1
Di
σi
̂+1
2δi
̂ +
Diρi
̂ Li
−τi
̂
Qi
,
Di > 0, Qi > 0, Li > 0, for i = 1,2, … … … . . n.
(7)
TAC1(D1, Q1, L1) TAC2(D2, Q2, L2)…. ……….TACn(Dn, Qn, Ln)
(D1
1
, Q1
1
, L1
1 ) TAC1
∗
(D1
1
, Q1
1
, L1
1 ) TAC2(D1
1
, Q1
1
, L1
1 )………..….TACn(D1
1
, Q1
1
, L1
1
)
(D2
2
, Q2
2
, L2
2 ) TAC1(D2
2
, Q2
2
, L2
2 ) TAC2
∗
(D2
2
, Q2
2
, L2
2 ) . … … … TACn(D2
2
, Q2
2
, L2
2
)
…… ….. ….. ………….. ………..
…… ….. ….. ………….. ………..
(Dn
n
, Qn
n
, Ln
n ) TAC1(Dn
n
, Qn
n
, Ln
n ) TAC2(Dn
n
, Qn
n
, Ln
n ) … … …. TACn
∗
(Dn
n
, Qn
n
, Ln
n
),
Let Uk
= max{TACk(Di
i
, Qi
i
, Li
i
), i = 1,2, … . , n} for k = 1,2, … . , n and
Lk
= TACk
∗
(Dk
k
, Qk
k
, Lk
k
)for k = 1,2, … . , n.
Hence Uk
, Lk
are identified, Lk
≤ TAPk(Di
i
, Qi
i
, Li
i
) ≤ Uk
, for i = 1,2, … . , n ; k =
1,2, … . , n.
(8)
μTACk
H
(TACk(Dk, Qk, Lk))
=
1
2
tanh
((
Uk + Lk
2
− TACk(Dk, Qk, Lk)) σk
)
+
1
2
.

233
Multi
item
inventory
model
include
lead
time
with
demand
dependent
production
cost
and
set-up-cost
in
fuzzy
environment
In this technique the problem is defined as follows:
After simplification the above problem can be written as
Now the above problem can be freely solved by suitable mathematical programming algorithm and then
we shall get the appropiet solution of the MOIM.
3.2| Fuzzy Non-Linear Programming (FNLP) Technique based on Max-Min
In this technique fuzzy membership function 𝜇𝑇𝐴𝐶𝑘
(𝑇𝐴𝐶𝑘(𝑄𝑘, 𝐷𝑘)) for the kth objective function
𝑇𝐴𝐶𝑘(𝐷𝑘, 𝑄𝑘, 𝐿𝑘) respectively for 𝑘 = 1,2, … . , 𝑛 are defined as follows:
In this technique the problem is defined as follows:
Max λ,
Subject to
1
2
tanh ((
Uk+Lk
2
− TACk(Dk, Qk, Lk)) σk) +
1
2
≥ λ ,
∑ wi
̂Qi ≤
n
i=1 W, λ ≥ 0, Dk >, Qk > 0, Lk > 0, for k = 1,2, … … … . . n.
Max y ,
Subject to
y + σkTACk(Dk, Qk, Lk) ≤
Uk+Lk
2
σk,
∑ wi
̂Qi ≤
n
i=1 W, y ≥ 0, Dk >, Qk > 0, Lk > 0 for k = 1,2, … … … . . n.
μTACk
(TACk(Dk, Qk, Lk))
=
{
1 for TACk(Dk, Qk, Lk) < Lk
Uk − TACk(Dk, Qk, Lk)
Uk − Lk
for Lk ≤ TACk(Dk, Qk, Lk) ≤ Uk
0 for TACk(Dk, Qk, Lk) > Uk
for k = 1,2, … . , n.
Max α′,
Subject to
TACk(Dk, Qk, Lk) + α′(Uk − Lk) ≤ Uk , for k = 1,2, … . , n,
∑ wi
̂Qi ≤
n
i=1 W, 0 ≤ α′ ≤ 1, Dk >, Qk > 0, Lk > 0.

234
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|J.
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(2020)
227-243
Now the above problem can be freely solved by suitable mathematical programming algorithm and then
we shall get the required solution of the MOIM.
3.3| Fuzzy Additive Goal Programming (FAGP) Technique Based on Additive
Operator
Using the above membership function, fuzzy non-linear programming problem is formulated as
Now the above problem can be solved by suitable mathematical programming algorithm and then we
shall get the solution of the MOIM.
4| Geometric Programming Technique
Let us consider a Multi Objective Geometric Programming (MOGP) problem is as follows
Where 𝑐𝑟𝑘, 𝑐𝑠0𝑘(> 0), 𝑟𝑘𝑗 and 𝑠0𝑘𝑗 (𝑗 = 1,2, … . , 𝑚; 𝑟 = 0,1,2, … . , 𝑝; 𝑘 = 1,2, . . … . 𝑙𝑟 ; 𝑠 =
1,2,3, … … … . . , 𝑛)are all real numbers. 𝑇𝑠0 is the number of terms in the 𝑠𝑡ℎ
objective function and 𝑙𝑟 is
the number of terms in the 𝑟𝑡ℎ
constraint.
Now introducing the weights 𝑤𝑖 (𝑖 = 1,2,3, … … … . . , 𝑛), the above MOGP converted into the single
objective geometric programming problem as following
Primal Problem.
Max ∑ Uk−TACk(Dk,Qk,Lk)
Uk−Lk
n
k=1 ,
Subject to
Uk − TACk(Dk, Qk, Lk) ≤ Uk − Lk,
∑ wi
̂Qi ≤
n
i=1 W, Dk >, Qk > 0, Lk > 0 for k = 1,2, … . , n.
Minimize gs0(t) = ∑ cs0k ∏ tj
s0kj
m
j=1
Ts0
k=1 , s = 1,2,3, … … … . . , n,
Subject to
gr(t) = ∑ crk ∏ tj
rkj
m
j=1
lr
k=1 ≤ 1 , r = 1,2,3, … … … . . , p,
tj > 0, j = 1,2, … , m.
Minimize g(t) = ∑ ws ∑ cs0k ∏ tj
s0kj
m
j=1
Ts0
k=1
n
s=1 , s = 1,2,3, … … … . . , n,
i.e. = ∑ ∑ wscs0k ∏ tj
s0kj
m
j=1
Ts0
k=1
n
s=1 ,
Subject to
gr(t) = ∑ crk ∏ tj
rkj
m
j=1
lr
k=1 ≤ 1 , r = 1,2,3, … … … . . , p,
tj > 0, j = 1,2, … , m,
∑ wi
n
i=1 = 1, wi > 0, i = 1,2,3, … … … . . , n.
(9)

235
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time
with
demand
dependent
production
cost
and
set-up-cost
in
fuzzy
environment
Let 𝑇 be the total numbers of terms ( including constraints), number of variables is 𝑚. Then the degree of
the difficulty (DD) is 𝑇 − (𝑚 + 1).
Dual Program.
The dual programming of Eq. (9) is given as follows:
Now here three cases may arises
Case I. 𝑇0 = 𝑚 + 1, (i.e. DD=0). So DP presents a system of linear equations for the dual variables. So
we have a unique solution vector of dual variables.
Case II. 𝑇0 > 𝑚 + 1, So a system of linear equations is presented for the dual variables, where the number
of linear equations is less than the number of dual variables. So it is concluded that dual variables vector
have many solutions.
Case III. 𝑇0 < 𝑚 + 1, so a system of linear equations is presented for the dual variables, where the number
of linear equations is greater than the number of dual variables. It is seen that generally no solution vector
exists for the dual variables here.
4.1| Solution Procedure of My Proposed Problem
Primal Problem.
Maximize v(θ) =
∏ ∏ (
wscs0k
θ0sk
)
θ0sk
∏ ∏ (
crk
θrk
)
θrk
lr
k=1
p
r=1
Tso
k=1
n
s=1 (∑ θrk
lr
k=1 )
∑ θrk
lr
k=1
,
Subject to
∑ ∑ θ0sk
Ts0
k=1
n
s=1 = 1, (Normality condition)
∑ ∑ rkjθrk + ∑ ∑ s0kjθ0sk
Ts0
k=1
n
s=1 = 0
lr
k=1
p
r=1 , ( j = 1,2, … . , m)
(Orthogonality conditions)
θ0sk, θrk > 0, (r = 0,1,2, … . , p; k = 1,2, . . … . lr ; s = 1,2,3, … … … . . , n).
(Positivity conditions)
Minimize TAC(D, Q, L)
= ∑ wi
′
n
i=1
(αi
̂ Di
(1−βi
̂)
+ hi
̂ (
Qi
2
+ kω√Li) +
γi
̂Qi
δi
̂ −1
Di
σi
̂+1
2δi
̂
+ Diρi
̂Li
−τi
̂
Qi
−1
) ,
Subject to
∑
wi
̂
W
Qi ≤ 1
n
i=1 ,
∑ wi
′
n
i=1 = 1, wi
′
> 0, i = 1,2,3, … … … . . , n .
(10)

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Dual Program.
The dual programming of Eq. (10) is given as follows:
Solving the above linear equations we have
𝜃𝑖1 =
𝜎𝑖
̂+1
𝛽𝑖
̂−1
𝑦𝑖 +
1
𝛽𝑖
̂−1
𝑥𝑖, 𝜃𝑖2 = 1 − {(1 + 2𝜏𝑖
̂) +
1
𝛽𝑖
̂−1
𝑥𝑖} −
𝜎𝑖
̂+𝛽𝑖
̂
𝛽𝑖
̂−1
𝑦𝑖, 𝜃𝑖3 = 2𝜏𝑖
̂𝑥𝑖, 𝜃𝑖4 = 𝑦𝑖, 𝜃𝑖5 = 𝑥𝑖,
𝜃𝑖1
′
= 1 − (2𝜏𝑖
̂ +
1
𝛽𝑖
̂ − 1
) 𝑥𝑖 − (
𝜎𝑖
̂ + 𝛽𝑖
̂
𝛽𝑖
̂ − 1
+ 𝛿𝑖
̂ − 1) 𝑦𝑖 .
Putting the above values in Eq. (11) we have
Now using the primal-dual relation we have
Maximize v(θ)
= ∏ (
wi
′
αi
̂
θi1
)
θi1
(
wi
′
hi
̂
2θi2
)
θi2
(
wi
′
kω
θi3
)
θi3
(
wi
′
γi
̂
2δi
̂
θi4
)
θi4
(
wi
′
ρi
̂
θi5
)
θi5
(
wi
̂
Wθi1
′ )
θi1
′
n
i=1
(∑ θi1
′
n
i=1
)
∑ θi1
′
n
i=1
,
Subject to
θi1 + θi2 + θi3 + θi4 + θi5 = 1,
(1 − βi
̂)θi1 + (σi
̂ + 1)θi4 + θi5 = 0,
θi2 + (δi
̂ − 1)θi4 − θi5 + θi1
′
= 0,
θi3
2
− τi
̂θi5 = 0 ,
∑ wi
′
n
i=1
= 1, wi
′
> 0,
θi1, θi2, θi3, θi4, θi5, θi1
′
≥ 0 for i = 1,2,3, … … … . . , n.
(11)
Maximize v(x, y)
= ∏ (
wi
′(βi
̂ − 1)αi
̂
(σi
̂ + 1)yi + xi
)
σi
̂+1
βi
̂−1
yi+
1
βi
̂−1
xi
n
i=1 (
wi
′
hi
̂
2 {1 − {(1 + 2τi
̂) +
1
βi
̂ − 1
xi} −
σi
̂ + βi
̂
βi
̂ − 1
yi})
1−{(1+2τi
̂)+
1
βi
̂−1
xi}−
σi
̂+βi
̂
βi
̂−1
yi
(
wi
′
kω
2τi
̂xi
)
2τi
̂xi
(
wi
′
γi
̂
2δi
̂
yi
)
yi
(
wi
′
ρi
̂
xi
)
xi
(
wi
̂
Wzi
)
zi
(∑ zi
n
i=1 )∑ zi
n
i=1 ,
∑ wi
′
n
i=1
= 1,
Where zi = 1 − (2τi
̂ +
1
βi
̂−1
) xi − (
σi
̂+βi
̂
βi
̂−1
+ δi
̂ − 1) yi,
xi, yi > 0, wi
′
> 0 and x = (x1, x2, x3, … . . , xn), y = (y1, y2, y3, … … … , yn).

237
Multi
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time
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demand
dependent
production
cost
and
set-up-cost
in
fuzzy
environment
TAC∗(D, Q, L) = n(v∗(x, y))
1/n
;
wi
′
αi
̂Di
∗(1−βi
̂)
= θi1
∗
(v∗(x, y))
1/n
;
wi
′
hi
̂Qi
∗
2
= θi2
∗
(v∗(x, y))
1/n
;
wi′hikωLi∗=θi3∗v∗x,y1/n, for i=1,2,3,………..,n.
5| Numerical Example
Here we consider an inventory system which consists of two items with following parameter values in
proper units. Total storage area 𝑊 = 500 𝑆𝑞. 𝑓𝑡. and 𝑘 = 3, 𝜔 = 5, 𝑤1
′
= 0.5, 𝑤1
′
= 0.5.
Table 1. Input imprecise data for shape parameters.
Approximate value of the above parameter is
Table 2. Defuzzification of the fuzzy numbers.
Parameters
Items
I II
αi
̃ (200,205,210,215; 0.9) (215,220,225,230; 0.8)
βi
̃ (4,5,6,7; 0.8) (5,6,7,8; 0.8)
hi
̃ (2,4,5,6; 0.9) (2,2.5,3,3.5; 0.8)
ρi
̃ (2,2.3,2.4,2.5; 0.9) (3,3.1,3.2,3.3; 0.9)
γi
̃ (90,95,100,105; 0.7) (92,95,98,102; 0.8)
δi
̃ (0.02,0.03,0.04,0.05; 0.8) (0.04,0.05,0.06,0.07; 0.8)
σi
̃ (0.2,0.3,0.4,0.5; 0.8) (0.2,0.3,0.4,0.5; 0.9)
wi
̃ (1.5,1.6,1.7,1.8; 0.7) (1.7,1.8,1.9,2.0; 0.9)
τi
̃
(
1
6
,
1
5
,
1
4
,
1
3
; 0.9) (
1
7
,
1
6
,
1
5
,
1
4
; 0.9)
Defuzzification of the Fuzzy Numbers Items
I II
αi
̂ 186.75 178
βi
̂ 4.4 5.2
hi
̂ 3.825 2.2
ρi
̂ 2.07 2.835
γi
̂ 68.25 77.4
δi
̂ 0.028 0.044
σi
̂ 0.28 0.315
wi
̂ 1.155 2.115
τi
̂ 0.21375 0.17089

238
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|J.
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(2020)
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Table 3. Optimal solutions of MOIM using different methods.
Fig. 2. Minimizing cost of both items using different methods.
From the above figure shows that GP, FPTHMF, FNLP, and FAGP methods almost provide the same
results.
6| Sensitivity Analysis
In the sensitivity analysis the optimal solutions have been found buy using FNLP method.
Table 4. Optimal solution of MOIM for different values of 𝛂𝟏, 𝛂𝟐.
Fig. 3. Minimizing cost of both items for different values of 𝛂𝟏, 𝛂𝟐.
From the Fig. 3 suggests that the minimum cost of both items is increased when values of 𝛼1, 𝛼2 are
increased.
Methods 𝐃𝟏
∗
𝐐𝟏
∗
𝐋𝟏
∗
𝐓𝐀𝐂𝟏
∗
𝐃𝟐
∗
𝐐𝟐
∗
𝐋𝟐
∗
𝐓𝐀𝐂𝟐
∗
FPTHMF 2.50 11.51 0.34 × 10−3 53.95 2.28 15.54 0.30 × 10−3 41.03
FNLP 2.50 11.51 0.34 × 10−3 53.95 2.30 15.57 0.33 × 10−3 41.03
FAGP 2.49 11.35 0.35 × 10−3 53.96 2.30 15.63 0.37 × 10−3 41.03
GP 2.58 10.07 0.27 × 10−3 54.58 2.20 17.58 0.42 × 10−3 41.52
Method 𝛂𝟏, 𝛂𝟐 𝐃𝟏
∗
𝐐𝟏
∗
𝐋𝟏
∗
𝐓𝐀𝐂𝟏
∗
𝐃𝟐
∗
𝐐𝟐
∗
𝐋𝟐
∗
𝐓𝐀𝐂𝟐
∗
FNLP
-20% 2.36 11.12 0.33 × 10−3 52.14 2.19 15.12 0.29 × 10−3 39.82
-10% 2.43 11.33 0.34 × 10−3 53.09 2.24 15.36 0.33 × 10−3 40.45
10% 2.56 11.68 0.35 × 10−3 54.75 2.34 15.78 0.37 × 10−3 41.55
20% 2.61 10.84 0.35 × 10−3 55.49 2.38 17.97 0.42 × 10−3 42.03
FPTHMF
FNLP
FAGP
GP
0
20
40
60
TAC1* TAC2*
FPTHMF
FNLP
FAGP
GP
-20%
-10%
10%
20%
0
20
40
60
TAC1* TAC2*
-20%
-10%
10%
20%

239
Multi
item
inventory
model
include
lead
time
with
demand
dependent
production
cost
and
set-up-cost
in
fuzzy
environment
Table 5. Optimal solution of MOIM for different values of 𝛃𝟏, 𝛃𝟐.
Fig. 4. Minimizing cost of 1st and 2nd items for different values of 𝛃𝟏, 𝛃𝟐.
From the Fig. 4 suggests that the optimal cost of both items is decreased when values of 𝛽1, 𝛽2 are increased.
Table 6. Optimal solution of MOIM for different values of 𝛄𝟏, 𝛄𝟐.
Fig. 5. Minimizing cost of both items for different values of 𝛄𝟏, 𝛄𝟐.
Iuyi
Method 𝛃𝟏, 𝛃𝟐 𝐃𝟏
∗
𝐐𝟏
∗
𝐋𝟏
∗
𝐓𝐀𝐂𝟏
∗
𝐃𝟐
∗
𝐐𝟐
∗
𝐋𝟐
∗
𝐓𝐀𝐂𝟐
∗
FNLP
-20% 2.93 12.73 0.37 × 10−3 62.82 2.67 17.19 0.32 × 10−3 47.24
-10% 2.69 12.05 0.35 × 10−3
57.77 2.46 16.30 0.31 × 10−3
43.71
10% 2.35 11.07 0.33 × 10−3
50.98 2.16 15.00 0.29 × 10−3
38.92
20% 2.22 10.70 0.32 × 10−3 48.59 2.06 14.52 0.28 × 10−3 37.23
Method 𝛄𝟏, 𝛄𝟐 𝐃𝟏
∗
𝐐𝟏
∗
𝐋𝟏
∗
𝐓𝐀𝐂𝟏
∗
𝐃𝟐
∗
𝐐𝟐
∗
𝐋𝟐
∗
𝐓𝐀𝐂𝟐
∗
FNLP -20% 2.57 10.58 0.40 × 10−3 49.70 2.34 14.26 0.35 × 10−3 37.60
10%- 2.53 11.06 0.37 × 10−3
51.89 2.32 14.94 0.32 × 10−3
39.36
10% 2.47 11.94 0.32 × 10−3
55.92 2.27 16.19 0.28 × 10−3
42.60
20% 2.45 12.35 0.30 × 10−3
57.78 2.25 16.77 0.26 × 10−3
44.11
-20%
-10%
10%
20%
0
20
40
60
80
TAC1*
TAC2*
-20%
-10%
10%
20%
-20%
-10%
10%
20%
0
20
40
60
TAC1* TAC2*
-20%
-10%
10%
20%

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From the above Fig. 5 suggests that the optimal cost of both items is increased when values of 𝛾1, 𝛾2
are increased.
Table 7. Optimal solutions of MOIM for different values of 𝛔𝟏, 𝛔𝟐.
Fig. 6. Minimizing cost of 1st and 2nd items for different values of 𝛔𝟏, 𝛔𝟐.
From the above Fig. 6 suggests that the minimum cost of both items is increased when values of 𝜎1, 𝜎2
are increased.
Table 8. Optimal solutions of MOIM for different values of 𝛒𝟏, 𝛒𝟐.
Table 9. Optimal solutions of MOIM for different values of 𝛕𝟏, 𝛕𝟐.
Method 𝛔𝟏, 𝛔𝟐 𝐃𝟏
∗
𝐐𝟏
∗
𝐋𝟏
∗
𝐓𝐀𝐂𝟏
∗
𝐃𝟐
∗
𝐐𝟐
∗
𝐋𝟐
∗
𝐓𝐀𝐂𝟐
∗
FNLP
-20% 2.54 11.36 0.35 × 10−3
52.92 2.33 15.35 0.31 × 10−3
40.18
-10% 2.52 11.44 0.35 × 10−3
53.44 2.31 15.46 0.31 × 10−3
40.60
10% 2.48 11.59 0.33 × 10−3 54.47 2.28 15.70 0.29 × 10−3 41.45
20% 2.46 11.67 0.33 × 10−3
54.99 2.26 15.81 0.29 × 10−3
41.87
Method 𝛒𝟏, 𝛒𝟐 𝐃𝟏
∗
𝐐𝟏
∗
𝐋𝟏
∗
𝐓𝐀𝐂𝟏
∗
𝐃𝟐
∗
𝐐𝟐
∗
𝐋𝟐
∗
𝐓𝐀𝐂𝟐
∗
FNLP
-20% 2.50 11.42 0.25 × 10−3
53.44 2.29 15.47 0.23 × 10−3
40.68
-10% 2.50 11.47 0.30 × 10−3
53.70 2.29 15.53 0.26 × 10−3
40.86
10% 2.50 11.55 0.39 × 10−3
54.20 2.29 15.64 0.34 × 10−3
41.19
20% 2.50 11.60 0.43 × 10−3
54.45 2.29 15.69 0.39 × 10−3
41.35
Method 𝛕𝟏, 𝛕𝟐 𝐃𝟏
∗
𝐐𝟏
∗
𝐋𝟏
∗
𝐓𝐀𝐂𝟏
∗
𝐃𝟐
∗
𝐐𝟐
∗
𝐋𝟐
∗
𝐓𝐀𝐂𝟐
∗
FNLP
-20% 2.50 11.40 0.15 × 10−3
53.15 2.29 15.47 0.14 × 10−3
40.58
-10% 2.50 11.46 0.23 × 10−3
53.54 2.29 15.52 0.21 × 10−3
40.80
10% 2.50 11.57 0.48 × 10−3
54.39 2.29 15.64 0.42 × 10−3
41.26
20% 2.50 11.62 0.67 × 10−3
54.83 2.29 15.69 0.57 × 10−3
41.50
-20%
-10%
10%
20%
0
20
40
60
TAC1*
TAC2*
-20%
-10%
10%
20%

241
Multi
item
inventory
model
include
lead
time
with
demand
dependent
production
cost
and
set-up-cost
in
fuzzy
environment
Fig. 7. Minimizing cost of both items for different values of 𝜌1, 𝜌2.
From the above Fig. 7 suggests that the minimum cost of both items is increased when values of 𝜌1, 𝜌2 are
increased.
Fig. 8. Minimizing cost of 1st and 2nd items for different values of 𝛕𝟏, 𝛕𝟐.
From the above Fig. 8 suggests that the minimum cost of both items is increased when values of 𝜏1, 𝜏2
are increased.
7| Conclusion
In this article, we have developed an inventory model of multi-item with limitations on storage space in a
fuzzy environment. Here we considered the constant demand rate and production cost is dependent on
the demand rate. Set-up- cost is dependent on average inventory level as well as demand. Lead time
crashing cost is considered the continuous function of leading time. Due to uncertainty all cost parameters
are taken as a generalized trapezoidal fuzzy number. The formulated problem has been solved by various
techniques like GP approach, FPTHMF, FNLP, and FAGP. Numerical example is given under considering
two items to illustrate the model. A numerical problem is solved by using LINGO13 software.
This paper will be extended by using linear, quadratic demand, ramp type demand, power demand, and
stochastic demand etc., introduce shortages, generalize the model under two-level credit period strategy
etc. Inflation plays a crucial position in Inventory Management (IM) but here it is not considered. So
inflation can be used in this model for practical. Also other types of fuzzy numbers like triangular fuzzy
numbers; PfFN, pFN, etc. may be used for all cost parameters of the model.
-20%
-10%
10%
20%
0
20
40
60
TAC1*
TAC2*
-20%
-10%
10%
20%
-20%
-10%
10%
20%
0
20
40
60
TAC1*
TAC2*
-20%
-10%
10%
20%

242
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|J.
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227-243
Acknowledgements
I would like to thank my wife Smt. Priya Das for his non-stop encouragement to write this research
paper. The authors are grateful to the reviewers for their comments and suggestions.
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Bit, A. K. (2004). Fuzzy programming with hyperbolic membership functions for multiobjective
capacitated transportation problem. Opsearch, 41(2), 106-120.
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