A multi-objective inventory management system under uncertain
conditions
A. Salimiana
salimian_atefeh@yahoo.com
S. Nodousta* (*corresponding author)
*Email: sara_arezoo_n@yahoo.com
A. Mirzazadeha
a.mirzazadeh@aut.ac.ir
a
Department of Industrial Engineering, University of Kharazmi, Tehran, Iran
Abstract
In this study, a multi-objective joint replenishment inventory system for deteriorating items
has been considered. The proposed model minimizes the total expected present value of costs
and the expected value of inventory level over the time horizon. We propose the ideal point
approach to formulate the model. The stochastic inflationary conditions with uniform pdf
over the time horizon have been considered and the demand rate is a linear function of
discount rate. This model incorporates both imperfect production quality and imperfect
inspection. For evaluation of the model, the numerical example has been provided.
Keywords: Production systems management; Inventory; multi-objective; stochastic;
deteriorating items.
1.
Introduction
Nowadays, almost every important real world problem involves more than one objective.
More so than ever before, decision makers find it imperative to evaluate solution alternatives
according to multiple conflicting objectives. These problems are modeled as multi-objective
decision making (MODM) problems identifying the types of measures that might be
said as "criteria". The thrust of these models is to design the "best" alternative by
considering the various interactions within the objectives and design constraints which
best satisfies decision makers (DMs) by way of attaining some acceptable levels of a set
of some quantifiable objectives.
Though multi-objective decision making (MODM) problems have been formulated and
solved in many other areas like air pollution, structural analysis, till now very few papers on
MODM have been published in the field of optimal inventory control. Padmanabhan and Vrat
[1] formulated an inventory problem of deteriorating items with two objectives minimization
of total average cost and wastage cost in crisp environment and solved by nonlinear goal
programming method. Roy and Maiti [2] formulated an inventory problem of deteriorating
items with two objectives, namely, maximizing total average profit and minimizing total
waste cost in fuzzy environment.
Guillen et al. [3] introduced the model which optimized profit, demand satisfaction, and
financial risk cost of a three echelon supply chain. Azaron et al [4] uses the goal attainment
technique to optimize total cost, total cost variance, and financial risk cost of a three echelon
supply chain. Chen et al. [5] uses a two-phase fuzzy decision-making to optimize profit
between the supply chain participants, customer service levels, and safe inventory level.
Liang [6] designed the original multi-objective linear programming which attempts to
simultaneously minimize total costs and total delivery time in relation to inventory levels,
available machine capacity and labor levels at each source, and forecast demand and available
warehouse space at each destination and total budget. Javadi et al. [7] solved a multiobjective mixed-model assembly line sequencing problem by a fuzzy goal programming
approach. Wee et al. [8] develop a fuzzy multi-objective joint replenishment inventory model
of deteriorating items .their model maximized the profit and return on inventory investment
(ROII) under fuzzy demand and shortage cost constraint.
Traditional inventory models tend to obtain an economic (optimal) order quantity (EOQ) or
economic production quantity (EPQ) based on the ordering/setup cost and the inventory
carrying cost. One assumption is that the items produced by the facility are all of a perfect
quality. A number of researchers have not used the perfect quality. Duffuaa and Khan [9]
suggested an inspection plan for these critical components where an inspector can commit a
number of misclassifications. Duffuaa and Khan [10] carried out a sensitivity analysis to
study the effect of different types of misclassifications on the optimal inspection plan. There
may be many sources of errors in inspection; one of which is inaccuracy in records. Kok and
Shang [11] discussed inaccuracies in inventory records. They proved that an inspection
adjusted base-stock policy is optimal for a single period problem, where inspection is
performed if the inventory recorded is less than a threshold level. Atali et al. [12] also
modeled the discrepancies between actual and the recorded inventories in retail and
distribution environments. They quantified the value of RFID (radio-frequency identification)
that reduces the amount of such discrepancies. We leave this issue here for future research.
Khan et al. [13] extends the work of Salamehand Jaber. In their model the defective items,
classified by the inspector and the buyer would be salvaged as a single batch that is sold at a
lower price.
The effect of inflation and time-value of money cannot be ignored in global economics.
Buzacott [14] first derived an EOQ model by considering the inflationary effect on costs.
Misra [15] then extended the EOQ model with different inflation rates for various associated
costs. Mirzazadeh and Sarfaraz [16] presented a multiple items inventory system with budget
constraint and the Uniform distribution for external inflation rate. These two recent models do
not consider shortages and deteriorating items. Later, Yang et al. [17] established various
inventory models with time varying demand patterns under inflation. Some researchers like
Moon and Lee [18], Yang [19], Soleimani et al. [20], Moon et al. [21] and Jolai et al. [22] –
derived different types of models under inflation and time-discounting. Mirzazadeh [23]
proposed a detailed comparison of the average annual cost and the discounted cost models
under stochastic inflationary conditions. Dey et al. [24] extended this type of model by
considering a two-storage system and dynamic demand under inflation. Chern et al. [25]
developed inventory lot size models for deteriorating items with fluctuating demand under
inflation. Recently, Sarkar et al. [26] discussed a finite replenishment model with increasing
demand under inflation. Sarkar et al. [27] studied the production inventory model with
variable demand under the effect of inflation. Sarkar et al. [28] developed an inventory model
with different types of stochastic demand pattern.
In this study we minimize the cost of inventory system and the inventory under stochastic
inflation rate with considering imperfect production and imperfect inspection for deteriorating
items. Inventory decreasing is important, because: (1) the quantity of the deteriorated goods is
related to inventory level so that decreasing in inventory causes decreasing destroyed good,
(2) decreasing in inventory level causes increasing company flexibility against changes in the
market conditions, customer needs and so on, (3) low inventory system causes faster
company adaptation with technology changes, (4) decreasing in inventory causes better cash
flow and rate of return.
The paper is organized as follows. Assumptions and notations are introduced in Section 2.
Section3 discusses the mathematical model. Section 4 explains the solution procedure. In
Section 5, we presented a numerical example. Finally, conclusions are given in Section 6.
2.
Assumptions and notations:
The following assumptions have been considered.
1.
The time horizon of the production system is finite.
2.
The demand rate is a linear function of the inflation rate.
3.
The lead time is zero.
4.
Shortages are permitted and fully backlogged except for the final cycle.
5.
The effect of inflation and time value of money is considered.
6.
A constant fraction of the on-hand inventory deteriorates per unit of time and there is
no repair of replacement of the deteriorated inventory.
7.
The inflation rate is stochastic with uniform distribution function.
8.
Production and inspection are not perfect.
9.
The distribution of R is continuous uniform. 𝑅~𝑈𝑛𝑖𝑓𝑜𝑟𝑚(𝑅1 , 𝑅2 )
10.
Assuming that inspection error will occur in production cycle and not occur in rework
cycle.
For convenience, the following notations are used throughout the entire paper:
i
the rate of inflation
r
the discount rate
R the discount rate net of inflation 𝑅 = 𝑟 − 𝑖
D the demand rate, which is a linear function of the inflation rate. (𝐷(𝑅) = 𝑎 − 𝑏𝑅, that a,
b>0).
Ψ probability that an item is defective
𝛼
probability of Type I error (classifying a non-defective item as defective)
β
probability of Type II error (Classifying a defective item as non-defective)
p
production and inspection rate of the lot y
𝑝𝑠 production and inspection rate of serviceable items. (𝑝𝑠 = [(1 − 𝜓)(1 − 𝛼) +
𝜓𝛽]𝑝)
𝑝𝑅 rework and inspection rate of the lot. y ((1 − 𝜓))( 1 − 𝛼))
𝑝𝑅𝑆 rework and inspection rate of serviceable items. 𝑝𝑅𝑆 = [𝜓(1 − 𝛽) + (1 − 𝜓)𝛼]𝛾𝑝𝑅 )
γ
probability that an item is non-defective after rework.
n
the number of cycles
θ
the constant deterioration rate per unit time, where (0< θ<1).
ℎ1
the holding cost per unit per unit time at time zero
ℎ2
the shortage cost per unit per unit time at time zero.
A
the ordering (Setup) cost per order at time zero.
c
the production cost per unit per unit time at time zero
c1
the inspection cost per unit per unit time at time zero.
In this study, we suppose that the manufacturer produces and simultaneously inspects a lot y
at a rate of p during its production run. Since the production process is not perfect the lot y
contains defective items of ψy along with non-defective items of (1-ψ) y, where a defect
proportion of ψ is constant. Further, since the inspection process of an entire lot screening is
also not perfect, it generates both Type I and Type II inspection errors , given their respective
proportion
of
𝛼 = 𝑝𝑟 (𝑖𝑡𝑒𝑚𝑠𝑠𝑐𝑟𝑒𝑒𝑛𝑑𝑎𝑠𝑑𝑒𝑓𝑒𝑐𝑡𝑠|𝑛𝑜𝑛 − 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝑖𝑡𝑒𝑚𝑠)
and𝛽 =
𝑝𝑅 (𝑖𝑡𝑒𝑚𝑠𝑛𝑜𝑡𝑠𝑐𝑟𝑒𝑒𝑛𝑒𝑑𝑎𝑠𝑑𝑒𝑓𝑒𝑐𝑡𝑠|𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝑖𝑡𝑒𝑚𝑠), (0 < 𝛼, 𝛽 < 1 𝑎𝑛𝑑𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡).
We assume 𝛼, 𝛽 are independent of a defect proportion ψ. Thus, in an entire lot inspection
process during T1 due to the Type I inspection error, (1-ψ) 𝛼y are falsely screened out and
treated as defects, leaving only (1-ψ) (1- 𝛼) y units of non-defects as serviceable items
.Therefore, due to the Type II inspection error, ψ𝛽𝑦units among the defective items of ψ y are
falsely not screened out as defects and regarded as serviceable items, and thus only ψ(1 −
𝛽)y units among the defects are successfully screened out. Therefore, only the serviceable
items (i.e., falsely unscreened defects and successfully unscreened non-defects) of [ψ𝛽 +
(1 − ψ)(1 − 𝛼)]y, excluding successfully screened defects and falsely screened non-defects
of [ψ(1 − 𝛽) + (1 − ψ)𝛼] y, among the lot y in each cycle T are used to satisfy continuous
customer demand at a rate of D, where it is assumed that D<ps <p. So those unscreened
defects of ψ𝛽𝑦 are passed on to customers along with non-defects of (1 − ψ) ( 1 − 𝛼) y.
Then, those customers who bought defects would detect quality problems and return them
due to quality dissatisfaction. In handling those screened and return items of [ψ + (1 − ψ)𝛼]
y in each cycle, the manufacturer reworks them. The rework items of [ψ + (1 − ψ)𝛼] y, are
reworked at a rate of 𝑝𝑅 during rework run time of T2-T1 right after T1 and. We assume that
rework is not perfect but inspection process of rework items is perfect. It is assumed that
D<𝑝𝑅𝑆 <𝑝𝑅 .
3.
The mathematical modeling and analysis
Inventory Level Of Serviceable Items
The inventory cycles of serviceable items is divided into five different parts. (Fig. 1)
Shortage Level
T4
T1 T2
T3
T
T+T1 T+T2
T+T4 2T
T+T3
(n-1)T (n-1)T+T1
T5
nT
Time
Inventory Level Of Screened Items
Figure1. Graphical representation of the inventory system of the serviceable items
T1 T2
T
T+T1 T+T2
(n-1)T (n-1)T+T1 T5
nT
Time
Figure2. Graphical representation of the inventory system of screened items
Inventory Level Of Returned Items
T1 T2
T3
T
T+T1
T+T3 T+T4
(n-1)T (n-1)T+T1
nT
Time
Figure 3.Graphical representation of the inventory system of returned items
Let 𝐼1𝑖 (𝑡𝑖 ) denote the inventory level at any time 𝑡𝑖 in the ith part of cycle(𝑖 = 1,2,3,4,5). By
using the following differential equation, we can govern the inventory level of serviceable
items at time 𝑡𝑖 .
𝑑𝐼11 (𝑡1 )
+
𝑑𝑡1
𝜃𝐼11 (𝑡1 ) = 𝑃𝑠 − 𝐷
0 ≤ 𝑡1 ≤ 𝑇1
(1)
Since the inventory level of serviceable items is positive in this interval, we have
deterioration.
During[𝑇1 ,𝑇2 ), the inventory level of serviceable items can be described as follows:
𝑑𝐼12 (𝑡2 )
+
𝑑𝑡2
𝜃𝐼12 (𝑡2 ) = 𝑃𝑅𝑆 − 𝐷
0 ≤ 𝑡2 ≤ 𝑇2 − 𝑇1
(2)
During[𝑇2 ,𝑇3 ), we don’t have production or reworking, and then the inventory level of
serviceable items is governed by:
𝑑𝐼13 (𝑡3 )
𝑑(𝑡3 )
== −𝐷 − 𝜃𝐼13 (𝑡3 )
0 ≤ 𝑡3 ≤ 𝑇3 − 𝑇2
(3)
During[𝑇3 ,𝑇4 ), we have no deterioration; therefore, the shortage level is governed by:
𝑑𝐼14 (𝑡4 )
𝑑(𝑡4 )
= −𝐷
0 ≤ 𝑡4 ≤ 𝑇4 − 𝑇3
(4)
Finally during[𝑇4 ,𝑇), the shortage level is representing by:
𝑑𝐼15 (𝑡5 )
𝑑(𝑡5 )
= 𝑃𝑠 − 𝐷
0 ≤ 𝑡5 ≤ 𝑇 − 𝑇4
(5)
In the last cycle, shortage are not allowed and the inventory level of serviceable items is
govern by the following differential equations (let 𝐼1𝑖 (𝑡𝑖 ) denote the inventory level of
serviceable items at any time 𝑡𝑖 in the (𝑖 − 5)𝑡ℎ part of last cycle that is 𝑖 = 6,7,8 )
𝑑𝐼16 (𝑡6 )
𝑑(𝑡6 )
= 𝑃𝑠 − 𝐷 − 𝜃𝐼16 (𝑡6 )
𝑑𝐼17 (𝑡7 )
+
𝑑(𝑡7 )
0 < 𝑡6 ≤ 𝑇1
𝜃𝐼17 (𝑡7 ) = −𝐷 + 𝑃𝑅𝑆 0 < 𝑡5 ≤ 𝑇5 − ((𝑛 − 1)𝑇 + 𝑇1 )
(6)
(7)
𝑑𝐼18 (𝑡8 )
𝑑(𝑡8 )
= −𝐷 − 𝜃𝐼18 (𝑡8 )
0 < 𝑡8 ≤ 𝑛𝑇 − 𝑇5
(8)
The solution of the above differential equation after applying the following boundary
conditions:
𝐼11 (0) = 0 , 𝐼12 (0) = 𝐼11 (𝑇1 ), 𝐼13 (𝑇3 − 𝑇2 ) = 0 , 𝐼14 (0) = 0 , 𝐼15 (𝑇 − 𝑇4 ) = 0 , 𝐼16 (0) = 0 , 𝐼17 (0) =
𝐼16 (𝑇1 ) , 𝐼18 (𝑛𝑇 − 𝑇5 ) = 0 , are:
𝐼11 (𝑡1 ) =
𝑃𝑠 −𝐷
(1
𝜃
− 𝑒 −𝜃𝑡1 )
0 ≤ 𝑡1 ≤ 𝑇1
(𝑃𝑠 −𝐷)(1−𝑒 −𝜃𝑇1 )−(𝑃𝑅𝑆 −𝐷)
𝐼12 (𝑡2 ) = (
𝜃
) 𝑒 −𝜃𝑡2 +
𝐷
𝐼13 (𝑡3 ) = − (1 − 𝑒 −𝜃(𝑡3 −(𝑇3 −𝑇2 ) )
𝜃
𝐼14 (𝑡4 ) = −𝐷𝑡4
𝑃𝑅𝑆 −𝐷
𝜃
(9)
0 ≤ 𝑡2 ≤ 𝑇2 − 𝑇1
0 ≤ 𝑡3 ≤ 𝑇3 − 𝑇2
0 ≤ 𝑡4 ≤ 𝑇4 − 𝑇3
𝑃𝑠 −𝐷
(1
𝜃
𝐼17 (𝑡7 ) = 𝑒 −𝜃𝑡7 (
− 𝑒 −𝜃𝑡6 )
(13)
0 < 𝑡6 ≤ 𝑇1
(𝑝𝑠 −𝐷)(1−𝑒 −𝜃𝑇1 )
𝜃
+
𝐷−𝑝𝑅𝑆
𝐷−𝑝
) − 𝜃 𝑅𝑆 , 0
𝜃
𝐷
𝐼18 (𝑡8 ) = − 𝜃 (1 − 𝑒 −𝜃(𝑡−(𝑛𝑇−𝑇5 ) )
(11)
(12)
𝐼15 (𝑡5 ) = (𝑃𝑆 − 𝐷)(𝑡5 − (𝑇 − 𝑇4 )) 0 ≤ 𝑡5 ≤ 𝑇 − 𝑇4
𝐼16 (𝑡6 ) =
(10)
< 𝑡7 ≤ 𝑇5 − (𝑛 − 1)𝑇 − 𝑇1
0 < 𝑡8 ≤ 𝑁𝑇 − 𝑇5
(14)
(15)
(16)
Using the above equation, we can calculate the values of 𝑇2 , 𝑇4 𝑎𝑛𝑑𝑇5 with respect to
Solving𝐼12 (𝑇2 − 𝑇1 ) = 𝐼13 (0)𝑓𝑜𝑟 𝑇2 𝑎𝑛𝑑 𝐼14 (𝑇4 − 𝑇3 ) =
𝑇1 , 𝑇3 𝑎𝑛𝑑𝑇.
𝐼15 (0)𝑓𝑜𝑟𝑇4 𝑎𝑛𝑑 𝐼18 (0) = 𝐼17 (𝑇5 − (𝑛 − 1)𝑇 − 𝑇1 )𝑓𝑜𝑟𝑇5 we have,
−e−θT1 ps +e−θT1 D+ps −pRS −De−θT1+θT3
) + 𝑇1
PRS
1
𝑇2 = 𝜃 ln (−
𝑇4 =
𝐷𝑇3
𝑃𝑆
+
(𝑃𝑆 −𝐷)𝑇
(17)
(18)
𝑃𝑠
−𝑒−𝜃𝑇1 𝑝𝑠 +𝑒−𝜃𝑇1 𝐷+𝑝𝑠 −𝑝𝑅𝑆 +𝐷𝑒−𝜃𝑇1 +𝜃𝑇
)+𝜃𝑇1 −𝜃𝑇
−𝑝𝑅𝑆 +2𝐷
𝜃𝑛𝑇+ln(
𝑇5 =
𝜃
(19)
The inventory cycle of screened items is divided into three different parts. (Figure 2)Let
𝐼2𝑖 (𝑡𝑖 ) denote the inventory level of screened items at any time 𝑡𝑖 in the ith part of cycle(𝑖 =
1,2,3). By using the following differential equation, the inventory level of screened items is
governed:
𝑑𝐼21 (𝑡1 )
+
𝑑𝑡1
𝜃𝐼21 (𝑡1 ) = 𝑃 − 𝑃𝑠 − 𝐷
𝑑𝐼22 (𝑡2 )
+
𝑑𝑡2
𝜃𝐼22 (𝑡2 ) = (1 − 𝛾)𝑃𝑅
0 ≤ 𝑡1 ≤ 𝑇1
(20)
0 ≤ 𝑡2 ≤ 𝑇2 − 𝑇1
(21)
As reworking starts at 𝑇1 , we have no screened item.
𝐼23 (𝑡3 ) = 0
0 ≤ 𝑡 ≤ 𝑇 − 𝑇2
(22)
After the applying the following boundary conditions𝐼21 (0) = 0, 𝐼22 (0) = 0, the solution of
the above differential equation are:
𝐼21 (𝑡1 ) =
𝑝−𝑃𝑠
(1 −
𝜃
𝐼22 (𝑡2 ) =
(1−𝛾)𝑃𝑅
(1
𝜃
𝑒 −𝜃𝑡1 )
− 𝑒 −𝜃𝑡2 )
𝐼23 (𝑡3 ) = 0
0 ≤ 𝑡1 ≤ 𝑇1
(23)
0 ≤ 𝑡1 ≤ 𝑇2 − 𝑇1
(24)
0 ≤ 𝑡 ≤ 𝑇 − 𝑇2
(25)
The inventory cycle of returned items is divided into three parts. (Figure 3) Let 𝐼3𝑖 (𝑡𝑖 ) denote
the inventory level of returned items at any time 𝑡𝑖 in the ith part of cycle(𝑖 = 1,2,3). We can
govern the inventory level of returned items by using the following differential equations.
𝑑𝐼31 (𝑡1 )
+
𝑑𝑡1
𝜃𝐼31 (𝑡1 ) =
𝜓𝛽𝑃𝑇1
𝑇3
𝑑𝐼32 (𝑡2 )
+
𝑑𝑡2
𝜃𝐼32 (𝑡2 ) =
𝜓𝛽𝑃𝑇1
𝑇3
𝐼33 (𝑡3 ) = ( 𝜓𝛽𝑃𝑇1
𝑇3 −𝑇1
𝑇3
0 ≤ 𝑡1 ≤ 𝑇1
(26)
0 ≤ 𝑡2 ≤ 𝑇3 − 𝑇1
(27)
− 𝜃 𝜓𝛽𝑃𝑇1
𝑇3 −𝑇1
)
𝑇3
0 ≤ 𝑡 ≤ 𝑇 − 𝑇3
(28)
Since for the first cycle, we have no returned items from the previous cycle,𝐼1 (0) =
0, 𝐼2 (0) = 0 then for the first cycle, we have:
𝐼311 (𝑡1 ) =
𝜓𝛽𝑃𝑇1
(1
𝑇3
𝐼321 (𝑡2 ) =
𝜓𝛽𝑃𝑇1
(1 −
𝑇3
𝐼331 (𝑡3 ) = ( 𝜓𝛽𝑃𝑇1
− 𝑒 −𝜃𝑡1 )
𝑒 −𝜃𝑡2 )
𝑇3 −𝑇1
𝑇3
0 ≤ 𝑡1 ≤ 𝑇1
(29)
0 ≤ 𝑡2 ≤ 𝑇3 − 𝑇1
− 𝜃 𝜓𝛽𝑃𝑇1
𝑇3 −𝑇1
)
𝑇3
(30)
0 ≤ 𝑡 ≤ 𝑇 − 𝑇3
(31)
For second, third,…, Nth cycle we have returned items from its previous cycle. So
𝐼312 (0) =
𝜓𝛽𝑃𝑇1
(𝑇3
𝑇3
− 𝑇1 ) − 𝜃 𝜓𝛽𝑃𝑇1
(𝑇3 −𝑇1 )
𝑇3
(32)
𝐼322 (0) = 0
(33)
Then the solution of above differential equations is:
𝐼312 (𝑡1 ) =
𝜓𝛽𝑃𝑇1
(1
𝑇3
𝐼322 (𝑡2 ) =
𝜓𝛽𝑃𝑇1
(1 −
𝑇3
𝐼332 (𝑡3 ) = ( 𝜓𝛽𝑃𝑇1
− 𝑒 −𝜃𝑡1 ) + 𝑒 −𝜃𝑡1 ( 𝜓𝛽𝑃𝑇1
𝑒 −𝜃𝑡2 )
𝑇3 −𝑇1
𝑇3
𝑇3 −𝑇1
𝑇3
− 𝜃 𝜓𝛽𝑃𝑇1
𝑇3 −𝑇1
)
𝑇3
0 ≤ 𝑡2 ≤ 𝑇3 − 𝑇1
− 𝜃 𝜓𝛽𝑃𝑇1
𝑇3 −𝑇1
)
𝑇3
0 ≤ 𝑡 ≤ 𝑇 − 𝑇3
, 0 ≤ 𝑡1 ≤ 𝑇1
(34)
(35)
(36)
The objectives of the problem are minimization of the total expected present value of costs
and minimization of the expected value of inventory over the time horizon. Consider ECP as
the expected present value of costs of production and inspection, ECH as the expected present
value of costs of holding, ECS as the EPV of costs of shortage and ECA as the EPV of costs
of setup. The total EPV of cost over the time horizon (ETC) is:
𝐸𝑇𝐶(𝑛, 𝑇1 , 𝑇3 ) = 𝐸𝐶𝑃 + 𝐸𝐶𝐻 + 𝐸𝐶𝑆 + 𝐸𝐶𝑅
And
(37)
𝑇
𝐼 (̅ 𝑇1 , 𝑇3 ) =
𝑇 −𝑇1
𝑇 −𝑇
𝐼12 (𝑡2 )𝑑𝑡2 +∫0 3 2 𝐼13 (𝑡3 )𝑑 𝑡3 )
(𝑛−1)(∫0 1 𝐼11 (𝑡1 )𝑑𝑡1 +∫0 2
𝑛
𝑇
𝑇 −(𝑛−1)𝑇−𝑇1
(∫0 1 𝐼16 (𝑡61 )𝑑𝑡6 +∫0 5
𝑛𝑇−𝑇5
𝐼17 (𝑡7 )𝑑𝑡7 +∫0
+
𝐼18 (𝑡8 )𝑑𝑡8 )
(38)
𝑛
The detailed analysis is given as follows:
3.1. The EPV of setup cost (ECA)
Consider CA as setup cost, therefore,
−𝑅((𝑗−1)𝑇+𝑇4 )
𝐶𝐴 = 𝐴 + ∑𝑁
𝑗=1 𝐴𝑒
(39)
By replacing equation (18) equation (39), we have
𝑅
𝐸𝐶𝐴 = ∫𝑅 2 𝐶𝐴
1
1
𝑑𝑅
𝑅2 −𝑅1
(40)
3.2. The EPV of production and inspection cost (ECP)
Expect the last period, in any cycle we have production at time 𝑗𝑇, 𝑗𝑇 + 𝑇1 , 𝑗𝑇 + 𝑇4 (𝑗 =
1, . . , 𝑛 − 2)
Consider CP as production cost, therefore
𝑇
𝑇 −𝑇1
𝐶𝑃 = ∑𝑛𝑗=1 [∫0 1(𝑐 + 𝑐1 ). 𝑝. 𝑒 −(𝑗−1)𝑅𝑇 𝑒 −𝑅𝑇1 𝑑𝑡1 + ∫0 2
𝑇−𝑇4
∑𝑛−1
𝑗=1 (∫0
(𝑐 + 𝑐1 ). 𝑝𝑅 . 𝑒 −(𝑗−1)𝑅𝑇 𝑒 −𝑅𝑇1 𝑒 −𝑅𝑡2 𝑑𝑡2 ] +
(𝑐 + 𝑐1 ). 𝑝𝑃. 𝑒 −𝑅𝑇4 𝑒 −𝑅𝑡5 𝑒 −(𝑗−1)𝑅𝑇 𝑑𝑡5 )
(41)
Notice in the last cycle production will occur at times(𝑛 − 1)𝑇, (𝑛 − 1)𝑇 + 𝑇1 .
Therefore the EPV of production and inspection cost is:
𝑅
1
−𝑅
2
1
𝐸𝐶𝑃 = ∫𝑅 2 𝐶𝑃 𝑅
1
𝑑𝑅
(42)
3.3. The EPV of holding cost (ECH)
Let 𝐸𝐶𝐻1 , 𝐸𝐶𝐻2 𝑎𝑛𝑑 𝐸𝐶𝐻3 be the EPV of the holding cost of serviceable items, screened
items and returned items respectively. Then
𝐸𝐶𝐻 = 𝐸𝐶𝐻1 + 𝐸𝐶𝐻2 + 𝐸𝐶𝐻3
(43)
The holding cost of serviceable items for (𝑁 − 1) cycles is
T
T −T1
1
−Rt1
𝐶𝐻11 = ∑n−1
dt1 + ∫0 2
j=1 [h1 ∫0 I11 (t1 )e
T −T2
∫0 3
h1 I12 (t 2 )e−RT1 e−Rt2 dt 2 +
h1 I13 (t 3 )e−RT2 e−Rt3 dt 3 ] . e−(j−1)RT
(44)
For nth cycle:
T
T −(n−1)T+T1
𝐶𝐻1𝑛 = ∫0 1 h1 I16 (t 6 )e−Rt6 e−R(n−1)T dt 6 + ∫0 5
nT−T5
∫0
h1 I18 (t 8 )e−RT7 e−Rt8 dt 8
h1 I17 (t 7 )e−R((n−1)T+T1 ) e−Rt7 dt 7 +
(45)
Then,
𝐶𝐻1 = CH11 + CH1n
And
(46)
𝑅
1
−𝑅
2
1
𝐸𝐶𝐻1 = ∫𝑅 2 𝐶𝐻1 𝑅
1
𝑑𝑅
(47)
In the similar way:
𝑇
𝐶𝐻2 = ∑𝑛𝑗=1 [∫0 1 ℎ1 𝐼21 (𝑡1 )𝑒 −𝑅𝑡1 𝑒 −(𝑗−1)𝑅𝑇 𝑑𝑡1 +
𝑇 −𝑇1
∫0 2
ℎ1 𝐼22 (𝑡2 )𝑒 −𝑅𝑇1 𝑒 −𝑅𝑡2 𝑒 −(𝑗−1)𝑅𝑇 𝑑𝑡2 ]
(48)
And
𝑅
𝐸𝐶𝐻2 = ∫𝑅 2 𝐶𝐻2
1
1
𝑅2 −𝑅1
𝑑𝑅
𝑇
(49)
𝑇 −𝑇1
𝐶𝐻3 = ∫0 1 ℎ1 𝐼311 𝑒 −𝑅𝑡1 𝑑𝑡1 + ∫0 3
𝑇
𝑇−𝑇3
ℎ1 𝐼321 (𝑡2 )𝑒 −𝑅𝑇1 𝑒 −𝑅𝑡2 𝑑𝑡2 + ∫0
𝑇 −𝑇1
1
−𝑅𝑡1 −(𝑗−1)𝑅𝑇
∑𝑁
𝑒
𝑑𝑡1 + ∫0 3
𝑗=2(∫0 ℎ1 𝐼312 𝑒
𝑇−𝑇3
∫0
ℎ1 𝐼331 (𝑡3 ) 𝑒 −𝑅𝑇3 𝑒 −𝑅𝑡3 𝑑𝑡3 +
ℎ1 𝐼322 (𝑡2 ) 𝑒 −𝑅𝑇1 𝑒 −𝑅𝑡2 𝑑𝑡2 +
ℎ1 𝐼332 (𝑡3 )𝑒 −𝑅𝑇3 𝑒 −𝑅𝑡3 𝑑𝑡3 )
(50)
And
𝑅
𝐸𝐶𝐻3 = ∫𝑅 2 𝐶𝐻3 𝑅
1
2 −𝑅1
1
𝑑𝑅
(51)
3.4. The EPV of shortage cost (ECS)
Shortages are not allowed in the last cycle. Consider CS as shortage cost, therefore,
𝑇 −𝑇3
4
𝐶𝑆 = ∑𝑛−1
𝑗=1 (∫0
𝑇−𝑇4
∫0
ℎ2 𝐼14 (𝑡4 )𝑒 −𝑅𝑇3 𝑒 −𝑅𝑡4 𝑑𝑡4 +
ℎ2 𝐼15 (𝑡5 )𝑒 −𝑅𝑇4 𝑒 −𝑅𝑡5 𝑑𝑡5 ) 𝑒 −(𝑗−1)𝑅𝑇
(52)
And
𝑅
𝐸𝐶𝑆 = ∫𝑅 2 𝐶𝑆.
1
1
𝑑𝑅
𝑅2 −𝑅1
(53)
So, the EPV of the total costs of inventory systems over time horizon is obtained using
Equations 40, 42, 43, 53 which is shown in equation 37.
4. Model analysis:
In this section, we first introduce the ideal point approach and then, we use this approach to
solve the model.
Consider the following multi-objective programming problem
Min𝑓𝑗 (𝑥); j=1, 2,..., k
s.t.: gi(x) <0; i=1… m
Where x is a n-dimensional decision vector,
For any 𝑓𝑗 (𝑥) , define the ideal point 𝑓𝑗 (𝑥 ∗𝑗 ) that 𝑥 ∗𝑗 minimize𝑓𝑗 (𝑥).𝑓𝑗 (𝑥 ∗𝑗 ) is the criteria
space, which is called the ideal point. Since, the measure of ‘’closeness”, LP-metric is used.
LP-metric defines the distance between two points 𝑓𝑗 (𝑥)𝑎𝑛𝑑 𝑓𝑗 (𝑥 ∗𝑗 ) in k-dimensional space
as
1
𝑑𝑑 = {∑𝑘𝑗=1 𝛾𝑗 [𝑓𝑗 (𝑥 ∗𝑗 ) − 𝑓𝑗 (𝑥)]𝑑 }𝑑
𝑤ℎ𝑒𝑟𝑒 𝑑 ≥ 1
(54)
Where𝛾𝑗 , j=1... K is relative importance (weights) of the objective function 𝑓𝑗 (𝑥).
The compromise solution, with parameter d, minimizes the dd-metric in (54).
Because of the incommensurability among objectives, it is impossible to use the mentioned
above distance family directly. To remove the effects of the incommensurability, we need to
normalize the distance family of (54) by using the reference point as
1
fj(x∗j)− fj(x) 𝑑 𝑑
] }
fj(x∗j)
𝑑𝑑 = {∑𝑘𝑗=1 𝛾𝑗 [
𝑤ℎ𝑒𝑟𝑒 𝑑 ≥ 0
(55)
Therefore, in this study, we have
Min 𝑑𝑑 (𝑛, 𝑇1 , 𝑇3 ) =
𝐸𝑇𝐶(𝑛∗ ,𝑇1∗ ,𝑇3∗ )−𝐸𝑇𝐶(𝑛,𝑇1 ,𝑇3 ) 𝑑
{𝛾1 [
]
𝐸𝑇𝐶(𝑛∗ ,𝑇1∗ ,𝑇3∗ )
1
+
𝐼(𝑇 ∗ ,𝑇2∗ )−𝐼(𝑇1 ,𝑇2 ) 𝑑 𝑑
𝛾2 [ 1 𝐼(𝑇
] }
∗ ∗
1 ,𝑇2 )
(56)
s.t. 0 < 𝑇1 < 𝑇3
The problem is to determine the optimal value of n, T1, T3 so as to minimize the total
expected inventory system cost and inventory separately and minimize dp. For given values of
n and T1, derive T3 from the following equations.
𝑑𝐸𝑇𝐶(𝑛, 𝑇1 , 𝑇3 )
=0
𝑑𝑇3
ETC (n, T1, T3*) is derived by substituting (n, T1, T3*) into Equation (37).
𝑑𝐼(𝑇1 , 𝑇3 )
=0
𝑑𝑇3
I (T1, T3*) is derived by substituting (T1, T3*) into Equation (38)
Therefore, for given values of n and T1we obtain ideal point of ETC (n, T1, T3) and I (T1,
T3).
For given values of n, T1, we use following equation to determine the optimum value of 𝑑𝑑
𝑑𝑑𝑑 (𝑛, 𝑇1 , 𝑇3 )
=0
𝑑𝑇3
Then, n increase and ETC (n, T1, T3*) and I (T1, T3*) and 𝑑𝑑 (𝑛, 𝑇1 , 𝑇3∗ ) derive again. For
different value of n and T1, the above stages repeat until the minimum 𝑑𝑑 (𝑛, 𝑇1 , 𝑇3 )can be
found.
5. Numerical example
For evaluation the model, the following numerical example is provided.
Table 1. Numerical examples
PARAMETERS
VALUS
UNIT
𝑝
8000
units/year
𝑝𝑠
6500
units/year
𝐴
100
$/unit
𝑐
25
$/unit
H
10
year
c1
1
$/unit
𝜃
0.03
-
𝑅
𝑅~𝑈𝑛𝑖𝑓𝑜𝑟𝑚(0.18,0.21)
-
𝛼
0.02
-
𝛽
0.05
-
a
2000
-
b
100
-
𝛾
0.08
-
𝛾1
3/4
-
𝛾2
1/4
-
Ψ
0.02
-
h1
4
$/unit/year
h2
7
$/unit/year
After calculating with the mentioned method we obtain following results:
Table 2. Experimental results
𝐸𝑇𝐶 𝑇3 ∗𝐸𝑇𝐶
𝐼 𝑇3 ∗𝐼
𝑇3 ∗𝐼
𝑇3 ∗𝑑
n
T
𝑇3 ∗𝐸𝑇𝐶
5
2
0.38
1.109 × 107
0.45
530
0.35
0.0055
10
1
0.18
1.358 × 106
0.22
621
0.16
0.0034
15
2/3
0.12
9.101 × 106
0.15
436
0.11∗
0.0001
20
1/2
0.088
1.436 × 107
0.09
584
0.081
0.004
6. Conclusion
𝑑
𝑑𝑑𝑇
3
∗
In this study a multi-objective inventory model with considering deteriorating items,
imperfect production and imperfect inspection under stochastic inflation rate is developed.
This study is the first paper that considers multi-objective inventory model under stochastic
inflation rate. We assumed that demand rate is a linear function of inflation rate. In this paper
we minimized total cost of inventory system and inventory level by using ideal point
approach. This approach is rarely used for multi-objective of inventory models. The proposed
model can be extended in various ways for future research. For instance, we could consider
(1) minimizing deteriorated items or minimizing dissatisfaction of customers as an object (2)
nonzero lead time (3) lost sales (4) costs of dissatisfaction of customers because of receiving
defective items.
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