Procurement Policy of Vulnerable Parts with Jointly Distributed Lifespan
Procurement Policy of Vulnerable Parts with Jointly Distributed Lifespan
Cun Rong Li 1, Bhaba R. Sarker 2*, Hui Zhi Yi2, and Meng Yu1
* Corresponding author: Tel.: (1) 225 578 5370; E-mail: bsarker@lsu.edu
1School of Mechanical & Electronics Engineering
Wuhan University of Technology
Wuhan, Hubei 430070, P. R. China
2Department of Mechanical & Industrial Engineering
Louisiana State University
Baton Rouge, LA 70803, USA
ABSTRACT
Vulnerable parts exist in most machines or equipment. Failure of those vulnerable parts may result in damage of
the machine or the product, and may sometimes cause unnecessary down-time within a production cycle, all of
which increases the production costs. Therefore, such failures should be minimized and the parts should be
replaced at a point before the end of their expected lifespan, which usually follow some kind of distributions and
can be known due to its mass usage attributes. On the other hand, too early replacement of the parts also means
increased maintenance costs, increased inventory, and increased parts purchasing cost. This study developed a
mathematical model to deal with the issue of determining the optimal stopping time of vulnerable tools. Under the
preemption situation, a periodic procurement policy is investigated simultaneously to find the optimal order cycle
and eventually minimize the total cost. A case study is given to show the feasibility of the model and several
parameters are analyzed to show the impact on the total cost and decision variables.
1. INTRODUCTION
Lifespan estimation has been studied in many areas especially for mechanical components in a manufacturing
company. Vulnerable parts of a machine or equipment are not constantly purchased in a short notice or request. When
the vulnerable parts lose effectiveness, it sometimes causes serious consequence. So it is important to dig into this
situation and try to make right decision on how to choose the right stopping point of part lifespan. Vulnerable parts in
a machine or equipment are usually changed with scheduled timetable. Under the situation of different stopping time,
optimal order size and order cycle of the vulnerable parts also have great significance in maximizing the profitability
and minimizing the costs. The goal of this paper is to propose a cost minimization model (CMM) considering the
jointly distributed lifespan of vulnerable parts which have similar function. The economical maximum allowable usage
time of the vulnerable part and optimal ordering cycle time can be resolved by applying the CMM model by minimizing
the total cost.
There are many researches that concentrated on the lifespan distribution of a product, mechanical part or assembly.
Hiromichi et al. (2010) analyzed the lifetime distribution of more than 0.7 million products sold in Japanese
supermarkets in the area of lifespan estimation. Using Weibull and the artificial neural networks approach, Mazhar et
al. (2007) built functional relationships between historical data and the parameters of the distribution of parts’
remaining lifespan, which made it possible to capture and interpret the existing complex stochastic distribution of the
remaining lifespan of components so that we can make best use of it. Li and Sarker (2013) investigated the traditional
metal cutting theory to predict the cutting tool lifespan efficiently based on the F.W.Taylor’s equation using fixed
cutting factors. Jun et al. (2008) studied an evaluation of the durability of vehicle parts for service life and safety in the
early stages of design, then they used the dynamic stress time history and the modal stress recovery method to predict
the lifespan and applied to the design of vehicle parts. As to the facet of cost assessment related to product lifespan,
Qian and Ben-Arich (2008) presented a methodology for integrating parametric cost estimation with activity-based
costing which can be applied to cost estimations of the design of machined parts in one manufacturing factory. From the
aspect of statistics, many researches put joint distribution into use of operational research (Laih, 2013, Conti et al.,
2013). Some contributions came from reliability engineering, like, used for joint redundancy and imperfect preventive
maintenance optimization with series–parallel multi-state systems (Levitin and Lisnianski, 1999). However, there is
rarely detailed discussion on influence of parts lifespan with procurement policy.
According to specific influence of the vulnerable parts on the machine, a dependent distribution of the machine
needs to be discussed. Former literature can be hardly found on the combination of joint distribution for vulnerable
Flexible Automation and Intelligent Manufacturing, FAIM2014
parts and its host machine. Further, based on the dependent distribution of the machine, the expected effective working
time of the machine need to be discussed which impose an important impact on the procurement policy of the
vulnerable parts. So this paper is to construct a procurement policy model on the vulnerable parts based on their
predetermined distributions.
2. FORMULATION FOR OPTIMAL COST ESTIMATION
The objective of this section is to formulate the cost estimation model with which the total cost of purchasing,
holding, penalty can be minimized.
2.1. ASSUMPTIONS AND NOTATIONS
The combined issue of preemption and procurement for vulnerable parts is quite complex. In order to make a
discreet discussion in this paper, several assumptions are taken into account during the formulation of the cost
minimization process.
Assumptions:
In order to model the issue in a relatively simple way, the following assumptions are necessary in this study:
1. If one vulnerable part fails, the machine encounters a downtime due to the key function of the part.
2. The demand of the product is continuous, and the machine works under a mass production system with
insignificant variation of working time.
3. All parts of the machine are independent so the failure of one part will not lead to the failure of the others.
4. There are multiple vulnerable parts of the same type in a single machine.
5. Failure of a vulnerable part damages the work piece, thus results in a penalty cost.
Notations and definitions summary:
All definitions of parameters, variables, throughout the paper are listed as below:
f c : Fixed cost for every order ($/order),
h c : Unit holding cost of a part ($/part/ year),
u c : Unit purchase (variable) cost of a part ($/part),
p c : Unit penalty cost of part failure ($/part),
f C : Fixed cost of order Replenishment per year ($),
h C : Annual holding cost ($),
pc C : Annual purchasing cost of parts ($),
pt C : Annual penalty cost ($),
p D : Demand of vulnerable parts required per year,
Q : Order size of parts (parts/order),
T : Lifespan of vulnerable part (in time-unit),
c T : Cycle length, the length of time between placement of replenishment orders (year),
M T : Working time for machine or equipment (in time-unit),
w T : Total working time for a single machine per year (or equipment) (time-units/year),
max
M T : Maximum allowable working time for machine or equipment (in time-unit),
TC : Total cost ($/year).
Procurement Policy of Vulnerable Parts with Jointly Distributed Lifespan
2.2. PROBLEM DEFINITION
Equipment or machine is a key factor in regular manufacturing process in manufacturing industry. The machine is
an assembly which consists of many functional subassembly or parts, and there usually exists several vulnerable parts
which play an important role in machining process. Under this situation, it is assumed that the lifespan of the vulnerable
part follows a general distribution: T ~ f (T) . If one of the parts fails, there is a need for change of this part, and will
also result in damage penalty, p c . On the contrary, if the parts with high probability of potential failure are changed
before any of the parts fail, no penalty occurs. However, the preemption of the vulnerable parts results in more
replenishment for changing of the parts, and increase the purchasing cost and holding cost accordingly. So this is a
balance issue for preemption and inventory control with the vulnerable part.
In this issue, every part has a purchasing cost of u c . Unit holding cost per year is given by h c , there exists a fixed
purchasing expense, f c for every order. Based on the optimal decision of the preemption of the machine (or
equipment), max
M T , with a rational order cycle, c T , as decision variables, for a designated period (like, a year),
minimizing the total cost, TC is the objective of this paper.
2.3. ESTIMATION OF EXPECTED LIFESPAN FOR A MACHINE WITH VULNERABLE PARTS
The vulnerable parts are assumed to be the same type of part, they vary from each other due to their individual minor
difference and usually follow a generic distribution T ~ f (T) . Under this circumstance, the maximum working length
of the machine depends on the shortest lifespan of all the vulnerable parts. The shortest lifespan of all the parts is the
bottleneck, and it follows the well-known cask theory. Here, i T , for i = 1,2,...n represents the practical lifespan of each
parts in the machine. The maximum working length for the machine, M T , can be easily expressed as
M T =min( ) i T , for i = 1,2,...n . (1)
Since each part follows the same distribution T ~ f (T) , and there are n vulnerable parts in the specific machine,
according to Equation (1), the distribution of the machine follows a distribution jointly depend on all the parts. The
distribution of the maximum working time for a machine is defined as ( ) M g T , where M T is the minimum lifespan of all
the vulnerable parts. The cumulative probability function for ( ) M g T is defined as ( ) M G T . Since each part of the
machine are independent variables, the distribution of ( ) M g T can be deducted with set theory.
Since M T = min( ) i T , for i = 1,2,...n , the cumulative probability, ( max )
M G T , can be given as
( max )
M G T = {min( ) max}
i M P T < T . By deduction, the final expression of ( max )
M G T illustrated in Figure 3 is described as
( max )
M G T =1- n
M [1− F(T max )] . (2)
By differentiation on equation (2), the probability density function of maximum working time for a machine is given
by
g(T) = n[1− F(T)]n−1 f (T) . (3)
Under the distribution described in equation (3), if a machine is assumed to be the stopping time for parts changing
at the time max
M T = T , there are some probability of machine failure before the stopping point, max
M T , and also probability
of regular running at the stopping time point. As a result, the expected effective working time for a machine composed
of two parts which is marked as area A and B in Figure 1.
For the portion with max
M T < T , marked area , A , as shown in Figure 1, the expected effective working time in the
range max
M T < T , can be expressed as
= ∫ =
max
max
0
( ) ( )
M
M
T
T T E T Tg T dT . (4)
Replace g(T) with equation (3) in equation (4), the effective expected working time in area A is given by
∫ −
= = −
max
max
0
( ) { [1 ( )] 1 ( )}
M
M
T
n
T T E T T n F T f T dT (5)
Flexible Automation and Intelligent Manufacturing, FAIM2014
Figure 1. Expected effective working time for a machine.
For the other portion marked as area B in Figure 3, the effective working length of the machine exceeds the
maximum allowable usage time, max
M T > T ; the practical working length is stopped at the time M T . So, the expected
time length for this preempted portion can be defined as
( max )
M T E = ∫ ∞
max
max ( )
TM
M T g T dT = max [1 ( )]
M M T −G T , (6)
where g(T) follows a distribution described in equation (3).
With the portion marked with B area, combine equation (6) with equation (2), the expected value of this area can be
given by
( max )
M E T = n
M M T max[1− F(T max )] . (7)
Since the total population of the machine is comprised of two groups, A and B as explained above, each group of the
probability has different expected working length for that machine. Under this situation, we now define the total
expected working length of both groups with an assumption of max
M M T < T as
A B E g T + [ ( )] = ( ) ( max )
T T max M E T E T
M
+ = . (8)
2.4. MODELING FOR MINIMIZATION OF TOTAL COST
Under an assumption of situation for massive production, the annual working time for a machine is fixed. As a
result, based on an approximate regular continuous working length (which has an expected time length expressed in
equation (8) once the maximum stopping time, max
M T , is determined) between the vulnerable parts changing, the
vulnerable parts is assumed to follow a fixed-period, c T , review policy where an order of Q parts/cycle is placed to
meet the total yearly demand of vulnerable parts. The total cost includes fixed cost of a replenishment order f C ,
holding cost h C , purchasing cost pc C and penalty cost pt C . Then the total cost for a purchasing cycle, TC can be
expressed as
f h pc pt TC = C + C + C + C . (9)
The total working time for a single machine in a year is w T . From equation (8), the expected continuous working
length without parts changing is A B E g T + [ ( )] . The number of vulnerable parts for changing every time is n because
there are totally n same parts in a machine. So, the total demand of vulnerable parts is
p D =
A B
w
E g T
nT
+ [ ( )]
(10)
c T is defined to be the purchasing cycle (in year), and f c is the fixed purchasing expense for every time, so the
annual total fixed purchasing cost can be calculated as
Optimal stopping lifespan, max
M T
Density
Working time, TM
A
B
g(T) = n[1− F(T)]n−1 f (T)
Expected working time, E(TM)
Procurement Policy of Vulnerable Parts with Jointly Distributed Lifespan
f C =
c
f
T
c
, (11)
Annual holding cost, h C , can be calculated as the unit holding cost h c multiplied by the average inventory, which
can be expressed as I = Q/ 2 = / 2 p c D T . Replacing p D with equation (10), the annual total holding cost is given by
h C =
A B
w h c
E g T
nT c T
+ 2 [ ( )]
(12)
For annual parts purchasing cost, it is only an issue with purchasing quantity, so, it can be easily expressed as
pc C =
A B
u wT
E g
nT c
+ [ ( )]
(13)
Annual penalty cost is related to the yearly total failures of the machine (or equipment). ( max )
M G T , the cumulative
probability of the machine failure that occurs before the stopping time, max
M T , is expressed in equation (2) as ( max )
M G T
=1- n
M [1− F(T max )] . The annual total penalty cost, pt C = ( max )
p p M c D G T , combined with equation (10), thus can be
rewritten as
pt C =
A B
n
w p M
E g T
nT c F T
+
− −
[ ( )]
{1 [1 ( max )] }
. (14)
The total cost, TC , by merging equation (9), (11), (12), (13) and (14), can be deducted as
( max , )
M c TC T T =
c
f
T
c
+
A B
n
w h c w u w p M
E g T
nT c T nT c nT c F T
+
+ + − −
[ ( )]
/ 2 {1 [1 ( max )] }
. (15)
As shown in equation (15), there are mainly two components, and one component (fixed purchasing cost) is a
function of order cycle, c T . The other one regarding to holding cost, part purchasing cost and penalty cost is related to
the expected maximum working length, A B E g T + [ ( )] . These relationships make the analysis more observable.
3. SOLUTION FOR CMM FUNCTION
There are two decision variables in equation (15), and the objective of this function is to minimize the total cost. In
order to find the stationary point of this equation, differentiation method can be employed to resolve this issue.
3.1. MATHEMATICAL SOLUTION
In order to find the stationary points for the objective function expressed in equation (15), partial differential
methods on both c T and max
M T are required to retrieve its close form expression. By differentiation on c T ,
M c c ∂TC(T max ,T ) / ∂T = 0 yields
c T =
h w
n
M M
T
n
f
nc T
c T n F T f T dT T F T
M
∫ − + − 2 { [1 ( )] − ( )} max[1 ( max )]
0
1
max
(16)
By differentiation on max
M T with objective function (15), let ( max , ) / max M c M ∂TC T T ∂T = 0, a final expression is given as
follows (See Appendix B for detailed deduction process).
∫ − + − − n
M M
T
n
p M nc f T T n F T f T dT T F T
M
( ) { [1 ( )] ( )} max[1 ( max )]
0
max 1
max
- ( / 2 {1 [1 ( max )] })[1 ( max )]
M
n
h c u p M c T + c + c − − F T − F T = 0 (17)
In order to eliminate one of the decision variables, by merging equation (16) and (17), a final expression with only
one decision variable, max
M T , is given by
− + −
− − ∫ − n
M M
T
n
M
w
f
p M F T T n F T f T dT T F T
nT
c
nc f T
M
( ) [1 ( )] { [1 ( )] ( )} max[1 ( max )]
0
max max 1
max
Flexible Automation and Intelligent Manufacturing, FAIM2014
- ( {1 [1 ( max )] })[1 ( max )]
M
n
u p M c + c − − F T − F T = 0. (18)
From equation (18), we cannot get a close form expression for the decision variable, max
M T . But using equation (15),
the computational amount can be decrease dramatically, so a minimum-oriented searching (MOS) algorithm is
employed to resolve its approximate solution. With further calculation, decision variable, c T , the order size, Q , which
depends on c T , and minimum total cost, TC , can be resolved accordingly.
3.2. MINIMUM-ORIENTED SEARCHING ALGORITHM
Given every value of max
M T , a corresponding result, c T , can be computed with equation (16), and then the minimum
value of ( max , )
M c TC T T is determined in this searching process. The simplified MOS algorithm can be illustrated as a
flow chart shown in Figure 2.
Figure 2. Simplified flow chart of MOS searching algorithm.
3.3. A CASE STUDY
In order to verify the CMM model by employing MOS searching Algorithm, a case study is conducted to show its
feasibility. In this example, normally distributed vulnerable parts are given to pursue this exploration. Experimental
equipment is designed for testing of electromagnetic valve. The equipment has 4 customized gear-racks which are key
functional parts for testing process, and the part is also a quick-wear part due to its frequent changing of testing work
piece. Under this situation, the lifespan of these vulnerable parts follows a normal distribution with p μ = 200 hours and
σ = 5. An equipment need to work for a total 3650 hours per year under a massive production requirement which
means a steady demand of working time. For parts purchasing process, there is a fixed order cost, f c = $30/order and
unit cost, u c =$20/unit. In testing process, unexpected failure for each of the 4 parts has potential damage for the testing
work piece and result in an average penalty cost, p c =$1/unit, consequently, all the 4 parts need to be changed to avoid
further damage. For every part in the stock, there is an annually holding cost, h c =$3/unit/year. Under this situation, a
maximum working length (or stopping time) for the equipment, max
M T , and order cycle, c T are to be resolved to
minimize the total cost, TC .
By running the MOS program with an initial step size Δ =10 and stopping criteria s = $0.01, Table 1 shows the
searching result of this vulnerable parts changing and procurement problem. The optimal decision variables,
max
M T =191.0006 hours, order cycle c T =0.382 years ≈139 days, order size Q =13.09 ≈13 units and the minimized total
cost TC* =$2,918.547.
max = max + Δ
M M T T
Yes
Δ ←Δ / 2 No
Start
Compute ( max , )
M c TC T T with equation (19)
( max )
M TC T - * ( max )
M TC T < s
max* max
M M T = T , * ( max , )
M c TC T T = ( max , )
M c TC T T , calculate *
c T and Q*
Stop
Initialization for f c , p c , h c , u c , s , w T , Δ
Set the starting value max = Δ
M T
Compute A B E g T + [ ( )] with equation (8) and c T
with equation (16)
Procurement Policy of Vulnerable Parts with Jointly Distributed Lifespan
Table 1 show that the MOS algorithm is efficient in finding the optimal solution under designated stopping criteria,
s . The convergence of this algorithm is proved to be robust in similar problem simulation.
Table 1. Searching result for CMM with MOS algorithm.
Step No. Stopping time, max
M T Stop criteria, s Step size Δ Total cost, TC
1 11.0000 474,238.335 10.00000 47,924.996
2 21.0000 22,691.014 10.00000 25,233.982
3 31.0000 8,071.243 10.00000 17,162.739
… … … .. …
16 191.1563 1.926 0.15625 2,920.467
17 191.0781 0.910 0.07813 2,919.451
18 191.0391 0.442 0.03906 2,918.983
… … … … …
22 191.0024 0.027 0.00244 2,918.568
23 191.0012 0.013 0.00122 2,918.554
24 191.0006 0.007 0.00061 2,918.547
4. PARAMETRIC ANALYSIS
The maximum working length of a machine, max
M T , is determined by the minimum lifespan of all the vulnerable parts
in this machine, so the number of parts, n , has an impact on regular working time. Besides, from the view of
management, holding cost, h c , and penalty cost, p c also impose significant influence on the order size and machine
stopping time, and eventually have impact on the total cost. TC . In this sector, insight analysis of h c , p c and the parts
number, n is conducted as influence on the total cost.
4.1. VARIATION OF PENALTY COST p c AND HOLDING COST h c
Penalty cost is a factor which influence the total cost when failure occurs, as a result, in order to decrease the
negative impact of penalty cost, an earlier preemption of the machine for parts changing is preferred. For verification,
a set of penalty cost, p c and holding cost, h c are tested to compare the difference of optimal value of m T . By adopting
the same example in Section 3.3, testing parameters, p c is set to 0.5, 1.0, 2.0 dollars per unit, and h c is set to 3.00,
4.00, 5.00 dollars per unit per year, respectively. The computational results with MOS algorithm are shown in Table 2.
Table 2. Comparison of impact on m T for different values of p c and h c .
Holding cost
h c ($/unit)
Penalty cost
p c ($/unit)
Stopping time
max*
M T (Hours)
Order cycle
*
c T (Days)
Order size
Q*
TC* ($)
3.00 0.50 191.0012 139 13 2,909.21
3.00 1.00 191.0006 139 13 2,918.55
3.00 2.00 191.0003 139 13 2,937.22
2.00 1.00 191.0006 171 16 2,889.72
4.00 1.00 191.0006 121 11 2,942.85
Table 2 also shows that increase of penalty cost, p c , resulting in earlier preemption for the working machine and it
is beneficial to the final result of total cost. Decease of order size is preferred when higher unit holding cost rises.
However, total cost get higher whenever what policy is adopted. In management aspect, reasonable management level
with rational value of p c and h c can thoroughly decrease the total cost to a relative low level.
4.2. INFLUENCES OF VARIANT PARTS, n
Equation (3) gives the probability density function of failure time for n parts, where n , represents the quantity of
the same vulnerable parts in a machine and have impact on the maximum allowable working time for the machine. It is
Flexible Automation and Intelligent Manufacturing, FAIM2014
depicted that with the increase of the vulnerable parts in a machine, the expectation of the function, g(T) , decreases.
However, the variation of the function falls into a smaller range. Since the machine is affected by n parts, the allowable
maximum working time decreases accordingly.
In order to reveal the influence of the quantity of the vulnerable parts, different quantity of parts ( 3, 4, 5, and 6) are
input into CMM model by computing with MOS algorithm. Table 3 shows the comparison of the impact of TC* with
different quantity n . With the increase of the quantity of vulnerable parts, the maximum allowable stopping time,
max*
M T decreases. The change with max*
M T also reflects the decrease of a jointly overall expected working time with the
vulnerable parts.
Table 3. Comparison of impact on TC* with different quantity n .
quantity
n (units)
Stopping time
max*
M T (Hours)
Order cycle
*
c T (Days)
Order size
Q*
TC* ($)
3 191.0195 161 15 2,196.12
4 191.0006 139 13 2,918.55
5 191.0003 124 12 3,649.37
6 191.0001 113 11 4,391.26
5. CONCLUSION
This paper studied the procurement policy for vulnerable parts which are key components in a machine. The
dependent distribution of the machine, which is jointly formulated from the generic distribution of the vulnerable parts,
is investigated. Under a preemption of machine working time, the expectation of the transformed distribution of the
machine is studied. Based on this research, a procurement CMM model was proposed to minimize the total cost. In
order to verify the feasibility of the CMM model, an example with normally distributed vulnerable parts was given to
simulate the computational process and the result illustrated the effectiveness of the model. Further, several parameters
are studied to analyze to what extent these parameters influence the final total cost.
ACKNOWLEDGEMENTS
This research work at Louisiana State University, Baton Rouge, LA 70803 was supported by China Scholarship
Council under the Grant Number: 2011-69-5508.
REFERENCES
[1] Conti, P. L., Marella, D., Scanu, M., (2013), “Uncertainty analysis for statistical matching of ordered categorical
variables,” Computational Statistics & Data Analysis, 68(12): 311–325.
[2] Hiromichi, U., Tsutomu, W., Misako, T., (2010), “Life-span distributions of supermarket products,” Journal of Physics,
221(2010)012018.
[3] Jun, K. J., Park, T. W., Lee, S. H., Jung, S. P., Yoon, J. W., (2008), “Prediction of fatigue life and estimation of its reliability
on the parts of an are suspension system,” International Journal of Automotive Technology, DOI
10.1007/s12239-008-0088-4.
[4] Laih, Y. W., (2013), “Measuring rank correlation coefficients between financial time series: A GARCH-copula based
sequence alignment algorithm,” European Journal of Operational Research, 232(2): 375–382.
[5] Levitin, G,. Lisnianski, A., (1999), “Joint redundancy and maintenance optimization for multistate series-parallel systems,”
Reliability Engineering & System Safety, 64(1): 33–42.
[6] Li, C. R., Sarker, B. R., (2013), “Lifespan prediction of cutting tools for high-value-added products,” International Journal
of Advanced Manufacturing Technology, 69(5-8): 1887–1894.
[7] Mazhar, M. I., Kara, S., Kaebernick, H., (2007), “Remaining life estimation of used components in consumer products: Life
cycle data analysis by Weibull and artificial neural networks,” Journal of Operations Management, 25(2007) 1184–1193.
[8] Qian, L., Ben-Arieh, D., (2008), “Parametric cost estimation based on activity-based costing: A case study for design and
development of rotational parts,” Int. J. Production Economics, 113 (2008): 805–818.