Quality Technology &
Quantitative Management
Vol. 10, No. 2, pp. 221-240, 2013
QTQM
© ICAQM 2013
A Dominant Maintenance Strategy Assessment Model
for Localized Third-Party Logistics Service under
Performance-based Consideration
Yi-Kuei Lin, Jong-Jang Lin and Ruey-Huei Yeh
Department of Industrial Management
National Taiwan University of Science and Technology, Taipei, Taiwan
(Received January 2012, accepted July 2012)
______________________________________________________________________
Abstract: The partnerships of performance-based contracting (PBC) between government and original
equipment manufacturer (OEM) have been well demonstrated in most studies, but very few attempts have
been made at such partnerships between a foreign government (FG) and a localized third-party logistics
(3PL) supplier while they operated the same system. This article constructs a principal-agent model to
support resource allocation, and then uses it to analyze commonly observed risk-aversion contracting
between a FG customer (principal) and a localized 3PL logistics supplier (agent) with cooperative game
combinations by fixed payment, cost-sharing incentive, as well as a performance incentive conditions.
Finally, a real military logistics service application in Taiwan is demonstrated by the assessment model to
generate the maximum utilities while under cost-sharing incentive condition by using offset obligation
between FG and OEM.
Keywords: Dominant strategy, performance-based contracting, principal-agent, risk-aversion, third-party
logistics (3PL).
______________________________________________________________________
Acronyms
3PL
third-party logistics
DCS
direct commercial sales
FB
first-best solution
FG
foreign government
FMS
foreign military sales
MRO
repair and overhaul
OEM
original equipment manufacturer
PBC
performance-based contracting
PBL
performance-based logistics
SCMP
strategic commercial maintenance policy
TG
Taiwan government
Notation
a
cost reduction effort
B
backorder
CT
total cost
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Cs
fixed cost
F(x)
cumulative distribution function (cdf)
f(x)
probability density function (pdf)
H
Bordered Hessian determinant
i
index to indicate ith subsystem
j
index to indicate jth subsystem, where i≠j
L
repair lead time distribution
n
distinct subsystems in a system
N
number of system
O
repair pipeline (on-order) inventory
r
risk aversion ratio
s
inventory of subsystem
TC+
cost-plus contract
TF
fixed price contract
TP
fixed price contract combined with penalty item
TPB
performance based contract

cost sharing weighting factor

uncertainty in total cost
δ
Lagrange multiplier
f
hardware cost
a
software cost
m
summation cost for all MRO actions
λ
Poisson rate
μ
mean repair lead time

penalty weighting rate
ρ
coefficient of variation
σ
standard deviation
 a 
incurring disutility

fixed payment
1. Introduction
T
here has been an increasing interest about the performance-based logistics services
contracting relationship between customer and service supplier in capital-intensive
industrial domains such as aerospace, defense and public transport since the last decade of
20th century [18]. In fact, performance-based contracting (PBC) is a major shift in logistics
for highly complex systems for offering a novel logistics service approach in the given
industries above. It is also known as “power-by-hour” in commercial airliners or
“performance-based logistics” (PBL) in defense industries by assessing industry best
A Dominant Maintenance Strategy Assessment Model
223
practices and developing a dominate maintenance strategy [6]. This effort especially
emphasized the importance of the best-value partnerships between government and
industries about the maximum operational effectiveness of the system, i.e. purchasing
system performance outcomes instead of individual parts or repair actions on a
predetermined level of availability to meet the customer’s objectives. As the result, there
exists a critical element of PBC about the clear separation between the customer’s
expectations and the supplier’s implementation. Furthermore, the PBC explicitly identifies
what is required, but the supplier concentrates their attentions on fulfilling the requirement
[11]. As a consequence of this flexibility, PBC should promote new and improved ways to
manage spare-parts inventory, reduce administrative overhead, negotiate contracts, and
make resource allocation decisions.
Over the past years, most studies about logistics services partnerships between
government and industries focused on customer who operates N identical-assembled
products (“systems”, which should be helicopters, tanks, frigates and other equipments etc.).
Each system is composed of n distinct major subsystems, e.g. a helicopter can represent as
avionics, engine(s), landing gear, main/tail rotor blades and/or fire-control (weapon) etc.
Meanwhile, each subsystem should be produced and maintained by a unique original
equipment manufacturer (OEM), and be simplified as a single composite item by ignoring
the indenture structure in the subsystem’s bill of materials [6]. The above-mentioned
framework has been well demonstrated in logistics services partnerships between customer
and OEM, e.g. the U.S. government and its defense industries.
There are three distinct models are blended together in this article: First, Sherbrooke
[16] introduced METRIC model for heuristic optimization algorithms in allocating
inventory resources for multi-echelon and multi-indentured environments. Subsequent
models were focused on improving computational efficiency and incorporating more
realistic modeling assumptions by enabling the management of service parts inventory
resources planning in real applications, e.g. aerospace and defense industry [17], as well as
developing a framework for software solutions in various industries [4]. Second, numerous
papers studied the principal-agent models and applied with various field, Scherer [15]
introduced the theory of contractual incentives to optimal cost sharing under the impact of
risk aversion condition in defense procurement, McAfee and McMillan [11] introduced an
analysis for government contract bidding under significant cost-related risks. Furthermore,
Kim et al. [6-7] focused on outcome-based reimbursement policies for risk aversion, and
studied cost-plus and fixed-price contracts in the context of after-sales support and compare
them with performance-based contracts between OEM and government. Finally,
Pasternack [14] showed that coordination between a customer and a supplier can be
achieved with buy-back contracts. Cachon [1] presented a survey about supply chain
coordination with contracts, which noted numerous contracts could be utilized to achieve
supply chain coordination and the subsequent profit allocations differ. Nagarajan and
Sošic´ [13] gave a survey on using cooperative bargaining models to allocate profit between
supply chain members. Zhao et al. [21] focused on the economics efficiency of options, and
deployed a cooperative game approach to address the coordination of supply chains by
option contract problems.
However, very few attempts have been made at such partnerships between a foreign
government (FG) and localized third-party logistics (3PL) supplier, and those observed
phenomenon real existed in many other countries where OEM does (and will) not locate
his branch MRO facilities. For example, a FG customer operates military related systems,
which were purchased by foreign military sales (FMS) or direct commercial sales (DCS)
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channels from U.S. government, therefore some of the subsystems are maintained by
localized 3PL Supplier(s), which are located at the same country or location with the FG
customer for the system’s requirements of availability. The above relationship result from
the 3PL supplier may award the logistics service contract under authorities from OEM and
some incentive conditions from the FG customer, e.g. request OEM transfer the MRO
technologies to 3PL supplier by using the obligations of offset or industrial cooperation
agreement between OEM and FG customer. The above-mentioned relationship between
government, OEM, FG and 3PL supplier can be illustrated in Figure 1.
Foreign Country
FG
Contracting for
Logistics Service
Country of OEM
FMS
Offset
Obligations
3PL Supplier
USG
Contracting for
Logistics Service
OEM
Authorities
Figure 1. Relationships between government, OEM, FG and 3PL supplier.
Such a radical variation in the contracting approach has caused confusion among
supplier of logistic services, as well as few academic literatures develop the frameworks
with respect to how such contracting should be evaluated. On the basis of an increasing
interest in the partnerships between FG customer and 3PL supplier, this article will
interpret as follows: First, we interpret the observed phenomenon about the major
difference of MRO sequence between OEM and 3PL supplier, then derive cost structure
and specify contracting terms by principal-agent model. Second, we present four
contracting types in order to determine the maximum expected utility of 3PL supplier by
using optimization with equality constraints method, under considering the 3PL supplier is
risk-averse and its actions can be completed observable, i.e. the first-best solution (FB), due
to the FG customer and 3PL supplier constitute a coalition by contract selection and
enforcement, and which may enforce cooperative behavior as a typical cooperative game.
Finally, a real contracting evaluation framework for military logistics service case study in
Taiwan is demonstrated. The result of our framework indicates the expected utility under
incentive condition, which is souring from the aforementioned offset obligations between
Taiwan government (FG customer) and OEM set, should constitute a dominant
maintenance strategy for both government and 3PL supplier parties through various
contracting assessment.
1.1. MRO Sequence via OEM and 3PL
The current studies employed a typical sequence of MRO for a subsystem i by OEM
[6] as the following steps:
A Dominant Maintenance Strategy Assessment Model
225
(1)
Failures of the subsystem i is assumed to occur at a Poisson rate λ and
independently from failures of any other subsystems j.
(2)
OEM sets up an inventory of spares, and maintains a repair facility for actions
of MRO, since the subsystem i was in develop phase for the system. Those
inventory and facility were controlled by a one-for-one base stock policy.
(3)
A failed assembly/component (unit) in subsystem i should be replaced by a
working unit (if it is available) from the inventory of supplier i immediately.
(4)
If the replacement is unavailable, a backorder Bi should be occurred, and results
in subsystem i becomes inoperable.
(5)
The failed unit enters the repair facility, modeled as an M/G/∞ queue and
ample capacity [17].
Figure 2 illustrates this sequence between customer and OEM.
Backorder
Inventory
Owned by
OEM
Subsystems
in Deployment
Repair Facility
Figure 2. The current typical MRO sequence by OEM [6].
This paper employed a few attempted to occur at a Poisson rate λ and independently
from failures of other subsystems j.
(1)
The 3PL supplier sets up and maintains a repair facility under the assistance
from OEM, and the FG customer maintains an inventory of spares. Those
inventory and facility were controlled by a one-for-one base stock policy.
(2)
A failed assembly/component (unit) in subsystem i should be replaced by a
working unit (if it is available) from the inventory of supplier i immediately.
(3)
If the replacement is unavailable, a backorder Bi should be occurred, and results
in subsystem i becomes inoperable.
(4)
The failed unit enters the repair facility, modeled as an M/G/∞ queue and
ample capacity.
Figure 3 illustrates this sequence between FG customer and localized third-party
supplier.
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Backorder
Owned by
FG Customer
Inventory
Owned by
3PL Supplier
Subsystems
in Deployment
authorities
OEM
Repair Facility
Figure 3. The MRO sequence by 3PL supplier.
From those two MRO sequences have been discussed above, we can recognize that
there existed major difference in step 2 as follows:
First, the OEM may ignore the cost of establishing the MRO facility in OEM sequence,
due to which has been apportioned charges previously in the development phase of the
subsystem; however, the cost of establishing the MRO facility and the technical assistances
cost must be considered in 3PL supplier sequence, and this circumstance let the 3PL
supplier seriously consider the risks to implement the MRO contracting for the subsystem
before not awarding the MRO authorities from OEM and export license from the country
of the OEM; and not having an outsourcing volition from FG customer.
Second, the OEM usually set up the inventory of spares in OEM sequence, differs
greatly from the FG customer set up the inventory of spares in 3PL supplier sequence. It
was resulted from the FG customer usually considered the uncertainties or risk factors
which the 3PL supplier may not have or out of the MRO authorities from OEM by export
control regulations or other business considerations; and the FG customer may not deal
contract(s) with the 3PL supplier/OEM for some reasons.
Table 1 summarizes the major difference between the current typical MRO sequence
by OEM and the MRO sequence by 3PL suppliers.
Table 1. Major difference in MRO sequence between OEM and 3PL suppliers.
Sequence by
Deference items
Cost to establish MRO facility
Cost for inventory of spares
OEM
(Typical)
Ignore
Set up by OEM
3PL supplier
Facility cost and technical
assistances cost from OEM
Set up by FG customer
2. PBC Analysis of 3PL Supplier
Consider the failure of the subsystem i is assumed to occur at a Poisson rate λ, and
independently from failures of any other subsystem j. The FG customer maintains an
inventory of spares and which is controlled by a one-for-one base stock policy, which allow
a failed subsystem may be replaced from the inventory immediately under the replacement
A Dominant Maintenance Strategy Assessment Model
227
is available; or a backorder may occur under the replacement above is unavailable. This
backorder is not only assumed to affect system inoperable, but the downtime of the
subsystem also leads the downtime of the system.
As regards the failure, we idealize the repair facility with ample capacity, i.e. infinite
number of servers; the defective subsystem enters the repair facility and modeled as an
M/G/∞ queue. It is considered a reasonable approximation in many circumstances, and
also leads to the repair lead times of different items are independent [17]. It takes on
average lead time Li to repair the failure of the subsystem, and once the task is completed,
the repaired subsystem is placed in the inventory. In addition, forward and return
transportation lead times are incorporated into the repair lead time, and which are assumed
to be independent of the customer location [20].
The backorder Bi of subsystem is a random variable which is observed at a random
point in time while steady state is reached. The FG customer chooses a spare stocking level
si for subsystem i, therefore these are related to each other through
Bi  Oi  si  .

(1)
The stationary random variable Oi represents the repair pipeline (on-order) inventory,
and Feeney and Sherbrooke [5] introduced Palm’s Theorem to state that Oi is Poisson
distributed for any repair lead time distribution with the mean of the subsystem i
i   Li .
(2)
Consider the subsystem is typically expensive and have long-turn operational life cycle,
no subsystem is discarded during the entire lifecycle of system. Finally the MRO sequence
is a close-loop sequence and FG customer owns a total of nN + si units of subsystem.
This failure rate is fixed as an approximation, due to the finite population means λ is
a function of the number of working units in this closed-loop sequence. Nevertheless, this
approximation is reasonable in our assumptions context is satisfied in practice for most
repairable subsystems on basis of the condition:
E  Bi si    Li  nN  si .
(3)
This condition ensures that, on average, the number of subsystems being repaired at
any given time is relatively smaller than the total of nN + si units of the subsystem, and the
correction due to state dependency can be ignored.
Afterward, the repair pipeline (on-order) inventory Oi is distributed continuously with
cumulative distribution function F(x) and probability density function f(x), which have
nonnegative conditions for  0,  and F (0)  0 . Therefore, the distribution of Bi can be
obtained from
Pr  Bi  x si   Pr O  x  si  ,
(4)
E  Bi si   si 1  F  x  dx .
(5)
Than we derive the differentiation from (5),
dE  Bi si 
 1  F  si   0,
ds
(6)
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d 2 E  Bi si 
dsi2
 f  si   0.
(7)
Hence the expected backorder Bi is decreasing and convex in inventory si.
2.1. 3PL Supplier Cost
We use fixed and variable components to derive the fixed cost Cs of the 3PL supplier.
The fixed components contain with building up cost term, e.g. hardware section such as
facility founding cost  f ; software sections such as authorities and technical assistance (by
OEM) cost a , and the summation cost for all MRO actions m through
C s   f  a  m .
(8)
The fixed cost of logistics may be reduced by an incentive item a, which is the 3PL
supplier’s cost reduction effort by itself. We set there exist an incurring disutility   a  of
the 3PL supplier, which is a convex increasing, therefore    a   0 and    a   0 . In
this article, we assume   a  can be assessed by the FG customer. This convention, we
effectively assume that the reduction effort a is the 3PL supplier’s own discretionary
decision and, hence, the FG customer does not subsidize the 3PL supplier’s internal cost
for it. In other words, the FG customer reimburses the undisputable direct cost only. In the
sequel, Chen [2] assumed a quadratic functional form
 a  
ka 2
, k  0.
2
(9)
On the basis of the above discussion, Laffont and Tirole [8] presented a relationship,
which is under an observable condition by the FG customer and a basis of reimbursement
conditions, about the total cost CT for the 3PL supplier through
CT  C s  a   .
(10)
Where the variable component ε represents the uncertainty in total cost that is
beyond 3PL supplier’s control, and it is uncorrelated with backorder Bi, we therefore obtain
Cov  , Bi   0 .
This assumption does not consider an alternative, whereby the 3PL supplier’s effort
impacts the reliability and/or repair capabilities of the subsystem with or without extra
technical assistance from OEM.
2.2. Contracting Terms and Utilities
We now consider one payment of FG customer to one 3PL supplier which is
comprised of a fixed payment i , a reimbursement for the 3PL supplier’s total cost CT ,
and a backorder-contingent incentive payment Bi . Specifically, it has the form
T CT , Bi     CT   Bi .
(11)
Where the parameters  ,  and  are determined by both FG customer and 3PL
supplier in the contract. Parameter  is the fixed payment,  is a cost sharing weighting
factor of 3PL supplier by FG customer, and  is the penalty weighting rate for backorders
Bi incurred by the 3PL supplier.
Furthermore, Scherer [15] demonstrated the risk-averse of the 3PL supplier by
expected mean-variance utility through
A Dominant Maintenance Strategy Assessment Model
229
E U  X    E  X  
rVar  X 
2
(12)
.
The risk aversion factor r with 0  r  , and r  R , such that greater r is the more
risk aversion the 3PL supplier has. This utility function has been widely used in recent
operations management literature because of its tractability.
Kim et al. [6] derived the expected utility function E U s (T (CT , Bi )  CT )   (a ) a, si 
of the 3PL supplier by a given contract T CT , Bi  through Equations (9) and (10)
ka 2
  (1   )(C s  a )   E  Bi si  
 r (1   ) 2
2
Var  
2
 r 2
Var  Bi si 
2
.
(13)
The first three terms together of (11) represent the expected net income of the 3PL
supplier, whereas the fourth term is internal disutility for exerting cost reduction effort. The
last two terms, respectively, represent risk premiums associated with cost and performance
uncertainties.
is
Similarly, the expected utility function E U c ( Ti (CT , Bi )) a, si  of the FG customer
(   (C s  a )   E  Bi si   r  2
Var  
2
 r 2
Var  Bi si 
).
2
(14)
The first three terms together of (12) represent the expected net payment of the FG
customer; the last two terms represent the penalties in case backorders Bi incurred in the
MRO sequence. The 3PL supplier is assumed to have fixed reservation utility in one
contract duration; therefore he can gain by not participating in the trade with the FG
customer.
Our representation of the logistics services support relationship is based on the
standard single-location, steady-state repairable model with a take-it-or-leave-it contract.
Under the assumptions of the model, the sequence of events is as follows:
(1) The FG customer acquired numbers of the system and set the base stock levels
of spares inventory;
(2) The FG customer offers the 3PL supplier a take-it-or-leave-it contract;
(3) The 3PL supplier accept or reject the contract, under the technical assistance by
OEM;
(4) Once the 3PL supplier accept the contract, who will set the maintenance facility
and take cost reduction measures by OEM;
(5) Realized costs and backorders are evaluated at the end of the contract horizon;
(6) 3PL supplier is compensated according to the contract terms.
3. Constitute Dominate Strategy
Consider the FG customer and 3PL supplier has submitted the above Sequence of
Contracting Events in a PBC environment, i.e. the FG customer and 3PL supplier
constitute a coalition by contract selection and enforcement, and which may enforce
cooperative behavior as a typical cooperative game [12]. Furthermore, the 3PL supplier is
risk-averse and its actions can be completed observable, i.e. under the first-best solution
(FB), hence we can evaluate the effectiveness of the most widely used contract forms by
controlling  and  in (8):
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(1)
A pure fixed price contract TF with   0 and   0 , thus the contracting term
TF CT , Bi    .
(2)
A common practices that combined TF with penalty item TP by 0    1 and
  0 , thus TP CT , Bi      Bi .
(3)
A cost-plus contract TC+ with full reimbursement by   0 and 0    1 , thus
TC  CT , Bi     CT .
(4)
A performance based contract TPB with
TPB CT , Bi     CT   Bi .
0  1
and 0    1, thus
The above proposition is simply demonstrated in real practices: first, the FG customer
keeps to minimum budget and risk for system sustainment, it results the FG customer
serious concern the penalty  in contract to ensure the claim can be effective protected.
Secondly, the 3PL supplier keeps to maximum utilities and it causes the 3PL supplier shall
make his best cost sharing effort  to ensure the maximum profit. Table 2 summarizes the
FG customer and 3PL supplier behaviors under all of these contract combinations, which
can be integrated as a typical normal form of game theory.
Table 2. Incentive effects of various contracting combinations.
Cost sharing
Penalty
FG
Customer
3PL Supplier
 0
TF  C , B   
TP C , B      B
 0
0  1
0  1
TC  C , B     C
TPB C , B     C   B
Therefore we can derive the expected utility function E U ( x ) of the FG customer
from (14) and the 3PL supplier from (13) with the above contracting type by weighting
parameters  and  . Table 3 summarizes the expected utility combinations of the FG
customer and 3PL supplier in the above contracting type.
Table 3. Expected utility of various contracting combinations.
Type
3PL Supplier E U s ( x )
FG customer
TF

TP
 (  EBi si   r 2
Var Bi s i
TC+
 (   (C s  a)  r 2
Var  
)
2

2
)
 (   (C s  a)  E Bｉ sｉ  r 2
TPB
 r 2
ka 2
Var  
r
2
2
ka 2
  (C s  a)  E Bi si  
2

Var
B


Var 
i si 
r
 r 2
2
2
ka 2
Var  
  ( 1   )( C s  a ) 
 r( 1   )2
2
2
ka 2
  (1   )(C s  a )  E Bｉ sｉ 
2
Var Bｉ sｉ
2 Var  
2
 r (1   )
 r
2
2
  (C s  a ) 
VarBｉ sｉ
)
2
Var 
2
3.1. Optimization with Equality Constraints
The primary purpose of imposing a constraint is to give due cognizance to certain
limiting factor present in the optimization problem under discussion. In real practices for
any contracting, the best utilities in some constrained conditions, i.e. finding the stationary
A Dominant Maintenance Strategy Assessment Model
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values, are always the most interested issue for every contracting party. When the constraint
is itself a complicated or when there are several constraints to consider in Table 2, we here
resort to a known method of Lagrange-multiplier [19] to convert the four contracting types
of constrained-stationary value into a form, hence we let  is the Lagrange multiplier to
observe the optimization with equality constraints by the following four theorems to specify
the conditions with constraint varies ( ,  ) in Table 2.
Theorem 1: (Utility of fixed pricing contracting TF) We have
U F    (c s  a ) 
Var  
ka 2
r
,
2
2
(15)
s.t. budget constraint of FG customer
Cg   .
(16)
By using Lagrangian, there exists none maximum expected utility U F* .
Proof. The Lagrangian for utility maximization problem is
U F*    (c s  a ) 
Var  
ka 2
r
  (C g   ).
2
2
(17)
By setting partial derivatives
(
U U U
)
,
,
 a 
as simultaneous equations, i.e. the first-order condition, which can be written as
U
 C g    0,

U
 1  ka  0,
a
Var  
U

 0.

2
(18)
The solution of simultaneous equation (18) can be find as
C g  ,
1
a ,
k
Var    0.
(19)
Hence we use the solution in (19) and find 3PL supplier’s utilities
U F*  (  C s ) 
1
.
2k
(20)
For a constrained extremum about the expected utilities of 3PL supplier E U  a,    ,
subject to constrained budget of FG customer, the second-order necessary-and-sufficient
conditions should be revolved around the algebraic sign of the second-order total
differential d 2U , evaluated at a stationary point by Bordered Hessian determinant H [19]
to check the second-order necessary-and-sufficient condition for a maximum by bordered
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Lin, Lin and Yeh
Hessian determinant
0
Ca
C
H F  Ca
*
U aa
U a*
C
U *a
*
U 
0
0
0
 0 1 0  0.
0 0 0
(21)
Thus we conclude the second-order sufficient condition is not satisfied for maximum
expected utility U F .
Theorem 2: (Utility of fixed pricing with penalty item contracting TP) We have
Var  Bi si 
Var  
ka 2
U P    (C s  a )   E  Bi si  
r
 r 2
2
2
2
(22)
s.t. budget constraint of FG customer
C g     E  Bi si   r 2
Var  Bi si 
.
2
(23)
By using Lagrangian, there exists a maximum expected utility U P due to the
second-order sufficient condition is satisfied the constrained optimum.
Proof. The Lagrangian for utility maximization problem is
Var  Bi si 
Var  
ka 2
r
 r 2
U P*    (C s  a )   E  Bi si  
2
2
2
Var  B i si 
  (C g     E  Bi si   r 2
).
2
(24)
By setting partial derivatives
(
U U U
,
,
)
 a 
as simultaneous equations (24) can be written as
Var  B s 
U
 C g     E  B s   r 2
 0,

2
U
 1  ka  0,
a
Var  B s 
U Var  

 1    2
 0.
2
2

(25)
The solution of simultaneous equations (25) can be find as

Var  B s   Var  
,
 2Var  B s 
1
a ,
k
2(C g     E  B s  )
 
.
 2Var  B s 
(26)
A Dominant Maintenance Strategy Assessment Model
233
Next, check the second-order necessary-and-sufficient condition for a maximum by
bordered Hessian determinant
0
Ca
C
H P  Ca
*
U aa
U a*
C
U *a
*
U 
0
0
 0 k
-1 0
-1
0  k  0.
0
(27)
Thus, we conclude the second-order sufficient condition is satisfied for a maximum
expected utility U P .
Theorem 3: (Utility of cost-plus contract TC+) We have
U C     (1   )(C s  a ) 
Var  
ka 2
 r (1   )2
,
2
2
(28)
s.t. budget constraint of FG customer
C g     (C s  a )  r  2
Var  
.
2
(29)
By using Lagrangian, there exists a maximum expected utility U C  due to the
second-order sufficient condition is satisfied the constrained optimum.
Proof. The Lagrangian for utility maximization problem is
U C*     (1   )(C s  a ) 
Var  
Var  
ka 2
 r (1   )2
  (C g     (C s  a )  r  2
). (30)
2
2
2
By setting partial derivatives
(
U U U
,
,
)
 a 
as simultaneous equations can be written as
Var  
U
 C g     (C s  a )  r  2
 0,

2
U
 ka    0,
a
Var  B s 
Var  
U
 (1   ) 2
  2
 0.

2
2
(31)
The solution of simultaneous equations (31) can be find as

 1   
2
,
2
1- 2 ,
a
k
2 C g     C s  a  
r
Var  
(32)
.
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Lin, Lin and Yeh
Next, check the second-order necessary-and-sufficient condition for a maximum by
bordered Hessian determinant
0
Ca
C
H  Ca
*
U aa
U a*
C
U *a
*

U 
 2Var  
0
1
1
k
2
0
0
0
 Var  

k 4 Var  
2
2
4
2
 0.
(33)
Thus, we conclude the second-order sufficient condition is satisfied for a maximum
expected utility U C  .
Theorem 4: (Utility of performance based contract TPB) We have
Var  Bi si 
Var  
ka 2
 r (1   ) 2
 r 2
,
U PB    (1   )(C s  a )   E  Bi si  
2
2
2
(34)
s.t. budget constraint of FG customer
C g     (C s  a )   E  B s   r  2
Var  B s 
Var  
 r 2
.
2
2
(35)
By using Lagrangian, there exists a maximum expected utility U PB due to the
second-order sufficient condition is satisfied the constrained optimum.
Proof. The Lagrangian for utility maximization problem is
Var  B s 
Var  
ka 2
U PB    (1   )(C s  a )   E  B s  
 r (1   ) 2
 r 2
2
2
2


Var  B s 
Var  
   C g     (C s  a )   E  B s   r  2
 r 2
.


2
2


(36)
By setting partial derivatives
(
U U U
,
,
)
 a 
as simultaneous equations can be written as
Var  Bi s i 
Var  
U
 C g     (C s  a )   E  Bi si   r  2
 r 2
 0,

2
2
U
 1  ka  0,
a
  2Var     2Var  B s  
Var  B s 
U
2 Var  
2
 1   

 
  0.



2
2
2


The solution of simultaneous equations (37) can be find as
(37)
A Dominant Maintenance Strategy Assessment Model
235
1



2  C g      C s     E  B s  
k


,
r 
 2Var     2Var  B s 
1
a ,
k

(38)
1   2 Var     2Var  B s 
.
 2Var     2Var  B s 
Next, check the second-order necessary-and-sufficient condition for a maximum by
bordered Hessian determinant
0
Ca
C
0
0
0
k
2
0
 2Var     2Var  B s 
0
0
C
*
H PB  C a U aa
U a* 
U *a U *

 2Var     2Var  B s 


k  2Var     2Var  B s 

4
2

(39)
2
 0.
Thus, we conclude the second-order sufficient condition is satisfied for a maximum
expected utility U PB .
4. A military logistics Example with Risk-Averse 3PL Supplier
In this section we present a real MRO service contracting evaluation with a risk-averse
3PL supplier in Taiwan for military logistics. This typical example demonstrates how our
model in this article can be applied in a real practice to support the long-term strategic
commercial maintenance policy (SCMP) and negotiations about an eight years contract
between Taiwan government (TG), i.e. FG customer, and a local 3PL supplier under
authorities from OEM in Taiwan.
Type T series engine is a family of turbo-shaft engine, which powered a current N=100
(or 190 at the end of 2015) military twin-engine helicopters are deployed in the fleet and
plus 10% engines of total fleet for spare by TG. Table 4 summarizes the budget constraint
considerations under a general situation, i.e. without critical mission requirements, about
the MRO capability requirements of type T series engine.
Table 4. Considerations about MRO capability requirements of type T series engine.
FG Customer
Contract duration: 8 years
Budget constraint  =$19.8M
Type T engine n=200, s=20
Type T engine unit cost c=$0.5M
Max. penalty rate =10%
3PL Supplier
Facility founding cost  f : $2.5M
Software cost a : $0.3M/yr for 3 years
Breakeven: 5 years
Unit MRO Cost: $0.2M per engine
Profit: 10% of unit MRO cost
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Lin, Lin and Yeh
Table 5 summarizes the estimated data by 3PL supplier about the yearly MRO and
backorder quantity in contract duration before contracting.
Table 5. Estimated MRO/ Backorder Data in Contract Duration by 3PL supplier.
Year
MRO
Backorder
1st
10
1
2nd
8
0
3rd
8
1
4th
9
0
5th
9
1
6th
9
0
7th
10
0
8th
10
1
Total
73
6
For this SCMP contracting under a selected availability target, an incentive term was
offered for supporting the software cost of 3PL supplier by using an offset obligation
between TG and OEM, based on a prerequisite which the 3PL supplier started to build the
MRO capacity of the Type T series engine system.
In this example, we first derive cost sharing weighting factor

a
Cs
 0.05.
(40)
In addition, this SCMP contracting value ($19.8 million) reaching the threshold for
large procurement definition in "Enforcement Rules of the Government Procurement" of
TG, thus we can derive   0.1 as the maximum penalty weighting rate.
Second, we determine values of parameters k and Var   by using the following
approach. Let K be the 3PL supplier’s fixed cost such that
K  E K   .
(41)
For 3PL supplier, we derive that the expected fixed cost is 36 times higher than the unit
cost cu. The maximum of cost reduction a  1/ k is assumed to be 1  1.5 E  K  , thus
k
7
.
36c u
(42)
For the sake of simplicity, we also assume that the coefficient of variation

Var  
E K 
.
(43)
We infer the risk aversion coefficient for 3PL supplier by using a simplistic market
capitalization of a representative manufacturer of such a Type T engine subsystem, e.g. if
OEM of helicopter is chosen as the customer and OEM as the supplier, we calculate the
risk aversion ratio of
r
rn
 6.
rN
(44)
Furthermore, we also infer a numerical coefficient of variation ρ, in order to
discriminate three scenarios with small (ρs), medium (ρm), and high (ρh) cost uncertainty.
These numerical values can be captured by setting cost uncertainty of small ρs  1  2.5 ,
medium ρm  1  2 and high ρh  1  1.5 , where  represents the numerical value of
standard deviation, which comes from the cumulative distribution function of the normal
distribution. Hence the numerical ρ can be calculated as
A Dominant Maintenance Strategy Assessment Model
237
 s  1  2.5  1  Pr(   2.5  x    2.5 )  1  0.9875  0.012
(45)
 m  1  2  1  Pr(   2  x    2 )  1  0.9544  0.045
(46)
 h  1  1.5  1  Pr(   1.5  x    1.5 )  1  0.8663  0.143
(47)
Therefore we can determine the optimal expected utilities of both parties, which are
presented in Table 6 by contracting types and cost uncertainty.
Table 6. Expected optimal contract utility solutions of FG and 3PL supplier.
Contracting
Type
TF
TP
TC+
TPB
ρs=0.012
FG
3PL
-19,800
3,002
-19,798
3,001
-19,029
3,782
-19,027
3,780
ρm=0.045
FG
3PL
-19,800
2,779
-19,798
2,777
-19,030
3,580
-19,028
3,579
Unit: $1,000 USD
ρh=0.143
FG
3PL
-19,800
2,121
-19,798
2,120
-19,033
2,987
-19,029
2,985
Furthermore, we summarize the priority about contracting choices for both parties in
Table 7, under the numerical value in Table 6, cost uncertainty and rational cooperative
game behavior conditions.
Table 7. Expected contracting choices priority for FG and 3PL supplier.
ρs=0.012
ρm=0.045
ρh=0.143
FG
TPB> TC+>TF> TP
TPB> TC+>TF> TP
TPB> TC+>TF> TP
3PL
TC+> TPB> TF> TP
TC+> TPB> TF> TP
TC+> TPB> TF> TP
In views of maximum profits for 3PL supplier, minimum budgeting for FG and
cooperative game behavior conditions, we observe the results in Table 6 and Table 7 as
follows:
(1)
As generally expected, the expected numerical solutions indicate higher
uncertainty results lower expected utilities for all four contracting types;
(2)
The expected contracting choice priority for 3PL supplier should be TC+, TPB, TF,
and TP, due to maximum profits consideration. However, same priority for FG
should be TPB, TC+, TF, and TP, due to minimum budgeting consideration; and
(3)
TPB and TC+ should be the dominant maintenance strategy for this real logistics
service contracting, due to
(a). the numerical solutions exhibit the TC+>TPB with little difference by
expected profits to 3PL supplier such as
TC   TPB  TF  TP ,
(48)
and TPB>TC+ with same little difference by expected budgeting to FG such as
TPB  TC   TF  TP .
(49)
(b). an incentive term for supporting the software cost to 3PL supplier from
OEM, by using offset obligation between FG and OEM.
238
Lin, Lin and Yeh
5. Conclusions
Good faith is always an essential ingredient to many contracting relationships [9]. This
paper has taken the view to present a dominant maintenance strategy approach based a
effective formulation to estimate the expected utilities for a selected military logistics
service contracts under SCMP in Taiwan such that both parties share expected utilities in a
bargaining way, which very few attempts have been made at such partnerships between a
FG customer and localized 3PL supplier.
By blending the classical problem of managing MRO service with principal-agent
model and cooperative game, we analyze the incentives with four commonly used
contracting arrangements, fixed-price (TF), fixed-price with penalty item (TP), cost-plus
(TC+), and performance-based (TPB). In the model studied, we assume that FG customer
provide a cost incentive condition by using offset obligation from OEM, the results
demonstrate that it does improve the expected utilities of 3PL supplier and reduce the
payment of FG customer under a given target availability. The innovation is in explicitly
modeling decision making for a dominant strategy and considering how FG consumer and
3PL supplier behave while facing uncertainties arising from both support costs and product
performance. Specifically, the model is not only discover the incentive and penalty terms in
the exhibit complementarily, i.e. the most concerned issue by 3PL supplier and FG
customer in the same direction as cost uncertainty changes, but also allows both parties to
make normative predictions with respect to how to evolve a dominant maintenance
strategy which was very few attempts in literatures.
Furthermore, this article recommend the TPB contracting type should be the first
dominant maintenance strategy for FG consumer and 3PL supplier, due to the expected
payment (budget) is less than other three contracting types under a given system availability
target; and almost same expected utilities (profits) with TC+ contracting in such real military
logistics service practice in Taiwan.
More generally, the model developed here may perhaps find application in other areas
insights, e.g. design for horizontal supply-chain integration (multi-agent) contracting and
vertical supply-chain integration contracting, but where the assumption that parties can
perfectly commit themselves via detailed written contracts is strained.
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Author’s Biographies:
Yi-Kuei Lin is a Chair Professor at the Industrial Management Department, National
Taiwan University of Science and Technology, Taiwan, Republic of China. He received the
B.S. in Applied Mathematics from National Chiao Tung University, Taiwan. He obtained
the M.S., and the Ph.D., respectively in Industrial Engineering and Engineering
Management Department at National Tsing Hua University, Taiwan. His research interest
includes stochastic network reliability, performance evaluation, project management, and
operations research. He has published numerous papers in refereed journals including IEEE
Transactions on Reliability, Reliability Engineering & System Safety, Information Sciences,
European Journal of Operational Research, Computers & Operations Research, and International
Journal of Production Economics.
240
Lin, Lin and Yeh
Jong-Jang Lin is a Ph.D. student and received his M.B.A. in the Industrial Management
Department, National Taiwan University of Science and Technology, Taiwan. His research
interest includes logistics supply chain management and decision analysis.
Ruey-Huei Yeh is a Professor at the Industrial Management Department, National Taiwan
University of Science and Technology, Taiwan. He received the B.S. in Industrial
Engineering and Engineering Management Department at National Tsing Hua University,
Taiwan, and M.S. & Ph.D. in the University of Michigan, Ann Arbor, U.S.A. His research
interest includes reliability analysis, decision analysis, quality control, and warranty
policies. He is currently an associate editor of IEEE Transactions on Reliability.