This paper solves the search for interior solutions to optimization problems using stochastic variables. This is done by way of some new properties of distribution functions with increasing failure rates as characterized in Barlow and Proschan (1965). Building upon Lariviere (2006), we show that an objective function of the type R(x) = F(x)+xF(x), where F(x) = 1-F(x), can also admit one interior maximal solution when the distribution function F has an increasing failure rate (IFR)

Brusset, Xavier

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MPRA
Munich Personal RePEc Archive
Properties of distributions with
increasing failure rate
Xavier Brusset
ESSCA
30. October 2009
Online at http://mpra.ub.uni-muenchen.de/18299/
MPRA Paper No. 18299, posted 3. November 2009 03:05 UTC
Properties of distributions with increasing failure rate
Xavier Brusset1,∗
ESSCA, 1, rue Lakanal, BP 40348, 49003 Angers, France,
Abstract
This paper solves the search for interior solutions to optimization problems
using stochastic variables. This is done by way of some new properties of
distribution functions with increasing failure rates as characterized in Bar-
low and Proschan (1965). Building upon Lariviere (2006), we show that an
objective function of the type 푅(푥) = 퐹 (푥)+푥퐹 (푥), where 퐹 (푥) = 1−퐹 (푥),
can also admit one interior maximal solution when the distribution function
퐹 has an increasing failure rate (IFR).
Keywords: optimization problem, probability distribution, increasing
failure rate
1. Introduction
The problem solved here emerges from a mechanism design approach
where a principal wishes to oﬀer a price for some product or service to an
agent without being aware of the agent’s willingness to pay for such product
∗Corresponding author, permanent address: 8, rue Michel Salles, 92210 Saint Cloud,
France, tel:+33-6-28 29 22 00
Email address: xavier.brusset@essca.fr (Xavier Brusset)
1I declare that this material is my own intellectual property and certify that I have the
right to deposit the contribution with MPRA because I am the sole author and copyright
owner of this paper.
Preprint submitted to MPRA November 2, 2009
or service. If the agent rejects the oﬀer, the principal is left with his reser-
vation utility. The seller is in a Bayesian setting of incomplete information
and must form a belief about the agent’s willingness to pay.
We now consider the case of a principal who wishes to sell to a agent but
is not informed of the agent’s willingness to pay. The principal has to form a
Bayesian belief about this willingness to pay 푋. This Bayesian belief can be
modeled as a random variable which follows a distribution 퐹 supported on
[푋,푋]. If the principal fails to estimate correctly the customer’s willingness
to pay, his revenue is reduced to 1. The principal wishes to determine the
point 푥 which maximizes his revenue within the range [푋,푋].
Let 푋 be a non-negative random variable with distribution 퐹 and let
퐹 (푋) = 1 − 퐹 (푋). Let 푓 be its probability density. We deﬁne a fail-
ure rate function as in Barlow and Proschan (1965): 푟(푋) = 푓(푋)/퐹 (푋).
The generalized failure rate is deﬁned in Lariviere and Porteus (2001) as
푔(푋) = 푋푓(푋)/퐹 (푋). 푋 has an increasing failure rate (IFR) or, equiva-
lently, 퐹 is an IFR distribution if 푟(푋) is weakly increasing for all 푋 such
that 퐹 (푋) < 1. The IFR distributions are of interest in operations and
supply chain management research because of the implications in the eval-
uation of some types of objective functions which model stochastic events.
In Lariviere (2006), the example cited is of a service’s pricing. One cus-
tomer arrives per period, and service takes one period. The cost of service
is zero. Customers privately observe their valuations, which are independent
and identically distributed according to 퐹 (푋). A ﬁrm posting price 푝 then
faces demand 퐷(푝) = 퐹 (푝) and sets 푝 to maximize revenue, 푅(푝) = 푝퐷(푝).
2
An optimal price 푝∗ must solve
푅′(푝∗) = 퐷(푝∗) + 푝∗퐷′(푝∗) = 퐹 (푝∗)(1− 푔(푝∗)) = 0, (1)
when 푅′′(푝∗) < 0.
The uniqueness of 푝∗ depends on the generalized failure rate 푔(푋). In
particular, as stated in Lariviere (2006), if 푔(푋) is increasing, it can equal
one at only a single point, and the unique 푝∗ must solve 푔(푝∗) = 1.
2. Problem
In the case of the principal, he must maximize his revenue function. This
revenue function can be spelled out as
푅(푥) = 퐹 (푥) + 푥퐹 (푥),
= 퐹 (푥)(1− 푥) + 푥. (2)
Let us demonstrate that when the distribution 퐹 is IFR, the revenue
function admits an interior unique point 푥 which maximizes it.
We ﬁrst mention a corollary of the increasing failure rate of interest in
what follows.
An increasing failure rate has equivalently the following property
푟′(푥) ≥ 0. (3)
This means that
푓 ′(푥)(1− 퐹 (푥)) + 푓(푥)2 ≥ 0. (4)
The F.O.C. requires that
푅′(푥) = 푓(푥)(1− 푥) + 퐹 (푥) = 0. (5)
3
If 푓 is such that 퐹 (1) = 1, then 푥 = 1 is solution and is also a maximum.
This case is when the distribution ranges over [0, 1].
For all cases such that 푓(푥) > 0, we can write
1− 푥 = −퐹 (푥)
푓(푥)
. (6)
The S.O.C. for a maximum requires that
푅′′(푥) = −2푓(푥) + (1− 푥)푓 ′(푥) < 0. (7)
In the case when 푓(푥) ∕= 0 When we replace (1 − 푥) from (6) in (7), we
obtain
− 2푓(푥)− 푓 ′(푥)퐹 (푥)
푓(푥)
< 0. (8)
Let us deﬁne
∀푥, ∣ 푓(푥) > 0, 푄(푥) = 푓 ′(푥)퐹 (푥)
푓(푥)
− 2푓(푥) (9)
We wish to demonstrate that
∃푥0 ∣푄(푥0) < 0. (10)
From (4), we can write
푓 ′(푥)퐹 (푥) ≤ 푓 ′(푥) + 푓(푥)2. (11)
When replacing 푓 ′(푥)퐹 (푥) in (10), it is equivalent to demonstrate that
푓 ′(푥)− 푓(푥)2 < 0, (12)
since 푓(푥) > 0. Let us call
퐻(푥) = 푓(푥)2 − 푓 ′(푥). (13)
4
Demonstrating (12) is equivalent to verifying the conditions for which
퐻(푥) > 0. (14)
From Barlow and Proschan (1965), we know that the distributions which
enjoy increasing failure rates include the gamma, Weibull, modiﬁed extreme
value and the truncated normal distribution when their parameters are the
commonly accepted ones.
Let us see if the most popular distributions who have increasing failure
rates comply.
[The poisson distribution also has an increasing failure rate, but the ex-
ponential, which has a constant failure rate, is not studied here.]
2.1. Gamma distribution
The parameters of the gamma distribution which allow for an IFR are
훼 > 1 and 휆 > 0.
푓(푥) =
푒−푥휆휆(푥휆)훼−1
Γ(훼)
(15)
and we obtain
퐻(푥) =
푒−2푥휆(푥휆)훼
(
(푥휆)훼 + 푒푥휆(−훼 + 푥휆+ 1)Γ(훼))
푥2Γ(훼)2
(16)
which is strictly positive for the domain of the parameters 훼 > 1 and 휆 > 0.
2.2. Weibull distribution
The parameters of the Weibull distribution are 휆 > 0, 훼 > 1 and 푥 > 0.
We have
푓(푥) = 푒−푥
훼휆푥훼−1훼휆. (17)
5
In this case,
퐻(푥) = 훼휆푒−2휆푥
훼
푥훼−2
(
훼휆푥훼 + 푒휆푥
훼(
훼 (휆푥훼 − 1) + 1)) . (18)
Given the domain of the parameters, 퐻(푥) > 0 when 푥 > 0.
2.3. Modiﬁed extreme value distribution
This distribution function has as pdf
푓(푥) =
푒(훼−푥)/훽−
푥−훼
훽
훽
(19)
and as failure rate function
푟(푥) =
푒−
푥−훼
훽
훽(푒푒
−푥−훼
훽 − 1)
(20)
for this distribution, we have
푄(푥) =
1
훽
(
푒−푒
−푥−훼
훽
(푒−
푥−훼)
훽 − 1)− 2푒−푥−훼훽 −푒
−푥−훼
훽
)
(21)
We observe that, when 훽 > 0
lim
푥→∞
푄(푥) = − 1
훽
. (22)
Further, for 훽 > 0
lim
푥→0
푄(푥) = −(1 + 푒
훼
훽 )푒−푒
훼/훽
훽
,
lim
훼→∞
푄(푥) = 0. (23)
And as
푄′(푥) = −푒
푒
−푥−훼
훽 − 2(푥−훼)
훽
훽2
< 0, (24)
we conclude that
∀푥 ≥ 0,∀훼 ∈ ℝ, 훽 ∈ ℝ∗+, 푄(푥) ≤ 0. (25)
When the parameter 훼 is not large, which would be the general case as this
parameter represents a shifting parameter, we have 푄(푥) < 0.
6
2.4. Truncated normal distribution
The parameters of the Normal distribution which enable to have an IFR
are 휇 > 0, 휎 > 0, for which we have
푓(푥) =
푒−
(푥−휇)2
2휎2√
2휋휎
(26)
but the ensuing
퐻(푥) =
√
2휋(푥− 휇) + 휎푒− (푥−휇)
2
휎2
2휋푒−
(푥−휇)2
휎2 휎3
(27)
is complicated to study.
Instead, we have to look at the initial inequality in (10) to see when it is
satisﬁed.
In fact, since 휇 is only a translation factor and the factor 휎 is a scale
factor, this function is always negative for 휇 > 0 and 휎 > 0. Figure 1
represents this function with 휇 = 0.5 and 휎 = 1. We obtain
푄(푥) = −
(푥− 휇)
(
erf
(
푥−휇√
2휎
)
+ 1
)
2휎2
−
√
2
휋
푒−
(푥−휇)2
2휎2
휎
, (28)
with erf(x) as the integral of the gaussian distribution 퐹 (푥) with 휇 = 0 and
휎 = 1.
3. Conclusion
This problem was numerically solved in Brusset (2010) as a particular
instance of the much broader problem presented here. We prove here that for
the parameters which enable the above distributions to be IFR distributions,
the objective function of the form 푅(푥) = 퐹 (푥) + 푥퐹 (푥), diﬀerent from the
ones contemplated in Lariviere (2006), admit one single interior point over
the range of the decision variable 푥 which maximizes it.
7
Figure 1: Evolution of function 푄(푥) when 휇 = 0.5 and 휎 = 1.
References
Barlow, R. E., Proschan, F., 1965. Mathematical theory of reliability. John
Wiley & Sons, pp. 9–18.
Brusset, X., 2010. Modeling contractual relationships in transport. Ph.D.
thesis, Louvain School of Management, Place des doyens, 1, B-1348 Lou-
vain la Neuve, Belgium.
Lariviere, M., 2006. A note on probability distributions with increasing gen-
eralized failure rates. Operations Research 54 (3), 602–604.
Lariviere, M. A., Porteus, E. L., Fall 2001. Selling to the newsvendor: an
analysis of price-only contracts. Manufacturing & Service Operations Man-
agement 3 (4), 293–305.
8


Properties of distributions with increasing failure rate

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Properties of distributions with increasing failure rate

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