Reliability evaluation methods

This section investigates how demand fluctuations from different markets influence the production of the manufacturing plants and affect their performance.

Based on Hsu and Wen (2002), this study revises the definition of the reliability as it copes with the characteristics of supply chain networks. The unreliability problem arises from the condition the proposed capacity cannot match customer demand due to abnormal events occur. When abnormal events lead to a shrunk demand, excess supply occurs, following increased production cost due to low capacity utilization. On the other hand, though demand expansion causes decreased production cost, revenue loss follows once proposed capacity cannot meet the excessive demand. Thus, the results of a supply chain network design, i.e., the proposed capacity and production allocation produce reliability for the manufacturer only when the demand fluctuates within ranges that allow the capacity utilization of the manufacturing plants to maintain cost economies and customer service level. This study defines the reliability as the probability that initially proposed capacity of the manufacturing plant will operate effectively under demand fluctuations.

This study assumes that the proposed capacities of the manufacturing plants resulting from supply chain network design in Chapter 3, v , associated with the _nk

average monthly customer demand, f , are initially reliable, where the capacity _ns utilization of the plant could maintain a break-even level or/and the production does not exceed the full-capacity production level. Let the capacity utilization be the basic criterion for evaluating the reliability of the manufacturing plant under demand fluctuations. The capacity utilization of manufacturing plant n with respect to _k random production amount _n^t

f k

~ in month t, )~ ( k

k n

n f

Y , is defined as:

*

~

~ ) (

k k k k

n t t n n

n v

f f

Y = (4.1)

Since the proposed capacity, v^*_nk , is fixed, ~ ) ( _n^t

nk f k

Y is directly proportion to the

realizations of ~f_n^t_k

for month t. Let f_n^t_k represent a random realization of ~f_n^t_k and a potential value of production amount of manufacturing plant n under all demand _k fluctuations over month t. If _n ( _n^t )=0

k

k f

Y , it implies that the potential production amount is zero, i.e. _n^t =0

f k ; and _n ( _n^t )≥1

k

k f

Y reveals that the manufacturing plant is under full-capacity production or the potential production amount exceeds its capacity, which further implies the capacity cannot satisfy the excess demand. This study assumes =1

nk

Y to be the maximally acceptable capacity utilization of manufacturing

plant n , where there exists a lowest unit-product production cost, while let _k Y be _nk the minimally acceptable capacity utilization, at which tolerable minimum revenue for the manufacturer is assumed.

When the proposed capacity applied under fluctuating demand, and if ~f_n^t_k

leads

manufacturing plant n is unreliable under demand fluctuations in month t. _k Specifically, the reliability of a specific manufacturing plant is defined as the probability that the capacity utilization falls between the acceptable limits, namely:

]

The impacts of fluctuating customer demand on the production amount of different manufacturing plants are further analyzed. Let

nk

θ be the proportion of the production from manufacturing plant n to that from all plants, which is the results of _k

initially proposed production allocation among the manufacturing plants, namely:

θ also implies the magnitude of the manufacturing plant to the manufacturer, in a way that the more substantial production a plant is, the more the manufacturer relies on its output to serve customers and

∑

∀n_k ^k

θn . Since the total production amount from all manufacturing plants is restricted to meet demands from all customers,

∑

∀n_k ^k

can be rewritten as:

∑

Substituting Eq. (4.4) for

nk

~f

in Eq. (4.2), Eq. (4.2) can be rewritten in terms of customer demand, restated:

~ ]

Assume that random variable _n^t f s

~ follows a normal distribution with parameters _n^t f s

~ for all customers is independent. Total fluctuating demand,

∑

fn

~ , is also a random variable, distributing with a normal distribution with mean

∑

σ2 . The reliability of manufacturing plant

nk can now be evaluated by using the cumulative distribution functions of normal distribution, namely:

) )

where )Φ(z is the cumulative distribution function of the standard normal distribution.

In practice, some abnormal events may occur at a particular market and continue for a period of time, such as finance crisis, war or natural disaster, economy recovery, which further cause demand from that market fluctuating. The network performance is affected in a way that the more fluctuating demand is different from the forecasted for a long period of time, the more accumulated revenue loss the manufacturer will surfer.

An abnormal state is one in which monthly customer demand values do not follow the normal demand distributions, estimated from all survey years, due to the occurrence of an abnormal event. For customer n_s, let K represent the set of all distinct states, _ns which occur on the market during the planning year and let

} gives the number of distinct abnormal states, and w_n⁰_s represents a normal state, in which no abnormal fluctuation occurs, respectively. Let Pr(wⁱ_ns) be the probability that state wⁱ_ns occurs during the planning year, where Pr(w_nⁱ_s)≥0 and variable. To simplify this study, v~_nⁱ_s is supposed to have a finite discrete distribution:

} planning year and I_nⁱ_s^,j represent the set of months belonging to the time interval within which an abnormal state w_nⁱ_s continues on the location of customer n_s , i.e.

monthly demand from customer n_s in abnormal state w_nⁱ_s follows another normal distribution with different parametric values. That is, the monthly demand associated with abnormal state ⁱ_n

ws follows another random variable, _n^t _ij f s_,

~ , _n^ij I s

t∈ ^,

∀ . Note

that the mean and standard deviation of the distribution _n^t _ij f s_,

~ is related to the effect

and duration of the event corresponding to state _nⁱ

ws. Consider different durations of abnormal state ⁱ_n

ws , _n^ij

v s , and their probabilities p_j , the average demand from customer n_s in month t given abnormal state _nⁱ

Furthermore, the expected fluctuating demand from customer n in month t, _s depending on the occurrence of abnormal states, is obtained as: _n^{t W}_{i n}ⁱ _n^t _i

s The reliability of manufacturing plant n in month t associated with abnormal demand _k further can be calculated using Eq. (5.6).