Reliability evaluation and adjustment for supply chain network design with demand fluctuations
4.3 Supply chain network adjustment model
4.3.1 Customer demand expansion
Demand expansion resulting from an abnormal state may lead the potential production amount to exceed the capacity in a manufacturing plant, given customer demand being satisfied. Because there is limited capacity, demand expansion usually accompanies unsatisfied customers. The excessive demand burdens the manufacturer with an even heavy revenue loss if the abnormal state lasts for a long period of time.
In accordance with these unreliable situations, this study proposes a production adjustment strategy, where it suggests the reliable manufacturing plants with remaining capacities to produce more or booking the capacity of outsourcing firms. The induced costs and benefits associated with adjustment decisions are also discussed. This study formulates a mathematical programming model for determining the optimal adjustment decisions in terms of production reallocations among all plants by minimizing total adjustment cost during months with excessive demand, given the sum of allocation cost, extra material purchase cost, difference in production cost, penalty cost and transportation cost.
Let t {Ini,j, ns, i}
s ∀ ∀
≡ represent the set of months belonging to the time interval within which excessive demand continues and n(t) is the number of months in t where
the adjustment is scheduled and executed. Let K be the set of the manufacturing plants operated by the manufacturer, J≡{n&k} be the set of the detected unreliable manufacturing plants, and n , k nk∈K−J, represents a reliable manufacturing plant,
where 1~ ) alternative outsourcing firm, where the product quality is indifferent from the manufacturer. For the sake of simplification, this study averages the total customer demands and denotes f as average monthly customer demand for the manufacturer s during n(t) months,
∑∑
The allocation cost includes fixed allocation cost and variable allocation cost.
The fixed allocation costs are those expenses associated with production schedule change costs, contract cost with outsourcing firms, etc and are incurred if the manufacturer once determines an adjustment. The variable allocation costs can be divided into two categories: outsourcing cost and compensation cost, where the former are costs charged by the outsourcing firms, while the latter reflects additional labor costs and extra utilities cost, etc because of over-production than scheduled in a reliable manufacturing plant. Let
mk
o be the unit-product outsourcing cost paid for outsourcing firm m and k
nk
h be the unit-production compensation cost for manufacturing plant n . The outsourcing cost reflects not only production and k material purchase costs borne by the outsourcing firm, but also a premium charged, and thus ≥
mk
o h can be concluded. The total allocation cost over n(t) months, G, is nk
formulated as:
∑ ∑
where O represents the fixed allocation cost, and k
mk
production amounts allocated from manufacturing plant n& to outsourcing firm k m k
and to reliable manufacturing plant n , respectively. Indicators k k
k
represent, respectively, whether there exist production allocation relationships between n& and kˆ m and between k n& and kˆ n . Moreover, k
∑
in Eq. (4.8) indicate the outsourcing amount from outsourcing firm m and additional k production amount from manufacturing plant n , respectively. k
The extra material purchase cost arises due to there could be no sufficient material to support the production, given considerable customer demands. Let p be the average unit-material purchase cost and p would be high since it is an emergency purchase and also due to there should be large material demanded in the market during high demand period. The extra material purchase cost over n(t) months, R, is given as:
∑∑
∀ ∀The difference in production cost discussed herein accounts for the benefits brought by production reallocation such that all manufacturing plants reach their full-capacity production. Let
nk
f& and
nk
f represent, respectively, the realized average monthly production amount of manufacturing plant n& and k n under demand k
expansion, i.e. unadjusted amounts, while fn&'k and fn'k are the adjusted ones, respectively. Then, the relationship between adjusted and unadjusted production amount can be represented as follows:
∑ ∑ ∑∑
∑
∀Note that the adjusted production amount is restricted by the capacity,
k all manufacturing plants, which do not reach their full-capacity production before production reallocation, the total difference in production cost over n(t) months, Q, can then be formulated as:
∑
∀Substituting Eq. (4.10b) for '
nk
where 0Q> reveals that there is always cost savings due to production reallocation and the total benefits are significant with considerable additional production amount,
∑
∀& , and the number of months with excessive demands, n(t).
The manufacturer could also decide to stay in status quo and do nothing such that the production allocation among the manufacturing plants is the same as initially proposed; however, revenue loss or customer service downgraded exist due to unsatisfied demands. The penalty cost is introduced to represent the expected loss.
The unit-product penalty cost can be estimated based on the unit-product price, P, and a proportion of penalty cost to price, φ. The total penalty cost over n(t) months, T, is
given by:
Production reallocation may avoid high penalty costs and result in a decreased production cost; nevertheless, it could lead to an increased transportation cost if the product is shipped from a distant manufacturing plant or/and outsourcing firm to customers in different regions. Let t and nk tmk be the average unit-product transportation costs from manufacturing plant n to customers and that from k outsourcing firm m to customers, respectively, then the total transportation cost over k n(t) months, E, can be formulated as:
)
From the discussion to date, the supply chain network adjustment model in response to customer demand expansion can then be determined by solving the following programming model (P2):
P2: min G+R+T +E−Q (4.15a)
Eq. (4.15a) is the objective function that minimizes total adjustment cost over n(t)
months. Eqs. (4.15b), (4.15c) and (4.15d) express the relationships between adjusted and unadjusted production amount of the manufacturing plants. Eqs. (4.15e) and (4.15f) constrain decision variable xmn&kk and ynn&kk to be binary. Finally, Eq. (4.15g) defines decision variables
k
k m
qn& , and
k k n n ,&
∆ to be non-negative integers.