An integrated plant capacity and production model with economies of scales
3.2 Model formulation
3.2.2 Cost functions
∑
+ ∀∀
=
s s k
k n n
s
k kk
n fn w , 1 f ∀k (3.3) where
∏
s +k wk,k 1 represents the total supply flows at echelon k when the customer demand is one unit. Eq. (3.3) shows that the total supply flows at the echelons are different and dependent upon customer demand.
3.2.2 Cost functions
The decision maker in this study is a manufacturer who operates multiple high-tech product manufacturing plants and serves customers in different regions. Therefore, the upper echelon of the manufacturing plants can be defined as the raw material vendors, and the lower echelon as the customers. This study assumes that the firm’s procurement and outsourcing decisions are centralized at the corporate headquarter rather than in the individual manufacturing plants. Let kˆ be the echelon of manufacturing plants and nkˆ represent a manufacturing plant at echelon kˆ ,
respectively, while kˆ −1 is the echelon of raw material vendors and n represents a kˆ−1 raw material vendor, respectively. Consequently, echelon kˆ +1 refers to customers, which can also be represented as echelon s.
Let ˆkˆ−1
k
n
fn and k
s
n
fn ˆ represent the amount of raw material shipped from raw material vendor n to manufacturing plant kˆ−1 nkˆ and the amount of final products produced by manufacturing plant nkˆ and transported to customer n , respectively. s And, let
nk
f ˆ be the production amount at manufacturing plant nkˆ , which is constrained by the plant’s capacity,
k
k n
n v
f ˆ ≤ ˆ. Then, the relationship between the total amount of raw material required by manufacturing plant nkˆ and the amount of raw
material shipped among raw material vendors can be revised according to Eq. (3.1) as:
δn represents that raw material is transported from a raw material vendor
ˆ−1
n to a manufacturing plant k nkˆ; otherwise, ˆkˆ−1 =0
k
n
δn . Moreover, the relationship between the amount of production produced by manufacturing plant nkˆ and the
amount of products shipped from manufacturing plant nkˆ to customer n can be s formulated according to Eq. (3.2), as:
∑
∀Since the total customer demand must be satisfied, then the following condition must hold, that is to say,
∑ ∑∑
ˆ β . Furthermore, the capacity utilization of manufacturing plant nkˆ can be defined as
Costs in this study can be classified as inbound, fixed, production and outbound costs. Inbound costs include raw material purchase and transportation costs, which relates to the movement of the flows from the raw material vendors to the manufacturing plants. The fixed costs represent all expenses required for the manufacturer to contract with active raw material vendors. Production costs incorporate both the capital cost and the variable production costs, where the capital cost includes costs related to the purchasing and installation of related equipments, and plant construction and land rental fee, etc. and differs among manufacturing plants due to different locations and sizes of the manufacturing plants. The variable production
cost includes those paid for input factors other than raw materials, such as labor, utility and insurance, etc. Outbound costs refer to the costs related to transporting the final products from the manufacturing plants to the customers.
The production costs of manufacturing plant nkˆ,
nk
c represent, respectively, the capital costs and variable production costs, which depend on capacity,
nk
v ˆ . Eq. (3.6) shows that the production cost increases with the increase in the production amount,
nk
f ˆ. Furthermore, the
average production cost per unit product of manufacturing plant nkˆ can be expressed as
On the contrary, Eq. (3.7) shows that the average production cost per unit product decreases with the increase in production amount; however, the extent depends upon the capital cost with respect to the capacity. Although a manufacturing plant with large-size capacity could operate efficiently, the manufacturer experiences a high average production cost when the production amount is low, and as a result the high capital cost cannot be absorbed as shown in Eq. (3.7). Figure 3.3 shows the relationship between the average production cost and the production amount for different sizes of capacity, where 1
nˆk
v , 2
nˆk
v , 3
nˆk
v are three different-size capacities for a
manufacturing plant nkˆ, 1
Figure 3.3 The relationship between the average production cost and the production amount
As shown in Figure 3.3, the average production cost per unit product decreases as the production amount for each of the three sizes of capacity increases. And, when the production amount is low, the largest-size capacity 1
nˆk
v yields the highest average production cost among the three different-size capacities. Nevertheless, as the production expands, the average production cost for the largest-size capacity 1
nˆk
v decreases with a maximum decreasing rate. When the production amount exceeds point A in Figure 3.3, the average production cost for the largest-size capacity 1
nˆk
v reaches the lowest among the three different-size capacities, implying that a manufacturing plant with the largest-size capacity 1
nˆk
v is the most advantageous in terms of saving costs. Since the production amount depends on the demand, a large-size capacity is suggested when there is a large demand.
For a manufacturer operating multiple manufacturing plants, the impact of scale
economies on the total average production cost depend not only on the total customer demand, but also on the production assignments among manufacturing plants.
Furthermore, the total average production cost per unit product for the manufacturer,
Hkˆ, can be formulated as
The denominator in Eq. (3.8) is fixed and given due to the fact that the total amount of production is constrained in order to satisfy the total customer demand. Eq. (3.8) shows that the total average production cost per unit product for the manufacturer depends on both the capacity and the production volumes of all manufacturing plants.
Let Vnkˆ−1 be the fixed cost of the manufacturer with raw material vendor n , kˆ−1 which is independent as to the amount of raw material procured and the total number of manufacturing plants served since the cost is mainly the results of the manufacturer contracting with the vendor. Then, the total fixed cost for the manufacturer can be expressed as
∑
γ is an indicator variable representing whether
raw material vendor n is an active vendor for the manufacturer, and where kˆ−1
ˆ−1
δn for all manufacturing plants, that is, }
active vendor for the manufacturer; otherwise,
ˆ−1
nk
γ is equal to 0. Then, the average fixed cost per unit raw material for the manufacturer can be formulated as
∑∑
The unit of Eq. (3.9) is measured based on the raw material. To be based on products, Eq. (3.9) can be revised according to Eq. (3.3), and shown as:
ˆ 1
where Vkˆ−1 represents the average fixed cost per unit product for the manufacturer.
With regards to the raw material purchase costs, the main influences affecting average raw material purchase cost are unit raw material production cost and a reasonable payment ratio determined by the raw material vendor. Regardless of the reasonable payment ratio, the unit raw material purchase cost is low when the market for that raw material is high due to the fact that there exists economies of scale. To simplify the problem, this study assumes two influences, i.e. unit raw material production cost and payment ratio, are exogenous, and denote unit raw material purchase cost from raw material vendor nkˆ−1 by
all the purchase costs paid to all raw material vendors, the total raw material purchase cost for the manufacturer can be formulated as:
∑
where Eqs. (3.10) and (3.11) are the costs resulting from the raw material procurement.
The transportation costs in this study capture the costs resulting from the spatial distance between two locations. High-tech products are usually characterized as
having a high market value and usually depreciate quickly. A nearby raw material vendor is usually selected, since the raw material delivery time is short and as a result the inventory can be kept at a low level. Transportation costs decrease if the manufacturing plant and the raw material vendor are located in the same location because the distance is short and a low-cost transportation mode, i.e. trucks, can be employed. Consequently transportation costs are high if the locations of the manufacturing plant and the raw material vendor are a long distance apart from each other. Let ˆkˆ−1
k
n
tn represent the average unit-distance transportation cost per unit of raw material between the locations of raw material vendor nkˆ−1 and manufacturing plant
nkˆ. The transportation cost for transporting raw material from raw material vendor
ˆ−1
fn are the average distance and the amount of raw material shipped from the location of raw material vendor nkˆ−1 to manufacturing plant nkˆ. Then, the total transportation cost of the raw material for the manufacturer can be formulated as
∑ ∑
∀ ∀ −Eq. (3.12) shows that the transportation costs vary with the different combinations of (nkˆ−1,nkˆ) due to the different distances and average transportation cost per unit of raw material between raw material vendor nkˆ−1 and manufacturing plant nkˆ.
Summing up Eqs. (3.11) and (3.12), the inbound cost for the manufacturer can be shown as:
Dividing Eq. (3.13) by the total amount of raw material supplied by raw material
fn , the average inbound cost per unit of raw material for the manufacturer, Hkˆkˆ−1, can be formulated as:
The average inbound cost per unit product for the manufacturer can be obtained as
ˆ 1
As suggested by Chopra (2003), products with a high value are suitable for a delivery network with direct shipping, that is, products are shipped directly from the manufacturing plant to the customer. In addition, manufacturing plants with a large production can serve many customers, yet this may lead to high transportation costs.
To provide better service and reduce transportation costs, manufacturing plants are advised to serve nearby customers. The outbound cost for the manufacturer can be expressed as
∑∑
tnˆ represents the average unit-distance transportation cost per unit product between the location of the manufacturing plant nkˆ and customer ns . Moreover, the average outbound cost per unit product for manufacturer, Tskˆ, can be shown as:
ˆ represents the total production amount. Since the total customer demand must be satisfied,
∑ ∑
∀
The total average cost per unit product for the manufacturer is the sum of the average inbound, fixed, production and outbound costs in the entire supply chain, and can be formulated as:
− + From the discussions above, the nonlinear MIP model for the supply chain network design can be formulated as follows.
Min − + Eq. (3.17a) is the objective function that minimizes the total average cost per unit product. Eq. (3.17b) states that the amount of raw material requested by
manufacturing plant nkˆ is the sum of the amount of raw material provided by its active raw material vendors. Eq. (3.17c) defines that the amount of products shipped from manufacturing plant nkˆ to the customers is equal to the amount of production. Eq.
(3.17d) constrains the total production amount to meeting the total customer demand.
Eq. (3.17e) is the supply limit of raw material vendor nkˆ−1, where
ˆ−1
nk
S represents the maximum amount of raw material supplied by vendor nkˆ−1. Eq. (3.17f) defines the capacity utilization of manufacturing plant nkˆ. Eq. (3.17g) constrains the decision variables ˆ , ˆ and ˆkˆ−1
k k
k s
n n n
n
n f f
v to be non-negative. Finally, Eqs. (3.17h) and (3.17i) define the decision variables ˆkˆ−1
k
n
δn and βnnskˆ to be binary. The decision variables are
nk
v ˆ ,
nk
f ˆ , ˆ , ˆkˆ−1
k k s
n n n
n f
f , ˆkˆ−1
k
n
δn and βnnskˆ. That is, the manufacturer can apply the model to optimally decide the size of the capacity as well as the production volumes for all manufacturing plants, the amount of raw material from the raw material vendors to manufacturing plants, and which manufacturing plants should produce how much production to serve customers in different regions. Furthermore, the optimal capacity utilization of manufacturing plants and the optimal number of active raw material vendors for the manufacturer can also be obtained from the model.
3.3 Algorithm
This chapter formulates a nonlinear MIP model to integrate capacity and production planning problems by considering economies of scale in the supply chain network design. Using the exact algorithm to solve the problem may require a considerable amount of time and can only solve small problems. In this study, we adopt the simulated annealing (SA) heuristic proposed by Kirkpatrick et al. (1983) to
solve the optimal problem. The SA algorithm is based on Metropolis et al. (1953), which was originally proposed as a means of finding the equilibrium configuration of a collection of atoms at a given temperature. It has been extensively used in solving very large scale integration (VLSI) layout and graph partitioning problems. The major advantage of the SA algorithm over other local search methods is the ability to avoid becoming trapped in the local optimal. The SA algorithm employs a random search, which not only accepts changes that decrease the objective function, but also accepts some changes that increase it. And, the latter are accepted with a probability of Boltzmann distribution. In this section, we first develop an approach to generate an initial solution, and then use the SA algorithm to develop the heuristic for improving the initial solution.