Problem Description

A MNC owns or can potentially build in future a set IF of processing facilities (f ∈ IF) in countries across the globe. We divide the facilities into two groups: EIF being the

set of existing facilities and FIF being the set of future (new) facilities such that IF = EIF ∪ FIF. These internal facilities of the MNC either manufacture useful products from some raw materials (or wastes) or simply treat wastes without producing any useful products. In addition to interacting with each other in terms of receiving/supplying materials to each other, they (f ∈ IF) also interact with another set EF of external facilities (f ∈ EF) that do not belong (or are external) to the MNC. We define F = IF ∪ EF, and assume that the location and the incoming and outgoing materials for each f ∈ F (whether existing or future, internal or external) are prefixed and known. Multiple facilities may exist at the same location or plant site. For instance, an existing plant site currently produces B and C from A, and E from C and D. The site has sufficient space to build two more processes: one to produce G from C and F, and the other to produce J from H. Then, we model this plant site as four separate facilities (f = 1, 2, 3, 4) as follows.

(f = 1) A
B + C (f = 2) C + D
E (f = 3) C + F
G (f = 4) H
I

Facilities 1 and 2 exist now, while 3 and 4 are new that the company may or may not build.

For each f ∈ F, we group its associated materials (raw materials, products, byproducts, wastes, etc.) into two sets. IMf denotes the set of incoming materials mi (i

∈ IMf) consumed by f, and OMf denoting the set of outgoing materials mi (i ∈ OMf) produced by f. Note that we include only the materials that are relevant in terms of interaction among the facilities. For instance, suppose that an external facility f produces C and D from A and B. However, the MNC neither supplies currently or

ponders supplying at any time A or B to f nor needs currently or ponders needing at any time D from f at any of its internal facilities. Then, we simply set IMf as a null set, and OMf = {C}. Similarly, IMf for an internal facility f may not include products (e.g.

waste products) that are inconsequential, unless we also treat the waste disposal site as a separate facility by itself. Finally, for each internal facility f (f ∈ IF), we designate one material π(f) as a primary material, and define the current production capacity (Qf0) of f as the rate (ton per fiscal year) at which f uses or produces π(f) at time zero. Note that π(f) can be an either incoming or outgoing material, and all future internal facilities (f ∈ FIF) have Qf0 = 0.

Considering a global problem, we let all facilities be located in N different nations (n = 1, 2, …, N) or countries, and define Fn as the set of facilities situated in nation n (f ∈ Fn, F1∪ F₂∪ … ∪ F_N = F, and Fn ∩ Fn′ = null set for n ≠ n′). The legislations of a host country n normally imposes several restrictions on the ownership, imports, exports, accounts, earnings, etc. of the facilities located in its jurisdiction (f ∈ F_n). The internal facilities of each country n (f ∈ IF ∩ Fn) pay corporate and other taxes collectively to the country’s revenue authorities at the end of each fiscal year.

Based on the sales forecast from the marketing division, the MNC wishes to develop an optimum, strategic, and global capacity expansion plan over a planning horizon of T fiscal years or periods (t = 1, 2, …, T). The objective of this plan is to maximize the net present value (NPV) of the company’s net cash flows over the planning horizon.

The desired expansion plan must determine:

(a) Time, location and amount of capacity expansion of each f ∈ IF (b) Actual flows of all materials to and from each f ∈ F during each t

We make the following assumptions for the above DCEP.

1. All business intelligence data that are crucial for generating a reliable capacity expansion plan are available. These include the forecasts for product demands, raw material requirements, raw material prices, product prices, transportation costs, operating costs, fixed and variable capacity expansion costs, capacity expansion limits, annual interest rates, import duties, and corporate taxes of all internal manufacturing facilities, and the capacities of all external supplier facilities over the T periods (fiscal years).

2. The fluctuations in currency exchange rates over the T periods are already accounted for in the business intelligence data. Hence, we express all expenditures and returns in terms of a numeraire currency.

3. Expansion-related construction activities do not affect the available production capacity of any internal facility f at any time.

4. All activities related to the capacity expansion or new plant construction at any f ∈ IF require δ(f) periods before the expanded or new capacity becomes available. For instance, if δ(f) = 3, and the capacity expansion or new construction begins at the start of t = 1, then the expanded or new capacity is available only during and after t

= 4.

5. An expansion or new construction cannot begin while an expansion or construction is underway. In other words, if δ(f) = 3 and an expansion or construction begins at t

= 1, then another expansion or construction cannot begin until after the end of t = 3.

6. The fixed costs for the expansion of an existing facility and for the construction of a new facility are different, but their linear variable costs are the same.

7. No inventory is carried forward from one period to the next at any internal facility f.

This is reasonable, since the length (one year) of each period in the planning horizon is sufficiently long.

8. Every internal facility f is liable for the tariffs on all its imports from facilities that are outside its own country. The import tariff is levied based on the cost, insurance, and freight (CIF) cost (see Karimi et al., 2002 for more detailed CIF description) of imports at f. This refers to the total value of goods including the purchase, insurance, and freight costs incurred in bringing them to the delivery facility.

9. The mass balance for each internal facility f is given by,

f f

if i if i

i i

m m

σ σ

∈ ∈

∑

∑

IM OM

f ∈ IF (2.1)

where, mi denotes material i that f consumes or produces, and σif is analogous to the stoichiometric coefficient of a species i in a reaction except that the above balance is in terms of mass (ton) rather than moles. For example, if a facility f consumes 2 kg of A and 1 kg of B to produce 1.8 kg of C and 1.2 kg of D, then σAf

= 2, σBf = 1, σCf = 1.8, and σDf = 1.2. This facility could have either any of A, B, C, and D as the primary designated material.

10. For both the expansion of an internal existing facility and the construction of a new internal facility, depreciation is computed using the same formula.

11. Each internal facility has a constant lower limit on its production rate over the entire planning horizon. Thus, a facility, once it exists, must operate at or above that rate, and cannot shut down.

12. Products are shipped directly from the internal facilities to the customers and the latter bear the costs of materials, insurance, freight, and import duties.

We now present a formulation for the above stated DCEP. Unless stated otherwise, the indexes (f, t, i, etc.) assume their full ranges of values.