Enhancing Corporate Value in the Optimal Design ... - ACS Publications

Ind. Eng. Chem. Res. 2007, 46, 7739-7757

7739

Enhancing Corporate Value in the Optimal Design of Chemical Supply Chains Jose´ Miguel Laıńez,† Gonzalo Guille´ n-Gosa´ lbez,‡ Mariana Badell,† Antonio Espun˜ a,† and Luis Puigjaner*,† Department of Chemical Engineering-ETSEIB, UniVersitat Polite` cnica de Catalunya, AVda. Diagonal, 647, G2, E-08028 Barcelona, Spain, and Department of Chemical Engineering, Carnegie Mellon UniVersity, 5000 Forbes AVenue, Pittsburgh, PennsylVania 15213

The tight profit margins under which chemical process industries (CPI) operate are forcing companies to pay more and more attention to the design and operation of their supply chains (SCs). Traditional approaches available in the process systems engineering (PSE) literature to address the design and operation of chemical SCs usually focus on the process operations side and neglect the financial part of the problem. This work deals with the design and retrofit of chemical SCs and proposes a novel framework to address this problem that consists of incorporating financial considerations at the strategic decision-making level. Within this framework, the decisions that have a long-lasting effect on the firm are assessed through integrated models that are capable of holistically optimizing the combined effects of process operations and finances. The proposed approach adopts the corporate value of the firm (CV) as the objective to be maximized. The main advantages of such a holistic approach are highlighted through a case study, in which a comparison with the traditional biased method that pursues a fairly simple performance indicator as objective and disregards the financial variables and constraints of the problem is carried out. The integrated solution guarantees the feasibility of the strategic decisions from the financial point of view by ensuring liquidity control. Furthermore, it also leads to a superior economic performance as it exhibits a greater capacity of improving the value of the firm. 1. Introduction The concept of the supply chain (SC), which first appeared in the early 1990s, has been the focus of growing interest, as the possibility of providing an integrated management of an SC can reduce the propagation of unexpected/undesirable events throughout the network and can markedly improve the profitability of all the involved parties. Supply chain management (SCM) aims to integrate plants with their suppliers and customers so that they can be managed as a single entity and to coordinate all input/output flows (of materials, information, and funds) so that products are produced and distributed in the right quantities, to the right locations, and at the right time.1 Therefore, SCM implies the handling of material and information flows throughout the entire SC, from suppliers to customers while encompassing warehouses and distribution centers (DCs), and usually including after-sales services, returns, and recycling.2 The main objective is to achieve acceptable financial returns together with the desired consumer satisfaction levels. The design and operation of a SC can both be posed as largescale dynamic decision-making problems.3 These problems are receiving increasing attention in both the operations research (OR) and the process systems engineering (PSE) communities, and as a result of this effort, significant progress has been made in problem formulations and decomposition strategies aiming at their solution. One of the major challenges in the chemical industry for the new millennium will be the optimization of the SC operation to achieve the value preservation and growth.4 Unfortunately, devising new solutions to address old unsolved problems is not an easy task. The SC models that currently exist usually assist * Corresponding author. E-mail: [email protected]. Tel.: +34 934 016 678. Fax: +34 934 010 979. † Universitat Polite ` cnica de Catalunya. ‡ Carnegie Mellon University.

practitioners in determining the location of the facilities embedded in the network, as well as the tasks that should be accomplished at each moment in order to maximize a given criteria. These kinds of operative models are no longer enough. If companies want to achieve a competitive advantage in the marketplace, they must implement tools that play a more interfunctional role and are able to provide integrated plans for the whole SC. These plans should include optimal financial as well as operative planning decisions in terms of value preservation. Fact-based strategic planning is increasingly desired and pursued by firms, but many issues remain about how to do it.5 This work focuses on the strategic level of the SCM problem. In simple terms, the SC design problem involves the identification of the combination of suppliers, producers, and distributors able to provide the right mix and quantity of products and services to customers in an efficient way.6 The main novelty of our work compared with other approaches devised to date lies in the inclusion of financial considerations at the strategic decision-making level. To tackle the resulting problem, we apply mixed-integer modeling techniques that can be used as a decision-support tool for strategic investment planning that takes into account financial concerns. The paper is organized as follows. Section 2 presents a critical review of previous work on the integration of process operations and finances. In Section 3, a formal definition of the problem of interest is given; In Section 4, the corresponding mathematical formulation is derived, and its performance is illustrated through a case study in Section 5. Finally, some conclusions about this work are drawn in Section 6. 2. Integrating Process Operations and Finances Recent advances in PSE have focused on devising enterprisewide modeling and optimization strategies that integrate decisions of distinct functions of a business into a global model. Nevertheless, despite the effort made in the area, almost all of

10.1021/ie070181e CCC: $37.00 © 2007 American Chemical Society Published on Web 10/04/2007

authors

Ahmed and Sahinidis25 Applequist et al.3 Berning et al.26 Badell and co-workers8,13,27 Bok et al.28 Cakravastia et al.29 Chen and Lee30 Fandel and Stammen31 Gjerdrum et al.32 Guilleń and co-workers13,16,33,34 Gupta and Maranas35 Gupta et al.36 Hugo and Pistikopoulos15 Iyer and Grossmann37 Jun-Hyung et al.38 Jung et al.39 Kallrath40 Lababidi et al.41 Lasschuit and Thijssen42 Lee et al.43 Liu and Sahinidis44 Mokashi and Kokossis45 Neiro and Pinto46 Oh and Karimi47 Papageorgiou et al.48 Perea-Lo´pez et al.49 Romero et al.8 Sabri and Beamon50 Seferlis and Giannelos51 Shulz et al.52 Sundaramamoorthy and Karimi53 Tsiakis et al.22 Tu¨rkay et al.54 Wan et al.55 Zhou et al.56

x

x

dividend

x

x

net revenue

Table 1. Supply Chain Models for the CPI

x

x x

x

x

x

x

x x x

x

x

profit

x x x

x x

x

x x x x x

x x

x

cost

economic basis

x

x

x

x x

x

x

x x

x

risk

NPV

x

x

equity

x

x

x

x

CSL

x

earliness

x

x

x

x

inventory level

x

x

setup

x

cleanup

efficiency basis makespan

performance measures

tardiness

customer service basis

x

facilities utilization

x

material consumption

x

energy consumption

x

waste generation

environmental basis

x

ecoindicator 99

7740 Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7741

Figure 1. Supply chain model structure.

Figure 2. Piecewise linear function of the discount factors.

the models developed to date focus on the process operations side and neglect decisions involving marketing campaigns, investment planning, corporate financial decisions, and many others aspects of enterprise planning related to SCM.5 However, the effective control of cash is one of the most important requirements of financial management, and its steady and healthy circulation throughout the entire business operation has repeatedly been shown to be the basis of business solvency.7 In fact, the availability of cash governs the production decisions taken in a company. For this reason, operative models should not consider cash as an infinite resource. A production plan cannot be implemented if it violates the minimum cash flow imposed by the firm (i.e., if the amount of raw materials and/ or utilities required cannot be purchased because of a temporary lack of cash). Moreover, assessing the feasibility of the scheduling/planning decisions from a financial point of view may not be enough for companies that want to achieve a competitive advantage in the marketplace. Fierce competition in today’s global markets is forcing companies to perform further analyses in order to find the best production-distribution decisions to be carried out in their SCs. If they wish to remain competitive, it is essential that they properly assess the different process operations alternatives in terms of their ability to markedly improve the value of the company. In fact, maximum-profit or minimumcost decisions may lead to poor financial results if their financial impact is not properly assessed prior to being implemented. Thus, managers should extend their analysis to include the more general objective of maximizing the value of the firm as opposed to the common optimization of traditional biased key performance indicators (KPIs) such as cost or profit. We believe that companies can create more value and achieve better performance by devising integrated approaches for SCM. Such strategies should be capable of holistically optimizing the combined effects of process operations and finances, thus exploiting the synergy between different management disciplines. The approaches that currently exist, however, still address the overall problem in a sequential fashion through the

optimization of partial KPIs. The use of these traditional sequential procedures is mainly motivated by the functional organizational structures of the firms. Today, most companies have separate departments for production, supply, logistics, service to customers, etc. In an environment of this type, each functional area’s plan is sequentially considered as input to the others according to a hierarchy. Thus, the models supporting the decision making in batch chemical processes operate in an isolated way. They optimize partial sets of decision variables, but they do not lead to real integration of relationships, despite promoting the sharing of information between different business entities.8 This partitioning of decision making in companies has been reflected in the goals of the studies and the optimization models developed to support them. The way in which software packages for enterprise planning have evolved has caused some problems and inefficiencies in SCM. Certainly, enterprise resource planning systems (ERPs) is not enough because its main core is a not-flexible nonredundant database that centrally organizes transactional information. Because of the lack of software capable to link corporate databases and optimization modeling systems, the material logic of the pioneer material requirements planning (MRP) system still remains as the kernel of most of the current commercial ERPs. Analytical IT has not incorporated the multifunctional planning models required to optimize the combined effects of the different variables involved in the problem. Under this scenario, an organization’s ability to respond to the market changes in an efficient manner is being hampered. Management policies supported by integrated operative plans and budgets are required to open new ways of making satisfactory overall decisions. Fortunately, there is an increasing awareness of the impact that chemical process production systems have on the financial areas of the firms, which has led to enterprise-wide management strategies that aim to provide a holistic view of the system. In fact, the need to extend the studies and analysis of process operations to incorporate financial considerations has been widely recognized in the literature.3-5,9,10 Thus, with the recent advances in optimization theory and software applications, there is no apparent reason why models for SCM that merge concepts from diverse areas of the firm cannot be constructed. However, although optimization models seem to offer an appealing framework for analyzing corporate financial decisions and constraints, as well as for integrating them with process operation decisions and constraints,9 relatively few integrated corporate financial models have been implemented so far.11-13 Thus, although planning and scheduling models for SCM should improve an overall business performance measure, they usually neglect the financial side of the problem and pursue a myopic key performance indicator as the objective to be optimized. In fact, almost all of the approaches devised to date account for the maximization of objectives that are based on the classical transactional analysis of costs and benefits, or in key operative parameters that act as intermediate costrelated performance measures. Table 1 shows the results of a review made to analyze the KPIs applied so far in supply chain modeling in the chemical process industries (CPI). The objective functions of these approaches have been classified into four categories: (i) economic basis, (ii) customer service basis, (iii) efficiency basis, and (iv) environmental basis. As one can see from Table 1, profit and cost have been the most exploited indicators, while financial matters have been systematically neglected in the models devised to date in the literature.

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Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007

Figure 3. Supply chain structure of case study. Table 2. Specific Volume of Products i (υi (10-5 m3/kg)) product

υi

P1 P2 P3

4.3 8.0 5.5

Table 4. Capacity Consumption of Plant Equipment j by Each Product i (θij (cu/kg)) equipment

Table 3. Mass Fractions for Consumption of Raw Materials by Plant Equipment j (rrij (adim.)) raw material product P1 P2

R1 0.59

R2

R3

0.35

j ) TA 0.18

R4

R5

R6

0.24

0.47

0.24

0.30

0.47

0.24 0.30

0.30

0.24 0.30

0.30

0.71

P1 P2 P3

0.59

0.30

0.24

0.77

P1 P3

0.59 0.24

0.35 0.71

TA

TB

TC

TD

P1 P2 P3

8.00 7.00 7.00

7.20 8.40 8.40

8.40 7.35 7.35

7.60 8.82 7.70

Table 5. Cost (mu/kg) and Maximum Availability (Aert (103 kg)) of Raw Material r at Each Period of Time t raw material

j ) TB P2 P3

product

j ) TC 0.18 0.53 j ) TD 0.12 0.24

Thus, the present article proposes a general framework for the design of SCs based on the development of holistic models that cover both areas of the company, the process operations and the finances. To achieve our goal (i.e., the integration of the decision making at different business levels), a mathematical formulation that utilizes mixed-integer modeling techniques and merges variables and constraints belonging to each of the abovementioned disciplines is derived and applied to a case study. The presented strategy considers the financial performance as a design objective and not merely as a constraint on operations. Furthermore, the corporate value of the firm (CV) is adopted as the objective to be maximized as an alternative to the commonly used profit or cost. The main advantages of this

availability cost

R1

R2

R3

R4

R5

R6

150.0 10.10

125.0 6.60

335.0 6.10

60.0 8.10

60.0 12.1

40.0 10.1

approach are highlighted through a case study in which our strategy is compared with the traditional approach that computes the maximum-profit or net present value (NPV) design without financial considerations. Numerical results show that significant benefits can be obtained if an integrated formulation accounting for the optimization of a suitable financial performance indicator is applied. 3. Problem Statement The intent of the SC network design problem is typically to determine the optimal manufacturing and distribution network for the entire product line of a company. The most common approach is to formulate a large-scale mixed-integer linear program (MILP) that captures the relevant fixed and variable operating costs for each facility and each major product family. The fixed costs are usually associated with the investment and/ or overhead costs for opening and operating a facility, or with placing a product family in a facility. The variable costs include not only the manufacturing, procurement, and distributions costs but also the tariffs and taxes that depend on the network design.

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7743 Table 6. Demand of Product i at Market m at Each Period t (Demimt (103 kg))

Table 10. Transportation Cost of Product i from Site s to Distribution Center w (mu/m3)

t product i

1-7

24-32

site s

8-20

21-23

33-47

48-56

57-60

105.0 50.8 43.8

150.0 72.5 62.5

105.0 50.8 43.8

P1 P2 P3

105.0 50.8 43.8

105.0 50.8 43.8

m ) M1 105.0 150.0 50.8 72.5 43.8 62.5

P1 P2 P3

105.0 119.0 133.0

600.0 680.0 760.0

m ) M2 105.0 150.0 119.0 170.0 133.0 190.0

105.0 119.0 133.0

150.0 170.0 190.0

105.0 119.0 133.0

35.0 220.5 150.5

m ) M3 35.0 50.0 220.5 315.0 150.0 215.0

35.0 220.5 150.5

50.0 315.0 215.0

35.0 220.5 150.0

70.0 294.0 147.0

100.0 420.0 210.0

70.0 294.0 147.0

140.0 171.5 49.0

200.0 245.0 70.0

140.0 171.5 49.0

P1 P2 P3

35.0 220.5 150.5

P1 P2 P3

70.0 294.0 147.0

400.0 1680.0 840.0

m ) M4 70.0 100.0 294.0 420.0 147.0 210.0

P1 P2 P3

140.0 171.5 49.0

140.0 171.5 49.0

m ) M5 140.0 200.0 171.5 245.0 49.0 70.0

Table 7. Equipment j Fixed Cost (FCFSjst (mu/mg)) and Investment (PriceFS jst (mu/kg)) equipment j fixed cost investment

TA

TB

TC

TD

25.0 3.26

37.0 3.91

50.0 2.93

45.0 4.04

Table 8. Distribution Centers Fixed Cost (FCFWwt (mu/m3)) and 3 Investment (PriceFS wt (mu/m ))

distribution center w

S1

W1 W2 W3 W4

0.02 0.48 0.19 0.21

W1 W2 W3 W4

0.04 0.90 0.36 0.38

W1 W2 W3 W4

0.03 0.62 0.25 0.26

fixed cost investment

W2

W3

W4

17.5 100.0

15.0 80.0

15.0 75.0

13.75 70.0

Table 9. Price of Product i at Each Market m and Period t (Priceimt (mu/kg)) product P1

P2

P3

16.15

m ) M1, M3, M4, M5 17.51

18.39

16.43

m ) M2 17.81

18.71

The network design problem focuses on the design of two or three major echelons in the SC for multiple products. Because of the nature of the problem being solved, network design is typically solved every 2-5 years.14 The deterministic design/planning problem of a multi-echelon SC with financial considerations can be stated as follows. We are given the following information. Process operations data: • a fixed time horizon; • a set of products; • a set of markets where products are available to customers and their demands; • a set of potential geographical sites for locating manufacturing plants and distribution centers; • a set of potential equipment for manufacturing the different products at the set of plants;

S3

0.48 0.02 0.36 0.60

0.27 0.18 0.24 0.24

0.90 0.04 0.67 1.12

0.49 0.34 0.45 0.45

0.62 0.03 0.46 0.77

0.34 0.23 0.31 0.31

i ) P1

i ) P2

i ) P3

Table 11. Transportation Cost of Product i from Distribution Center w to Market m (mu/m3) market m distribution center w

M1

M2

M3

M4

M5

W1 W2 W3 W4

0.02 0.48 0.21 0.21

i ) P1 0.48 0.02 0.39 0.60

0.21 0.39 0.04 0.17

0.15 0.60 0.18 0.02

0.11 0.22 0.22 0.22

W1 W2 W3 W4

0.04 0.90 0.38 0.38

i ) P2 0.90 0.04 0.72 1.12

0.38 0.72 0.07 0.32

0.27 1.12 0.34 0.04

0.20 0.40 0.40 0.40

W1 W2 W3 W4

0.03 0.62 0.26 0.26

i ) P3 0.62 0.03 0.49 0.77

0.26 0.49 0.05 0.22

0.19 0.77 0.23 0.03

0.14 0.28 0.28 0.28

distribution center w W1

S2

• lower and upper bounds for the increment of the capacity of the equipment and the distribution centers; • product recipes (mass balance coefficients and consumption of capacity); • capacity of the suppliers; • minimum utilization rate of the capacity. Financial data: • direct cost parameters such as production, handling, transportation, and raw material costs; • prices of products at each market during the time horizon; • coefficients for investment and sales of marketable securities; • discount factors for prompt payment and amount purchased to suppliers; • relationship between capital investment and capacity of plants and distribution centers; • relationship between indirect expenses and capacity of the equipment of plants and distribution centers; • pledging costs; • tax rate and number of depreciation time intervals; • interest rate for the long- and short-term debt; • salvage value; • shareholders risk premium data. The goal is to determine: • the facilities to be opened;

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Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 Table 15. Maximum Net Present Value Network Designa

Table 12. Production Cost of Product i Manufactured in Plant Equipment j at Site s (mu/kg)

manufacturing sites

site s plant equipment j

S1

TA TB TC TD

0.63 0.51 0.72 0.77

A B C D

0.55 0.59 0.63 0.90

A B C D

0.55 0.59 0.63 0.78

S2

S3

0.59 0.48 0.69 0.73

0.71 0.58 0.82 0.88

0.52 0.56 0.60 0.85

0.62 0.67 0.72 1.02

0.52 0.56 0.60 0.74

0.62 0.67 0.72 0.89

s

j

t)0

t)1

t ) 24

t ) 48

S1

TA TB TC TD TA TB TC TD TA TB TC TD

10 000.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 50 000.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 500 000.0 0.0 500 000.0 500 000.0 329 830.5 0.0 496 610.2 478 813.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

i ) P1 S2

i ) P2

S3

i ) P3

distribution centers

Table 13. Fraction of Sales That Are Receivable at n Time Periods after Sales (δmtt′) time periods between execution and maturing period of sales (n ) t′ - t) market m

0

1

2

3

4

5

6

M1, M3, M5 M2 M4

0.05 0.00 0.00

0.10 0.00 0.05

0.25 0.00 0.05

0.60 0.00 0.15

0.00 0.05 0.75

0.00 0.20 0.00

0.00 0.75 0.05

capacity increment (cu) for time period (t)

s

j

t)0

t)1

t ) 24

t ) 48

S1


10 000.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 50 000.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 500 000.0 407 796.6 500 000.0 500 000.0 0.0 0.0 496 610.2 478 813.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

S2

S3


capacity increment (m3) for time period (t)

w

t)0

t)1

t ) 24

t ) 48

W1 W2 W3 W4

4 000.0 0.0 0.0 0.0

0.0 2 000.0 0.0 0.0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0

a Total profit ) 121 653 714.87 mu; net present value (NPV) ) 47 476 865.35 mu; and corporate value (CV) ) 36 392 463.16 mu.

• the increment of the capacity of the equipment of the plants and the distribution centers at each time period; • the assignment of the manufacturing and distribution tasks to the nodes of the network; • the amount of final products to be sold; • the investments and sales of marketable securities; • the amount of accounts receivable pledged at each period; • the schedule of payments to suppliers at each time period; • the long- and short-term debt acquired and repaid at each time period, such that corporate value evaluated at the end of the planning horizon is maximized.


w

t)0

t)1

t ) 24

t ) 48

W1 W2 W3 W4

4 000.0 0.0 0.0 0.0

0.0 2 000.0 0.0 0.0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0


Table 16. Maximum Corporate Value Network Designa manufacturing sites


s

j

t)0

t)1

t ) 24

t ) 48

S1


10 000.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 50 000.0 0.0 0.0 0.0

500 000.0 395 796.6 500 000.0 500 000.0 0.0 0.0 0.0 0.0 0.0 0.0 496 610.2 478 813.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Table 14. Maximum Profit Network Designa manufacturing sites


S2

S3



w

t)0

t)1

t ) 24

t ) 48

W1 W2 W3 W4

4 000.0 0.0 0.0 0.0

2 000.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0


4. Mathematical Formulation: Integrated Model The mathematical formulation derived to address the problem described above is next presented. The variables and constraints of the model can be roughly classified into two groups. The first one concerns the process operations constraints given by the SC topology. The second one deals with the financial area. Both sets of equations are next described in detail. 4.1. Design-Planning Formulation. The structure of the SC taken as reference to develop the mathematical model is illustrated in Figure 1. The design/planning mathematical formulation is based on the work of Hugo and Pistikopoulos15 in which the authors presented a mathematical methodology that included life-cycle assessment criteria as an additional objective to be optimized at the strategic level of SCM. The model has been enhanced to allow the storage of products and to include distribution-center nodes in the SC network. The equations of the model are described in detail in the next sections. 4.1.1. Mass Balance Constraints. The mass balance must be satisfied in each of the nodes that integrate the SC network.


Figure 4. Net present value-profit Pareto curve.

Figure 7. Sales carried out in each market for each optimal SC network configuration. Table 17. Value Accumulation at Each Quarter for Maximum Profit SC Network Structure (mu)

Figure 5. Corporate value-profit Pareto curve.

quarter

profit after sales

change in capital invested

discounted free cash flows

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.00 4 979 375.42 5 251 954.60 5 234 607.12 5 244 008.20 5 146 001.25 5 380 627.78 5 330 766.91 5 379 379.61 5 375 959.78 5 374 603.12 5 333 591.15 5 401 312.47 5 377 798.88 5 385 400.36 5 354 411.10 5 396 571.81 5 365 391.15 5 416 973.99 5 336 731.59 5 337 915.65

-30 351 256.18 -10 526 125.59 -7 009 339.22 -6 949 517.30 -1 927 388.18 1 013 250.10 2 683 607.13 -6 945 423.04 -6 977 691.44 -6 936 971.06 -6 899 941.88 -6 805 316.21 -6 867 048.93 -6 797 135.46 -6 772 325.30 -6 693 320.93 -6 717 360.69 -6 637 153.40 -6 193 005.48 14 026 624.90 25 062 282.92

-30 351 256.18 -5 399 781.52 -1 641 946.02 -1 539 054.85 2 809 642.48 5 110 179.65 6 506 897.48 -1 230 594.10 -1 169 368.75 -1 096 604.41 -1 028 686.82 -953 202.67 -911 170.65 -847 169.56 -794 905.94 -736 657.13 -697 712.49 -645 028.00 -380 958.86 9 058 103.45 13 618 620.48

the initial inventory kept at the site must equal the inventory at the end of the period t plus the quantity consumed by the manufacturing tasks. Let us note that this equation should be only applied to those products consuming raw material r (i ∈(Ij ∩ Ir)) and manufactured in the equipment j of plant s.

∑ Purcherst + SIrst-1 ) SIrst + ∑j ∑

RrijPijst

i∈(Ij∩Ir)

e∈Er

∀r,s,t (1)

The mass balance for final products i in each manufacturing site s is enforced via eq 2. This expression states that the amount of final product manufactured at each site during a given time period t plus the initial stock of the product must equal the final inventory of the product plus the amount transported from the site to the distribution centers w. Figure 6. Corporate value-net present value Pareto curve.

Pijst + SOist-1 ) SOist + ∑ Qiwst ∑ j∈J w

∀i,s,t

(2)

i

4.1.1.a. Manufacturing Sites. Equation 1 represents the mass balance for each raw material r consumed at each manufacturing site s in every time period t. Thus, this equation states that the purchases of raw material r provided by suppliers e (Er) plus

4.1.1.b. Distribution Centers. Equation 3 expresses the mass balance for the distribution centers w. Thus, this equation states that the total amount of final product i coming from all the sites

7746


Table 18. Capital Investment Needed Composition at Each Quarter for Maximum Profit SC Network Structure (mu) quarter 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

∆ accounts receivable

∆ accounts payable

net investment

other expenses

0.00 0.00 -30 351 256.18 0.00 -13 726 925.38 3 473 127.79 758 781.39 -1 031 109.42 -7 203 031.10 9 582.22 758 781.39 -574 671.75 -7 177 150.07 25 747.27 758 781.39 -556 895.90 -2 187 730.79 27 638.42 758 781.39 -526 077.22 1 081 190.49 -15 596.95 758 781.39 -811 124.83 5 931 473.28 45 959.03 758 781.39 -4 052 606.57 2 856 836.98 -91 896.50 758 781.39 -10 469 144.93 2 157 420.14 -19 103.95 758 781.39 -9 874 789.04 1 554 279.24 16 663.76 758 781.39 -9 266 695.47 -1 939 231.32 -20 241.00 758 781.39 -5 699 250.95 267 221.61 69 729.45 758 781.39 -7 901 048.67 -988 169.32 -79 620.83 758 781.39 -6 558 040.18 2 971 595.32 26 524.00 758 781.39 -10 554 036.18 -5 258 865.45 20 792.55 758 781.39 -2 293 033.81 34 377.20 -2 725.35 758 781.39 -7 483 754.20 -6 640 156.05 60.11 758 781.39 -836 046.16 -2 724 314.59 57 580.84 758 781.39 -4 729 201.08 -6 582 340.22 -79 188.38 758 781.39 -290 258.28 16 801 324.19 -3 465 032.47 758 781.39 -68 448.22 20 772 195.85 3 462 181.79 758 781.39 69 123.87

s plus the initial inventory of the product kept at the distribution center must equal the final inventory plus the sales in the final markets m.

Salesiwmt ∑s Qiwst + SWiwt-1 ) SWiwt + ∑ m

∀i,w,t (3)

4.1.1.c. Marketplaces. Unlike other works in the literature, our model assumes that part of the demand can actually be left unsatisfied because of limited production capacity. Thus, eq 4 forces the sales of product i carried out in market m during time period t to be less than or equal to the demand.

∑w Salesiwmt e Demimt

∀i,m,t

(4)

The ability of responding to customer requirements turns out to be one of the most basic functions of SCM.16 Thus, customer service level (CSL) should also be taken into consideration when formulating a SC model.17 As proposed by Guilleń et al.,16 constraint 5 can be considered to explicitly take into account the demand satisfaction strategy of the enterprise. This equation imposes a minimum target for the demand satisfaction (MinCLS), which must be attained in all time periods t.

Salesiwmt ∑i ∑w ∑ m Demimt ∑i ∑ m

formulated by fixing at time period t ) 0 the value of the variables representing the capacity of the facilities according to the topology of the initial network. Equations 6 and 7 are added to control the changes in the capacities of the facilities over time. These constraints include the binary variables Vjst and Xwt, which take a value of 1 if the facility being represented (either the equipment j at manufacturing site s or the distribution center w) is expanded in capacity and 0 otherwise. Let us note that the increments in capacity are bounded in the range [FSELjst, FSEUjst], which represents the realistic physical interval in which they must fall.

VjstFSELjst e FSEjst e VjstFSEUjst ∀j,s,t

(6)

XwtFWELwt e FWEwt e XwtFWEUwt ∀w,t

(7)

Equations 8 and 9 are added to update the total capacity (FSjst and FWwt) by the amount increased during planning period t (FSEjst and FWEwt).

FSjst ) FSjst-1 + FSEjst ∀j,s,t

(8)

FWwt ) FWwt-1 + FWEwt ∀w,t

(9)

Equations 10-12 are included to calculate the planning period t when a manufacturing site s initiates its operations. SBst is a binary variable that takes the value of 1 if the facility is opened at period t and 0 otherwise. Equation 10 enforces the necessary conditions to define the new binary variable. Thus, if the binary variable that represents the increment in the capacity of any equipment j at site s in period t (Vjst) equals 1, the summation of the new binary variable from the initial period to the current one must also equal 1. Equation 12 is the reformulation of the previous logic condition, and it is obtained by replacing the implication by its equivalent disjunction (see eq 11). Equation 13, which is similar to constraint 12, is applied to enforce the definition of the binary variable SWwt, which must equal 1 if the warehouse w is opened in period time t and 0 otherwise. t

∑ SBst′ ) 1

Vjst ) 1 w

∀j,s,t

(10)

∀j,s,t

(12)

∀w,t

(13)

t′)1

t

g MinCLS ∀t

(5)

4.1.2. Capacity and Facilities Location Constraints. These constraints are also inspired in the work of Hugo and Pistikopoulos.15 Two different variables are defined, FSjst and FWwt, that represent the total capacity of equipment j in manufacturing sites s and distribution centers w, respectively, during time period t. Furthermore, variables FSEjst and FWEwt denote the expansion in capacity of the different facilities of the network during time period t. Thus, the establishment of a facility takes place in the first period of time in which these variables take a nonzero value. Let us note that the model is general enough to address not only the design of a new SC but also the retrofitting of an existing network. In the latter case, the problem should be

1 - Vjst +

∑ SBst′ g 1

t′)1 t

1 - Xwt +

∑ SWwt′ g 1 t′)1

Equation 14 forces the total production rate in each plant to be greater than a minimum desired production rate (βsjFSjst-1) and lower than the available capacity (FSjst-1). In this equation, θij represents the capacity consumption of equipment j by product i and βsj is the minimum percentage of utilization of equipment j at site s.

βsjFSjst-1 e

θijPijst e FSjst-1 ∑ i∈I

∀j,s,t

(14)

j

Equation 15 is analogous to constraint 14. Here, υi and γw constitute the specific volume of product i and the minimum

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7747 Table 19. Value Accumulation at Each Quarter for Maximum NPV SC Network Structure (mu)

Table 21. Value Accumulation at Each Quarter for Maximum Corporate Value SC Network Structure (mu)

quarter

profit after sales



quarter

profit after sales



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.00 4 953 811.54 5 225 813.08 5 210 405.11 5 214 225.00 5 106 984.43 5 356 914.78 5 307 053.90 5 355 666.60 5 352 246.76 5 350 890.10 5 309 878.15 5 377 599.46 5 354 085.86 5 361 687.35 5 330 698.09 5 372 858.81 5 341 678.14 5 393 260.97 5 313 018.57 5 314 202.64

-29 831 967.69 -10 500 320.15 -6 983 320.68 -6 926 855.39 -1 897 410.08 1 191 581.43 2 567 876.84 -6 925 888.16 -6 958 764.83 -6 918 652.70 -6 882 231.79 -6 788 214.37 -6 850 555.37 -3 620 617.94 -3 194 311.46 1 056 013.43 825 049.90 40 816.93 728 735.48 17 765 251.56 25 155 673.52

-29 831 967.69 -5 399 907.20 -1 642 061.77 -1 540 432.30 2 809 340.17 5 225 507.52 6 394 729.61 -1 233 777.20 -1 172 869.86 -1 100 393.07 -1 032 734.46 -957 482.61 -915 657.86 1 016 356.68 1 229 831.56 3 514 669.57 3 271 124.69 2 732 678.02 2 979 004.33 10 780 841.81 13 649 889.65

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.00 5 011 489.12 5 113 371.13 5 046 557.50 5 015 568.86 5 028 760.55 5 033 686.60 5 004 625.53 5 059 867.89 5 048 530.93 5 053 842.03 5 010 711.83 5 083 550.98 5 046 907.60 5 064 769.27 5 031 243.42 5 067 950.77 5 045 804.16 5 094 725.49 5 026 092.55 1 385 805.44

-30 344 381.18 -10 569 963.96 -2 811 321.27 -1 348 925.44 636 919.04 653 190.87 642 948.40 687 214.73 722 748.36 760 316.83 797 349.87 835 274.25 869 457.34 907 859.08 944 339.90 982 440.41 1 018 619.38 1 056 943.58 1 092 237.46 1 131 188.39 19 944 730.99

-30 344 381.18 -5 408 516.84 2 149 140.23 3 293 572.68 4 868 929.93 4 698 549.76 4 514 324.20 4 337 648.06 4 230 787.39 4 080 541.16 3 946 019.10 3 785 786.07 3 700 572.78 3 554 088.53 3 443 648.05 3 308 348.84 3 214 915.00 3 094 838.98 3 011 937.40 2 877 900.69 9 553 795.99

Table 20. Capital Investment Needed Composition at Each Quarter for Maximum NPV SC Network Structure (mu) quarter 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



net investment

other expenses

0.00 0.00 -29 831 967.69 0.00 -13 684 371.26 3 473 127.79 745 799.19 -1 034 875.87 -7 171 296.18 9 582.22 745 799.19 -567 405.91 -7 148 467.65 25 747.27 745 799.19 -549 934.20 -2 150 888.31 27 638.42 745 799.19 -519 959.38 1 230 780.21 -15 596.95 745 799.19 -769 401.01 5 392 964.67 45 959.03 745 799.19 -3 616 846.05 3 161 990.20 -91 896.50 745 799.19 -10 741 781.06 -1 238 315.11 -19 103.95 745 799.19 -6 447 144.95 -3 653 922.85 16 663.76 745 799.19 -4 027 192.80 5 695 774.12 -20 241.00 745 799.19 -13 303 564.11 -6 120 293.01 69 729.45 745 799.19 -1 483 450.02 -6 572 419.73 -79 620.83 745 799.19 -944 313.98 -4 028 962.07 26 524.00 745 799.19 -363 979.06 -452 349.67 -3 468 512.68 745 799.19 -19 248.30 240 078.78 0.00 745 799.19 70 135.45 -27 471.00 0.00 745 799.19 106 721.72 -850 409.63 0.00 745 799.19 145 427.37 -195 941.51 0.00 745 799.19 178 877.80 16 801 324.19 0.00 745 799.19 218 128.18 20 772 195.85 3 462 181.79 745 799.19 175 496.70

percentage of capacity utilization of distribution center w, respectively.

γwFWwt-1 e

∑i υiSWiwt e FWwt-1

∀w,t

(15)

The model assumes a maximum availability of raw materials. Thus, eq 16 forces the amount of raw material r purchased from supplier e at each time period t to be lower than an upper bound given by physical limitations (Aert). In this expression, Re denotes the set of raw materials provided by supplier e.

∑s Purcherst e Aert

∀e,r∈Re,t

(16)

4.2. Financial Formulation: Cash Management. In the presented approach, the management of cash associated with the operation of the SC is analyzed by extending the mathematical formulation developed by Guilleń et al.13 Such formulation is thus connected to the process operations variables and constraints through the periods and sizes of purchases of raw

Table 22. Capital Investment Needed Composition at Each Quarter for Maximum Corporate Value SC Network Structure (mu) quarter 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



net investment

other expenses

0.00 0.00 -30 344 381.18 0.00 -13 996 346.26 3 473 127.79 758 609.52 -805 355.02 -1 555 940.35 -1 523 731.25 758 609.52 -490 259.19 0.00 -1 949 396.54 758 609.52 -158 138.44 0.00 0.00 758 609.52 -121 690.49 0.00 0.00 758 609.52 -105 418.66 0.00 0.00 758 609.52 -115 661.13 0.00 0.00 758 609.52 -71 394.81 0.00 0.00 758 609.52 -35 861.16 0.00 0.00 758 609.52 1 707.31 0.00 0.00 758 609.52 38 740.34 0.00 0.00 758 609.52 76 664.73 0.00 0.00 758 609.52 110 847.79 0.00 0.00 758 609.52 149 249.56 0.00 0.00 758 609.52 185 730.38 0.00 0.00 758 609.52 223 830.88 0.00 0.00 758 609.52 260 009.85 0.00 0.00 758 609.52 298 334.06 0.00 0.00 758 609.52 333 627.92 0.00 0.00 758 609.52 372 578.86 15 552 286.61 3 462 181.79 758 609.52 171 653.05

materials and utilities from suppliers, the sales of final products to customers, and the investment costs. As a result of the application of this integrated model, optimal SC design and financial decisions can be computed simultaneously. Therefore, payments to providers, short- and long-term borrowing, pledging decisions, and the buying/selling of securities are planned in conjunction with manufacturing and distribution tasks. The financial side of the problem is then tackled through the inclusion of a set of constraints that accommodate the aforementioned economical issues. Such constraints are next described in detail. 4.2.1. Budgeting Model. The budgeting variables and constraints of the model should be determined according to the specific applicable rules (depreciation), legislation (taxes), etc. This may lead to different formulations depending on the case being analyzed. To overcome this problem, a set of general equations that intend to reflect a general case have been developed. Nevertheless, the mathematical formulation is general enough to be extended to other particular cases. The budgeting model considers the same t planning periods applied in the strategic SC formulation that cover the whole

7748


time horizon. This assumption allows an easy integration of both sets of constraints into a single holistic model. The cash balance for each planning period is the following:

CLinet e CLinemax ∀t CLinet ) CLinet-1(1 + irSD t ) + Borrowt - Repayt ∀t

t

Casht ) Casht-1 + ECasht + NetCLine t

∑e t′)1 ∑

Payet′t -

- FAssett + Capitalt + NetLDebt + FCostt + NetMS t t Othert ∀t (17) The cash at each period t (Casht) is a function of the available cash at period t - 1 (Casht-1); the exogenous cash from the sales of products or, in general, from any other inflow of cash (ECasht); the amount borrowed or repaid to the short-term credit ); the raw materials, production, and transport line (NetCLine t payments on accounts payable incurred in any previous or actual period t (Payett′); the payments of the fixed cost (FCostt); the sales and purchases of marketable securities (NetMS t ); the amount invested on facilities (FAssett); the capital supported by the shareholders of the company (Capitalt); the amount ); borrowed or repaid to the long-term credit line (NetLDebt t and, finally, other expected outflows or inflows of cash (Othert). A certain proportion of the accounts receivable may be pledged at the beginning of a period. Pledging is the transfer of a receivable from the previous creditor (assignor) to a new creditor (assignee). Therefore, when a firm pledges its future receivables, it receives in the same period only a part, normally 80%, of their face value. Thus, it can be assumed that a certain proportion of the receivables outstanding at the beginning of a period is received during that period through pledge, as stated by eq 18. t′

∑

t′

Pledtt′′ e

t′′)t-dmax M

∑

ASalest′′t

t′′)t-dmax M

∀t,∀t - dmax M e t′′ e t (18) dmax M ) max {dm}

In this equation, the variable Pledtt′ represents the amount pledged within period t′ on accounts receivable maturing in period t, while ASalest′t represents the accounts receivable associated with the sales of products executed in period t′ and maturing in t. Here, the parameter dmax M denotes the maximum maturing period at the markets. Let us note that pledging represents a very expensive way of getting cash that will only be used when no more credit can be obtained from the bank. t

ECasht )

∑

t′)t-dmax M

ASalest′t -

∑

t′)t-dmax M

Pledtt′ +

Finally, the exogenous cash is computed by means of eq 20 as the difference between the amount of accounts receivable maturing in period t and incurred in previous periods t′ (ASalest′t) minus the amount of receivables pledged in previous periods on accounts receivable maturing in period t plus the amount pledged in the actual period on accounts receivable maturing in future periods. In this expression, φt′t represents the face value of the receivables being pledged.

(23)

Repayt g irSD t CLinet-1 ∀t

(24)

Payett′Coefett′ ) EPurchet ∑ t′)t

∀e,t e T - de

(25)

With regard to the accounts payable, which are due to the consumption of raw materials, production, and transport services, let us note that our formulation assumes that the financial officer, at his option, may stretch or delay payments on such accounts. Discounts for prompt payment can be obtained if purchases are paid in a short time and cannot be taken if the payments are stretched. Thus, eq 25 forces the payments executed in period t′ on accounts payable to supplier e incurred in period t to equal the total amount due. In this expression, technical coefficients (Coefett′) that multiply the payments executed in periods t′ on accounts payable incurred in t are introduced in the formulation in order to take into account the terms of the raw materials, production, and transport credits (i.e., 2%-one week, net-28 days). t+de

Payett′Coefett′ e EPurchet ∑ t′)t

∀e,t > T - de

(26)

The payment constraints belonging to the last periods of time are formulated as inequalities (eq 26), as it is not reasonable to require that total accounts payable be zero at the end of the planning period. T

) SMS NetMS t t

T

MS MS Yt′t + ∑ Zt′t + ∑ t′)t+1 t′)t+1

t-1

∑ t′)1

∑ φt′tPledt′t t′)t+1 ∀t (20)

NetCLine ) Borrowt - Repayt ∀t t

t+de

t+dmax M

t-1

(22)

A short-term financing source is represented by an open line of credit with a maximum limit imposed by the bank (eq 21). Under an agreement with the bank, credits can be obtained at the beginning of any period and are due after 1 year at a given interest rate (irSD t ) that depends on the specific agreement reached with the bank. Equations 22 and 23 make a balance on borrowings, considering for each period the updated debt (CLinet-1) from the previous periods, the balance between ), and the interest of the borrows and repayments (NetCLine t credit line (irSD CLine ). Moreover, the bank regularly ret-1 t quires a repayment greater than or equal to the interests accumulated in previous periods, as is stated by eq 24.

(19)

m

(21)

t-1

MS MS (1 + Dtt′ )Ytt′ -

t′

∑ t′′)1

MS MS (1 + Ett′ )Ztt′ ∑ t′)1

∀t (27)

t′-1

MS MS Ztt′′ (1 + Ett′′ ) e SMS + t

MS MS (1 + Dtt′′ )Ytt′′ ∑ t′′)1

∀t,t′ < t (28)

Equation 27 makes a balance for marketable securities. The portfolio of marketable securities held by the firm at the beginning of the first period includes several sets of securities with known face values maturing within the time horizon (SMS t ). All marketable securities can be sold prior to maturity at


Figure 8. Optimal profit financial Gantt chart.

a discount or loss for the firm, as stated by eq 27. Revenues and costs associated with the transactions in marketable securiMS MS ties are given by technical coefficients (Dtt′ and Ett′MS). Yt′t is the cash invested at period t on securities maturing at period t′. MS is the cash income obtained through the security sold at Zt′t period t maturing at period t′. Equation 28 is applied to the constraint in each period that the total amount of marketable securities sold prior to maturity be lower than the available ones (those belonging to the initial portfolio plus the ones purchased in previous periods minus those sold before).

FAssett ) LBorrowt + Capitalt ∀t

(29)

Equation 29 balances the investment with the capital supported by shareholders (Capitalt) and the amount borrowed from banks as long-term debt (LBorrowt) at each time period t.

LDebtt ) LDebtt-1(1 + irLD t ) + LBorrowt - LRepayt ∀t (30) NetLDebt ) LBorrowt - LRepayt ∀t t

(31)

LRepayt g irLD t LDebtt-1 ∀t

(32)

Equations 30-32 reflect the payment conditions associated with the long-term debt. Let us note that these constraints are similar to those associated with the short-term credit line, as, in practice, both types of debts can be treated in a similar way. Nevertheless, let us note that, in the case of the long-term debt, the amount repaid in each period of time LRepayt remains usually constant in every time period.

Casht g MinCash ∀t

(33)

Equation 33 limits the cash in each period (Casht) to be larger than a minimum value (MinCash). A minimum cash is usually required to handle uncertain events, like delays in customer payments, thus ensuring enterprise liquidity. The bank requires a compensating balance, normally higher than 20% of the

amount borrowed. Therefore, the minimum cash (MinCash) has to be higher than the compensating balance imposed by the bank. 4.3. Objective Function. While the more common objectives used by the PSE community are minimum makespan, maximum profit, net present value (NPV), maximum revenue, and minimum cost, the finance community has been making financial business decisions for years by taking into account other indicators such as market-to-book value, liquidity ratios, leverage, capital structure ratios, return on equity, sales margin, turnover ratios, and stock security ratios, among others. Nevertheless, nowadays the maximization of the shareholder’s value (SHV) of the firm seems to be the main priority of the firms and what really drives their decisions. The use of SHV as the objective to be maximized is mainly motivated by the fact that it reflects in a rather accurate way the capacity that the company has to create value. The SHV of the firm can indeed be improved by maximizing its corporate value (CV). Specifically, according to Weissenrieder,18 the market value of a company is a function of four factors: (i) investment, (ii) cash flows, (iii) economic life, and (iv) capital cost. Specifically, our work applies the discounted-free-cash-flow method (DFCF) to compute the corporate value of a company. 4.3.1. Discounted-Free-Cash-Flow Method (DFCF). Our strategy applies the discounted-free-cash-flow method (DFCF) to assess the decisions undertaken by a firm. This method has recently become the most preferred approach for the valuation of companies given its capacity of properly assessing the four main factors that contribute to create the market value of a firm. In fact, the DFCF method is well-entrenched in finance theory, and its use is gaining wider acceptance in industrial scenarios. The DFCF method values a project or an entire company by determining the present value of its future cash flows and discounting them, taking into account the appropriate capital cost during the time horizon for which it is defined19 (economic life). According to financial theory, the enterprise market value of a firm is given by the difference between the discounted stream of future cash flows during the planning horizon and the net total debt at the end of its life time (NetDebtT), as it is stated

7750


by constraint 34. The final total debt includes both the shortand long-term debt and also the cash (eq 35).

CV ) DFCF - NetDebtT

(34)

NetDebtT ) CLineT + LDebtT - CashT

(35)

In the calculation of the DFCF, one must discount the free cash flows of each period t and the salvage value (SV) at a rate equivalent to the capital cost (see eq 36). The salvage value could be calculated as a percentage of the total investment or by any other applicable method. The capital cost reflects the time value of the money and also the risk of the investment. In fact, the capital cost can be regarded as the expected return required to attract funds to a particular investment.20

FCFt

T

DFCF )

∑ t)0

(1 + WACCt)t

(1 + WACCt)T

E(ROE) ) r0 + φRe

∑

t′)t-dmax s +1

t′+dmax s

t+dmax s

t

∑ ASalest′t′′ - t′)t+1 ∑ ∑ t′′)t+1

t′′)t′-dmax s

Pledt′t′′ ∀t (43)

∆ARect ) ARect - ARect-1 ∀t

(44)

(37) (38)

∆Invt )

FCFt ) Profitt(1 - trate) - NetInvestt - ∆NWCt ∀t (39) Cash flows at every period t (FCFt) are given by the profit after taxes, the net change in investments, and the change in net working capital. Specifically, the free cash flows are the difference between the net operating profit after taxes (NOPAT) and the increase in capital invested. From this definition, it follows that there will be value creation if the incoming value (Profitt(1 - trate)) is greater than the consumed value (∆NWCt - NetInvestt) as shown in eq 39.

Profitt ) ESalest - (

∀t (40)

Equation 40 is applied to compute the profit at each period t from the incomes associated with the sales of final products (ESalest), the production costs, the cost of the transport services (EPurchet), the fixed costs (FCostt), and the change in inventory (∆Invt).

NetInvestt ) FAssett - Dept ∀t

t

Equation 43 computes the accounts receivables corresponding to period t from the sales executed in the actual or any earlier period and maturing in periods beyond the present one minus the receivables pledged in previous periods, including those associated with the actual one. Equation 44 determines the change in accounts receivable at time period t.

To compute the expected return on equity, which is denoted by E(ROE), we apply eq 38. In this expression, E(ROE) is computed as the sum of a risk-free rate r0 and a risk premium (φRe). The former term represents the rate of return of an investment free of default risk available in the market and is usually equal to the yield to maturity offered by a government security. The latter represents the expected amount of return above the risk-free rate in exchange for a given amount of variance.3,20 One of the most commonly employed methods to estimate the risk premium is the capital asset pricing model (CAPM) (for more details regarding this topic, the reader is referred to the book by Sharpe21).

∑e EPurchet + FCostt - ∆Invt)

(42)

The change in net working capital associated with period t (NWCt) is computed from the change in accounts receivables, plus the change in inventory, minus the change in accounts payable, plus any other financial expenses or incomes (FExt), as stated by eq 42.

(36)

The capital cost can be determined through the weightedaverage method. This method considers the total capital structure of the company, including the overall equity and the debt, as is shown in eq 37. In this expression, λt denotes the proportion of equity over the total capital investment.

WACCt ) λtE(ROE) + irt(1 - λt)(1 - trate) ∀t

∆NWCt ) (∆ARect + ∆Invt - ∆APayt + FExt) ∀t

ARect )

SV

+

The net investment at each period t represents the monetary value of the fixed assets acquired in that period minus the depreciation. As mentioned before, the depreciation term should be computed according to the specific applicable rules (straight line, sum-of-years digits, declining balance, etc.)

(41)

∑r ∑s IuRM rt (SIrst - SIrst-1) + ∑i IuFPit [∑s (SOist - SOist-1) + ∑w (SWiwt - SWiwt-1)]

∀t (45)

Equation 45 expresses changes in inventory including raw material stocks in manufacturing sites, as well as final product stocks in distribution centers and manufacturing sites. IuRM rt and IuFP represent the inventory values for each raw material r and it product i at each planning period t, respectively. t

APayt )

t

t

∑e t′)1 ∑ EPurchet′ - ∑e t′′)1 ∑ t′)t′′ ∑ Coefett′Payet′′t′

∀t (46)

∆APayt ) APayt - APayt-1 ∀t

(47)

The accounts payable (APayt) are determined as the difference between all the purchases executed in previous periods (EPurchet′) minus all payments done until the actual period (Payet′′t′), as stated in eq 46. This constraint also takes into account the discounts for prompt payments (Coefett′). The change in accounts payable at time period t is, thus, represented by constraint 47. t+dmax s

FExt )

t

∑ (1 - φt′t)Pledt′t - ∑e t′)1 ∑ Payet′t(Coefet′t - 1) + t′)t+1 ∑t′ Ett′MSZtt′MS - ∑t′ Dtt′MSYtt′MS ∀t (48)

Finally, eq 48 computes other financial expenses and incomes (FExt) associated with the SC operation at every time period t. This term includes pledging costs ((1 - φt′t)Pledt′t), discounts for prompt payments to suppliers (Payet′t(Coefet′t - 1)), and also


Figure 9. Optimal NPV financial Gantt chart.

Figure 10. Optimal CV financial Gantt chart. MS MS MS MS the earnings (Dtt′ Ytt′ ) and the expenses (Ett′ Ztt′ ) associated with the transactions of marketable securities. 4.4. Integration between Models. The integration between both formulations is carried out through the sales of products, the purchases of raw materials, the transport services and utilities to final providers, the fixed cost associated with the operation of the network, and the total investment in capital. Thus, the accounts receivable incurred in any period t and maturing in period t′ can be easily computed from the sales of products executed in period t, the fraction of these sales that will be collected in period t′, and the prices of the products sold, as is stated in eq 49. Here, δmtt′ denotes the fraction of sales carried out in market m in period t that will be paid in period t′.

ASalestt′ )

Salesiwmtδmtt′Priceimt ∑i ∑w ∑ m

∀t,t′ > t

(49)

tr prod ∀e,t (50) EPurchet ) Purchrm et + Purchet + Purchet

The external purchases from supplier e at every period t (EPurchet), which are computed through eq 50, include the purchases of raw materials and transport and production utilities.

The external purchases to supplier e at every period t (EPurchet) can be then computed through eq 51.

Purchrm et )

∑r ∑s Purcherstψert

∀e,t

(51)

Furthermore, constraints 52-54 can be added to model the quantity discounts, i.e., price reductions offered by the suppliers to induce large orders. Certainly, the relationship between the discount factor offered by external supplier e for raw material r (DFerd) and the amount of raw material purchased can be modeled as a piecewise linear function (see Figure 2). The inclusion of these constraints allows the potential benefits of reduced purchase prices and fewer orders to be traded-off against the increase in inventory costs. Specifically, we define a set of discounts intervals Der for each raw material r for which supplier e offers quantity discounts (r∈DRe). Each interval corresponds to a different discount factor DFerd. The limits of interval d∈Der rm are denoted as Purchertd-1 and Purchertd. We introduce a new set of binary variables Fertd that take the value of 1 if the amount of raw material r purchased from supplier e in period t by all

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the sites falls into discount interval d and the value of 0 otherwise:

{

rm rm 1 if Purchrm ert ∈ [Purchertd-1, Purchertd] Fertd ) 0 otherwise

}

KPI

To enforce the above definition, the following linear constraints are applied:22

∑ Fertd ) 1 d∈D

∀e∈DRe,r,t

Table 23. KPI Values of Optimal Configurations for Second Case Study (mu)

(52)

er

network configuration

total profit

NPV

CV

optimal total profit optimal NPV optimal CV

116 597 744.59 116 377 423.52 105 955 397.81

45 374 556.23 45 516 838.30 42 068 183.09

89 386 052.53 110 018 635.40 145 023 155.58

of the fixed costs associated with the current plant equipment j already installed in every site s (FCFSjstFSjst-1) plus the sum of the fixed costs of each distribution center w (FCFWwtFWwt-1).

FCostt ) rm rm e Purchrm Fertd-1Purchertd-1 ertd e FertdPurchertd ∀e∈ DRe,r,t,d∈Der (53)

Purchrm ert )

∑

Purchrm ertd ∀e∈DRe,r,t

(54)

d∈Der

Constraint 52 forces each order to fall into a single interval, i.e., only one of the variables Xertd (say, for d ) d*) takes a value of 1, with all others being 0. Constraint 53 allocates each of the orders to its corresponding interval using the defined binary variable Fertd. Such an equation forces the auxiliary continuous variable Purchrm ertd to equal 0 for all d * d*, while rm also bounding Purchertd* in the range [Purchertd*-1 ,Purchrm ertd*]. Finally, constraint 54 expresses the condition for which the summation of the auxiliary variable Purchrm ertd over d must equal the variable Purchert. This implies that Purchrm ert ) rm rm rm rm Purchertd* and, therefore, Purchert∈ [Purchertd*-1, Purchertd*], as desired. Let us note that similar constraints could be easily derived to account for other types of quantity discounts, i.e., in utilities, transportation services, and so forth. Taking into account the discounts offered by the suppliers e for raw material r for a set of discount intervals d, which are denoted by DFerd, the total amount of money to be invested in raw materials is the following:

Purchrm et )

∑r d∈D ∑

er

Purchrm ertdψertFertd(1 - DFerd) ∀e∈DRe,t (55)

On the other hand, the external purchases of transport services and production utilities are determined through eqs 56 and 57. tr2 Here, Ftr1 eiws and Feiwm denote the unitary transport cost associated with sending products from plants to warehouses and from warehouses to markets, respectively. Furthermore, τut1 ijse represents the unitary production cost associated to the plants, ut3 ut4 whereas τut2 rse τise τiwe represents the inventory costs.

Purchtret )

SalesiwmtFtr2 ∑i ∑j ∑s QiwstFtr1eiws + ∑i ∑w ∑ eiwm m

∀e,t (56)

Purchprod et )

ut2 Pijstτut1 ∑i ∑j ∑ ijse + ∑ ∑ SIrstτrse + ijs r s ut3 SO τ + ∑i ∑s ist ise ∑i ∑w SWiwtτut4 iwe ∀e,t

(57)

The total fixed cost of operating a given SC structure in every time period t can be computed by means of eq 58 as the sum

∑j ∑s FCFSjstFSjst-1 + ∑w FCFWwtFWwt-1

∀t (58)

Finally, the total investment in capital or fixed assets is computed through eq 59. This term includes the investment made to expand the capacity of equipment j in manufacturing site s in period t (PriceFS jst FSEjst), plus the investment required to open a manufacturing plant, in case it is opened at period t (ISstSBst), plus the investment required to support distribution center w capacity increase (PriceFW wt FWEwt), plus the investment required to set a distribution center if it is opened at period t (ISstSBst).

FAssett )

∑s (∑j PriceFSjst FSEjst + ISstSBst) + W ∑w (PriceFW wt FWEwt + IwtSWwt)

∀t (59)

The overall problem can be, therefore, mathematically posed as follows:

maximize CV subject to eqs 17-59. 5. Case Study The capabilities of the proposed approach are illustrated by solving a retrofitting problem of a SC comprising several manufacturing sites, distribution centers, and markets located in different European countries. A set of potential technologies are assumed to be available in the manufacturing sites. Furthermore, several potential locations for the manufacturing sites and the distribution centers, from which the products should be transported to the final markets, are also considered, as is depicted in Figure 3. The potential locations for the plants embedded in the SC (S1, S2, and S3) are Barcelona (B), Budapest (Bu), and Milan (Mi). These plants can manufacture three different products (P1, P2, and P3) with four different technologies (TA-TD). These final products must be transported to the distribution centers prior to being sent to the final markets (M1-M5), where they become available to customers. We assume an existing installed capacity of TA in S1 and S3 of 10 000 and 50 000 cu, respectively. The investment cost associated with the establishment of a manufacturing site is equal to 8 800 000 mu. The potential locations for the distribution centers (W1-W4) are B, Bu, Manchester (M), and Lisbon (Li), whereas for the final markets they are are B, Bu, London (L), Li, and Toulouse (T). We assume that, at time zero, W1 has an installed capacity of 4 000 m3. The investment needed to open a distribution center is equal to 2 500 000 mu. The specific volumes of the products and raw materials are shown in Table 2. The amount of each type of raw material


required to manufacture each product depends on the specific equipment being applied in each case (see Table 3). Moreover, the capacity coefficients for each equipment and product are shown in Table 4. Table 5 gives the costs of the raw materials and their maximum availability at each period, which is assumed to remain constant within the whole planning horizon. The demand data are listed in Table 6. The initial inventories are supposed to be equal to zero for all products and raw materials. The upper bound imposed to the increment in the capacity of the technologies at each manufacturing site is 500 000 cu, and the lower bound is 50 000 cu. Table 7 shows the fixed and investment costs associated with each equipment. The upper and lower bounds imposed to the increment in capacity of the distribution centers are 2 000 and 30 000 m3, respectively. The rest of the data associated with the DCs can be found in Table 8. The capacities of the facilities can only be increased every 2 years. The salvage value is considered negligible at the end of the planning horizon. The availability of utilities is assumed to be unlimited. With regard to the financial matters, it is assumed that the firm has, at the beginning of the planning horizon, an initial portfolio of marketable securities. Specifically, the firm owns 15 000 mu in marketable securities maturing in period 2 and 18 000 mu maturing at period 3. The initial cash is assumed to be equal to the minimum cash allowed, which is 125 000 mu. Under an agreement with a bank, the firm has an open line of short-term credit at a 15% annual interest with a maximum debt allowed of 4 000 000 mu. The initial debt is assumed to be equal to zero, and the prices of the materials kept as inventories at the end of the time horizon are assumed to be 85% of their market prices for final products and a 100% of their market prices for raw materials. The market prices of the final products are assumed to remain constant during the whole planning horizon and are provided in Table 9. The SC under study has three external suppliers, the first one providing raw materials, the second one providing transportation services, and the third one providing labor utilities. Liabilities incurred with the raw materials supplier must be repaid within 1 month according to the terms of the credit (2%same period, net-28 days for the raw materials supplier). The supplier of raw materials offers discounts for large orders. Thus, a 3% discount is applied for orders of raw material R1 higher than 45 000 kg, and a 5% discount is applied for orders above 80 000 kg. The payments associated with the transport services and the labor tasks cannot be stretched and must be fulfilled within the same time period in which the purchase incidence takes place. Transportation costs are given in Tables 10 and 11, whereas production costs are shown in Table 12. The technical coefficients associated with the set of marketable securities that the firm has agreed to purchase and sell have been computed by considering a 2.8% annual interest for purchases and a 3.5% annual interest for sales. We consider outflows of cash equal to 5.0, 7.5, and 10.0 millions of mu in periods 13, 15, and 17 because of wages, rents, and dividends, respectively. We also assume that the ratio between the longterm debt and the equity must always be kept equal to 0.41. With regard to the long-term debt, let us note that the firm can access a long-term credit at a 10% annual interest. Shareholders expect an annual ROE equal to 30%. The taxes rate is equal to 30%. Depreciation is calculated by means of the straight-line method applied over a time horizon of 10 years. Finally, we assume that receivables on sales in any period are paid with a delay according to the proportions given in Table

13 and may be pledged at 80% of their face value regardless of their maturing period. Sixty-one planning periods with a length of 1 month each are considered. The implementation in GAMS23 of the integrated formulation leads to a MILP model with 40 306 equations, 46 916 continuous variables, and 252 discrete variables. It takes 185 CPU s to reach a solution with a 0% integrality gap on a AMD Athlon 3000 computer using the MIP solver of CPLEX (10.0). The integrated model is first solved by maximizing the corporate value of the firm. To explicitly show the tradeoff between the corporate value and the standard KPIs that neglect the cost associated with the financial transactions (i.e., NPV and profit), we next add a new constraint to force the model to seek solutions with higher NPVs and profits, respectively. This is indeed equivalent to applying the -constraint method proposed by Haimes et al.24 to our problem, taking into account in each case the above commented objectives, CV and NPV and CV and profit, respectively, at the same time. The objective of this analysis is to further explore the tradeoff between an integrated holistic solution and a sequential myopic one. Thus, the NPV and the profit are computed by applying eqs 61 and 62, respectively.

FFt ) (Profitt - FAssett) TProfit ) T

NPV )

∑ t)1

∑t FFt

(60) (61)

FFt (1 + RR)t

(62)

The SC network configurations obtained by following this procedure are summarized in Tables 14-16. The Pareto curves obtained by applying the aforementioned procedure are shown in Figures 4-6 . Numerical results show that the solutions computed by maximizing profit or NPV as single objectives are far away from the optimal one (i.e., the best solution in terms of CV). Certainly, the maximum corporate value solution is almost 300% higher than the one computed by maximizing profit and 267% higher than the one accomplished when maximizing NPV. On the other hand, the maximum profit and NPV solutions are quite similar. Moreover, from these results, it is clear that, in both cases, a conflict exists between the different objectives (i.e., maximum corporate value and maximum profit or NPV). Thus, numerical results will show that an improvement in the profit or NPV is only possible if the decision maker is willing to compromise the corporate value of the firm. Certainly, SC configurations with better profits or NPVs can only be achieved at the expense of a reduction in the corporate value of the firm. It is worth mentioning that this case study represents a very specific situation where there is one market (M2) in which the product prices are higher in comparison with the others (see Table 9). At such a market, accounts receivable are due within a large time period. Under this assumption, the design-planning model that accounts for the maximization of a myopic KPI (either profit or NPV) that neglects the financial side of the problem decides to configure a supply chain network capable of easily fulfilling the demand of market M2 as much as possible (see Figure 7). The profit and the NPV are indeed blind KPIs in the sense that they are not capable of properly assessing the financial impact associated with the shortages of cash. As a

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result, accounts receivables increase in certain periods of time in which the firm has to face important payments. The budgeting model is then forced to pledge them, mainly during months 1836 (quarters 6-12). Pledging is indeed a very expensive way of getting cash, and because of that, the firm reduces its capacity of creating value when applying it. On the other hand, the integrated approach that accounts for the maximization of a suitable financial objective function (i.e., corporate value) is able to properly assess the trade-off between the increment in profit that can be achieved by fulfilling demand at market M2 and the increment in net working capital that is required to carry out this decision. Hence, the integrated approach computes a SC configuration that does not fulfill the demand in M2 because of the poor payment conditions associated with its customers. Consequently, the net working capital needed is reduced significantly, thus increasing the value accumulated during the whole planning horizon. This can be observed in Tables 17-22, which depict the behavior of the corporate value and the structure of the consumed value during the planning horizon for each optimum SC network configuration. For the purpose of facilitating their interpretation, the planning periods (months) have been aggregated into quarters in these tables. In Figures 8-10, financial Gantt charts for each optimal SC configuration are shown. They describe how the cash is composed and utilized during the first 25 planning periods. To demonstrate the advantages of using CV and its robustness to assess strategic decisions under different scenarios, the previous case study has been modified and solved. Two modifications have been done: (i) prices payed by market M2 are equal to those payed by the rest of the markets and (ii) due time of account receivables in market M2 has the same conditions offered by market M4. The optimal SC network configurations obtained are quite similar to those resulting in the previous case study. In Table 23, the values of each KPI are shown for each optimal configuration. Even for this second case, it can be observed that the maximum CV solution is 62% better than the one computed by maximizing profit and 32% superior than the one accomplished when maximizing NPV. It is clear from these two examples that the integrated model introduced in this work has a great potential when addressing the challenge of designing a SC capable of preserving and improving the firm’s value. 6. Conclusions This article has addressed the design and retrofit of chemical SCs taking into account financial concerns. The proposed framework applies mixed-integer modeling techniques to develop holistic mathematical models for SCM that are able to optimize the process operations decisions in conjunction with the finances. The main advantages of this approach have been highlighted through a case study, in which the integrated model has been compared with the traditional method. The former approach pursues the maximization of a suitable financial key performance indicator that is able to properly assess the expenses associated with the shortages of cash and the value penalty associated to increments in net working capital (i.e., the corporate value of the firm at the end of the time horizon). On the other hand, the latter strategy accounts for the maximization of a biased key performance indicator that is unable to assess the costs of the financing sources. Numerical results show that the integrated solution not only ensures the feasibility of the strategic decisions from the financial viewpoint but also leads to a superior

economic performance given its higher capacity of creating value for the firm. The framework suggested in this article is, thus, in consonance with the new trends in PSE, which is going toward an enterprisewide optimization framework that aims to integrate all the functional decisions into a global model that should optimize an overall key performance measure. Here, the corporate value measured through the discounted-free-cash-flow method is proposed as a suitable first approach to this posed requirement. In our opinion, it is not enough to have a static planning model reviewed only in certain periods. It will be necessary to maintain a continuously updated enterprise modeling system that should connect the SC strategic, tactical, and operative levels. This modeling system must include the financial management of capital investments at the strategic level. The bottom level must incorporate a realistic cash-flow management including working-capital analysis and also short-term budget. The maximum corporate value as the target and the holistic model itself constitute a first step forward in creating this ongoing planning system. To create an appropriate framework for this challenge, we envisage a flexible optimization model that will allow the deployment of an agent-based tool for integrated SCM. To the best of our knowledge, a system based on an integrated model that takes into account a detailed financial analysis has not been implemented so far. The mathematical model created in this paper could be a premise to found the absent paradigm of this economic era. Further work will focus on taking into account uncertainty and risk analysis into the aforementioned integrated models. Acknowledgment Financial support received from “Generalitat de Catalunya” (FI programs) and European Community (Project PRISMMRTN-CT-2004-512233) is fully appreciated. Besides, financial support from Xartap (I0898) and ToleranT (DPI2006-05673) Projects is gratefully acknowledged. G.G.-G. also expresses his gratitude for the financial support received from the Fulbright/ Spanish Ministry of Education and Science visiting scholar program. Notation Indices e ) suppliers d ) discount interval i ) products j ) plant equipment m ) markets r ) raw materials s ) manufacturing sites t ) planning periods w ) distribution centers Sets DRe ) raw materials for which supplier e offers quantity discounts Edisc ) set of suppliers that offer quantity discount Er ) set of suppliers e that provide raw material r Ij ) products that can be processed in plant equipment j Ir ) products that consumed raw material r Ji ) equipment that can process product i Re ) set of raw materials provided by supplier e τ ) planning periods in which investments on facilities are allowed


Parameters Aert ) maximum availability of raw material r in period t associated with supplier e CLinemax ) upper bound of short-term credit line Coefett′ ) technical discount coefficient for payments to external supplier e executed in period t′ on accounts incurred in period t de ) maximum delay on payments of supplier e dm ) maximum delay in receivables at market m dmax M ) maximum delay in receivables at all markets MS Dtt′ ) technical coefficient for investments in marketable securities DFerd ) discount factor associated with discount interval d for raw material r offered by external supplier e Demimt ) demand of product i at market m in period t discerd ) discount offered by supplier e for raw material r at interval d MS ) technical coefficient for sales of marketable securities Ett′ E[ROE] ) expected return on equity FCFSjst ) fixed cost per unit of capacity of plant equipment j at site s in period t FCFWwt ) fixed cost per unit of capacity of distribution center w in period t gert ) cost of raw material r provided by supplier e in period t irLD ) interest rate of long-term debt t irSD t ) interest rate of short-term debt ISst ) investment required to open site s in period t IW wt ) investment required to open distribution center w in period t IuRM rt ) value of inventory of raw material r in period t IuFP it ) value of inventory of product i in period t MinCash ) lower bound of cash MinCSL ) lower bound of customer service level Othert ) other expected outflows or inflows of cash in period t Priceimt ) price of product i at market m in period t PriceFS jst ) investment required per unit of capacity of equipment j increased at site s in period t PriceFW wt ) investment required per unit of capacity of distribution center w increased in period t rf ) risk-free rate of return rp ) risk premium rate RR ) return rate SMS t ) marketable securities of the initial portfolio maturing in period t trate ) tax rate T ) length of planning horizon Binary Variables Fertd ) 1 if the amount of raw material r purchased from supplier e in period t is within discount interval d, 0 otherwise SBst ) 1 if site s is opened in period t, 0 otherwise Vjst ) 1 if the capacity of equipment j is increased at site s in period t, 0 otherwise WBwt ) 1 if distribution center w is opened in period t, 0 otherwise Xwt ) 1 if the capacity of distribution center w is increased in period t, 0 otherwise Continuous Variables APayt ) amount of accounts payable in period t ARect ) amount of accounts receivable in period t ASalestt′ ) sales executed in period t and receivables in period t′

Borrowt ) total amount borrowed from the short-term credit line in period t Capitalt ) capital supported by shareholders in period t Casht ) cash in period t CLinet ) short-term debt in period t CSL ) customer service level calculated at the end of the planning horizon CV ) corporate value at the end of the planning horizon Dept ) depreciation in period t DFCF ) sum of discounted free cash flows at the end of the planning horizon ECasht ) exogenous cash in period t EPurchet ) economic value of purchases executed in period t to supplier e ESalest ) economic value of sales carried out in period t FAssett ) increment in fixed assets in period t FCFt ) free cash flows in period t FCostt ) fixed cost in period t FExt ) other financial expenses and incomes in period t FFt ) funds flow in period t FSjst ) total capacity of plant equipment j during period t at site s FSEjst ) capacity increment of plant equipment j during period t at site s FWjst ) total capacity at distribution center w during period t FWEjst ) capacity increment at distribution center w during period t LBorrowt ) total amount of money borrowed from the longterm credit line in period t LDebtt ) long-term debt in period t LRepayt ) total amount repaid to the long-term credit line in period t NetCLine ) total amount of money borrowed or repaid to the t short-term credit line in period t ) total amount of money borrowed or repaid to the NetLDebt t long-term credit line in period t NetMS ) total amount received or paid in securities transact tions in period t NetDebtT ) net total debt at final period T NetInvestt ) net investment in period t NPV ) net present value computed for whole planning horizon Pijst ) production rate of product i in equipment j at site s in period t Paytt′ ) payments to external supplier e executed in period t′ on accounts payable incurred in period t Pledtt′ ) amount pledged within period t′ on accounts receivable maturing in period t Profitt ) profit achieved in period t Purchrm et ) amount of money payable to supplier e in period t associated with consumption of raw materials Purchtret ) amount of money payable to supplier e in period t associated with consumption of transport services Purchpr et ) amount of money payable to supplier e in period t associated with consumption of production utilities Purcherst ) amount of raw material r purchased to supplier e at site s in period t Purchrm et ) amount of raw material r purchased in period t qerdt ) amount of material r within discount interval d purchased to supplier e in period t Qiwst ) amount of product i sent from site s to distribution center w in period t

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Repayt ) total amount repaid to the short-term credit line in period t Salesiwmt ) amount of product i sold from distribution center w in market m in period t SIrst ) amount of stock of raw material r at site s in period t SOist ) amount of stock of product i at site s in period t SV ) salvage value of facilities at the end of planning horizon SWiwt ) amount of stock of product i at distribution center w in period t TProfit ) total profit achieved at end of planning horizon WACCt ) weighted average cost of capital in period t MS ) cash invested in period t′ in marketable securities Ytt′ maturing in period t MS Ztt′ ) security sold in period t′ maturing in period t ∆APayt ) change in amount of accounts payable in period t ∆ARect ) change in amount of accounts receivable in period t ∆Inv ) change in inventory value in period t ∆NWCt ) change in net working capital in period t Greek Symbols Rrij ) fixed coefficient for consumption of raw material r by product i βsj ) minimum utilization of plant equipment j capacity allowed at site s γw ) minimum utilization of distribution center w capacity allowed δmtt′ ) fraction of sales carried out in period t that are receivable in period t′ in market m θij ) capacity consumption of plant equipment j by product i λt ) proportion of equity over total capital investment in period t Ftr1 eiws ) unitary transport costs of product i from plant s to warehouse w payable to external supplier e tr2 Feiwm ) unitary transport costs of product i from warehouse w to market m τut1 ijse ) cost of the utilities associated with product i manufactured with equipment j in site s and payable to external supplier e τut2 rse ) cost of the utilities associated with handling the inventory of raw material r in site s and payable to external supplier e τut3 ise ) cost of the utilities associated with handling the inventory of final product i in site s and payable to external supplier e τut3 iwe ) cost of the utilities associated with handling the inventory of final product i in warehouse w and payable to external supplier e υi ) specific volume of product i φtt′ ) face value of accounts maturing in period t pledged in period t′ ψert ) price of raw material r offered by external supplier e in time period t Superscripts L ) lower bound U ) upper bound Literature Cited (1) Simchi-Levi, D.; Kaminsky, P.; Simchi-Levi, E. Designing and Managing the Supply Chain: Concepts, Strategies, and Case Studies; Irwin/ McGraw-Hill: New York, 2000. (2) Silver, E. A.; Pyke, D. F.; Peterson, R. InVentory Management and Production Planning and Scheduling; John Wiley & Sons, Inc.: New York, 1998.

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ReceiVed for reView January 31, 2007 ReVised manuscript receiVed July 17, 2007 Accepted July 27, 2007 IE070181E

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