Water desalination supply chain modelling and optimization

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Water Desalination Supply Chain Modelling and Optimization Malak T. Al-Nory#1, Stephen C. Graves#2 #

Engineering Systems Division, Massachusetts Institute of Technology Cambridge, MA, USA 1 2

[email protected] [email protected]

Abstract— The desalination industry has been growing progressively in the last few decades. A large number of new plants are contracted every year. Strategic decisions related to plant locations and capacity, the selection of the desalination technology, and many other technical decisions related to the plant design and operation are very critical to these strategic investments. Viewing the desalination industry network as a supply chain provides a holistic view allowing decision makers to perform optimization of water desalination operations end to end. The proposed methodology provides a set of modular simulation components to allow the creation of complex models to optimize the entire water desalination supply chain quickly and easily. The optimization is based on mathematical programming (MP) models that can be solved by external MP solvers.

I. INTRODUCTION Water covers three quarters of the earth’s surface, yet the shortage of fresh water is considered one of the most critical global issues to be resolved. According to United Nations reports [1], more than one-fifth of the world’s population (1.2 billion people) lives in areas of physical water scarcity and almost one quarter of the world’s population (1.6 billion people) faces economic water shortage where they do not have the appropriate infrastructure to extract water from sources. Even countries that do not suffer from water scarcity at this time may experience water shortages in the near future due to population growth (and the resulting increased water demand) and due to worldwide climate change such as drought and desertification. Therefore, additional fresh water sources are vital to be considered today by all countries. Water desalination has been realized worldwide as a viable solution to water shortage problems to process different types of water such as seawater, brackish water, or ground water. At the time of writing this paper, the total global capacity of desalination reached 71.9 million m3/d from about 16,000 desalination facilities around the world using a range of desalination technologies such as Multi Stage Flash (MSF), Multi Effect Distillation (MED), Reverse Osmosis (RO) and Electrodialysis (MED). The desalination industry has been growing steadily in the last few decades. New plants are contracted each year. The total capacity of plants contracted in 2010 was 5.9 million m³/d, which increased to 6.7 million m³/d in 2011, that is 747 new plants. [2]

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Many key decisions have to be taken before, during, and after the desalination plants are contracted and built. For example, there are strategic decisions related to the location and capacity of these plants, the selection of the desalination technology, and many other technical decisions related to the plant design and operation such as the number of stages and the water recovery rate. These decisions should be carefully studied because of their great implications on the plant investment and operation costs, efficiency, reliability and availability, scalability, and flexibility and many other important aspects. Great attention has been given in the water desalination literature to support decisions to optimize the unit operations of a single desalination plant and its processes. However, little or no attention is paid to support decisions to optimize the whole supply chain of water desalination entailing a network of plants. In fact, the term “water desalination supply chain” has not been defined yet in the desalination literature. We propose a methodology to support strategic decision making for the water desalination supply chain. Viewing the water desalination process as a supply chain provides a holistic view allowing decision makers to perform optimization of water desalination operations end to end. It also allows utilizing supply chain theories, models, and existing supply chain standards for performance measurement to increase overall efficiencies. Our methodology provides a modular framework to allow decision makers to create complex models to optimize the entire water desalination supply chain quickly and easily. The following sections are organized as follows. Section II describes related work. Section III defines the water desalination supply chain and provides a brief description of major water desalination processes. Section IV describes the proposed methodology using an example. Section V concludes and discusses future directions. II. RELATED WORK The performance of desalination processes is affected by many factors which can be studied and modelled to support decisions. The selection of the desalination technology and the optimal design of the desalination plant are considered the most important decisions in desalination systems. Structural optimization can be used to provide a detailed economical model for the complex design of the hybrid desalination plants such as the model proposed in [3]. The deterministic mixed-

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integer non-linear programming model is used to optimize operational parameters and process structure of a hybrid plant using Reverse Osmosis (RO) and Multi-Effect Distillation (MED) technologies. Process modelling and simulation are common tools for the optimization of the operations of desalination plants too. The model proposed in [4] examines the operating parameters and the performance of two RO plants. The statistical model proposed in [5] uses a neural network application to determine the optimal operating conditions of two types of commercial desalination plants (i.e., MSF and RO). The model identifies major independent variables to optimize the operation conditions for the desalination process. The work on process design modelling and optimization in general focuses on the process within the plant which is limited to the mechanical/thermal process decisions related to the operational level decisions. Optimization of water resource management work usually considers higher level of decisions, particularly the strategic decisions related to the optimal water supply and distribution to meet the overall demand. The linear programming model proposed in [6] solves the problem of matching different user demand such as for agriculture, industrial, and urban water with insufficient supply from different water sources such as desalination, water transported from other places, ground reservoirs, and dams. The problem objective is to optimally allocate water resources considering the value and priorities of the water usage in areas with limited water resources. The case study discussed in [7] provides means to use Vendor Management Inventory (VMI) control theory to water inventory control of the node lake of the study taking into account the possibility of flooding. Using a calculation model of optimal water order quantity under different water inventory control strategies increased water utilization efficiency and improved the total profit of the system. Perhaps the most related work to the present work is the optimization-based decision support tools for desalination systems. RESYSpro DESAL tool [8] provides support for technical and economical decisions using simulated system components. The main objective is to simulate the use of renewable energy components for desalination to help engineers in predicting the consequences of both the system design decisions (e.g., pump outlet pressure, ambient temperature) and the economical factors and design costs (e.g, investment for reference well pump, civil work for pumping or power subsystem). However, the tool provides a set of design components the user must choose from which makes available configurations very limited. In addition, the decisions supported are limited to operational parameters. The CoJava language [9] which is a Java-based programming language provides a process model that offers both the advantages of simulation-like process modelling, and the capabilities of true decision optimization by mathematical programming. SC-CoJava [10] [11] extends CoJava and provides a framework and an extensible library of simulation modelling components to support decision making for the supply chain. SC-CoJava also employs recourse stochastic models for decision making under uncertainty. [12] Stochastic

programming provides robust models for supply chain problems with uncertain parameters and is very applicable to the case of planning for water desalination supply chains. [13] However, water desalination processes require explicit definitions of desalination processes that the generic supply chain components of SC-CoJava do not support. Our methodology provides modular simulation models for desalination systems. These models allow simple creation of complex models for desalination supply chains, for example, consisting of desalination plants of different desalination technologies and different storage options, and considering strategic and operational level parameters. The optimization is based on mathematical programming models which are solved by external solvers such as CPLEX [14] or MINOS [15]. The optimization objective is defined by the modeller, for example to reduce energy consumption or reduce overall costs of the water desalination supply chain. III. WATER DESALINATION SUPPLY CHAIN The water desalination supply chain is a network of nodes with the objective of matching desalinated water demand with desalinated water supply. The water desalination supply chain activities consists of acquiring of raw water, raw materials, and chemicals needed for the desalination processes, manufacturing and shipping of needed components, construction and building of desalination plants and reservoirs, maintaining equipment, treating the discharged chemicals, storage of product water, distribution network activities, and delivering the final product to the end users. In addition, when water treatment activities exist, they represent product re-entry to the supply chain. The objective of this supply chain might be to minimize costs or energy consumption or maximize the amount of water produced given certain constraints. Among the limitations on different available resources used in the process, the amount of energy required represents the most challenging impediment and thus is considered the most important constraint. Figure 1 below shows a simplified example of the activities of the water desalination supply chain.

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Raw Water Feed

Supplies and Chemicals

Conversion Process • • •

Pre-treatment Treatment Post-treatment

Storage

Distribution

Brine Disposal

Fig. 1 A simplified example of water desalination supply chain

In Fig. 1 the demand on water from different water sources in different locations is met collectively by the produced and distributed desalinated water. The conversion process takes place in the desalination plants with different desalination technologies. Desalination technologies have been advancing rapidly providing many options for the water desalination industry. The main technologies commercially used for largescale production of desalinated water are described in the following. • Multi Stage Flash (MSF): A tube heats the seawater and evaporation results from a flow of brine flashing in the plant stages to produce vapour. • Multi Effect Distillation (MED): A seawater film evaporating in contact with a heat transfer surface. • Reverse Osmosis (RO): Industrial RO processes are pressure based where electric energy is used to pump saline through a series of semi permeable membranes. Table I provides a summary on the main characteristics of each of these technologies. TABLE I MAIN DESALINATION TECHNOLOGIES, ADOPTED FROM [16]

MSF

MED

RO

thermal evaporation large and very large seawater

thermal evaporation medium and large seawater

Product quality TDS

5-30 ppm (almost pure)

5-50 ppm (almost pure)

Energy source Energy requirement

heat

heat

14-25 kwh/m3

4-4.5 kwh/m

3.5-4.5 kwh/m3

Main issues

scaling, chemicals handling, large capital costs and energy consumption combined with power generation plants

large capital costs and energy consumption

33-37.5 %

33-37.5 %

reliability in high temperature and high salinity, fouling and bio fouling stand-alone plant or combined with MSF or MED 35-43 %

Family Main drive Plant size Feed source

Competitive settings

Recovery ratio

3

combined with power generation plants

membrane pressure medium, large and very large seawater and brackish water 20- 500 ppm (depending on number of passes) electricity

In addition to the variability in the requirements and performance as described in Table I above, different cost schemes of desalination systems exist because of the level of feed water salinity, energy sources and costs, capacity of plant, pre-treatment costs, disposal costs and other important site factors. [17, 18] These considerations influence the optimal design of the desalination systems. It is worth mentioning that hybrid systems are becoming very common. Hybrid systems might refer to the combination of power plants and desalination plants. Such hybrid systems have been introduced to balance the mismatch of demand on power and water. Many Middle East countries experience significant peak of demand on power in summer time. A significant percentage of the power plant remains idle in winter time. The steam of the power plant can be used in an MED or MSF plant to provide the energy needed to drive the thermal process. When demand on power is low, the turbine is bypassed to generate steam directly by boilers causing inefficiencies in power plant. Hybrid systems might also refer to using thermal (MSF or MED) and membrane (RO) technologies for the production of the desalination capacity. Adding a stand-alone RO plant to an existing MSF or MSF/power plant provides many advantages such as reducing water intake, extending membrane life, blending product water to achieve required quality, and reducing power to water ratio. Hybridization allows utilizing excess power in winter time to perform RO desalination and to continue operating the thermal desalination plant. In case of excess water production, the water can be stored for future use. [16] The efficient design and operation of the supply chain are considered critical components of the planning activities at the strategic level which involves deciding on the configuration of the network, including the number, location, capacity and technology of the desalination plants and storage systems. The tactical planning level of supply chain operations involves deciding the flow of material for acquiring, processing, and distribution. The operational planning level involves determining the operating conditions such as pressure, concentration, and feed flow rate. IV. METHODOLOGY BY EXAMPLE To understand the proposed methodology let us consider the following simplified example of a water desalination supply chain. • We have various desalination plants of different technologies to supply desalinated water; each one has a maximum capacity with certain known cost to produce 1 cubic meter of desalinated water. • We have various reservoirs to store desalinated water for later use; each one has a certain capacity with certain known cost to store 1 cubic meter of desalinated water. Total storage has to meet the national security levels. • We have various cities; each one has a certain demand level.

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The objective is to support the strategic decisions on which plants and reservoirs to build or open and in how to optimally allocate desalinated water production to each of the demand cities and to storage sites. A mathematical formulation is required to solve the supply chain design problem described above. The following deterministic mathematical model can be employed to solve the problem for a single time period.

Subject to yi = {0,1} i P ∪ R ∑ xi = ∑ xij xin i P xi ≤ si * yi i P xij ∑ xjn j R xj = lj + ∑ xj ≤ sj * yj j R ∑ xjn ≤ lj j R ∑ xin ≥ d n ∑ xj ≥ l xi , xj , xij , xin , xjn ≥ 0 i P, j R, n D

A. Mathematical Formulation for the Example Consider a supply chain network G=(N,A) N represents the set of nodes which includes P the set of candidate plants, R the set of candidate reservoirs, and D the set of demand cities. A represents the set of arcs connecting the plants to reservoirs or cities and reservoirs to cities. Note that not each node is connected to all the other nodes. i.e., not all plants can deliver to all the cities or all the reservoirs, thus the set in A is a discrete set of arcs known as the distribution network (existing pipelines). For the supply chain strategic decisions on which plant or reservoir to build we use a binary variable yi such that yi =1 if a plant or a reservoir is built at site i, and 0 otherwise, for all i For the operational decisions for each plant P ∪ R. or reservoir we use xi to denote the quantity produced at each plant or stored at each reservoir at site i. To model the flow of desalinated water from the plants to the reservoirs or to the cities, and from the reservoirs to the cities we use xij with two subscripts that denote the arc from i to j. The first subscript is the origination node of the network from which water is transported and the second subscript is the destination node of the network to which water is transported. The objective of the model minimizes a. Total investment costs (cost of building/opening new plants + cost of building/opening new reservoirs) min ∑ ci yi

Equations (1) to (9) describe the following constraints. (1) The decision to build a plant or a reservoir at site i is binary. (2) The quantity produced at a plant at site i must be transported out. (3) The quantity produced at a plant at site i is limited to its maximum capacity si. The constraint also guarantees that a plant can produce only if opened. (4) The quantity stored at a reservoir at site j R is defined as its starting level lj plus the quantity transported into the reservoir less the quantity transported out. (5) Total quantity stored at a reservoir at site j R is limited to its capacity sj. The constraint also guarantees that a reservoir can store water only if opened. (6) Total quantity transported from reservoir at site j R is limited to its current level lj. Note that we assume that the amount transported to be stored at a reservoir xj cannot or should not be transported out in the same time period. (7) Total quantity of desalinated water delivered to a city satisfies the demand of this city d n . (8) Total quantity of desalinated water stored by all reservoirs satisfies the national security level l . (9) The quantity produced at a plant, stored at a reservoir, or transported from a node to another node is non-negative.

Where ci denotes the investment cost for building/opening a plant or a reservoir at site i P R b.

min ∑ ∑

Total operational costs (cost of water production at plants + cost of water storage at reservoirs + cost of transporting water from plants to cities directly + cost of transporting water from plants to reservoirs + cost of transporting water from reservoirs to cities) ∑

q i xi + ∑ qjn xjn



qij xij





qin xin

Where qi denotes the cost of producing or storing 1 m3 of desalinated water at plant or reservoir at site i qij, qin, qjn denote the cost of transporting 1 m3 of desalinated water from plant at site i to reservoir at site j, from plant at site i to city at location n, from reservoir at site j to city at location n respectively

(1) (2) (3) (4) (5) (6) (7) (8) (9)

The above model is formulated as a single period, but it is structured so that an extension to multiple periods is straight forward. It is also a very simple model which skips the operational level parameters; however, formulating such a detailed model requires certain level of expertise in mathematical modelling. In addition, the problem gets very complex when adding operational level parameters for each of the nodes. Our modular modelling methodology provides the non-specialized users with decoupled modelling components to allow easy construction of complex optimization models. B. Conceptual Modelling Components Fig. 2 shows a hierarchy of Java classes each of which represents a component of the desalination supply chain. The supply chain network nodes are represented by the abstract class Node which is extended in the hierarchy by

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concrete classes such as Source, Supplier, Plant, Reservoir, maximum raw water salinity and location longitude and Disposal, and User. Each of these classes is further extended latitude. by specialized types. For example, Plant node is extended by types representing MSF plant, MED plant, RO plant, ED plant, public class Water extends Flow { a portable unit, and a Hybrid plant. double double double double double

TDS; temperature; pH; pressure; quantity;

Water(int fID, String fDes, double tds, double temp, double ph, double prsr, double qty) { super(fID, fDes); this.TDS = tds; this.temperature = temp; this.pH = ph; this.pressure = prsr; this.quantity = qty; } }

Fig. 3 Water class attributes

Using the Java programming language a Node class only encodes the transformation describing the relationships between its incoming flow and its outgoing flow. For example Fig. 4 shows two simple methods encoding how electrical energy and water requirements relate to the recovery ratio and performance ratio in a reverse osmosis plant. Similarly, MSF plant would encode the water and energy requirements but would also include the thermal energy requirements. protected double determEnergyReq(double perfRatio){ double elecEnergy = perfRatio * outFlowWtr.quantity; return elecEnergy; } protected double determWaterReq(double recovRatio){ double wtrQty = 1+ recovRatio * outFlowWtr.quantity; return wtrQty; }

Fig. 4 Two methods of the ROPlant class

Fig. 2 Hierarchy of Desalination Supply Chain modelling components

Conceptually, a node represents a transformation from incoming flow to outgoing flow. For example, the reverse osmosis plant ROPlant node transforms Flow of type Material (chemicals and equipment), Water (raw water), and Energy (electrical) to Flow of type Water which is desalinated and ready for distribution or storage. Desalination water is characterized by its pH, total dissolved solid (TDS), temperature, and pressure which are the Water class attributes (members) as shown in Fig. 3. A Node must be associated with one or more metrics of type ObjMetric to represent the objective of the Node such as assets, cost, flexibility, reliability, and responsiveness which are the standard metrics defined by Supply Chain Operations Reference. [19] Operational parameters are provided to the node by its NodeInfo class. For example ROPlant class has ROPlantInfo as a class member which in turn has members representing the plant performance ratio, recovery ratio, number of passes,

To formulate a decision problem, a user defines a subclass of the class Node and use the method Nd.choice(double min, double max) to indicate any unknown choice constants which become the decision variables of the problem. For example, the decision variable representing the water quantity to be produced by each plant among a set of possible plants to meet the demand can be defined using the Nd.choice method. The assert(Boolean) construct is used to indicate the problem constraints that must be satisfied described as Boolean conditions. For example, the constraint that the quantity produced by a plant must not exceed its maximum capacity can be defined using the assert construct. The Nd.minimize(double) or Nd.maximize(double) are used to indicate the objective to be minimized or maximized. C. Building the Supply Chain Model To build the supply chain model in the example above the user needs to create a new class which extends the Node class and encodes the plants, the reservoirs, and the cities by

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invoking the representing classes in the components library. The user then passes the parameters using the corresponding classes of the NodeInfo type. Compiling and running the Java classes provides a simulation procedure which leaves the decision variables open (i.e., non-deterministic). Then the Cojava compiler [9] translates this procedure into an equivalent decision problem in AMPL modelling language using a reduction algorithm. The resulting AMPL problem is then sent to an external MP solver for Linear Programming (LP) or Mixed Integer Linear Programming (MILP) as appropriate. AMPL is one available mathematical modelling language. The Cojava compiler [9] translates the Java code into AMPL constraints and this is the reason we provide the direct AMPL formulation of the problem (i.e., not the Cojava compiler product) in the appendices. Appendix I shows the dataset for the example in AMPL data format and Appendix II shows the direct AMPL model. We assume three plants (two existing and one new), four water reservoirs (three exiting and one new), and two cities with specific demand level for each. We also assume a minimum storage level at reservoirs to satisfy a national security level. The model includes binary variables and thus it was solved for MILP using Gurobi solver on NEOS Server [20]. The results obtained from the solver are shown in Table II below. TABLE II SOLVER RESULTS

Gurobi 4.5.1: optimal solution; objective 5645750 15 simplex iterations plus 26 simplex iterations for intbasis Build/Open Variables R1 1 R2 1 R3 1 RN 0 Production/Storage Variables P1 5450 R1 450 P2 10000 R2 0 PN 10000 R3 250 RN 0 Transportation Variables Plant to storage P1-R1 450 P2-R1 0 PN-R1 P1-R2 0 P2-R2 0 PN-R2 P1-R3 0 P2-R3 0 PN-R3 P1-RN 0 P2-RN 0 PN-RN Plant to users P1- U1 5000 P2-U1 0 PN-U1 P1- U2 0 P2-U2 10000 PN-U2 Reservoirs to users R1-U1 0 R2-U1 0 R3-U1 R1-U2 100 R2-U2 50 R3-U2 P1 P2 PN

1 1 1

AMPL model solved above. While the direct AMPL model is usable only for this specific problem, the simulation components can be used to solve a wide variety of problems to support decisions on different levels. The Java classes are designed with multiple constructors each allowing the user to invoke a class (to instantiate an object) using different set of parameters. For example, ROPlant class can be invoked with or without the operational level parameters such as recovery rate and energy consumption quantities. This allows additional flexibility in the usage of these classes. The main intuition behind the proposed methodology is to solve real world problems. The Cojava compiler [9] should be able to translate any problem size as long as the Java code size within the available Java compiler buffer. The buffer size can be adjusted by the user but it would be the bottleneck of the size issue. In addition, most solvers come with presolve option which if turned on all unnecessary constraints are eliminated to make the problem of a manageable size. However, the user needs to be aware of the number of variables the code would generate and the capability of the solver in use. V. CONCLUSIONS AND FUTURE WORK We provide modular simulation model components for desalination systems. These reusable and extensible model components allow simple creation of complex models for desalination supply chains easily and quickly. The generated optimization models are based on MP and thus can be solved by external MP solvers. Some of the interesting directions involve modelling the relationship between power and water in cogeneration and hybrid plants of more than one desalination technology. Other interesting directions involve modeling for multi-objective and multi-criteria planning in particular for cases when we have multiple decision makers each with different optimization criterion. ACKNOWLEDGMENT The authors would like to thank the King Fahd University of Petroleum and Minerals in Dhahran, Saudi Arabia, for funding the research reported in this paper through the Center for Clean Water and Clean Energy at MIT and KFUPM under the Ibn Khaldun Fellowship. The authors would like also to thank Effat University in Jeddah, Saudi Arabia, for the sabbatical assistance provided to the first author.

0 0 250 0

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0 9750

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The set of AMPL constraints provided by the Cojava compiler when running the user class for the supply chain problem represents an equivalent decision problem to the

[5]

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United Nations. (2012) Water for Life Decade: Water scarcity. [Online]. Available: http://www.un.org/waterforlifedecade/scarcity.shtml. IDA Desalination Yearbook 2011-2012, International Desalination Association. Media Analytics Ltd, Oxfored, United Kingdom, 2012. M. Skiborowski, et al., “Model-based structural optimization of seawater desalination plants,” Desalination,vol. 292, pp. 30-44, 2012 T. Kaghazchi, et al., “A methematical modeling of two industrial seawater desalination plants in the Persian Gulf region,” Desalination, vol. 252, pp. 135-142, 2010. K. Al-Shayji, “Modeling, Simulation, and Optimization of Large-Scale Commercial Desalination Plants,”. Chemical Engineering. Doctor of Philosophy thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 1998.

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APPENDIX I

param tij: P1 P2 PN param tin: P1 P2 PN param tjn: R1 R2 R3 RN

R1 2 20 11 U1 22 200 50 U1 10 22 30 12

R2 12 3 10 U2:= 120 70 100; U2:= 17 27 11 25;

#constraints parameters param demand:= U1 U2 param level:= 700; param psupply:= P1 P2 PN param rsupply:= R1 R2 R3 RN param rcurrent:=R1 R2 R3 RN

R3 1 12 2

RN:= 6 7 3;

5000 20000; 11000 10000 10000; 500 100 1000 2000; 100 50 100 0;

APPENDIX II AMPL Model File set plants; set reservoirs; set users; #investment costs of plants and reservoirs param pc{i in plants}; param rc{j in reservoirs}; #operational costs of plants and reservoirs param pq{i in plants}; param rq{j in reservoirs}; #transportation costs param tij{i in plants, j in reservoirs}; param tin{i in plants, n in users}; param tjn{j in reservoirs, n in users}; #constraints parameters param demand{n in users}; param level; param psupply{i in plants}; param rsupply{j in reservoirs}; param rcurrent{j in reservoirs};

AMPL Data File set plants:= P1 P2 PN; set reservoirs:= R1 R2 R3 RN; set users:= U1 U2;

#investment decision variables var py{i in plants} binary; var ry{j in reservoirs} binary;

#investment costs of plants and reservoirs param pc:= P1 0 P2 0 PN 1200000; param rc:= R1 0 R2 0 R3 0 RN 150000;

#production and storage decision variables var px{i in plants}; var rx{j in reservoirs}; #transportation decision variables var ijx{i in plants, j in reservoirs}; var inx{i in plants, n in users}; var jnx{j in reservoirs, n in users};

#operational costs of plants and reservoirs param pq:= P1 120 P2 100 PN 100; param rq:= R1 1 R2 2 R3 3 RN 1;

#objective function minimize Cost: sum{i in plants} pc[i]*py[i] + sum{j in reservoirs} rc[j]*ry[j]+ sum{i in plants} pq[i]*px[i]+ sum{j in reservoirs} rq[j]*rx[j]+

#transportation costs

179

sum{i in plants, j in reservoirs} tij[i,j]*ijx[i,j]+ sum{i in plants, n in users} tin[i,n]*inx[i,n]+ sum{j in reservoirs, n in users} tjn[j,n]*jnx[j,n]; #constraints subject to Positive1{i in plants, j in reservoirs}: ijx[i,j]>=0; subject to Positive2{i in plants, n in users}: inx[i,n]>=0; subject to Positive3{j in reservoirs, n in users}: jnx[j,n]>=0; subject to Positive4{i in plants}: px[i]>=0; subject to Positive5{j in reservoirs}: rx[j]>=0; subject to PSupply{i in plants}: px[i]