Journal of Microcomputer Applications (1987) 10,41-54

A multi-objective personal computer model for energy conservation planning

B. Malakooti

Systems Engineering Department, Center for Automation and Intelligent Systems, Case Western Reserve University, Cleveland, Ohio 44106, USA

In this paper a microcomputer software package is implemented to help the decision maker (DM) to obtain the best energy conservation policy in the melting phase of glass production. A multiple objective linear programming formulation for the melting phase is developed that demonstrates how different energy conservation options can be incorporated into the model. An interactive paired comparison method as a decision support system is discussed. The paired comparison method is implemented by the DM to find the best compromise solution given the conflicting objectives. A report of the implementation and the DM’s interactions with the method is provided.

1. Introduction In this paper we implement an interactive software package developed for microcomputers for selection of energy conservation options for the melting phase of

glass production. There have been several efforts to use computers for planning energy conservation in the United States and other countries (Haddock, 1984; Michaux & Tomas, 1985; Tabucanon, 1985). This study differs from others in that it focuses on one

phase of production in one type of industry and introduces the technological options in terms of energy conservation. We also demonstrate how such models can be formulated,

implemented, and solved by a multiple criteria approach. Our formulation of the problem considers different fuel substitutions, conservations, and their associated costs.

We present this model as a case study for the whole United States, but our intention is that it can be used as a pilot study for glass plants. We implement the model for the melting phase of production because the melting phase consumes the majority of the energy.

In this paper we develop a framework of multiple criteria decision making (MCDM) to consider different objectives such as maximizing profit, minimizing energy consumptions of different fuels, and maximizing the quality of the molten glass produced. The motivation for using the multiple criteria analysis is given in the next section.

Natural gas, electricity, and oil are main energy fuels consumed in the glass industry. Natural gas is the prime energy consumption in the current glass industry. In 1978,

Congress passed the Natural Gas Policy Act (NGPA). The NGPA brought intrastate gas under federal regulation by imposing complex price controls on gas production for the first time. In a study conducted by DOE in November 1981 (DOE, 1981), three alternatives to the natural gas policy act of 1978 were proposed: (1) extension of price controls, (2) full decontrol in 1982, and (3) phased decontrol. After a detailed analysis, the DOE study concluded that all three policies make natural gas cheaper in most

41 7138-0745/87/010041+ 14 $03.00/O 0 1987 Academic Press Inc. (London) Limited

42 B. Malakooti

regions than oil or electricity. This conclusion of the report implies that the transition from natural gas to any other fuels is not essential in the next five or ten years. It should be considered, however, in case of an emergency gas shortage in short-term or long-run planning.

Currently, feasible alternative fuels to natural gas are oil and electricity. Since oil is one of the main energy imports, the national policy is directed toward decreasing its consumption. The important resources that could be used in the generation of electricity are coal and nuclear power, but these fuels are only marginally feasible at the present time. For one thing, electricity depends critically on whether or not fast breeder reactors are developed. Strong uncertainties remain, however, and safety, waste disposal, and proliferation problems (which imply social and environmental risks) have not yet been resolved for nuclear power. Another option for generation of electricity is coal, although coal, whether it is directly combusted or converted to liquid, causes environmental problems. In general, coal-generated power is cheaper than nuclear power. In the next section we discuss how the energy problem can be formulated as multiple criteria decision making.

This paper consists of seven sections. In Section 2 the background and importance of multiple criteria decision making for energy analysis is discussed. We also describe the work done in this area at Brookhaven National Laboratory. In Section 3 the melting phase of glass production and its associated energy conservation options are explained. Section 4 discusses a linear programming formulation of the problem. In Section 5 the interactive paired comparison method and its recently developed microcomputer software package are explained. Section 6 reports the experiments with the computer package and finding the best compromise solution through interactions with the DM. Section 7 is the conclusions.

2. The background and importance of MCDM energy analysis

In most energy system analyses, the whole energy system has been programmed by the measure of total system cost. A single objective minimization problem, however, is not sufficient for current energy systems’ decision making problems. For example, the issues of environmental quality and foreign oil dependence compete in priority with cost production. Therefore, it is essential,to identify and employ approaches that characterize and evaluate an energy system by multiple criteria decision making.

Since the goal of multi-objective decision making is to identify and quantify the trade-offs between different economical or social objectives, it will aid the policy makers in achieving the best possible compromise between conflicting objectives. It is necessary, however, to provide the decision makers, both in the public and private sectors, with a more easily understood tool and a more easily defensible methodology on which trade-offs between certain sensitive and competing energy issues can be based.

Here follows a brief discussion and outline of the background of multi-objective energy systems related to Brookhaven National Laboratory (BNL) energy systems and to other technical developments of the last decade.

The Brookhaven energy system optimization model (BESOM) is a linear programming model that captures the numerous attributes of the United States’ energy

PC model for energy conservation planning 43

system (Kydes, 1978). Essentially, BESOM decides which energy forms are to satisfy energy service demands. In one of the analyses by the BNL, the following eight conflicting objectives were identified and quantified according to this model (Cherniavsky, 1979): (1) total annual cost; (2) annual cost excluding end-use devices; (3) capital requirements; (4) oil imports; (5) total crude oil; (6) resource use; (7) environmental index; and (8) total nuclear fuel. Also, the criteria of cost, environmental effect, crude oil, and nuclear fuel were considered according to BESOM in the year 2000 (Ho, 1979). So far, there have been five different multi-objective decision making analyses done with BESOM, two of them on generating efficient alternatives, one on using utility functions, and two on using interactive methods.

Generating efficient alternatives (Cherniavsky, 1979; Hoffman, Beller, Cherniavsky & Fisher, 1976) used the technique of determining trade-off curves. The procedure relies on the construction of trade-offs between different pairs of competing objectives. In using the trade-off information, the decision maker derives his best solution by forcing several of the target objectives to satisfy judgmentally determined levels while optimizing another objective. For example, in Hoffman et al. (1976) and Kydes (1978) trade-offs between several pairs of objective functions were investigated, including oil imports versus capital investment, environmental index versus total cost, and oil imports versus environmental index. Thereafter the best alternative was chosen by the DM. These processes proved computationally burdensome, however.

The utility function approach (see Hwang & Masad, 1979, for references) employed utility functions. Three forms of utilities were developed: a minimax form, a quadratic form, and a product form. In all three forms, the distance of any objective function from its optimal value was minimized according to the utility form. The utility function approach assumes a restrictive and inflexible format about the DM’s preferences.

The first interactive method selected to be tried was that of Zionts and Wallenius for linear programming problems (see Hwang & Masad, 1979, for overviews). Zionts & Deshpande (1978) discussed the progressive report of this method both in applying multi-objective energy systems and in developing the concave objective function case of the Zionts and Wallenius method. The last interactive method, by J. Ho using the holistic preference method (HOPE) (Ho, 1979), has been experimentally implemented by him in multiple criteria energy policy analysis. The algorithm uses parametric linear programming to solve the multiple criteria optimization.

In general, interactive methods are more flexible than the other two above techniques because the DM can respond to preference questions with some degree of flexibility aud choose the best alternative while searching through the set of alternatives. In the process of interaction the DM learns more about his/her preferences as he/she examines the partial set of alternatives. This learning process is then captured by the computer and is used to generate possibly more preferred alternatives. Finally, the optimal point may be obtained without the DM being required to enumerate the whole set of alternatives, which is cumbersome (as in the generative methods) or to construct the complete and specific utility function, which is restrictive (as in the utility function methods).

Some other efforts using multi-objective analysis are by Elchak & Raphael (1977) for the interrelationships of various supply, conversion, and consumption sectors based on a linear flow for Pennsylvania; they used a goal programming technique by setting target levels to achieve for objectives. In Tabacanon & Mukyangkoon (1985) a

44 B. Malakooti

microcomputer-based approach using goal programming is discussed for different production planning problems. Summers (1978) developed a multi-goal approach based on goal programming to solve the national energy planning.

Although goal programming is very well known for solving multi-objective programming problems, it is less flexible than interactive approaches because the goal or target levels are fixed and must be givlen. Furthermore, priority on objectives is lexicographical, implying no trade-off among objectives; for example, the best objective is optimized completely, then the second objective is optimized, and so forth.

Recently, Haddock & Sparrow (1985) developed a long-range energy plan determining what energy resources and conversion systems should be used and implemented by the year 2000 in Puerto Rico. The objectives were cost minimization, imports (coal, uranium, and oil) minimization, and the minimization of the environmental effects. To minimize the three objectives they used the global criterion method; see Hwang & Masad (1979) for references. The global criterion method assumes a given form of utility function to be optimized, i.e. it can be categorized as the utility function approached.

3. Glass melting energy analysis

3.1 General description

Of the three major glass types (flat glass, glass containers, and pressed and blown glass), the production of glass containers consumes more than 50% of all the energy used in the entire United States glass industry (Battelle Lab., 1975a, b). The production of glass containers (or any other type of glass) consists of five phases of production: preparation of batch mix. melting, fining, forming, post forming, and product handling. The melting/fining phase uses about 85% of the total energy. In this paper we consider only the melting phase of the glass container production; our analysis can be extended to all different types of glass and to different phases of production.

The input to the furnace in the melting process is either raw materials (weighted and proportionally mixed) or scrap glass (cullet). The main raw materials are sand, soda ash, feldspar, lime, and dolomite. Cullet is either brought from outside or recycled from other phases of production in the plant. The increase of cullet can increase impurity and hence lower the quality. In the glass melter, the raw materials and cullet are melted down. The molten glass is fined, that is, residual trapped gases are removed, and then the temperature of the glass is lowered so that it can be handled in the forming phase.

The glass container industry uses 11.0 x lo6 BTU of purchased fuel and electricity (excluding batch handling, space heating and cooling, and product handling) to produce a ton of finished glass (Battelle Lab., 1975a). The glass industry has historically depended on natural gas and electricity to supply over 90% of its energy needs. Large variations occur in the amount of energy used in the melting phase, which depends on many factors, such as furnace design, furnace age, furnace size, and glass composition. In 1974-75, the lack of natural gas availability significantly increased the use of fuel oil. Substitution of fuel for natural gas occurs basically in the melting operation and the heating of boilers. Electricity can also serve as a heat source in some phases, but this usually increases operating costs.

3.2 Energy conservation options in melting

We consider four options.

PC model for energy conservation planning 4.5

A. Energy substitution. Natural gas is the dominant energy source and since it is becoming less available, concern is expressed for what fuels can be substituted for it now and in the future. The substitution of oil or electricity for natural gas is the primary focus of attention. Basically, oil and propane are available for substitution. Electricity can also be used in a limited way. The indirect use of coal or nuclear energy may be considered for generation of electricity. The substitution of these fuels cannot totally replace natural gas in the glassmaking process.

B. Oxygen enrichment. Since increasing or enriching the air with extra oxygen decreases the amount of nitrogen heated, the result is a higher flame temperature and higher available heat release within the furnace per unit of fuel. It has been shown that increasing the oxygen content from 21% to 23% increases the theoretical flame temperature from 1970°C to 2151°C. This could increase outputs as much as 12% (Boone & Rosenberg, 1967).

C. Increase in c&et (scrap glass). Experience has shown that an increase of 1% results in 0.25% reduction in energy consumption. This would give one of the best options for energy conservation if the scrap glass is available and cheap enough to be purchased. The impurity of glass increases as the cullet increases.

D. Znspirated air to tip cooling. The use of regenerative devices, as currently implemented, is an effective method for recovering waste heat. By the use of regenerators, heat recovery for air preheating improves the efficiency of fuel utilization. With regenerators the air for combustion can be preheated to lOOO-1100°C or more, according to the construction and size of the regenerator and to the operating temperature of the tank. A prediction of 14% energy conservation was reported (Boehner, 1979) by researchers implementing this option.

4. A model for energy conservation

In this section we discuss a linear programming formulation of the problem and how the energy conservation options can be introduced into it. Our primary motivation is to illustrate how such a model can be constructed so that the private industry as well as other organizations would be able to construct similar models. The data were gathered for a five-year period for molten glass production in the United States. The model consists of 12 constraints and 28 variables. In this section we demonstrate the single-objective model for profit maximizing and in Section 6 we discuss the multi-objective model. The linear programming formulation is as follows.

MAX. -11 X,-1.87 X,-9.5 X,-2.33 X,-9.5 X,-2.33 X,-18 X,-20 X,,-9.99 X,, - 0.99 X, + 125.28 X,, + 119.06 X,, + 110.28 X,,

SUBJECT TO

(6) -X,-O.8 X,-O.7 X,+5.8 X,,+5.58 X,,+5.2 X,,+S X,,+4.8 X,,+4.5 X,,+5.8 X,,+5.58 X,*+5*2 X,,+5 z&,+4.8 X*,+4.5 X,,=O

(7) -x,+0*09 x,,+o*o9 x,*+0.09 x,,+o.o9 x,+0.09 x1,+0.09 x,=0 (8) -X,,+X,,+X,,+X,,+X,+X,,+X,,=O

46 B. Malakooti

(9) -X,+0.927 X,,+O.7725 X,,+O.515 X,,+O.927 X,,+O.7725 X,,+O.515 X,,+O.927 X,,+O*7725 X,,+O.515 X,,+0.927 X,+0.7725 X,,+O.515 X,,=O

(10) -X,,+O.103 X,,+O.2575 X,*+0.515 X,,+O.103 X,,+O.2575 X,,+0.515 X,,+O.103 X,,+O.2575 X,,+O.515 X,,+O.103 X,+0.2575 X,,+O515 X,,=O

(11) X,+X,,-o~12 X,,-0.12 X,*-O.12 X,,-0.12 X*,--O.12 X,,--0.12 X,,+X,= 112.0168

(12) X,, + X, = 32.40655 (13) X2,+X2,=41.61562 (14) X, to x,,ao

All the variables are listed below. They are all associated with the production of one

million tons of molten glass. Input materials are also measured in millions of tons. All the energies are in lOI BTU. All the costs are in millions of dollars.

X, = Slack, variable X, = Labour, operation and management costs X, = Slack variable X, = Gas consumption X, = Electricity consumption X, = Oil consumption X, = Electricity substitution for gas X8 = Oil substitution for gas X, = Batch mix (raw materials)

X,, = Cullet X,, = 10% cullet, no inspiration, no oxygen X,, = 25% cullet, no inspiration, no oxygen X,, = 50% cullet, no inspiration, no oxygen X,4= 10% cullet, inspiration, no oxygen X,, = 25% cullet, inspiration, no oxygen X,, = 25% cullet, inspiration, no oxygen X,, = 10% cullet, no inspiration, oxygen X,, = 25% cullet, no inspiration, oxygen X,, = 50% cullet, no inspiration, oxygen X,, = 10% cullet, inspiration, oxygen X,, = 25% cullet, inspiration, oxygen X,, = 50% cullet, inspiration, oxygen X,, = Inspirated air usage X,, = Oxygen usage X,, = Melting, production, demand I X, = Melting, production, demand II X,, = Melting, production, demand III X,, = Slack variable

See Figure 1 for an illustration of the model. Total molten glass production is x,, + x2,+ X2,. Constraint (2) implies that one million tons of molten glass requires 0.1 unit of labour, operation, and management where each unit costs $11 million. Note that

(11) is the coefficient of X2 in the objective function (1). Constraints (3) and (4) are electricity and oil consumption. Constraint (5) implies that the total sold molten glass is equivalent to the production of molten glass by all different energy consumption options. Constraint (6) is the gas .consumption for different energy conversion options. Energy substitution variables, X, and X8, can be used instead of gas, X,. However, substituting electricity or oil for gas entails an energy loss. These conversion coefficients are 0.8 and 0.7 in constraint (6).

The energy conservation options are formulated as follows. For simplicity we assumed only three levels of cullet percentages of 10, 25, and 50, given that minimum

PC model for energy conservation planning 47

input

Lobour and energy fuels

JJ-J-J-

Xl1 x9 x12 _

Melting phase t

moteriols production through

%O different options

x22 _

Waste

t t t t x23 x24 X7 X6

Energy options and substitutions

X 26 '26

x27

Total output

Figure 1. Demonstration of the model.

and maximum cullet are 10% and 50%. The cullet is either recycled from the plant or purchased from the outside. Given the three levels of cullet and two other different options (inspirated air and oxygen enrichment), we have 12 different options. They are variables X,, to X,,. For example, if variable X,, is positive in the final solution it indicates that we should use 25% Gullet, use the inspirated air option, but not use the oxygen enrichment option. Constraint (6) demonstrates the gas consumption for each of the above 12 options. Constraints (7) and (8) sum up the oxygen use and inspirated air usage of different types of production options by variables X,, and Xz3, respectively. Constraints (9) and (10) enforce the material requirements proportion of cullet versus other materials. For example, coefficients for X,, are 0.927 and 0.103 in constraints (9) and (10). This implies that 90% batch mix and 10% cullet are used. Note that O-927 + 0.103 = l-03, implying that 3% waste is considered. That is, l-03 million tons of input materials can produce one million tons of molten glass.

Constraint (10) is the capacity limitation. Note that since the energy conservation option of oxygen enrichment increases the throughput by 12%, the coefficient of variables X,, to X,, in (10) are 0.12. That is, if any of these variables are used they increase the capacity use by 12% of their amount.

Constraints (11) and (12) are related to the linear piecewise demand function. As we produce more glass the price per unit decreases. Variables X15, Xz6, and X,, are associated with three levels of production. Since the profit of X,, is higher than X,, and X,, is higher than X,, [see (l)], X,, cannot assume any positive value until X,, becomes positive and similarly X,, before X,,.

If the total production is between 0 and 32 million tons the price of the molten glass is $12528 million. From 32 to 41 million tons of production the price is $119 million. For

more than 41 million tons the price is $110 million per million tons. If we maximize the objective function using (1) to (14) the solution is as follows:

X,= 12.30 A’, = 640-09 A’,= 27.08

48 B. Malakooti

X,= 18.46 X,= 63.39

X,,= 63.39 Jr,,= 123.09 xz4= Il.07 Xzs= 32.40 X,,= 41.61 x,,= 49.07

This solution implies that we should use the options associated with X,s, i.e. use 50% Gullet and use the oxygen enrichment option. However, energy substitution or inspirated air options are not recommended. The total production, 32.40 + 41*61+ 49.07 = 123.07, is more than the capacity, I 12.01, because the oxygen enrichment increases the throughput and hence the total profit is increased. Although the above solution maximizes the profit, it may not be the best policy in terms of other objectives. For example, 50% cullet may increase impurity and hence reduce the quality of the glass. In terms of energy consumption the model mainly recommends gas; compare X, with X, and with X,, which is partially an imported fuel.

5. A summary of the interactive paired comparison method and the microcomputer package

This method, developed by Malakooti (19856), is an extension of the work of Malakooti & Ravindran (1985/6; Malakooti, 1985~~). Malakooti (1986) used Malakooti & Ravindran (1985/6) and Malakooti (1985~~) to solve all the phases of the glass container industry problem. Here we use Malakooti (1985b) to solve the melting phase of the glass production. The questions asked in this method are all in the form of paired comparison of alternatives, i.e. one alternative can be preferred, not preferred, or indifferent to another alternative. If the DM wishes, he can also respond by indicating strong or weak preference; that is, he may prefer one alternative to the other strongly or weakly. The strength of preference questions would help to converge to the optional point faster (with fewer questions) because more information is obtained from the DM in each question session.

Malakooti’s method (Malakooti, 19856) is developed to solve the multiple objective linear programming (MOLP) problem. It is assumed that there is a non-empty set of alternatives defined by linear constraints on the decision variables. All decision variables are non-negative. These are the number of objectives,fi,f*, . . ., f,, each being a linear function of decision variables. Since the objectives are usually non-commensurate and conflicting, we require a methodology to resolve the conflict by finding the best compromise solution. To find the best compromise, we recommend using an interactive method. During this interaction process with the DM we gather information about his preferences. We use the information obtained from the DM to decrease the number of possible alternatives, and to predict the best compromise solution and present it to him. As we proceed, either the DM is satisfied with the point presented to him (and we stop the process), or the number of possible alternatives reduces to one, i.e. the latest point would be the best compromise solution. In order to reduce the number of alternatives and predict a good solution, we assume that there is an unknown utility function that represents the DM’s preferences. This utility function, U, is a function of all objectives,

i.e. U&f2, . . .,f,). We can assume that its structure is (a) linear, or(b) quasi-concave, or (c) unknown.

PC model for energy conservation planning 49

The quasi-concave is more flexible than the linear or concave functions, and therefore less restrictive in terms of making assumptions regarding the behaviour (preferences) of the DM. As we gather information from the DM, we construct partial information on his utility function and based on this partial information we can decide if the utility function is linear, quasi-concave, or unknown. We also use the partial information to reduce the set of alternatives and try to generate better points. Since different DMs have different utility functions (although not known), their best compromise solutions differ.

Malakooti’s approach (Malakooti, 1985b) consists of three phases. In phase I, it assumes that the utility function is linear and tries to find the best solution. Since the utility function is assumed to be linear, the optimal point is one of the extreme points of the LP problem and hence we generate only extreme points and quickly find the best compromise solution. If the DM is not satisfied with the point we go to the next phase, which is more time consuming.

In phase II, the DM’s utility function is assumed to be quasi-concave and a number of questions are asked to find the best point. In addition to the extreme points of the MOLP problem, non-extreme points are also generated. This lets the DM examine many new points. The non-extreme points are generated by a mechanism called the one-dimensional search, in which we generate a finite number (usually equal to the number of criteria) of points associated with the line connecting the current point to an extreme point. If the DM is satisfied with the obtained solution we can stop. Otherwise we go to phase III.

In phase III, we assume that the DM’s utility function structure is not known or may not even exist. Hence we try different approaches to generate and examine new points. This is done through a decision support system in which the DM can control the program directly and explore different types of ‘if-then’ questions. In this step, for example, the DM can provide a set of weights (the importance of different objectives) and the algorithm can generate the point associated with the weights. Or the DM can provide a trial point and the algorithm informs the DM whether the trial point is feasible. If it is feasible, the method generates a better point (if possible) and presents it to the DM. If not, the method generates the closest feasible point and presents it to the DM. The method also has a subroutine for inconsistency resolution. If the DM provided inconsistent responses, the method identifies those inconsistent responses and presents them to the DM. If the DM wishes, he can correct the inconsistencies.

A computer package has been developed based on Malakooti’s (1985) paper. The package has several interesting features that can be used as subroutines. Each of the following features can be used independently or in conjunction with other subroutines.

(1) Ranking and assessment through strength of preference; for paired comparisons of alternatives the DM can introduce strong or weak preferences (has similarities to the Malakooti & Ravindran, 1985/6, approach).

(2) Assuming an implicit linear utility function (has similarities to the Zionts and Wallenius approach; see Hwang & Masad, 1979).

(3) Assuming an implicit quasi-concave utility function. (4) Search of the weight space (centre and upper boundaries) and generation of their

associated alternatives (has similarities to the Steuer interval approach; see Steuer & Schuler, 1978; Hwang & Masad, 1979).

(5) Inconsistency, modification, and a resolution. (6) Trial and error of weight (generation of efficient and extreme points). (7) Trial and error of alternatives; generation of feasible efficient non-extreme points

50 B. Malakooti

associated with the trial point (has similarities to setting a goal, reference point, or an ideal point; also has similarities to the goal programming and the compromise programming of Zeleny; see Hwang & Masad, 1979).

(8) One-dimensional search, searching the efficient facet, i.e. non-extreme points (has similarities to the Geoffrion, Dyer & Feinberg, 1972, approach; see also Hwang & Masad, 1979).

(9) Linear approximation of non-linear utility functions for finding the best direction based on paired comparison of alternatives.

The program is now available for IBM-PC (XT) for experimental purposes. The computer package is written in BASIC and it is operational on IBM XT using 512 RAM. The package may be obtained from the author.

6. The multi-objective model and experiments with the computer package

Now we consider the multi-objective model. The objectives are all to be maximized. The MOLP problem can be stated as follows:

Maximize profit; Maximize gas consumption; Maximize electricity conservation; Maximize oil conservation; Maximize quality;

subject to constraints (2) to (14).

f, = Function (1) in million $ f, = X, in lOI* BTU f,= --X,-X, in lO’*BTU f,= -X,--X, in 10”BTU f,= -X,, in million tons

The quality of the product is proportional to the percentage of the cullet used in the mixture of the molten glass. Hence we use X,, (total cullet) as the measure of the quality with respect to total production. This proportion is either lo%, 25%, or 50% of total production.

Table 1 depicts the best possible values for each objective.

Table 1. The range for each objective

Range f I .fi f, f4 fs

Worst 0 -713 - 827 - 932 -63 Best 10,373 0 0 0 0

Now we explain how the method was used to solve the problem. The starting point was point 0 and the candidate point was point # 1. The DM was asked to choose one of the two points. Table 2 demonstrates the sequence of points generated by the method as we proceeded. Each point is presented in terms off, fi, f,, f,, and f,, the left-hand side of Table 2.

Comparing # 0 to # 1, we note that the oil and electricity consumptions are the same. But if we prefer # 1, the trade-off is that the profit decreases, gas consumption increases, but the quality increases substantially. The DM preferred #0 to # 1 weakly.

As extra information and to provide an intuitive feeling of how the method works, the computer package also displays the weight values associated with the points. That is, if we

PC model for energy conservation planning 51

Alt f, f* f, f4 f, WI w2 W3 W4 W5

0 10,373 -640 -27 -18 -63 0.2 0.2 0.2 0.2 0.2 1 10,344 - 699 -27 -18 -22 1 0 0 0 0 2 9,440 0 -27 -932 -63 0 1 0 0 0 3 0 0 0 0 0 0 0 I 0 0 4 3,969 0 - 827 -18 -63 0.04 0.59 0 1 0 5 10,337 -713 -27 -18 -12 0.01 0 0 0 0.99 6 5,933 - 370 -16 -11 -7 0.01 0.18 0.27 0.27 0.27 7 9,291 -615 -27 -18 -12 0.02 0.2 0.03 0.12 0.64

Table 2. The sequence of generated points

8 6,555 - 429 -16 -11 -7 9 3,194 0 -919 -18 -12

10 9,295 0 -27 -1,038 -12 11 10,344 - 699 -27 -18 -22 12 10,351 - 684 -27 -18 -32 13 10,359 - 669 -27 -18 -43 14 10,366 -654 -27 -18 -53

Maximize Z = w,f, + w2 f, + w3f3 + w4 f, + w5 f, Subject to constraints (2) to (14)

the solution would be the generated point. These weights are shown in the right-hand side of Table 2. These generated points are always extreme points.

In the next step the algorithm generated point # 2. The DM was presented with # 0 (since it was preferred it remained as the basis of comparison) versus alternative # 2. The DM preferred #0 to #2 strongly. In the next step point #3 was generated by the method and the DM was questioned about #O versus # 3. The DM preferred #0 strongly. In the next step point #4 was generated, to which the DM preferred #0 weakly. Then point # 5 was generated. The DM preferred # 5 to # 0 weakly. Hence # 5 became the basis of comparison, i.e. it was the best point so far. The DM selected #5 over #0 because he considered the substantial improvement in the quality worth the increase in gas consumption while the profit almost remained constant. In the next step point #6 was generated. The DM preferred # 5 to # 6 strongly. Also, # 7 was generated, and the DM preferred #5 to #7 strongly.

At this stage the program declared that the current point # 5 was the best compromise solution with respect to an assumed linear utility function, i.e. phase I of the method was completed, but the DM wanted to continue the method with an assumed non-linear utility function, i.e. phase II of the method. Two lines in Table 2 indicate that we entered phase II, and in this phase weights are not used. In the next three steps, points # 8, #9, and # 10 were generated. The DM preferred # 5 to each of them strongly. At this stage the program went through a one-dimensional search. The points generated were associated with the line connecting # 5 to #O. These points were # 5, # 11, # 12, # 13, # 14, and #O. The DM was asked to choose the best one of these points. The DM selected point # 11 as the best. At this stage the current point was declared the best compromise solution by the DM. Note that this point is not an extreme point, i.e. there is no weight that can generate it.

52 B. Malakooti

In the next step the DM wished to examine some of the decision support system subroutines to find a better point. In each step the algorithm asked the following question: ‘Do you wish to see (1) your responses, and (2) the eliminated points?’ Once the DM wished to examine his previous responses in case he wanted to change them. The algorithm brought all the questions back one by one, but the DM did not change his preferences. Had the DM been inconsistent, the algorithm would have informed him and the same subroutine could have been implemented to correct the previous responses. To use the three different subroutines the algorithm also provided the following options in each step: ‘Enter P for Trial Point, L for Trial Weights, A for One-Dimensional Search.’

To use the trial weights (L), the DM provided (1, 0, 3,3, 1) as the weights for the objectives. For these weights the algorithm generated point #5. We note that the weights can only generate extreme points.

To use the trial point (P) subroutine of the algorithm, we asked whether point (5800, - 360, - 20, - 15, - 40) was feasible or not. The algorithm declared that the trial point was feasible, but that there existed a better point-(5941.65, = 333.09, = 16.28, - 11.10, - 38.12). The DM still preferred his best point, # 11, to this point.

The final solution values (point # 11) are as follows:

X,= 12.30 X,=699.18 A’,= 27.07 ,I’,= 18.46 x, = 103.93

X,,,= 22.82 xz4= Il.05 X,,= 32.40 A’,,= 41.61 x2,= 49.04 x,,= 77.11 X,8= 34.12 X,,= Il.60

If we compare this to the final solution of the maximizing profit of Section 4 (alternative 0 is the same as the maximizing profit solution), we realize that more gas consumption (X,) is recommended, but the quality (- X,,) is increased substantially and profit is reduced by a small amount (compare point 0 with point 11). Different combinations of three energy conservation options (X,7, X,*, X,,) are recommended by the model. They all recommend using oxygen enrichment options, but not inspirated air. Energy substitutions are not recommended. The final solution recommends producing 77.11 million tons with 10% cullet, 34.12 million tons with 25% cullet, and 11.60 million tons with 50% cullet. Although this solution could cause some problems in terms of quality consistency of the output, it may be possible to arrange different qualities, i.e. lo%, 25%, or 50% of cullet for different products. If we wish to have only one type of quality, we should use an integer programming package requiring that only one of the variables X,, to X,, be positive and all the rest be zero. The problem can be formulated as a mixed integer program and there are procedures to solve it.

7. Conclusions In this paper we implemented an interactive paired comparison method to obtain the best energy conservation policy for the melting phase of glass production. The problem

PC model for energy conservation planning 53

was formulated by multiple objective linear programming and solved through a computer package using an IBM PC microcomputer. Our experiments with the package and the method were satisfactory because the DM was able to interact with the package in a friendly way and the package led the DM to better alternatives. The questions were observed to be simple to answer (in the form of paired comparisons) and the DM was satisfied with the performance of the package. Several subroutines were used to provide more flexibility for the DM to examine or generate new alternatives. The energy conservation model demonstrated how different energy conservation options can be formulated through linear programming.

We found that microcomputers are effective and convenient tools for planning when combined with supportive friendly interactions in terms of decision making. However, being limited to 512 RAM (random access memory), we had to simplify and reformulate the LP problem to reduce it to 12 constraints and 28 variables so that the microcomputer package could be executed. Of course, a larger microcomputer would be helpful. Another option would be to rewrite our package in batch form to accommodate larger problems.

Acknowledgements

The author is grateful for the comments of anonymous referees and Professor Benjamin Hobbs, who acted as the Decision Maker and provided interesting feedback on the microcomputer package. Thanks to Mr Satoshi Araki and Mr Evan Tandler, who performed several experiments with the model and provided useful feedback.

This research was supported in part by the Brookhaven National Laboratory, Upton, New York, USA; Grant No. 0061-5&12875.

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B. Malakooti, PhD (Industrial Engineering), 1982, MSIE (industrial engineering), 1979 MS (Economics), 1978, Purdue University, West Lafayette, BSE (engineering), 1977, Hamadan College. He is an assistant professor of systems engineering at Case Western Reserve University. He worked for four years as a systems analyst and engineer for Computer Science Corporation Systems International and Vidgan Consulting Engineers Company. He is consultant to several manufacturing companies in Cleveland. His areas of research interest are: computer aided manufacturing and multiple criteria decision making. He is a member of ORSA, TIMS, senior member of SME, IIE, and board member of IIE Cleveland Chapter.