Energy Systems
5.6 Integer Programming Model for Energy Investment Options
This section presents an integer programming formulation as another type of OR model-ing for energy decision makmodel-ing. Plannmodel-ing a portfolio of energy investments is essential in resource-limited operations. The capital rationing example presented here (Badiru and Pulat, 1995) involves the determination of the optimal combination of energy investments so as to maximize present worth of total return on investment. Suppose an energy analyst is
5-6 Operations Research Applications givenN energy investment options, X1,X2,X3, . . ., XN, with the requirement to determine the level of investment in each option so that present worth of total investment return is maximized subject to a specified limit on available budget. The options are not mutually exclusive.
The investment in each option starts at a base levelbi (i = 1, 2, . . ., N) and increases by variable incrementskij (j = 1, 2, 3, . . ., Ki), where Ki is the number of increments used for optioni. Consequently, the level of investment in option Xi is defined as
xi=bi+
For most cases, the base investment will be 0. In those cases, we will have bi= 0. In the modeling procedure used for this example, we have:
Xi=
1 if the investment in option i is greater than zero 0 otherwise
and
Yij =
1 if the increment of alternative i is used 0 otherwise.
The variablexiis the actual level of investment in optioni, while Xiis an indicator variable indicating whether or not option i is one of the options selected for investment. Similarly, kij is the actual magnitude of the jth increment, while Yij is an indicator variable that indicates whether or not the jth increment is used for option i. The maximum possible investment in each option is defined asMi such that
bi≤ xi≤ Mi
There is a specified limit,B, on the total budget available to invest such that
i
xi≤ B
There is a known relationship between the level of investment,xi, in each option and the expected return, R(xi). This relationship is referred to as the utility function,f(.), for the option. The utility function may be developed through historical data, regression analysis, and forecasting models. For a given energy investment option, the utility function is used to determine the expected return,R(xi), for a specified level of investment in that option.
That is,
where rij is the incremental return obtained when the investment in option i is increased bykij. If the incremental return decreases as the level of investment increases, the utility function will be concave. In that case, we will have the following relationship:
rij− ri, j+1≥ 0 Thus,
Yij ≥ Yi, j+1
y 1 y 2 y 3 y 4 h 1
h 2
h 3
h 4
Return
R(x) curve
FIGURE 5.1 Utility curve for investment yield.
or
Yij− Yi, j+1 ≥ 0
so that only the firstn increments (j = 1, 2, . . ., n) that produce the highest returns are used for projecti. Figure 5.1 shows an example of a concave investment utility function.
If the incremental returns do not define a concave function,f(xi), then one has to intro-duce the inequality constraints presented above into the optimization model. Otherwise, the inequality constraints may be left out of the model, as the first inequality,Yij ≥ Yi,j+1, is always implicitly satisfied for concave functions. Our objective is to maximize the total investment return. That is,
Maximize:Z =
i
j
rijYij
Subject to the following constraints:
xi =bi+
j
kijYij ∀i bi≤ xi≤ Mi ∀i Yij≥ Yi, j+1 ∀i, j
i
xi≤ B
xi ≥ 0 ∀i
Yij= 0 or 1 ∀i, j
Now suppose we are given four options (i.e.,N = 4) and a budget limit of $10 million. The respective investments and returns are shown inTables 5.4through5.7.
All the values are in millions of dollars. For example, in Table 5.7, if an incremental investment of $0.20 million from stage 2 to stage 3 is made in option 1, the expected incremental return from the project will be $0.30 million. Thus, a total investment of $1.20 million in option 1 will yield present worth of total return of $1.90 million. The question addressed by the optimization model is to determine how many investment increments
5-8 Operations Research Applications TABLE 5.4 Investment Data for Energy Option 1
Incremental Level of Incremental Total
TABLE 5.5 Investment Data for Energy Option 2 Incremental Level of Incremental Total
TABLE 5.6 Investment Data for Energy Option 3 Incremental Level of Incremental Total
TABLE 5.7 Investment Data for Energy Option 4 Incremental Level of Incremental Total
should be used for each option. That is, when should we stop increasing the investments in a given option? Obviously, for a single option we would continue to invest as long as the incremental returns are larger than the incremental investments. However, for multiple investment options, investment interactions complicate the decision so that investment in one project cannot be independent of the other projects. The IP model of the capital rationing example was solved with LINDO software. The model is
Maximize: Z = 1.4Y11 + .2Y12 + .3Y13 + .1Y14 + .1Y15 + 6Y21 + .3Y22 + .3Y23 +.2Y24 + .1Y25 + .05Y26 + .05Y27 + 4.9Y31 + .3Y32 + .4Y33 + .3Y34 +.2Y35 + .1Y36 + .1Y37 + .1Y38 + 3Y41 + .5Y42 + .2Y43 + .1Y44 +.05Y45 + .15Y46
Subject to:
.8Y11 + .2Y12 + .2Y13 + .2Y14 + .2Y15 − X1 = 0
3.2Y21 + .2Y22 + .2Y23 + .2Y24 + .2Y25 + .2Y26 + .2Y27 − X2 = 0
2.0Y31 + .2Y32 + .2Y33 + .2Y334 + .2Y35 + .2Y36 + .2Y37 + .2Y38 − X3 = 0 1.95Y41 + .2Y42 + .2Y43 + .2Y44 + .2Y45 + .2Y46 + .2Y47 − X4 = 0
X1 + X2 + X3 + X4<= 10 Y12− Y13 >= 0
Y13− Y14 >= 0 Y14− Y15 >= 0 Y22− Y23 >= 0
· · · ·
Y26− Y27 >= 0 Y32− Y33 >= 0 Y33− Y34 >= 0 Y35− Y36 >= 0 Y36− Y37 >= 0 Y37− Y38 >= 0 Y43− Y44 >= 0 Y44− Y45 >= 0 Y45− Y46 >= 0
Xi>= 0 for i = 1, 2, . . ., 4 Yij = 0, 1 for all i and j
The solution indicates the following values for Yij. 5.6.1 Energy Option 1
Y11 = 1, Y12 = 1, Y13 = 1, Y14 = 0, Y15 = 0
Thus, the investment in option 1 isX1 = $1.20 million. The corresponding return is $1.90 million.
5.6.2 Option 2
Y21 = 1, Y22 = 1, Y23 = 1, Y24 = 1, Y25 = 0, Y26 = 0, Y27 = 0
Thus, the investment in option 2 isX2 = $3.80 million. The corresponding return is $6.80 million.
5.6.3 Option 3
Y31 = 1, Y32 = 1, Y33 = 1, Y34 = 1, Y35 = 0, Y36 = 0, Y37 = 0
Thus, the investment in option 3 isX3 = $2.60 million. The corresponding return is $5.90 million.
5.6.4 Option 4 Y41 = 1, Y42 = 1, Y43 = 1
Thus, the investment in option 4 isX4 = $2.35 million. The corresponding return is $3.70 million.
5-10 Operations Research Applications
1.2 3.8
2.6 2.35
1.9 6.8
5.9
3.7 10
8
6
4
2
Option 1
Option 4 Option 2
Option 3
Investments Returns
(In millions of dollars)
FIGURE 5.2 Comparison of investment levels.
150
125
100
75
50
25
0
Option 1 58.3%
Option 2 78.95%
Option 4 57.45%
Option 3 126.92%
Return on investment
(%)
FIGURE 5.3 Histogram of returns on investments.
The total investment in all four options is $9,950,000. Thus, the optimal solution indicates that not all of the $10,000,000 available should be invested. The expected present worth of return from the total investment is $18,300,000. This translates into 83.92% return on investment. Figure 5.2 presents histograms of the investments and the returns for the four options. The individual returns on investment from the options are shown graphically in Figure 5.3.
The optimal solution indicates an unusually large return on total investment. In a prac-tical setting, expectations may need to be scaled down to fit the realities of the investment environment. Not all optimization results will be directly applicable to real-world scenarios.
Possible extensions of the above model of capital rationing include the incorporation of risk
and time value of money into the solution procedure. Risk analysis would be relevant, partic-ularly for cases where the levels of returns for the various levels of investment are not known with certainty. The incorporation of time value of money would be useful if the investment analysis is to be performed for a given planning horizon. For example, we might need to make investment decisions to cover the next 5 years rather than just the current time.