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Inexact de Novo programming for water resources systems planning
European Journal of Operational Research 199 (2009) 531–541
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Innovative Applications of O.R.
Inexact de Novo programming for water resources systems planning Y.M. Zhang a, G.H. Huang a,b,*, X.D. Zhang a a b
Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 Chinese Research Academy of Environmental Science, Beijing Normal University, Beijing 100012-100875, China
a r t i c l e
i n f o
Article history: Received 6 November 2007 Accepted 14 November 2008 Available online 28 November 2008 Keywords: Uncertainty Management Water resources Decision analysis Multi-objective Environment de Novo
a b s t r a c t This study presents an interval de Novo programming (IDNP) approach for the design of optimal waterresources-management systems under uncertainty. The model is derived by incorporating the existing interval programming and de Novo programming, allowing uncertainties represented as intervals within the optimization framework. The developed IDNP approach has the advantages in constructing optimal system design via an ideal system by introducing the flexibility toward the available resources in the system constraints. A simple numerical example is introduced to illustrate the IDNP approach. The IDNP is then applied to design an inexact optimal system with budget limit instead of finding the optimum in a given system with fixed resources in a water resources planning case. The results demonstrate that the developed method efficiently produces stable solutions under different objectives. Optimal supplies of good-quality water are obtained in considering different revenue targets of municipal–industrial–agricultural competition under a given budget. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In modern society, systems analysis and design have become important managerial and operational issues confronting many countries and regions in the world. It is an active process that seeks a portfolio of resource levels and optimizes the objective function by allocating a budget according to a resource price, where resource levels are considered as decision variables (Zeleny, 1990). In water resources systems planning, decision makers are greatly concerned about decisions for determining resources availabilities and/or identifying desired management facilities. Therefore, designing a sound water resources system for modeling various water resources issues can help decision makers allocate water effectively and efficiently. The classical de Novo programming method nominated by Zeleny (1981, 1986, 1990) was effective for dealing with optimal design problems. Since the resource availability level is determined as a priori, the result of linear multicriteria programming is the optimal given system. In contrast, with unknown resource availability, the multicriteria de Novo programming (MDNP) seeks a portfolio of resource availability level to optimize multiple objective functions by allocating a budget according to the resource price. This approach results in an optimal system design. Zeleny (1990) proposed a basic method to construct the optimal system design for solving a de Novo problem via an ideal system design. One of the important issues in MNDP was to determine an optimum-path ratio for enforcing a particular budget level of resources so as to establish the optimal system design. In Zeleny’s basic method, the optimum-path ratio for achieving the multicriteria performance of the ideal system design related to a given budget level was used to determine the optimal system design. Bare and Mendoza (1990) illustrated tile potential use of designing optimal forest systems in the face of conflicting objectives. Both the generation and evaluation of compromise solutions under de Novo conditions were discussed. Shi (1995) introduced several optimum-path ratios for enforcing different budget levels of resources so as to find alternative optimal system designs for solving MNDP problems. Shi (1999) applied MDNP to formulate and solve problems of system design that involved multiple decision makers and a possible debt. Zeleny (2005) investigated the evolution of optimality of single or multiobjective programming, and summarized the eight major optimality concepts according to a dual classification: single versus multiple criteria versus the extent of the ‘given’, ranging from ‘all-but’ to ‘none except’. Previously, de Novo programming has been applied in some cases. Bare and Mendoza (1988) applied de Novo programming to single and multi-objective forestry land management problems. This problem considered constraints on labor, picnic sites, and hiking trails. It was proved that the system could be designed to perform in an ideal fashion within a constant budget level. Bare and Mendoza (1990) * Corresponding author. Address: Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada S4S 0A2. Tel.: +1 306 585 4095; fax: +1 306 585 4855. E-mail address: [email protected] (G.H. Huang). 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.11.019
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then considered another case of forestry land management where several compromise solutions were generated, proving that de Novo programming did improve compromise performance levels. Finally, decision maker was responsible for accepting or rejecting a decision alternative. Kotula (1997) mentioned the difficulty in construction of multipurpose reservoirs for achieving sustainable development, taking into account the grand scale and complex nature of project impacts on society, environment, and the economy. He applied the de Novo programming in considering the control and adjustment of reservoir design and operation characteristics which resulted in optimal or near optimal system performance throughout the life of the reservoir. The de Novo programming could shift the traditional approach of optimization, from optimization of a given reservoir system, to the design of the optimal reservoir system. Actually, in real world de Novo programming problems, uncertainties may exist in related costs, system objectives and parameters. Generally, uncertain parameters can be represented as fuzzy, interval or stochastic numbers, resulting in uncertain de Novo programming. Li and Lee (1990) extended Zeleny’s basic method to identify fuzzy system designs for de Novo problems by considering the fuzziness in coefficients. Li and Lee (1993) further treated fuzzy goals and fuzzy coefficients simultaneously, depending on a numerical approach which could be solved as either linear or nonlinear problems. Kim et al. (1993) formulated a de Novo 0–1 bicriteria linear programming with interval coefficients under generalized upper bounding structure. Interval coefficients were transformed into a fuzzy state by the fuzzy transformation based on the degree of satisfying inequality relationship and order relationship between intervals. This transformation was limited and hard to be applied to a generalized de Novo model. Sasaki et al. (1995) proposed an implementation of the genetic algorithm for solving DNP problems with fuzzy goal and constraints. The proposed approach had the flexibility to obtain better solutions compared to crisp constraints. Considering random distribution of budget, Chen and Hsieh (2006) presented a fuzzy multi-stage de Novo programming which was viewed as a fuzzy dynamic programming problem. Uncertain parameters were presented as intervals with unknown distributions, leading to an inexact de Novo programming. Nevertheless, the previous studies on de Novo programming could seldom deal with interval parameters. Actually, variations of system performance and decision variables of the inexact programming can be investigated by solving relative submodels. By using inexact programming, equivalent information can be obtained in a faster and simpler way (Huang et al., 1992). In practical problems, uncertainty often exists due to the imprecise and vague nature of system parameters. The quality of available information is often not satisfactory enough to be represented as deterministic values; unavailability of required data with probability distribution over the planning horizon often leads to the difficulties in application of stochastic programming to practical water resources management problems, and; the incomplete data and the decision makers’ subjectivity make it difficult to generate the appropriate and reasonable fuzzy membership functions. Instead, the decision makers can only provide the lower and upper bounds of the parameters, without knowing any distribution forms or crisp values. Thus, the obtained data may be represented as interval numbers, which are often more intuitive. Therefore, the objective of this study is to develop an inexact de Novo programming (IDNP) approach for designing a water resources management system under uncertainty, where the modeling parameters can be represented as intervals. The developed approach can effectively deal with both single and multi-objective problems. The IDNP will then be applied to a hypothetical case study of water resources systems planning, where efficient water allocation to domestic, agricultural and industrial customers will be guaranteed. 2. Methodology 2.1. De Novo programming A de Novo programming model can be formulated as follows (Zeleny, 1990):
Max z ¼ Cx s:t: AX b 6 0; pb 6 B;
ð1Þ
x P 0; where p is the given unit prices of m resources, p = p1, p2, . . . , pm; B is total given variable budget; b is the resulting (optimal) portfolio of resources b = b1, b2, . . . , bm, which can be considered as design decision variables in this problem; x is the production levels, x = x1, x2, . . . , xm, which can be regarded as management decision variables. This model aims at allocating budget to each b, determining x, and maximizes P the benefit value z, where z ¼ Cx ¼ kj ckj xkj ; k ¼ 1; 2; . . . ; q. Objective functions zk are to be maximized simultaneously. Actually, truly optimal portfolio of a decision maker should more appropriately maximize the difference (cx pb) rather than cx. Problem (1) can be transformed into:
Max Z ¼ Cx s:t: Vx 6 B;
ð2Þ
x P 0; where Z = (z1, . . . , zq) and V = (V1, . . . , Vn) = pA. By the methodology of de Novo optimal design programming, problem (2) can be solved, for x and b, with respect to each of objective functions zk, respectively. Let zk ¼ max z; k ¼ 1; 2; . . . ; q, subject to constraints of (2). Let z ¼ ðz1 ; z2 ; . . . ; zk Þ denote the performance of the ideal design under a given B. Obviously, z* must be attainable for a given budget level B. For each of k values of z*, there is a corresponding zk -optimal portfolio b* calculated as in the previous section. As vector z* represents the metaoptimal performance, the corresponding x* and b* can be obtained by solving the following metaoptimum model (Zeleny, 1990):
Min B ¼ Vx s:t: Cx P Z ;
ð3Þ
x P 0: Problem (3) identifies the minimum budget B* at which the metaoptimum performance z* can be realized through x* and b*. The solution of problem (3), B* and x* can be designated as metaoptimum solution. It is obvious that metaoptimum budget B* must exceed any given budget B. For any particular budget level B, the optimum-path ratio r can be used
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r ¼ B=B :
ð4Þ *
*
*
The final solutions are as follows: x = rx , b = rb , z = rz . The optimum-path r provides an efficient tool for virtually instantaneous optimal redesign of even large-scale systems. It is a powerfully competitive tool which is able to maintain complex multicriteria decision systems by expanding of budget B along optimal path. The effects of de Novo programming on the feasible regions of decision alternatives are better illustrated through an example. Suppose that a given system contains two objective functions Z1 and Z2 which are to be maximized, four linear constraints on available resources decision variables, and two nonnegative decision variables X1 and X2. The de Novo programming approach designs the resource portfolio by modifying the slope and intercept of each soft constraint. The new feasible region will be optimally modified to satisfy the metaoptimum constraints. The optimal de Novo solutions can determine the minimum budget necessary to achieve the metaoptimum. If the metaoptimum constraints are of inequality type, such that the de Novo model aspires to at least achieve the metaoptimum, it is possible to over achieve the metaoptimum (Kotula, 1997). 2.2. Inexact de Novo Programming Before an inexact de Novo programming model is formulated, some useful definitions need to be introduced. Let x denotes a closed and bounded set of real numbers, an inexact number x± is defined as an interval with known upper and lower bounds but unknown distribution information (Huang, 1994):
x ¼ ½x ; xþ ¼ ft 2 xjx 6 t 6 xþ g;
ð5Þ
+
±
+
±
+
where x and x are the lower and upper bounds of x respectively. When x = x , x becomes a deterministic number, i.e. x = x . For x±, we define Sign(x±) as follows:
Signðx Þ ¼ 1;
if x P 0;
ð6Þ
if x < 0:
¼ 1;
ð7Þ
±
Its absolute value jx j is defined as follows:
jx j ¼ x ;
if x P 0;
ð8Þ
if x < 0:
ð9Þ
jxj ¼ x ; if x P 0; ¼ xþ ; if x < 0;
ð10Þ ð11Þ
and jxjþ ¼ xþ ;
ð12Þ
¼ x ; Thus, we have:
if x P 0;
if x < 0:
¼ x ;
ð13Þ
The single or multi-objective inexact de Novo programming for optimal system design can be formulated as follows:
Max z ¼ C x
s:t: A x b 6 0;
ð14Þ
p b 6 B ;
x P 0; where C 2 Rkj and A 2 Rij are matrices of dimensions h j and i j, respectively. c kj is the element of Matrix C ; aij is the element of Matrix A±; b± is vector of design decision variable, b ¼ ðb1 ; b2 ; . . . ; bi Þ; x± is vector of management decision variables, x ¼ ðx 1 ; x2 ; . . . ; xj Þ; ± ± p is the vector of the unit prices of i resources, p ¼ ðp1 ; p2 ; . . . ; pi Þ; and B is the given total available budget. These definitions are essential because there are two kinds of decision variables including design decision variables b± and management decision variables x± in the inexact de Novo program, while most optimization problems only have management decision variables. Moreover, all the decision variables in IDNP are inexact numbers, which are especially true for the practical problems. Solving problem (14) means finding the optimal allocation of B so that the corresponding resource portfolio b maximizes simultaneously the values Z = Cx of the product mix x. For all the coefficient are interval variables, this inexact linear programming model (1) can be solved by transformed into two sets of deterministic submodels, which correspond to the lower and upper bounds of the desired objective function value. This transformation process is based on an interactive algorithm, which is different from normal interval analysis and best/worst case analysis (Huang and Moore, 1993). The solutions provide stable intervals for the objective functions and decision variables with different levels of risk in violating the constraints. They can be easily interpreted for generating decision alternatives. For model (14), the submodel (I) corresponding to z+, which provides the first step of the solution process when the objective function is to be maximized, can be formulated as follows (assuming that bi > 0):
Maximize zþk ¼
t X
cþkj xþj
j¼1
s:t:
X
X xj
n X
cþkj xj ;
j¼tþ1
jaij j Signðaij Þxþj þ þ
8k X
jpi j Signðpi Þbi 6 Bþ ;
P 0;
8j;
ð15aÞ þ
jaij jþ Signðaþij Þxj 6 bi ;
8i;
ð15bÞ ð15cÞ ð15dÞ
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where x j ; j ¼ 1; 2; . . . ; t, are inexact variables with positive coefficients, and xj , j = t + 1,t + 2, . . . , n, are inexact variables with negative coefficients. Eqs. (15b) and (15c) can be transformed into:
X
jpi j Signðpi Þ
X
X X jaij j Signðaij Þxþj þ jaij jþ Signðaþij Þxj 6 Bþ : jpi j Signðpi Þ
ð15eÞ
þ þ þ þ þ Let zþ k ¼ maxzk ; k ¼ 1; . . . ; q, be the optimal value for kth objective of submodel (I) with Eq. (15). Let z ¼ ð z1 ; z2 ; . . . ; zq Þ be the q-objec+ tive value for the ideal system with respect to B . Then, the metaoptimum submodel I can be constructed as follows:
X
Minimize Bþ ¼ t X
s:t:
X X X jpi j Signðpi Þ jaij j Signðaij Þxþj þ jaij jþ Signðaþij Þxj jpi j Signðpi Þ n X
cþkj xþj
j¼1
cþkj xj P zþk ;
8k;
ð16aÞ ð16bÞ
j¼tþ1
xj P 0;
ð16cÞ
* + * + * + Solving Problem (I-META) yields x j , B , b , and r where:
þ
b ¼
X
jaij j Signðaij Þ xþj þ
X
jaij jþ Signðaþij Þ xj ;
ð17aÞ
þ þ
þ
r ¼B = B :
ð17bÞ
* +
* +
þ zk
* +
+ * +
The value B is the minimum budget to achieve through x j and b . Since B P B , r identifies the upper bound of optimum-path ratio þ þ for achieving the ideal performance *z+ for a given budget level B+. The optimal system design can be established as (x opt ; bopt ; zopt Þ, where þ þ þ þ þ þ þ þ þ þ xopt ¼ r xj ; ðj ¼ 1; 2; . . . ; tÞ; xopt ¼ r xj ðj ¼ t þ 1; t þ 2; . . . ; nÞ; bopt ¼ r b , and zopt ¼ r zk . The optimum-path ratio *r+ provides an effective and fast tool for optimal redesign of large-scale systems. The submodel (II) corresponding to z, which provides the second step of the solution process based on the optimal variable solution, þ zk , from submodel (I), can be formulated as follows (assuming that bi > 0):
Maximize zk ¼ X
s:t:
X zk xj
t X
ckj xj
j¼1
n X j¼tþ1
jaij jþ Signðaþij Þxj þ jpi j
6
þ
Signðpþi Þbi
þ zk ;
P 0;
ckj xþj ; X
8k
ð18aÞ
jaij j Signðaij Þxþj 6 bi ;
8i;
6B ;
ð18bÞ ð18cÞ
8k; 8j;
ð18dÞ ð18eÞ
where x j ; j ¼ 1; 2; . . . ; t, are inexact variables with positive coefficients, and xj ; j ¼ t þ 1; t þ 2; . . . ; n, are inexact variables with negative coefficients. Similarly, Eqs. (18b) and (18c) can be transformed into:
X
jpi jþ Signðpþi Þ
X
X X jpi jþ Signðpþi Þ jaij jþ Signðaþij Þxj þ jaij j Signðaij Þxþj 6 B :
ð18fÞ
Let z k ¼ max zk ; k ¼ 1; 2; . . . ; q, be the optimal value for kth objective of submodel (II) with Eq. (18). Let z ¼ ð z1 ; z2 ; . . . ; zq Þ be the q objective value for the ideal system with respect to B . Then, the metaoptimum submodel II can be formulated as follows:
Minimize B ¼ t X
s:t:
X
X X X jpi jþ Signðpþi Þ jpi jþ Signðpþi Þ jaij jþ Signðaþij Þxj þ jaij j Signðaij Þxþj
ckj xj
j¼1
n X
ckj xþj P zk ;
8k;
ð19aÞ ð19bÞ
j¼tþ1
xj P 0; X X X X þ jpi jþ Signðpþi Þ jpi j Signðpþi Þ jaij jþ Signðaþij Þxj þ jaij j Signðaij Þxþj 6 rþ ; B =
ð19dÞ
xj xþj
ð19cÞ
6
þ xj ;
j ¼ 1; 2; . . . ; t;
ð19eÞ
6
xj ;
j ¼ t þ 1; t þ 2; . . . ; n:
ð19fÞ
* Solving Problem (II-META) yields x j ; B ; b , and r where:
b ¼
X
jaij jþ Signðaþij Þ xj þ
r ¼ B = B :
X
jaij j Signðaij Þ xþj ;
ð20aÞ ð20bÞ
* + * * The value *B is the minimum budget to achieve z k through xj and b . Since B P B , r identifies the optimum-path ratio for achiev * ing the ideal performance z for a given budget level B . The optimal system design can be established as ðx opt ; bopt ; zopt Þ, where þ þ ¼ r x ðj ¼ 1; 2; . . . ; tÞ; x ¼ r x ðj ¼ t þ 1; t þ 2; . . . ; nÞ; b ¼ r b , and z ¼ r z . x opt opt opt opt j k j þ þ Therefore, bopt ; zþ opt ; xopt ðj ¼ 1; 2; . . . ; tÞ and xopt ðj ¼ t þ 1; t þ 2; . . . ; nÞ can be obtained by solving the submodel (I) defined by Eqs. (15) þ (17), whereas bopt ; z opt ; xopt ðj ¼ 1; 2; . . . ; tÞ and xopt ðj ¼ t þ 1; t þ 2; . . . ; nÞ can be obtained from Eqs. (18)–(20). Thus, final solutions can be þ þ þ ¼ ½z ; z ; x ¼ ½x ; x and optimal design solution bopt ¼ ½bopt ; bopt . obtained with z opt opt opt opt opt opt Fig. 1 shows the scheme of the modeling methodology. In conclusion, the solution algorithm of the IDNP model with the objective being maximized is presented as follows:
Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
Right hand side variability
Inexact data
de Novo programming
Interval-parameter programming INDP lower bound submodel
535
INDP model
INDP upper bound submodel
First lower INDP model
First upper INDP model
Meta lower INDP model
Meta upper INDP model
Solutions
Generation of decision Fig. 1. Schematic of the INDP methodology.
Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. Step 7. Step 8. Step 9. Step 10.
Formulate IDNP model (14). Transform the IDNP model into two submodels, where the upper bound of the z± is desired since the objective is to maximize z±. Formulate and solve z+ submodel (15) and obtain zþ k. * + þ þ Formulate and solve z+ metaoptimum submodel I (16) and obtain x j ; B ; b , and r . þ þ Calculate xopt ; bopt and zopt . Formulate and solve z submodel (18) and obtain zþ k. * Formulate and solve z metaoptimum submodel II (19) and obtain x j ; B ; b , and r . Calculate xopt ,bopt and zopt . þ þ þ bopt ¼ ½bopt ; bopt and x Solutions of the IDNP model are: z opt ¼ ½zopt ; zopt ; opt ¼ ½xopt ; xopt . Stop.
2.3. A numerical example To illustrate the method, a detailed example is introduced. Consider a case where the decision makers want to maximize the economic benefit and environmental protection benefit of two districts in a big agricultural farm by allocating necessary electric and water resources. However, the total budget is limited within [40, 48] million dollars. Let x 1 ; x2 represent the farming area of two districts and b1 ; b2 rep resent the total electric and water cost consumed by x1 ; x2 , respectively with the price of 1/unit and 0.8/unit under the total resource-consuming budget. The detailed description of this example is as follows:
Max z1 ¼ ½1:5; 2x1 þ ½2; 3x2
ð21aÞ
ðEconomic benefit objectiveÞ Max z2 ¼ ½2; 3x1 þ ½0:6; 1x2
ð21bÞ
ðEnvironmental benefit objectiveÞ
s: t: ½2; 2:5x1 þ ½1:5; 2x2 6 b1 ;
ð21cÞ
ðWater consuming constraintsÞ
½1:5; 2x1 þ ½2; 2:5x2 6 b2 ;
ð21dÞ
ðElectric consuming constraintsÞ
b1 þ 0:8b2 6 ½40; 48;
ð21eÞ
ðTotal budget constraintsÞ x1 6 7:2;
ð21fÞ
ðMaximum land available of distinct 1Þ x2 6 9:
ð21gÞ
ðMaximum land available of distinct 2Þ + It is assumed that the objective functions of z 1 and z2 are equally important. From Eq. (15), the submodel (I) corresponding to z can be formulated. Then, the corresponding values of all the symbols of the upper bound model are listed in Table 1.
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Max zþ1 ¼ 2xþ1 þ 3xþ2 Max zþ2 ¼ 3xþ1 þ xþ2 þ
s:t: 2xþ1 þ 1:5xþ2 6 b1 ; þ
1:5xþ1 þ 2xþ2 6 b2 ; þ b1 xþ1 xþ2
þ
þ 0:8b2
ð22Þ
6 48;
6 7:2; 6 9:
+ þ By solving submodel I (22), we can obtain zþ 1 ; z2 ¼ ð39:56; 29:65Þ with respect to a given B value of 48. From the results of the submodel (I), a metaoptimum submodel can be constructed as follows:
Min Bþ ¼ 1 ð2xþ1 þ 1:5xþ2 Þ þ 0:8 ð1:5xþ1 þ 2xþ2 Þ 2xþ1 þ 3xþ2 P 39:56; 3xþ1 þ xþ2 P 29:65;
ð23Þ
xþ1 6 7:2; xþ2 6 9: þ þ * + þ The feasibility of zþ 1 ; z2 can only be assured by the metaoptimum solutions v x1 ; x2 ¼ ð7:06; 8:48Þ with the cost B of 48.87. þ þ þ Because the optimal-path ratio *r+ = 48/48.87 = 98.2%, the resulting opt xþ ; x ¼ ð6:93; 8:33Þ and z ; z ¼ ð38:85; 29:11Þ. The optiopt opt opt 1 2 1 2 þ þ mal portfolio bopt , with respect to B+ = 48, can be calculated by substituting opt xþ 1 ; opt x2 into the constraints of model (23). The results are þ þ þ þ þ þ opt b1 ¼ 98:2% ð2x1 þ 1:5x2 Þ ¼ 26:35 and opt b2 ¼ 98:2% ð2x1 þ 1:5x2 Þ ¼ 27:06. þ The next step is to consider the lower bound model. Based on the optimal variable solutions of opt zþ 1 and opt z2 , and Eq. (18), the submodel (II) corresponding to z can be formulated as follows. The values of symbols are shown in Table 2.
Max z1 ¼ 1:5x1 þ 2x2 Max z2 ¼ 2x1 þ 0:6x2 s:t: 2:5x1 þ 2x2 6 b1 ;
2x1 þ 2:5x2 6 b2 ; b1 x1 x2 z1 z2
þ
0:8b2
6 40;
ð24Þ
6 7:2; 6 9; 6 zþ1 ; 6 zþ2 :
Similarly, the obtained optimal solutions are z 1 ¼ 19:46 and z2 ¼ 15:97, while the metaoptimum are x1 ¼ 6:54 and x2 ¼ 4:83, with the * + * cost B of 46.12. The optimal-path ratio r = 40/46.12 = 86.7%, the resulting opt x1 ; opt x2 ¼ ð5:67; 4:19Þ and opt z1 ; opt z2 ¼ ð14:63; 13:85Þ. The optimal portfolio bopt , with respect to B = 40, can be calculated by substituting opt x 1 ; opt x2 into the constraints of model (24). The results are opt b1 ; opt b2 ¼ ð22:54; 21:80Þ. Then the final solutions are: opt x1 ¼ ½5:67; 6:93; opt x2 ¼ ½4:19; 8:33; opt b1 ¼ ½22:54; 26:35; opt b2 ¼ ½21:80; ¼ ½14:63; 38:85; z ¼ ½13:85; 29:11. The results show that the planned farming areas of two districts are [5.67, 6.93] km2 27:06; opt z opt 2 1 2 and [4.19,8.33] km . Under the budget of [40, 48] million dollars, the allocated funds to electric and water consumption are [21.80, 27.06] million dollars and [21.80, 27.06] million dollars, respectively. The economic and environmental benefits are [14.63, 38.85] million dollars and [13.85, 29.11] million dollars, respectively.
Table 1 Values of symbols in upper bound model I. zþ k zþ 1 ðk ¼ 1Þ zþ 2 ðk ¼ 2Þ þ bi ði ¼ 1Þ þ
bi ði ¼ 2Þ
xþ ðj ¼ 1Þ j cþ kj cþ kj
jaijj Signða ij Þ jaijj Signða ij Þ
cþ 11 cþ 21
ja11j Signða 11 Þ ja21j Signða 21 Þ
xþ ðj ¼ 2Þ j 2 3 2 + 1.5 +
cþ 12 cþ 22 ja12j Signða 12 Þ ja22j Signða 22 Þ
3 1 1.5 + 2 +
Table 2 Values of symbols in upper bound model II. z k z 1 ðk ¼ 1Þ z 2 ðk ¼ 2Þ bi ði ¼ 1Þ
bi ði ¼ 2Þ
x j ðj ¼ 1Þ c kj c kj
+
jaijj Þ Signðaþ ij jaijj+ Þ Signðaþ ij
c 11 c 21
+
ja11j Signðaþ 11 Þ ja21j+ Signðaþ 21 Þ
x j ðj ¼ 2Þ 1.5 2 2.5 + 2 +
c 12 c 22 ja12j+ Signðaþ 12 Þ ja22j+ Signðaþ 22 Þ
2 0.6 2 + 2.5 +
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3. Application to water resources planning For many decades, water resources planning problems, such as the efficient allocation of water and reasonable water treatment plant design, have challenged water resource managers in generating optimal or compromise decisions (Yin et al., 1999; Maqsood et al., 2005). It is essential for the decision makers to gain a better understanding of system components and their interrelationships within water resources system so as to meet their objectives. In many practical cases, the complexity of the system and the limited knowledge are the main obstacles to obtaining appropriate decision alternatives in a conflicting objective environment. Actually, most of the previous planning problems focused on optimal water allocation and did not consider the initial planning problems such as design of water resources systems including determining the maximum treatment ability of water plants, capability of reservoirs and etc. de Novo programming is an effective means of seeking a portfolio of capability levels and optimizing the objective functions by allocating a budget according to a unit capability price, where capacity levels (such as capability of reservoirs) are considered as decision variables. In urban or regional water resources planning problems, optimal supplies of high-quality water will be emphasized due to conflicting revenue target of municipal–industrial–agricultural balance under a given construction budget of water treatment facilities. It is desired that water treatment facilities for municipality, industry and agriculture be effectively designed for supporting sustainable management of available water resources. In order to balance the earnings of the competitive water users under the limited water budget, it is recognized that the different objectives of water users should be considered in relation to each other. Moreover, in various real world cases, uncertainties are associated with a number of impact factors such as cost and water treatment efficiency (Huang and Liu, 2008). As a result, how to optimally design water resources systems under uncertainty is a main concern facing the decision makers. Generally, they pay more attentions to the optimal system design at the original planning stage instead of just selecting the optimal allocation alternatives from predetermined data sets. All of these require an integrated approach for planning, provision, and management of urban water resources systems. Therefore, the work in this paper is an extension of the previous efforts in optimization for water resources systems planning. The developed inexact de Novo programming approach could effectively deal with the optimal system design problems, where design parameters could be optimally determined under uncertainty. Consider a case in which a water manager is responsible for allocating water from a water abundant reservoir for two co-developing cities to three users: a municipality, an industrial unit, and an agricultural sector (Fig. 2). The total water usage budget is [550, 700] 106 dollars. In order to consider the maximum earnings for three water users, the three criteria will be calculated. The unit benefit of each water user is listed in Table 3. In urban water system design, it is necessary to guarantee that the water allocation must meet the least demand of each city and each water user (Wang et al., 2003). The minimum total water needs and minimum proportions of total allocated water to city for three water user are deliberated (Table 3). To guarantee quality or water quality, the different water treatment facilities for different water users are also investigated. The limited water usage budget includes water supply cost and water treatment cost. The price for each of them is listed in Table 4. The water resource planning model is formulated as follows:
Reservoir
WP 1
City 1
Industry
City 2 WP 3 Industry
Agriculture
WP2
Municipal Agriculture
WP 4
Municipal
Fig. 2. Schematic of water allocation to multiple users. Table 3 Benefit and least water demand from different water-using facilities of each city. Benefit (dollars/m3)
City 1
City 2
Industry Municipal Agriculture
[80, 90] [76, 80] [45, 50]
[92, 100] [70, 75] [40, 43]
Least total water demand (106 m3)
[55, 60]
[65, 70]
For users 6
3
Least water demand (10 m ) Least allocation proportion
Industry
Municipal
Agriculture
Industry
Municipal
Agriculture
[13, 15] [0.30, 0.32]
[13, 15] [0.25, 0.27]
[13, 15] [0.35, 0.36]
[15, 17] [0.35, 0.37]
[15, 17] [0.20, 0.23]
[15, 17] [0.30, 0.32]
538
Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
Table 4 Water treatment efficiency and least water demand coefficients. Water treatment efficiency
Industry
Municipal
Agriculture
City 1 City 2
[0.79, 0.81] [0.85, 0.88]
[0.65, 0.70] [0.7, 0.75]
[0.88, 0.90] [0.93, 0.95]
Total budget (106 dollars)
Water supply cost
Water treatment cost
[550, 700] Price/unit water
City 1 [2.5, 2.7]
City2 [2, 2.3]
Industry [1.5, 1.8]
Municipal [2, 2.1]
Agriculture [1,1.3]
Maximize benefits for each water user
Max zj ¼
m X
cij xij ;
8j
ð25aÞ
i¼1
Subject to: Treated water demand of city: n X
aij xij 6 bi ;
8i
ð25bÞ
j¼1
Total water needs of end user: m X
xij 6 bj ;
8j
ð25cÞ
i¼1
Budget limit for water system design: m X
pi bi þ
n X
i¼1
pj bj 6 B
ð25dÞ
j¼1
Minimum water allocation to city: m X
xij P li ;
8i
ð25eÞ
i¼1
Minimum proportions of total allocated water to end users in city i
xij P r ij
m X
xij ;
8i; j
ð25fÞ
i¼1
Minimum water allocation to end users in city i
xij P mij ;
8i; j
ð25gÞ
c ij
li
a ij
where is the unit benefit of water user j in city i, i = 1, 2, j = 1, 2, 3. is water treatment efficiency for water usage j in city i. is the least total water allocation to city i, m ij and r ij are the minimum water needs and minimum proportions of total allocated water to city for user i. x ij is decision variables of water allocation plan to user j in city i. bi is the design variables for total water allocation to city i which is cor responding to unit cost pi within the total water usage budget. bj are the design variables of water treatment capacity for water user i, which correspond to unit cost pj within the total water usage budget. The solution process is as follows. Firstly, the submodel (I) based on Eq. (15) is formulated and solved, for x and b, with respect to each þ þ + þ þ þ þ objective function zþ j separately. For the upper level of maximum objective, aij ; pi ; pj ; cij ; xij ; bi ; li ; mij ; r ij and B are considered. Let þ þ þ þ þ zj ¼ max zþ ; j ¼ 1; 2; 3, be the optimal value for jth objective of submodel (I) and Let z ¼ ð z ; z ; z Þ be the j-objective value for the 1 2 3 j ideal system with respect to B+. With *z+ = (9724.2, 5622.4, 4366.9), the metaoptimum submodel (I-META) can be constructed based on þ þ þ Eq. (16). The solutions of IM xþ j ; M B ; M b , and M r can be obtained by solving this model, as shown in Table 5. þ þ þ The minimum budget M B ¼ 886:06 under which the metaoptimum performance zþ i can be realized through IM xj and M b . The solu+ þ þ þ tions of (I-META), M B and IM xj can be designated as metaoptimum solutions. It is obvious that M B is greater than B . To enforce any particular budget level B, we use the optimum-path ratio M r þ of 0.79 and obtain the final solutions (shown in Table 6), which are þ þ þ þ þ þ þ þ þ xþ opt ¼ M r IM xj ; ðj ¼ 1; 2; 3Þ; bopt ¼ M r M b , and zopt ¼ M r zk . The maximum benefits zopt to each water user are 7682.1, 6 4441.7, and 3449.8 (10 dollars), respectively. Table 5 Solution of upper and lower levels of the first de Novo model. Water allocate
IM
þ þ xj M B
¼ 886:06
City 1 City 2 Water allocate City 1 City 2
IIM
xj M B
¼ 696:3
Industry
Municipal
Agriculture
18.0 81.1
15.0 58.9
27.0 70.1
Industry
Municipal
Agriculture
17.6 46.8
15.0 30.0
22.4 49.8
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Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541 Table 6 Decision variables of water allocation and capacity design. Water allocate X*
Industry
Municipal
Agriculture
City 1 City 2
[13.90, 14.22] [36.99, 63.99]
[11.85, 11.85] [23.68, 45.82]
[17.70, 21.33] [39.20, 55.30]
Objectives Benefits
Industry [4121.4, 7682.1] Water supply design bi
Municipal [2558.2, 4441.7] Water treatment limit bj
Agriculture [1927.2, 3449.8]
City 1
City 2
Industry
Municipal
Agriculture
Design variable
[43.5, 47.4]
[99.87, 164.1]
[43.8, 65.7]
[26.1, 44.3]
[53.3, 70.3]
Similarly, the inexact de Novo submodel (II) corresponding to z, which provides the second step of the solution process based on the , from submodel (I) can be formulated based on Eq. (15). For the lower level of maximum objective, optimal variable solution, zþ k þ þ aþ ij ; pi ; pj ; cij ; xij ; bi ; li ; mij ; r ij and B are considered. Let zj ¼ max zj ; j ¼ 1; 2; 3, be the optimal value for jth objective of submodel (II) and* + ; z ; z Þ With z = (5216.9, 3238.2, 2439.5) be the j-objective value for the ideal system with respect to B. Then, a metaopLet z ¼ ð z 1 2 3 timum submodel (II-META) can be constructed based on Eq. (19). þ The value of M B is equal to 696.3, which is the minimum budget to achieve z j through IIM xj and M b . M r (Table 3) is the opti * mum-path ratio for achieving the ideal performance z for a given budget level B . The optimal system design (Table 4) can be established þ as ðx opt ; bopt ; zopt Þ, where xopt ¼ M r IM xj ðj ¼ 1; 2; 3Þ; bopt ¼ M r M b , and zopt ¼ M r zk . The maximum benefits zopt to each water user is (4121.4, 2558.2, 1927.2) 106 dollars. Table 4 indicates that when the budget reaches the upper bound, allocated water for city 1 is mainly used to guarantee water demand of agriculture, then industry and municipality in turn; allocated water for city 2 is mainly for industry, then agriculture and municipality in turn. This is because industry users in city 2 bring the highest benefit when water demand is satisfied; whereas municipality and agricultural users in city 1 correspond to lower benefits. Another important thing is that the total water allocation to city 2 (164.1 106m3) is much higher than that to city 1 (47.4 106m3). This is due to the higher earnings, higher water treatment efficiency and lower water supply cost to city 2. On the other hand, when the budget is reduced to the lower bound, allocated water for city 2 is mainly for agriculture, then industry and municipality in turn; while city 2 keeps the same order as that when the budget reaches the upper bound. Table 4 indicates that the total water supplies to cities 1 and 2 are [43.5, 47.4] and [99.87, 164.1], respectively. Total water allocation to city 2 (99.87) is less than that to city 1 (43.55) although the benefits from city 1 is higher that that of city 2. This is because the decision makers should focus on the basic demand of the water users in each city when the budget is low; whenever the budget is abundant, the decision makers can pay more attention to earnings of each water user. The total treating water target for industrial use is [43.8, 65.7]. This can help decision makers design and equip proper water treatment facilities in water purifying industry. In three objectives of three water users, the maximum earning belongs to industry ([4121.4, 7682.1]). It provides two extremes of the system benefit. As actual values of the variables and parameters vary within their bounds, the system benefit may correspondingly change between 4121.4 and 7682.1 with varied reliability levels. The developed IDNP approach is especially useful to design an optimal system rather than optimize a given system. It can effectively deal with the system design problems involving multiple objectives. In the conventional multiobjective programming problems, trade-offs exist among multiple objectives, especially conflicting ones. Scenarios are often analyzed and a set of decision results are generated for reflecting the trade-offs. Generally, it is difficult for the decision makers to select the final or ‘‘best” decision alternatives from a number of solution options, especially for the problems of designing a new optimal system where the budget is unknown. The IDNP can eliminate the trade-offs through designing a better and preferably optimal system. Consequently, there are no trade-offs in a new designed system using the IDNP. This can allow the decision makers to incrementally reduce the number of the objectives through transforming the objectives into the constraints, and thus greatly improve the computational efficiency. Moreover, the obtained results can provide effective decision support for the decision makers to design an optimal system. The conventional planning problems using fuzzy and/or stochastic program mostly focused on optimizing a given system subject to a series of constraint resources which were assumed to be fixed or known. However, in real world problems, the constraint resources have imprecise features, which are difficult to or can not be determined precisely. Compared with the above traditional methods, the de Novo programming approach can effectively handle the optimal system design problems within the available resources. These resources are considered as decision variables in de Novo program subject to the budgetary constraints, which can thus affect the values of the objective values. In stochastic programming, the quality of information on some uncertainties is often not good enough to be presented as probability density functions (PDFs). Even if these functions are available, it is extremely hard to solve a large-scale stochastic programming problem with all uncertain parameters being expressed as PDFs. In fuzzy programming, the decision makers need to determine the aspiration or preference levels and tolerances of the objective functions and/or the constraints. Those are often assumed based on the decision makers’ subjectivity or for the simplicity. In real world problems, uncertainties may exist as ambiguous intervals because planners and engineers typically find it more difficult to specify distributions than to define fluctuation ranges. For example, a water allocation target can hardly be expressed as a PDF; however, it can be easily defined as an interval. The conventional fuzzy programming approach is unable to directly communicate uncertain input information expressed as intervals into the optimization processes, thus leading to the reduction of the optimal solutions’ accuracy. Moreover, the fuzzy membership functions also involve the knowledge of interval analysis under a given alpha-cut. Instead, the developed method makes intuitive sense since total water allocation to the cities should depend on some budgetary constraints. Based on the proposed IDNP, the minimum water allocation to each city can be determined, which means a tight system in terms of resources identification; at the same time, a higher objective function value can be obtained. Thus, the proposed method covers the expression of system uncertainties by the conventional fuzzy and stochastic programming when the PDFs and fuzzy membership functions can not effectively reflected.
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Compared with interval mathematical programming (IMP), the proposed method can effectively deal with the real world water resources system design problems, where uncertainties of parameters and decision variables are expressed as intervals. The developed IDNP shifted the traditional IMP approach for optimizing a given water resources system to the design of an optimal system under a given budget level of resources, which can not be achieved through the conventional IMP methods. The conventional IMP is to find the optimal solutions subject to fixed or known resources, the right-hand side of the constraints. Based on the results from the conventional IMP, the planners can not produce the optimal resource levels for helping make the appropriate decisions at the initial system design stage. The proposed IDNP method can provide a higher objective function value and a tighter system, where the levels of resources are optimally produced rather than prespecified by the planners. The levels of resources are considered as decision variables and the uncertainties are effectively handled, which makes the model more practical in reflecting the real world systems. Compared with multiobjective programming, the IDNP method is able to obtain design decision variables and management variables. Most of multiobjective programming approaches have limitations in dealing with true issue of designing optimal systems. The resource level (right-hand side of the model) is determined as a known number. Although 0–1 programming was already used for dealing with expansion problem considering right-hand side flexibility, it can only choose to expand or not a given option but not how much to expand. For example, in a three-period landfill expansion problem, the obtained results indicate that the landfill shall be expanded in the second and third periods. Actually, the decision makers would prefer to expand the landfill once instead of twice. Therefore, the developed IDNP approach has the advantages in constructing optimal system design via an ideal system by introducing the flexibility toward the available resources in the system constraints. 4. Conclusions An interval de Novo programming (IDNP) approach was developed for designing an optimal system under uncertainty. The model integrated interval programming and de Novo programming within an optimization framework. In its solution process, the IDNP model was transformed into two deterministic submodels, which corresponded to the lower and upper bounds, respectively. For each submodel, the design variables in the IDNP model were constrained by the total budget. By function transformation, the maximum value of each objective function could be obtained, corresponding to the performance of the ideal design under a given B. Lastly, a metaoptimum model was constructed and solved to get optimal design solutions. In water resources systems planning, the IDNP could be an effective tool for achieving sustainable development. The developed IDNP approach was applied to a hypothetical case of water resources systems planning, where the parameters were represented as intervals. Uncertainties of parameters and decision variables are effectively handled, which makes the model more real. In this water resources allocation problem, decision makers are confronted with control and adjustment of water disposal facility design and operation characteristics. IDNP is such an integrated approach for planning, provision, and management of urban water systems. It can shift the traditional interval programming approach for optimization of a given water resources system to the design of an optimal system, which can not be achieved through the conventional interval or fuzzy programming methods. Moreover, the IDNP method could generate an exclusive result preferred by the decision makers, including design decision variables and management decision variables. Within multiobjective decision making framework, IDNP may allow the decision makers to achieve a metaoptimal system performance and improve the performance of compromise solutions. The obtained results indicated that the IDNP method was also applicable to other practical problems involving optimal environmental systems design. Acknowledgement This research was supported by the Major State Basic Research Development Program of MOST (2005CB724200 and 2006CB403307), the Canadian Water Network under the Networks of Centers of Excellence (NCE), and the Natural Science and Engineering Research Council of Canada. We would like to thank Prof. Nurullah Umarusman for pointing out some errors which occurred in an earlier version of this paper. Appendix A x j is vector of management decision variables. bi is vector of design decision variable. c kj is parameter of decision variables in the objective function. a ij is parameter of decision variables in the constraints. p i is the unit prices parameter of design decision variable. B± is the given total available budget. z k is the objective function. Sign(*) is the sign of parameter *; ¼ a ij ; pi . j*j is the absolute value of parameter ; ¼ a ij ; pi . þ zk is the optimal value for kth objective of submodel ðmax zþ k Þ. xj is the value of management decision variables solved by metaoptimum submodel. * ± b is the value design decision variable solved by metaoptimum submodel. * ± * + B is the minimum budget to achieve zþ k through xj and b . * ± r is optimum-path ratio for achieving the ideal performance.*r± = B±/*B±. x opt is the optimal value management decision variables, xopt ¼ r xj . þ þ þ bopt is the optimal value design decision variables, bopt ¼ r þ b . þ þ þ zþ opt is the optimal value of the objectives, zopt ¼ r zk .
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References Bare, B.B., Mendoza, G.A., 1988. A soft optimization approach to forest land management planning. Canadian Journal of Forest Research 18, 545–552. Bare, B.B., Mendoza, G.A., 1990. Designing forest plans with conflicting objectives using de Novo programming. Journal of Environmental Management 31 (3), 237. Chen, Y.W., Hsieh, H.E., 2006. Fuzzy multi-stage de Novo programming problem. Applied Mathematics and Computation 181 (2), 1139–1147. Huang, G.H., 1994. Grey mathematical programming and its application to municipal solid waste management planning. Ph.D. Thesis. Department of Civil Engineering, McMaster University. Huang, Y.T., Liu, L., 2008. A hybrid perturbation and morris approach for identifying sensitive parameters in Surface Water Quality Models. Journal of Environmental Informatics 12 (2), 150–159. Huang, G.H., Moore, R.D., 1993. Grey linear programming, its solving approach, and its application. International Journal of Systems Science 24, 159–172. Huang, G.H., Baetz, B.W., Patry, G.G., 1992. A grey linear programming approach for municipal solid waste management planning under uncertainty. Civil Engineering Systems 9, 319–335. Kim, H., Ida, K., Gen, M., 1993. A de Novo approach for bicriteria 0–1 linear-programming with interval-coefficients under gub structure. Computers and Industrial Engineering 25 (1–4), 17–20. Kotula, A.S., 1997. Toward sustainable reservoir design: Application of de Novo programming. Department of Civil and Geological Engineering. Master of Science. University of Manitoba, Winnipeg. Li, R.J., Lee, E.S., 1990. Multicriteria de Novo programming with fuzzy parameters. Computers and Mathematics with Applications 19, 13–20. Li, R.J., Lee, E.S., 1993. De Novo programming with fuzzy coefficients and multiple fuzzy goals. Journal of Mathematical Analysis and Applications 172, 212–220. Maqsood, I., Huang, G.H., Yeomans, J.S., 2005. An interval-parameter fuzzy two-stage stochastic program for water resources management under uncertainty. European Journal of Operational Research 167 (1), 208–225. Sasaki, M., Gen, M., Yamashiro, M., 1995. A method for solving fuzzy de Novo programming problem by genetic algorithms. Computers and Industrial Engineering 29, 507– 511. Shi, Y., 1995. Studies on optimum-path ratios in multicriteria de Novo programming-problems. Computers and Mathematics with Applications 29 (5), 43–50. Shi, Y., 1999. Optimal system design with multiple decision makers and possible debt: A multicriteria de Novo programming approach. Operations Research 47 (5), 723–729. Wang, L.Z., Fang, L., Hipel, K.W., 2003. Water resources allocation: a cooperative game theoretic approach. Journal of Environmental Informatics 2 (2), 11–22. Yin, Y.Y., Huang, G.H., Hipel, K.W., 1999. Fuzzy relation analysis for multicriteria water resources management. Journal of Water Resources Planning and Management–ASCE 125 (1), 41–47. Zeleny, M., 1981. A case study in multiobjective design: De Novo programming. In: Nijkamp, P., Spronk, J. (Eds.), Multiple Criteria Analysis: Operational methods. Gower publishing Co., Hampshire, England. Zeleny, M., 1986. Optimal system design with multiple Criteria: De Novo programming approach. Engineering Costs and Production Economics 10, 89–94. Zeleny, M., 1990. Optimizing given systems vs. designing optimal systems: The de Novo programming approach. General Systems 17, 295–307. Zeleny, M., 2005. The evolution of optimality: De Novo programming. Evolutionary Multi-Criterion Optimization, vol. 3410. Springer-Verlag, Berlin. pp. 1–13.
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Innovative Applications of O.R.
Inexact de Novo programming for water resources systems planning Y.M. Zhang a, G.H. Huang a,b,*, X.D. Zhang a a b
Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 Chinese Research Academy of Environmental Science, Beijing Normal University, Beijing 100012-100875, China
a r t i c l e
i n f o
Article history: Received 6 November 2007 Accepted 14 November 2008 Available online 28 November 2008 Keywords: Uncertainty Management Water resources Decision analysis Multi-objective Environment de Novo
a b s t r a c t This study presents an interval de Novo programming (IDNP) approach for the design of optimal waterresources-management systems under uncertainty. The model is derived by incorporating the existing interval programming and de Novo programming, allowing uncertainties represented as intervals within the optimization framework. The developed IDNP approach has the advantages in constructing optimal system design via an ideal system by introducing the flexibility toward the available resources in the system constraints. A simple numerical example is introduced to illustrate the IDNP approach. The IDNP is then applied to design an inexact optimal system with budget limit instead of finding the optimum in a given system with fixed resources in a water resources planning case. The results demonstrate that the developed method efficiently produces stable solutions under different objectives. Optimal supplies of good-quality water are obtained in considering different revenue targets of municipal–industrial–agricultural competition under a given budget. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In modern society, systems analysis and design have become important managerial and operational issues confronting many countries and regions in the world. It is an active process that seeks a portfolio of resource levels and optimizes the objective function by allocating a budget according to a resource price, where resource levels are considered as decision variables (Zeleny, 1990). In water resources systems planning, decision makers are greatly concerned about decisions for determining resources availabilities and/or identifying desired management facilities. Therefore, designing a sound water resources system for modeling various water resources issues can help decision makers allocate water effectively and efficiently. The classical de Novo programming method nominated by Zeleny (1981, 1986, 1990) was effective for dealing with optimal design problems. Since the resource availability level is determined as a priori, the result of linear multicriteria programming is the optimal given system. In contrast, with unknown resource availability, the multicriteria de Novo programming (MDNP) seeks a portfolio of resource availability level to optimize multiple objective functions by allocating a budget according to the resource price. This approach results in an optimal system design. Zeleny (1990) proposed a basic method to construct the optimal system design for solving a de Novo problem via an ideal system design. One of the important issues in MNDP was to determine an optimum-path ratio for enforcing a particular budget level of resources so as to establish the optimal system design. In Zeleny’s basic method, the optimum-path ratio for achieving the multicriteria performance of the ideal system design related to a given budget level was used to determine the optimal system design. Bare and Mendoza (1990) illustrated tile potential use of designing optimal forest systems in the face of conflicting objectives. Both the generation and evaluation of compromise solutions under de Novo conditions were discussed. Shi (1995) introduced several optimum-path ratios for enforcing different budget levels of resources so as to find alternative optimal system designs for solving MNDP problems. Shi (1999) applied MDNP to formulate and solve problems of system design that involved multiple decision makers and a possible debt. Zeleny (2005) investigated the evolution of optimality of single or multiobjective programming, and summarized the eight major optimality concepts according to a dual classification: single versus multiple criteria versus the extent of the ‘given’, ranging from ‘all-but’ to ‘none except’. Previously, de Novo programming has been applied in some cases. Bare and Mendoza (1988) applied de Novo programming to single and multi-objective forestry land management problems. This problem considered constraints on labor, picnic sites, and hiking trails. It was proved that the system could be designed to perform in an ideal fashion within a constant budget level. Bare and Mendoza (1990) * Corresponding author. Address: Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada S4S 0A2. Tel.: +1 306 585 4095; fax: +1 306 585 4855. E-mail address: [email protected] (G.H. Huang). 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.11.019
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then considered another case of forestry land management where several compromise solutions were generated, proving that de Novo programming did improve compromise performance levels. Finally, decision maker was responsible for accepting or rejecting a decision alternative. Kotula (1997) mentioned the difficulty in construction of multipurpose reservoirs for achieving sustainable development, taking into account the grand scale and complex nature of project impacts on society, environment, and the economy. He applied the de Novo programming in considering the control and adjustment of reservoir design and operation characteristics which resulted in optimal or near optimal system performance throughout the life of the reservoir. The de Novo programming could shift the traditional approach of optimization, from optimization of a given reservoir system, to the design of the optimal reservoir system. Actually, in real world de Novo programming problems, uncertainties may exist in related costs, system objectives and parameters. Generally, uncertain parameters can be represented as fuzzy, interval or stochastic numbers, resulting in uncertain de Novo programming. Li and Lee (1990) extended Zeleny’s basic method to identify fuzzy system designs for de Novo problems by considering the fuzziness in coefficients. Li and Lee (1993) further treated fuzzy goals and fuzzy coefficients simultaneously, depending on a numerical approach which could be solved as either linear or nonlinear problems. Kim et al. (1993) formulated a de Novo 0–1 bicriteria linear programming with interval coefficients under generalized upper bounding structure. Interval coefficients were transformed into a fuzzy state by the fuzzy transformation based on the degree of satisfying inequality relationship and order relationship between intervals. This transformation was limited and hard to be applied to a generalized de Novo model. Sasaki et al. (1995) proposed an implementation of the genetic algorithm for solving DNP problems with fuzzy goal and constraints. The proposed approach had the flexibility to obtain better solutions compared to crisp constraints. Considering random distribution of budget, Chen and Hsieh (2006) presented a fuzzy multi-stage de Novo programming which was viewed as a fuzzy dynamic programming problem. Uncertain parameters were presented as intervals with unknown distributions, leading to an inexact de Novo programming. Nevertheless, the previous studies on de Novo programming could seldom deal with interval parameters. Actually, variations of system performance and decision variables of the inexact programming can be investigated by solving relative submodels. By using inexact programming, equivalent information can be obtained in a faster and simpler way (Huang et al., 1992). In practical problems, uncertainty often exists due to the imprecise and vague nature of system parameters. The quality of available information is often not satisfactory enough to be represented as deterministic values; unavailability of required data with probability distribution over the planning horizon often leads to the difficulties in application of stochastic programming to practical water resources management problems, and; the incomplete data and the decision makers’ subjectivity make it difficult to generate the appropriate and reasonable fuzzy membership functions. Instead, the decision makers can only provide the lower and upper bounds of the parameters, without knowing any distribution forms or crisp values. Thus, the obtained data may be represented as interval numbers, which are often more intuitive. Therefore, the objective of this study is to develop an inexact de Novo programming (IDNP) approach for designing a water resources management system under uncertainty, where the modeling parameters can be represented as intervals. The developed approach can effectively deal with both single and multi-objective problems. The IDNP will then be applied to a hypothetical case study of water resources systems planning, where efficient water allocation to domestic, agricultural and industrial customers will be guaranteed. 2. Methodology 2.1. De Novo programming A de Novo programming model can be formulated as follows (Zeleny, 1990):
Max z ¼ Cx s:t: AX b 6 0; pb 6 B;
ð1Þ
x P 0; where p is the given unit prices of m resources, p = p1, p2, . . . , pm; B is total given variable budget; b is the resulting (optimal) portfolio of resources b = b1, b2, . . . , bm, which can be considered as design decision variables in this problem; x is the production levels, x = x1, x2, . . . , xm, which can be regarded as management decision variables. This model aims at allocating budget to each b, determining x, and maximizes P the benefit value z, where z ¼ Cx ¼ kj ckj xkj ; k ¼ 1; 2; . . . ; q. Objective functions zk are to be maximized simultaneously. Actually, truly optimal portfolio of a decision maker should more appropriately maximize the difference (cx pb) rather than cx. Problem (1) can be transformed into:
Max Z ¼ Cx s:t: Vx 6 B;
ð2Þ
x P 0; where Z = (z1, . . . , zq) and V = (V1, . . . , Vn) = pA. By the methodology of de Novo optimal design programming, problem (2) can be solved, for x and b, with respect to each of objective functions zk, respectively. Let zk ¼ max z; k ¼ 1; 2; . . . ; q, subject to constraints of (2). Let z ¼ ðz1 ; z2 ; . . . ; zk Þ denote the performance of the ideal design under a given B. Obviously, z* must be attainable for a given budget level B. For each of k values of z*, there is a corresponding zk -optimal portfolio b* calculated as in the previous section. As vector z* represents the metaoptimal performance, the corresponding x* and b* can be obtained by solving the following metaoptimum model (Zeleny, 1990):
Min B ¼ Vx s:t: Cx P Z ;
ð3Þ
x P 0: Problem (3) identifies the minimum budget B* at which the metaoptimum performance z* can be realized through x* and b*. The solution of problem (3), B* and x* can be designated as metaoptimum solution. It is obvious that metaoptimum budget B* must exceed any given budget B. For any particular budget level B, the optimum-path ratio r can be used
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Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
r ¼ B=B :
ð4Þ *
*
*
The final solutions are as follows: x = rx , b = rb , z = rz . The optimum-path r provides an efficient tool for virtually instantaneous optimal redesign of even large-scale systems. It is a powerfully competitive tool which is able to maintain complex multicriteria decision systems by expanding of budget B along optimal path. The effects of de Novo programming on the feasible regions of decision alternatives are better illustrated through an example. Suppose that a given system contains two objective functions Z1 and Z2 which are to be maximized, four linear constraints on available resources decision variables, and two nonnegative decision variables X1 and X2. The de Novo programming approach designs the resource portfolio by modifying the slope and intercept of each soft constraint. The new feasible region will be optimally modified to satisfy the metaoptimum constraints. The optimal de Novo solutions can determine the minimum budget necessary to achieve the metaoptimum. If the metaoptimum constraints are of inequality type, such that the de Novo model aspires to at least achieve the metaoptimum, it is possible to over achieve the metaoptimum (Kotula, 1997). 2.2. Inexact de Novo Programming Before an inexact de Novo programming model is formulated, some useful definitions need to be introduced. Let x denotes a closed and bounded set of real numbers, an inexact number x± is defined as an interval with known upper and lower bounds but unknown distribution information (Huang, 1994):
x ¼ ½x ; xþ ¼ ft 2 xjx 6 t 6 xþ g;
ð5Þ
+
±
+
±
+
where x and x are the lower and upper bounds of x respectively. When x = x , x becomes a deterministic number, i.e. x = x . For x±, we define Sign(x±) as follows:
Signðx Þ ¼ 1;
if x P 0;
ð6Þ
if x < 0:
¼ 1;
ð7Þ
±
Its absolute value jx j is defined as follows:
jx j ¼ x ;
if x P 0;
ð8Þ
if x < 0:
ð9Þ
jxj ¼ x ; if x P 0; ¼ xþ ; if x < 0;
ð10Þ ð11Þ
and jxjþ ¼ xþ ;
ð12Þ
¼ x ; Thus, we have:
if x P 0;
if x < 0:
¼ x ;
ð13Þ
The single or multi-objective inexact de Novo programming for optimal system design can be formulated as follows:
Max z ¼ C x
s:t: A x b 6 0;
ð14Þ
p b 6 B ;
x P 0; where C 2 Rkj and A 2 Rij are matrices of dimensions h j and i j, respectively. c kj is the element of Matrix C ; aij is the element of Matrix A±; b± is vector of design decision variable, b ¼ ðb1 ; b2 ; . . . ; bi Þ; x± is vector of management decision variables, x ¼ ðx 1 ; x2 ; . . . ; xj Þ; ± ± p is the vector of the unit prices of i resources, p ¼ ðp1 ; p2 ; . . . ; pi Þ; and B is the given total available budget. These definitions are essential because there are two kinds of decision variables including design decision variables b± and management decision variables x± in the inexact de Novo program, while most optimization problems only have management decision variables. Moreover, all the decision variables in IDNP are inexact numbers, which are especially true for the practical problems. Solving problem (14) means finding the optimal allocation of B so that the corresponding resource portfolio b maximizes simultaneously the values Z = Cx of the product mix x. For all the coefficient are interval variables, this inexact linear programming model (1) can be solved by transformed into two sets of deterministic submodels, which correspond to the lower and upper bounds of the desired objective function value. This transformation process is based on an interactive algorithm, which is different from normal interval analysis and best/worst case analysis (Huang and Moore, 1993). The solutions provide stable intervals for the objective functions and decision variables with different levels of risk in violating the constraints. They can be easily interpreted for generating decision alternatives. For model (14), the submodel (I) corresponding to z+, which provides the first step of the solution process when the objective function is to be maximized, can be formulated as follows (assuming that bi > 0):
Maximize zþk ¼
t X
cþkj xþj
j¼1
s:t:
X
X xj
n X
cþkj xj ;
j¼tþ1
jaij j Signðaij Þxþj þ þ
8k X
jpi j Signðpi Þbi 6 Bþ ;
P 0;
8j;
ð15aÞ þ
jaij jþ Signðaþij Þxj 6 bi ;
8i;
ð15bÞ ð15cÞ ð15dÞ
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Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
where x j ; j ¼ 1; 2; . . . ; t, are inexact variables with positive coefficients, and xj , j = t + 1,t + 2, . . . , n, are inexact variables with negative coefficients. Eqs. (15b) and (15c) can be transformed into:
X
jpi j Signðpi Þ
X
X X jaij j Signðaij Þxþj þ jaij jþ Signðaþij Þxj 6 Bþ : jpi j Signðpi Þ
ð15eÞ
þ þ þ þ þ Let zþ k ¼ maxzk ; k ¼ 1; . . . ; q, be the optimal value for kth objective of submodel (I) with Eq. (15). Let z ¼ ð z1 ; z2 ; . . . ; zq Þ be the q-objec+ tive value for the ideal system with respect to B . Then, the metaoptimum submodel I can be constructed as follows:
X
Minimize Bþ ¼ t X
s:t:
X X X jpi j Signðpi Þ jaij j Signðaij Þxþj þ jaij jþ Signðaþij Þxj jpi j Signðpi Þ n X
cþkj xþj
j¼1
cþkj xj P zþk ;
8k;
ð16aÞ ð16bÞ
j¼tþ1
xj P 0;
ð16cÞ
* + * + * + Solving Problem (I-META) yields x j , B , b , and r where:
þ
b ¼
X
jaij j Signðaij Þ xþj þ
X
jaij jþ Signðaþij Þ xj ;
ð17aÞ
þ þ
þ
r ¼B = B :
ð17bÞ
* +
* +
þ zk
* +
+ * +
The value B is the minimum budget to achieve through x j and b . Since B P B , r identifies the upper bound of optimum-path ratio þ þ for achieving the ideal performance *z+ for a given budget level B+. The optimal system design can be established as (x opt ; bopt ; zopt Þ, where þ þ þ þ þ þ þ þ þ þ xopt ¼ r xj ; ðj ¼ 1; 2; . . . ; tÞ; xopt ¼ r xj ðj ¼ t þ 1; t þ 2; . . . ; nÞ; bopt ¼ r b , and zopt ¼ r zk . The optimum-path ratio *r+ provides an effective and fast tool for optimal redesign of large-scale systems. The submodel (II) corresponding to z, which provides the second step of the solution process based on the optimal variable solution, þ zk , from submodel (I), can be formulated as follows (assuming that bi > 0):
Maximize zk ¼ X
s:t:
X zk xj
t X
ckj xj
j¼1
n X j¼tþ1
jaij jþ Signðaþij Þxj þ jpi j
6
þ
Signðpþi Þbi
þ zk ;
P 0;
ckj xþj ; X
8k
ð18aÞ
jaij j Signðaij Þxþj 6 bi ;
8i;
6B ;
ð18bÞ ð18cÞ
8k; 8j;
ð18dÞ ð18eÞ
where x j ; j ¼ 1; 2; . . . ; t, are inexact variables with positive coefficients, and xj ; j ¼ t þ 1; t þ 2; . . . ; n, are inexact variables with negative coefficients. Similarly, Eqs. (18b) and (18c) can be transformed into:
X
jpi jþ Signðpþi Þ
X
X X jpi jþ Signðpþi Þ jaij jþ Signðaþij Þxj þ jaij j Signðaij Þxþj 6 B :
ð18fÞ
Let z k ¼ max zk ; k ¼ 1; 2; . . . ; q, be the optimal value for kth objective of submodel (II) with Eq. (18). Let z ¼ ð z1 ; z2 ; . . . ; zq Þ be the q objective value for the ideal system with respect to B . Then, the metaoptimum submodel II can be formulated as follows:
Minimize B ¼ t X
s:t:
X
X X X jpi jþ Signðpþi Þ jpi jþ Signðpþi Þ jaij jþ Signðaþij Þxj þ jaij j Signðaij Þxþj
ckj xj
j¼1
n X
ckj xþj P zk ;
8k;
ð19aÞ ð19bÞ
j¼tþ1
xj P 0; X X X X þ jpi jþ Signðpþi Þ jpi j Signðpþi Þ jaij jþ Signðaþij Þxj þ jaij j Signðaij Þxþj 6 rþ ; B =
ð19dÞ
xj xþj
ð19cÞ
6
þ xj ;
j ¼ 1; 2; . . . ; t;
ð19eÞ
6
xj ;
j ¼ t þ 1; t þ 2; . . . ; n:
ð19fÞ
* Solving Problem (II-META) yields x j ; B ; b , and r where:
b ¼
X
jaij jþ Signðaþij Þ xj þ
r ¼ B = B :
X
jaij j Signðaij Þ xþj ;
ð20aÞ ð20bÞ
* + * * The value *B is the minimum budget to achieve z k through xj and b . Since B P B , r identifies the optimum-path ratio for achiev * ing the ideal performance z for a given budget level B . The optimal system design can be established as ðx opt ; bopt ; zopt Þ, where þ þ ¼ r x ðj ¼ 1; 2; . . . ; tÞ; x ¼ r x ðj ¼ t þ 1; t þ 2; . . . ; nÞ; b ¼ r b , and z ¼ r z . x opt opt opt opt j k j þ þ Therefore, bopt ; zþ opt ; xopt ðj ¼ 1; 2; . . . ; tÞ and xopt ðj ¼ t þ 1; t þ 2; . . . ; nÞ can be obtained by solving the submodel (I) defined by Eqs. (15) þ (17), whereas bopt ; z opt ; xopt ðj ¼ 1; 2; . . . ; tÞ and xopt ðj ¼ t þ 1; t þ 2; . . . ; nÞ can be obtained from Eqs. (18)–(20). Thus, final solutions can be þ þ þ ¼ ½z ; z ; x ¼ ½x ; x and optimal design solution bopt ¼ ½bopt ; bopt . obtained with z opt opt opt opt opt opt Fig. 1 shows the scheme of the modeling methodology. In conclusion, the solution algorithm of the IDNP model with the objective being maximized is presented as follows:
Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
Right hand side variability
Inexact data
de Novo programming
Interval-parameter programming INDP lower bound submodel
535
INDP model
INDP upper bound submodel
First lower INDP model
First upper INDP model
Meta lower INDP model
Meta upper INDP model
Solutions
Generation of decision Fig. 1. Schematic of the INDP methodology.
Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. Step 7. Step 8. Step 9. Step 10.
Formulate IDNP model (14). Transform the IDNP model into two submodels, where the upper bound of the z± is desired since the objective is to maximize z±. Formulate and solve z+ submodel (15) and obtain zþ k. * + þ þ Formulate and solve z+ metaoptimum submodel I (16) and obtain x j ; B ; b , and r . þ þ Calculate xopt ; bopt and zopt . Formulate and solve z submodel (18) and obtain zþ k. * Formulate and solve z metaoptimum submodel II (19) and obtain x j ; B ; b , and r . Calculate xopt ,bopt and zopt . þ þ þ bopt ¼ ½bopt ; bopt and x Solutions of the IDNP model are: z opt ¼ ½zopt ; zopt ; opt ¼ ½xopt ; xopt . Stop.
2.3. A numerical example To illustrate the method, a detailed example is introduced. Consider a case where the decision makers want to maximize the economic benefit and environmental protection benefit of two districts in a big agricultural farm by allocating necessary electric and water resources. However, the total budget is limited within [40, 48] million dollars. Let x 1 ; x2 represent the farming area of two districts and b1 ; b2 rep resent the total electric and water cost consumed by x1 ; x2 , respectively with the price of 1/unit and 0.8/unit under the total resource-consuming budget. The detailed description of this example is as follows:
Max z1 ¼ ½1:5; 2x1 þ ½2; 3x2
ð21aÞ
ðEconomic benefit objectiveÞ Max z2 ¼ ½2; 3x1 þ ½0:6; 1x2
ð21bÞ
ðEnvironmental benefit objectiveÞ
s: t: ½2; 2:5x1 þ ½1:5; 2x2 6 b1 ;
ð21cÞ
ðWater consuming constraintsÞ
½1:5; 2x1 þ ½2; 2:5x2 6 b2 ;
ð21dÞ
ðElectric consuming constraintsÞ
b1 þ 0:8b2 6 ½40; 48;
ð21eÞ
ðTotal budget constraintsÞ x1 6 7:2;
ð21fÞ
ðMaximum land available of distinct 1Þ x2 6 9:
ð21gÞ
ðMaximum land available of distinct 2Þ + It is assumed that the objective functions of z 1 and z2 are equally important. From Eq. (15), the submodel (I) corresponding to z can be formulated. Then, the corresponding values of all the symbols of the upper bound model are listed in Table 1.
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Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
Max zþ1 ¼ 2xþ1 þ 3xþ2 Max zþ2 ¼ 3xþ1 þ xþ2 þ
s:t: 2xþ1 þ 1:5xþ2 6 b1 ; þ
1:5xþ1 þ 2xþ2 6 b2 ; þ b1 xþ1 xþ2
þ
þ 0:8b2
ð22Þ
6 48;
6 7:2; 6 9:
+ þ By solving submodel I (22), we can obtain zþ 1 ; z2 ¼ ð39:56; 29:65Þ with respect to a given B value of 48. From the results of the submodel (I), a metaoptimum submodel can be constructed as follows:
Min Bþ ¼ 1 ð2xþ1 þ 1:5xþ2 Þ þ 0:8 ð1:5xþ1 þ 2xþ2 Þ 2xþ1 þ 3xþ2 P 39:56; 3xþ1 þ xþ2 P 29:65;
ð23Þ
xþ1 6 7:2; xþ2 6 9: þ þ * + þ The feasibility of zþ 1 ; z2 can only be assured by the metaoptimum solutions v x1 ; x2 ¼ ð7:06; 8:48Þ with the cost B of 48.87. þ þ þ Because the optimal-path ratio *r+ = 48/48.87 = 98.2%, the resulting opt xþ ; x ¼ ð6:93; 8:33Þ and z ; z ¼ ð38:85; 29:11Þ. The optiopt opt opt 1 2 1 2 þ þ mal portfolio bopt , with respect to B+ = 48, can be calculated by substituting opt xþ 1 ; opt x2 into the constraints of model (23). The results are þ þ þ þ þ þ opt b1 ¼ 98:2% ð2x1 þ 1:5x2 Þ ¼ 26:35 and opt b2 ¼ 98:2% ð2x1 þ 1:5x2 Þ ¼ 27:06. þ The next step is to consider the lower bound model. Based on the optimal variable solutions of opt zþ 1 and opt z2 , and Eq. (18), the submodel (II) corresponding to z can be formulated as follows. The values of symbols are shown in Table 2.
Max z1 ¼ 1:5x1 þ 2x2 Max z2 ¼ 2x1 þ 0:6x2 s:t: 2:5x1 þ 2x2 6 b1 ;
2x1 þ 2:5x2 6 b2 ; b1 x1 x2 z1 z2
þ
0:8b2
6 40;
ð24Þ
6 7:2; 6 9; 6 zþ1 ; 6 zþ2 :
Similarly, the obtained optimal solutions are z 1 ¼ 19:46 and z2 ¼ 15:97, while the metaoptimum are x1 ¼ 6:54 and x2 ¼ 4:83, with the * + * cost B of 46.12. The optimal-path ratio r = 40/46.12 = 86.7%, the resulting opt x1 ; opt x2 ¼ ð5:67; 4:19Þ and opt z1 ; opt z2 ¼ ð14:63; 13:85Þ. The optimal portfolio bopt , with respect to B = 40, can be calculated by substituting opt x 1 ; opt x2 into the constraints of model (24). The results are opt b1 ; opt b2 ¼ ð22:54; 21:80Þ. Then the final solutions are: opt x1 ¼ ½5:67; 6:93; opt x2 ¼ ½4:19; 8:33; opt b1 ¼ ½22:54; 26:35; opt b2 ¼ ½21:80; ¼ ½14:63; 38:85; z ¼ ½13:85; 29:11. The results show that the planned farming areas of two districts are [5.67, 6.93] km2 27:06; opt z opt 2 1 2 and [4.19,8.33] km . Under the budget of [40, 48] million dollars, the allocated funds to electric and water consumption are [21.80, 27.06] million dollars and [21.80, 27.06] million dollars, respectively. The economic and environmental benefits are [14.63, 38.85] million dollars and [13.85, 29.11] million dollars, respectively.
Table 1 Values of symbols in upper bound model I. zþ k zþ 1 ðk ¼ 1Þ zþ 2 ðk ¼ 2Þ þ bi ði ¼ 1Þ þ
bi ði ¼ 2Þ
xþ ðj ¼ 1Þ j cþ kj cþ kj
jaijj Signða ij Þ jaijj Signða ij Þ
cþ 11 cþ 21
ja11j Signða 11 Þ ja21j Signða 21 Þ
xþ ðj ¼ 2Þ j 2 3 2 + 1.5 +
cþ 12 cþ 22 ja12j Signða 12 Þ ja22j Signða 22 Þ
3 1 1.5 + 2 +
Table 2 Values of symbols in upper bound model II. z k z 1 ðk ¼ 1Þ z 2 ðk ¼ 2Þ bi ði ¼ 1Þ
bi ði ¼ 2Þ
x j ðj ¼ 1Þ c kj c kj
+
jaijj Þ Signðaþ ij jaijj+ Þ Signðaþ ij
c 11 c 21
+
ja11j Signðaþ 11 Þ ja21j+ Signðaþ 21 Þ
x j ðj ¼ 2Þ 1.5 2 2.5 + 2 +
c 12 c 22 ja12j+ Signðaþ 12 Þ ja22j+ Signðaþ 22 Þ
2 0.6 2 + 2.5 +
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Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
3. Application to water resources planning For many decades, water resources planning problems, such as the efficient allocation of water and reasonable water treatment plant design, have challenged water resource managers in generating optimal or compromise decisions (Yin et al., 1999; Maqsood et al., 2005). It is essential for the decision makers to gain a better understanding of system components and their interrelationships within water resources system so as to meet their objectives. In many practical cases, the complexity of the system and the limited knowledge are the main obstacles to obtaining appropriate decision alternatives in a conflicting objective environment. Actually, most of the previous planning problems focused on optimal water allocation and did not consider the initial planning problems such as design of water resources systems including determining the maximum treatment ability of water plants, capability of reservoirs and etc. de Novo programming is an effective means of seeking a portfolio of capability levels and optimizing the objective functions by allocating a budget according to a unit capability price, where capacity levels (such as capability of reservoirs) are considered as decision variables. In urban or regional water resources planning problems, optimal supplies of high-quality water will be emphasized due to conflicting revenue target of municipal–industrial–agricultural balance under a given construction budget of water treatment facilities. It is desired that water treatment facilities for municipality, industry and agriculture be effectively designed for supporting sustainable management of available water resources. In order to balance the earnings of the competitive water users under the limited water budget, it is recognized that the different objectives of water users should be considered in relation to each other. Moreover, in various real world cases, uncertainties are associated with a number of impact factors such as cost and water treatment efficiency (Huang and Liu, 2008). As a result, how to optimally design water resources systems under uncertainty is a main concern facing the decision makers. Generally, they pay more attentions to the optimal system design at the original planning stage instead of just selecting the optimal allocation alternatives from predetermined data sets. All of these require an integrated approach for planning, provision, and management of urban water resources systems. Therefore, the work in this paper is an extension of the previous efforts in optimization for water resources systems planning. The developed inexact de Novo programming approach could effectively deal with the optimal system design problems, where design parameters could be optimally determined under uncertainty. Consider a case in which a water manager is responsible for allocating water from a water abundant reservoir for two co-developing cities to three users: a municipality, an industrial unit, and an agricultural sector (Fig. 2). The total water usage budget is [550, 700] 106 dollars. In order to consider the maximum earnings for three water users, the three criteria will be calculated. The unit benefit of each water user is listed in Table 3. In urban water system design, it is necessary to guarantee that the water allocation must meet the least demand of each city and each water user (Wang et al., 2003). The minimum total water needs and minimum proportions of total allocated water to city for three water user are deliberated (Table 3). To guarantee quality or water quality, the different water treatment facilities for different water users are also investigated. The limited water usage budget includes water supply cost and water treatment cost. The price for each of them is listed in Table 4. The water resource planning model is formulated as follows:
Reservoir
WP 1
City 1
Industry
City 2 WP 3 Industry
Agriculture
WP2
Municipal Agriculture
WP 4
Municipal
Fig. 2. Schematic of water allocation to multiple users. Table 3 Benefit and least water demand from different water-using facilities of each city. Benefit (dollars/m3)
City 1
City 2
Industry Municipal Agriculture
[80, 90] [76, 80] [45, 50]
[92, 100] [70, 75] [40, 43]
Least total water demand (106 m3)
[55, 60]
[65, 70]
For users 6
3
Least water demand (10 m ) Least allocation proportion
Industry
Municipal
Agriculture
Industry
Municipal
Agriculture
[13, 15] [0.30, 0.32]
[13, 15] [0.25, 0.27]
[13, 15] [0.35, 0.36]
[15, 17] [0.35, 0.37]
[15, 17] [0.20, 0.23]
[15, 17] [0.30, 0.32]
538
Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541
Table 4 Water treatment efficiency and least water demand coefficients. Water treatment efficiency
Industry
Municipal
Agriculture
City 1 City 2
[0.79, 0.81] [0.85, 0.88]
[0.65, 0.70] [0.7, 0.75]
[0.88, 0.90] [0.93, 0.95]
Total budget (106 dollars)
Water supply cost
Water treatment cost
[550, 700] Price/unit water
City 1 [2.5, 2.7]
City2 [2, 2.3]
Industry [1.5, 1.8]
Municipal [2, 2.1]
Agriculture [1,1.3]
Maximize benefits for each water user
Max zj ¼
m X
cij xij ;
8j
ð25aÞ
i¼1
Subject to: Treated water demand of city: n X
aij xij 6 bi ;
8i
ð25bÞ
j¼1
Total water needs of end user: m X
xij 6 bj ;
8j
ð25cÞ
i¼1
Budget limit for water system design: m X
pi bi þ
n X
i¼1
pj bj 6 B
ð25dÞ
j¼1
Minimum water allocation to city: m X
xij P li ;
8i
ð25eÞ
i¼1
Minimum proportions of total allocated water to end users in city i
xij P r ij
m X
xij ;
8i; j
ð25fÞ
i¼1
Minimum water allocation to end users in city i
xij P mij ;
8i; j
ð25gÞ
c ij
li
a ij
where is the unit benefit of water user j in city i, i = 1, 2, j = 1, 2, 3. is water treatment efficiency for water usage j in city i. is the least total water allocation to city i, m ij and r ij are the minimum water needs and minimum proportions of total allocated water to city for user i. x ij is decision variables of water allocation plan to user j in city i. bi is the design variables for total water allocation to city i which is cor responding to unit cost pi within the total water usage budget. bj are the design variables of water treatment capacity for water user i, which correspond to unit cost pj within the total water usage budget. The solution process is as follows. Firstly, the submodel (I) based on Eq. (15) is formulated and solved, for x and b, with respect to each þ þ + þ þ þ þ objective function zþ j separately. For the upper level of maximum objective, aij ; pi ; pj ; cij ; xij ; bi ; li ; mij ; r ij and B are considered. Let þ þ þ þ þ zj ¼ max zþ ; j ¼ 1; 2; 3, be the optimal value for jth objective of submodel (I) and Let z ¼ ð z ; z ; z Þ be the j-objective value for the 1 2 3 j ideal system with respect to B+. With *z+ = (9724.2, 5622.4, 4366.9), the metaoptimum submodel (I-META) can be constructed based on þ þ þ Eq. (16). The solutions of IM xþ j ; M B ; M b , and M r can be obtained by solving this model, as shown in Table 5. þ þ þ The minimum budget M B ¼ 886:06 under which the metaoptimum performance zþ i can be realized through IM xj and M b . The solu+ þ þ þ tions of (I-META), M B and IM xj can be designated as metaoptimum solutions. It is obvious that M B is greater than B . To enforce any particular budget level B, we use the optimum-path ratio M r þ of 0.79 and obtain the final solutions (shown in Table 6), which are þ þ þ þ þ þ þ þ þ xþ opt ¼ M r IM xj ; ðj ¼ 1; 2; 3Þ; bopt ¼ M r M b , and zopt ¼ M r zk . The maximum benefits zopt to each water user are 7682.1, 6 4441.7, and 3449.8 (10 dollars), respectively. Table 5 Solution of upper and lower levels of the first de Novo model. Water allocate
IM
þ þ xj M B
¼ 886:06
City 1 City 2 Water allocate City 1 City 2
IIM
xj M B
¼ 696:3
Industry
Municipal
Agriculture
18.0 81.1
15.0 58.9
27.0 70.1
Industry
Municipal
Agriculture
17.6 46.8
15.0 30.0
22.4 49.8
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Y.M. Zhang et al. / European Journal of Operational Research 199 (2009) 531–541 Table 6 Decision variables of water allocation and capacity design. Water allocate X*
Industry
Municipal
Agriculture
City 1 City 2
[13.90, 14.22] [36.99, 63.99]
[11.85, 11.85] [23.68, 45.82]
[17.70, 21.33] [39.20, 55.30]
Objectives Benefits
Industry [4121.4, 7682.1] Water supply design bi
Municipal [2558.2, 4441.7] Water treatment limit bj
Agriculture [1927.2, 3449.8]
City 1
City 2
Industry
Municipal
Agriculture
Design variable
[43.5, 47.4]
[99.87, 164.1]
[43.8, 65.7]
[26.1, 44.3]
[53.3, 70.3]
Similarly, the inexact de Novo submodel (II) corresponding to z, which provides the second step of the solution process based on the , from submodel (I) can be formulated based on Eq. (15). For the lower level of maximum objective, optimal variable solution, zþ k þ þ aþ ij ; pi ; pj ; cij ; xij ; bi ; li ; mij ; r ij and B are considered. Let zj ¼ max zj ; j ¼ 1; 2; 3, be the optimal value for jth objective of submodel (II) and* + ; z ; z Þ With z = (5216.9, 3238.2, 2439.5) be the j-objective value for the ideal system with respect to B. Then, a metaopLet z ¼ ð z 1 2 3 timum submodel (II-META) can be constructed based on Eq. (19). þ The value of M B is equal to 696.3, which is the minimum budget to achieve z j through IIM xj and M b . M r (Table 3) is the opti * mum-path ratio for achieving the ideal performance z for a given budget level B . The optimal system design (Table 4) can be established þ as ðx opt ; bopt ; zopt Þ, where xopt ¼ M r IM xj ðj ¼ 1; 2; 3Þ; bopt ¼ M r M b , and zopt ¼ M r zk . The maximum benefits zopt to each water user is (4121.4, 2558.2, 1927.2) 106 dollars. Table 4 indicates that when the budget reaches the upper bound, allocated water for city 1 is mainly used to guarantee water demand of agriculture, then industry and municipality in turn; allocated water for city 2 is mainly for industry, then agriculture and municipality in turn. This is because industry users in city 2 bring the highest benefit when water demand is satisfied; whereas municipality and agricultural users in city 1 correspond to lower benefits. Another important thing is that the total water allocation to city 2 (164.1 106m3) is much higher than that to city 1 (47.4 106m3). This is due to the higher earnings, higher water treatment efficiency and lower water supply cost to city 2. On the other hand, when the budget is reduced to the lower bound, allocated water for city 2 is mainly for agriculture, then industry and municipality in turn; while city 2 keeps the same order as that when the budget reaches the upper bound. Table 4 indicates that the total water supplies to cities 1 and 2 are [43.5, 47.4] and [99.87, 164.1], respectively. Total water allocation to city 2 (99.87) is less than that to city 1 (43.55) although the benefits from city 1 is higher that that of city 2. This is because the decision makers should focus on the basic demand of the water users in each city when the budget is low; whenever the budget is abundant, the decision makers can pay more attention to earnings of each water user. The total treating water target for industrial use is [43.8, 65.7]. This can help decision makers design and equip proper water treatment facilities in water purifying industry. In three objectives of three water users, the maximum earning belongs to industry ([4121.4, 7682.1]). It provides two extremes of the system benefit. As actual values of the variables and parameters vary within their bounds, the system benefit may correspondingly change between 4121.4 and 7682.1 with varied reliability levels. The developed IDNP approach is especially useful to design an optimal system rather than optimize a given system. It can effectively deal with the system design problems involving multiple objectives. In the conventional multiobjective programming problems, trade-offs exist among multiple objectives, especially conflicting ones. Scenarios are often analyzed and a set of decision results are generated for reflecting the trade-offs. Generally, it is difficult for the decision makers to select the final or ‘‘best” decision alternatives from a number of solution options, especially for the problems of designing a new optimal system where the budget is unknown. The IDNP can eliminate the trade-offs through designing a better and preferably optimal system. Consequently, there are no trade-offs in a new designed system using the IDNP. This can allow the decision makers to incrementally reduce the number of the objectives through transforming the objectives into the constraints, and thus greatly improve the computational efficiency. Moreover, the obtained results can provide effective decision support for the decision makers to design an optimal system. The conventional planning problems using fuzzy and/or stochastic program mostly focused on optimizing a given system subject to a series of constraint resources which were assumed to be fixed or known. However, in real world problems, the constraint resources have imprecise features, which are difficult to or can not be determined precisely. Compared with the above traditional methods, the de Novo programming approach can effectively handle the optimal system design problems within the available resources. These resources are considered as decision variables in de Novo program subject to the budgetary constraints, which can thus affect the values of the objective values. In stochastic programming, the quality of information on some uncertainties is often not good enough to be presented as probability density functions (PDFs). Even if these functions are available, it is extremely hard to solve a large-scale stochastic programming problem with all uncertain parameters being expressed as PDFs. In fuzzy programming, the decision makers need to determine the aspiration or preference levels and tolerances of the objective functions and/or the constraints. Those are often assumed based on the decision makers’ subjectivity or for the simplicity. In real world problems, uncertainties may exist as ambiguous intervals because planners and engineers typically find it more difficult to specify distributions than to define fluctuation ranges. For example, a water allocation target can hardly be expressed as a PDF; however, it can be easily defined as an interval. The conventional fuzzy programming approach is unable to directly communicate uncertain input information expressed as intervals into the optimization processes, thus leading to the reduction of the optimal solutions’ accuracy. Moreover, the fuzzy membership functions also involve the knowledge of interval analysis under a given alpha-cut. Instead, the developed method makes intuitive sense since total water allocation to the cities should depend on some budgetary constraints. Based on the proposed IDNP, the minimum water allocation to each city can be determined, which means a tight system in terms of resources identification; at the same time, a higher objective function value can be obtained. Thus, the proposed method covers the expression of system uncertainties by the conventional fuzzy and stochastic programming when the PDFs and fuzzy membership functions can not effectively reflected.
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Compared with interval mathematical programming (IMP), the proposed method can effectively deal with the real world water resources system design problems, where uncertainties of parameters and decision variables are expressed as intervals. The developed IDNP shifted the traditional IMP approach for optimizing a given water resources system to the design of an optimal system under a given budget level of resources, which can not be achieved through the conventional IMP methods. The conventional IMP is to find the optimal solutions subject to fixed or known resources, the right-hand side of the constraints. Based on the results from the conventional IMP, the planners can not produce the optimal resource levels for helping make the appropriate decisions at the initial system design stage. The proposed IDNP method can provide a higher objective function value and a tighter system, where the levels of resources are optimally produced rather than prespecified by the planners. The levels of resources are considered as decision variables and the uncertainties are effectively handled, which makes the model more practical in reflecting the real world systems. Compared with multiobjective programming, the IDNP method is able to obtain design decision variables and management variables. Most of multiobjective programming approaches have limitations in dealing with true issue of designing optimal systems. The resource level (right-hand side of the model) is determined as a known number. Although 0–1 programming was already used for dealing with expansion problem considering right-hand side flexibility, it can only choose to expand or not a given option but not how much to expand. For example, in a three-period landfill expansion problem, the obtained results indicate that the landfill shall be expanded in the second and third periods. Actually, the decision makers would prefer to expand the landfill once instead of twice. Therefore, the developed IDNP approach has the advantages in constructing optimal system design via an ideal system by introducing the flexibility toward the available resources in the system constraints. 4. Conclusions An interval de Novo programming (IDNP) approach was developed for designing an optimal system under uncertainty. The model integrated interval programming and de Novo programming within an optimization framework. In its solution process, the IDNP model was transformed into two deterministic submodels, which corresponded to the lower and upper bounds, respectively. For each submodel, the design variables in the IDNP model were constrained by the total budget. By function transformation, the maximum value of each objective function could be obtained, corresponding to the performance of the ideal design under a given B. Lastly, a metaoptimum model was constructed and solved to get optimal design solutions. In water resources systems planning, the IDNP could be an effective tool for achieving sustainable development. The developed IDNP approach was applied to a hypothetical case of water resources systems planning, where the parameters were represented as intervals. Uncertainties of parameters and decision variables are effectively handled, which makes the model more real. In this water resources allocation problem, decision makers are confronted with control and adjustment of water disposal facility design and operation characteristics. IDNP is such an integrated approach for planning, provision, and management of urban water systems. It can shift the traditional interval programming approach for optimization of a given water resources system to the design of an optimal system, which can not be achieved through the conventional interval or fuzzy programming methods. Moreover, the IDNP method could generate an exclusive result preferred by the decision makers, including design decision variables and management decision variables. Within multiobjective decision making framework, IDNP may allow the decision makers to achieve a metaoptimal system performance and improve the performance of compromise solutions. The obtained results indicated that the IDNP method was also applicable to other practical problems involving optimal environmental systems design. Acknowledgement This research was supported by the Major State Basic Research Development Program of MOST (2005CB724200 and 2006CB403307), the Canadian Water Network under the Networks of Centers of Excellence (NCE), and the Natural Science and Engineering Research Council of Canada. We would like to thank Prof. Nurullah Umarusman for pointing out some errors which occurred in an earlier version of this paper. Appendix A x j is vector of management decision variables. bi is vector of design decision variable. c kj is parameter of decision variables in the objective function. a ij is parameter of decision variables in the constraints. p i is the unit prices parameter of design decision variable. B± is the given total available budget. z k is the objective function. Sign(*) is the sign of parameter *; ¼ a ij ; pi . j*j is the absolute value of parameter ; ¼ a ij ; pi . þ zk is the optimal value for kth objective of submodel ðmax zþ k Þ. xj is the value of management decision variables solved by metaoptimum submodel. * ± b is the value design decision variable solved by metaoptimum submodel. * ± * + B is the minimum budget to achieve zþ k through xj and b . * ± r is optimum-path ratio for achieving the ideal performance.*r± = B±/*B±. x opt is the optimal value management decision variables, xopt ¼ r xj . þ þ þ bopt is the optimal value design decision variables, bopt ¼ r þ b . þ þ þ zþ opt is the optimal value of the objectives, zopt ¼ r zk .
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