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Incorporating additional meta-objectives into the extended lexicographic goal programming framework
European Journal of Operational Research xxx (2013) xxx–xxx
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Incorporating additional meta-objectives into the extended lexicographic goal programming framework Dylan Jones a,⇑, Mariano Jimenez b a b
Logistics and Management Mathematics Group, Department of Mathematics, University of Portsmouth, United Kingdom Department of Applied Economics I, University of The Basque Country (UPV-EHU), Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 24 April 2012 Accepted 27 December 2012 Available online xxxx
This paper introduces two new meta-objectives into the extended goal programming framework. The first meta-objective is the number of unmet goals and the second is a measure of closeness to the pairwise comparisons given by the decision maker. These complement the original two meta-objectives of the weighted sum of deviations and the maximal weighted deviation to provide a flexible four metaobjective framework. Lexicographic and non-lexicographic representations of the framework are developed. An example relating to transportation is solved in both the lexicographic and non-lexicographic forms. Weight sensitivity analysis is applied to the meta-weight parameters for the non-lexicographic case in order to find a range of available distinct solutions. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Extended goal programming Pairwise comparisons Multiple objective programming Weight sensitivity analysis
Subject to : uli nli þ v li pli 6 kl
1. Introduction The extended lexicographic goal programming (ELGP) is introduced by Romero (2001) with the purpose of providing a general framework which covers and allows the combination of the most common goal programming variants. It is also encompasses several other distance-based MCDM techniques. This work is further extended by Romero (2004) who provides a more generalised form of the achievement function and by Arenas et al. (2004) who extend the framework to include fuzzy models. The ELGP framework has been recently applied by De Andres et al. (2010) to a performance appraisal model. The general ELGP model is given with the linear form of the achievement function as proposed in (Romero, 2004) is given as:
2
(
X
)!
3
;...;7 þv 6 a1 k1 þ ð1 a1 Þ 7 6 i2H1 7 6 ( ) ! 7 6 7 6 X 6 ;...; 7 al kl þ ð1 al Þ uli nli þ v li pli Min a ¼ 6 7 7 6 i2Hl 6 ( )! 7 7 6 7 6 X 5 4 aL kL þ ð1 aL Þ uLi nLi þ v Li pLi u1i n1i
1 1 i pi
ð0:0Þ
i2HL
⇑ Corresponding author. Address: Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace, Portsmouth PO1 3HF, United Kingdom. Tel.: +44 2392846362; fax: +44 2392846353. E-mail addresses: [email protected] (D. Jones), [email protected] (M. Jimenez).
i 2 Hl ; l ¼ 1; . . . ; L;
f i ðxÞ þ ni pi ¼ bi
i ¼ 1; . . . ; q
ni ; pi P 0 i ¼ 1; . . . ; q
ð0:1Þ ð0:2Þ ð0:3Þ
where the model is defined as having L priority levels and q objectives. fi(x) is the achieved value of the ith objective which has an associated target value of bi. Deviational variables ni and pi denote the negative and positive deviations from the ith target value respectively. The maximal weighted deviation from amongst the set of unwanted deviations is denoted by kl. The sets Hl, l = 1, . . . , L include the objectives with an least one of their deviational variables to be penalised in priority level l. The weights uli and v li are associated with the relative level of importance associated with the minimisation of the negative and positive deviational variables from the ith target value in the lth priority level respectively. Unwanted deviations are given a positive weight and deviations which are not desired to be minimised in that priority level are given a zero weight. al is a parameter which controls the relative importance of efficiency and equity in the lth priority level. The ELGP formulation allows for the inclusion and combination of the optimisation, balancing, ordering, and satisficing underlying philosophies (Jones and Tamiz, 2010, Chapter 1). The lexicographic ordering philosophy is available via the priority structure of the achievement function. The satisfying philosophy is evident in the set of goals. The optimising philosophy can be achieved through the use of Pareto efficiency detection and restoration techniques (Jones and Tamiz, 2010, Chapter 6), or in the case of a single priority level (L = 1) through the setting of sufficiently high target goal values. The balancing philosophy is achieved through the inclusion
0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.12.025
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
of the maximal deviation terms (kl) in each priority level. Furthermore, the balance between optimisation (efficiency) and balance (equity) can be controlled at each priority level through the parameter al which can be varied between complete emphasis on optimisation (a = 0) and complete emphasis on balance (a = 1). The actual value of a chosen depends on the decision maker’s attitude towards balance and optimisation. It is recommended that some form of sensitivity or parametric analysis is carried undertaken on a, such as the algorithm presented in Section 4 of this paper. The ELGP framework is therefore a comprehensive tool for the inclusion of many types of underlying philosophies into the goal programming framework. 2. Meta-objectives This section details each of the proposed meta-objectives and makes linkages to previous work within the field of goal programming. 2.1. Meta objective 1 – Minimisation of the maximal unwanted deviation This is the first of the two meta-objectives taken from the Romero (2001, 2004) extended goal programming model. It is based around the L1 distance metric which links to the Rawlsian theory of social justice (Rawls, 1973). The minimisation of the maximal deviation is first used in a goal programming context by Flavell (1976). It underlies the variant of Chebyshev (or Minmax) goal programming. It ensures a balance between the objectives is maintained. In the context of a multi-stakeholder situation in can be used to ensure that a balance between stakeholders is achieved, thus facilitating the achievement of a consensus. It is hence associated with the concepts of fairness, equality, and social justice. 2.2. Meta objective 2 – Minimisation of average unwanted deviation This is the second of the two meta-objectives taken from the Romero (2001, 2004) extended goal programming model. It is based around the L1 distance metric which underlies the original weighted and lexicographic goal programming models as proposed by Charnes and Cooper (1961). The vast majority of the literature to date has utilised the L1 distance metric. As it is solely concerned with the minimisation of the average without regard for the distribution between individual objectives it is associated with the principles of optimisation (sometimes termed ruthless optimisation due to the above property) and efficiency (Jones and Tamiz, 2010, Chapter 1).
programming in the meta-goal programming formulation of Rodriguez et al. (2002). It is also suggested for pattern recognition using goal programming by Jones et al. (2007) The proposed formulation in Section 3 allows the incorporation of this metric into a parametric goal programming framework where the trade-offs with other meta-objectives can be effectively investigated. 2.4. Meta objective 4 – Minimisation of distance from given pairwise comparisons This paper proposes this novel meta-objective, which can be used for goal programmes where the decision maker preferences are expressed via a pairwise comparison matrix. It minimises the distance between the preferences expressed by the decision maker and those achieved by the goal programming solution. It is thus a measure of consistency with the preferences (this is not the same as consistency of the preferences which is a separate measure). The synergy between goal programming and pairwise comparisons is first noted by Gass (1986) who used the Analytical Hierarchy Process (AHP) to produce a weight set for a weighted goal programming model. Since then the linkage between goal programming and pairwise comparisons has been exploited in two ways. Firstly, goal programming has been used as a substitute to the Eigenvalue method (Saaty, 1980) as a means of generating the weight vectors from the pairwise comparison matrix. Secondly goal programming has been used in the manner of Gass (1986) above, that is the use of goal programming to generate the weights of a goal programme. This is noted by Jones (2011) to produce a valid starting point for a goal programming sensitivity analysis. However, due to the nature of the goal programming optimisation process (especially in the linear weighted case where the solution algorithm is often simplex-based) under current methodologies there is no explicit link between the final solution and the initial preferences. The proposed meta-objective rectifies this situation by allowing the connection to be explicitly measured and optimised. The consistency of the preferences can be measured in the standard fashion and the pairwise comparison matrix returned to the decision maker for re-consideration if the inconsistency level is too high. Thus the four meta-objectives outlined above allow for a comprehensive range of considerations including optimisation, efficiency, equality, fairness, social justice, target achieving, and consistency with preferences to be measured. This paper will use the extended goal programming framework to build a formulation which allows these considerations to be optimised and traded-off in a parametric manner.
3. Formulation of four meta-objective extended goal programming framework
2.3. Meta objective 3 – Minimisation of the number of unmet goals 3.1. Lexicographic version This is the first of the two meta-objectives introduced in this paper. It is based around the distance metric which has the following properties in the context of goal programming:
8 > < 0 if the ith unwanted dev iation v ariable has v alue zero si ¼ 1 if the ith unwanted dev iation v ariable has a positiv e > : v alue The weighted sum of these binary measures of whether the goals have been met or not will give a measure of the level of achievement of goals in the model. It is useful in situations where rewards are given if targets are met or penalties applied for unmet targets. This meta-objective is thus associated with target setting and achieving behaviour. This L0 metric is first seen in the context of goal
This model extends the number of meta-objectives in the extended goal programming model from two to four. The following meta-objectives are used at each priority level: (1) Minimisation of the normalised (L1) (maximum unwanted deviations from the set of goals in that priority level-given a weight al in the lth priority level. (2) Minimisation of the normalised (L1) (weighted sum of unwanted deviations from the set of goals in that priority level-given a weight bl in the lth priority level. (3) Minimisation of the number of unmet goals (L0) (from the set of goals in that priority level-given a weight vl in the lth priority level.
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
expressed level of achievement of the ith and jth unwanted deviational variables respectively.
(4) Minimisation of the discrepancy between the expressed pairwise preferences of the decision maker and the actual preferences indicated by the solution-given a weight d1 in the lth priority level.
3.2. Non-lexicographic version
It is also suggested that for convenience the meta-objective parameter weights sum to one at each priority level (al + bl + cl + dl = 1 l = 1, . . . , L).
8
9 X v 1 pi =
8
The non-lexicographic version is relevant when the decision maker does not have a clear ordering of goals in their preferences but rather wishes to investigate the trade-offs between the
9 =
91 3 = i 6 @a1 k1 þ b þ l1 s þ t1i ti þ d1 ðNij þ Pij Þ A; . . . ; 7 þ v1 1 7 6 ki ; : 1 : 1 i i ; : ; 7 6 1 1 1 i2Q i2Q i2Q i2Q ði;jÞ2Q 7 6 1 2 1 2 3 8 9 8 9 8 91 7 60 7 6 < = < = < = X X X X X l l 7 6 ui ni v p i l l i @al kl þ bl A 7 þ v l s þ t t ðN þ P Þ ; . . . ; þ þ d Min a ¼ 6 ij ij l l i i ki ; 7 6 : l ki : l i i ; : ; 7 6 l l l i2Q i2Q i2Q i2Q ði;jÞ2Q 1 2 1 2 3 7 60 8 9 8 9 8 91 7 6 7 6
u1i ni ki
i2Q 1
Subject to :
i2Q 2
i2Q 1
uli ni v li pi þ 6 kl i ¼ 1; . . . ; q l ¼ 1; . . . ; L ki ki f i ðxÞ þ ni pi ¼ bi i ¼ 1; . . . ; q f i ðxÞ þ Msi P bi
i 2 Q l1 l ¼ 1; . . . ; L Q l2
X
i2Q 2
ð1:1Þ ð1:2Þ ð1:3Þ
l ¼ 1; . . . ; L f i ðxÞ Mt i 6 bi i 2 ui ni nj þ Nij Pij ¼ 0 i; j 2 Q L1 i < j uj ki kj
ð1:4Þ
l ¼ 1; . . . ; L ui ni pj þ Nij Pij ¼ 0 i 2 Q l1 ; j 2 Q l2 ; v j ki kj
ð1:5Þ
l ¼ 1; . . . ; L
v i pi pj þ Nij Pij ¼ 0 v j ki kj
8 < X
ði;jÞ2Q 3
objectives and between the meta-objectives in the model. The algebraic version of the non-lexicographic version is given as:
(
Min a ¼ ak þ b ( þv
i; j 2
Subject to :
i
l ¼ 1; . . . ; L x2F
ð1:7Þ ð1:8Þ
ni ; pi P 0 i ¼ 1; . . . ; q; Nij ; Pij P 0 i; j ¼ 1; . . . qji < j si ; t i binary i ¼ 1; . . . ; q
X ui ni
i2Q 1
X
i2Q 1
ð1:6Þ Q L2
ð1:9Þ
where all existing notation is in common with the ELGP model introduced in Section 1 and the following additional notation is defined: Q l1 is the ordered set of the indices of the unwanted negative deviational variables in priority level l. Q l2 is the ordered set of the indices of the unwanted positive deviational variables in priority level l and Q l3 is the set of pairs of unwanted deviational variables indices in priority level l defined by:
Q l3 ¼ i; j 2 Q l1 ji < j [ i 2 Q l1 ; j 2 Q l2 [ i; j 2 Q l2 ji < j si is a binary variable that takes the value 1 if the achieved value of ith goal is less than the target value and value 0 otherwise. ti is a binary variable that takes the value 1 if the achieved value of ith goal is greater than the target value and value 0 otherwise. The si and ti variables represent whether the goals have been met for the cases of unwanted negative and positive deviations respectively. lli and tli are relevant the relative weights representing the penalty applied in the lth priority level for not meeting the ith goal on the negative and positive direction respectively. M is a large positive constant. The normalisation constant of the ith objective is given by ki. Nij and Pij are the deviations from the decision maker
ð1:0Þ
ki
li si þ
þ
X v 1p i
ki )
i2Q 2
X i2Q 2
mi ti
)
i
(
q 1 X ðN ij þ Pij Þ þd jQ 3 j ði;jÞ2Q
) ð2:0Þ
3
ui ni v i pi þ 6 k i ¼ 1; . . . ; q ki ki f i ðxÞ þ ni pi ¼ bi i ¼ 1; . . . ; q f i ðxÞ þ Msi P bi i 2 Q 1 f i ðxÞ Mt i 6 bi i 2 Q 2 ui ni nj þ Nij Pij ¼ 0 i; j 2 Q 1 i < j uj ki kj ui ni pj þ N P ¼ 0 i 2 Q 1; j 2 Q 2 v j ki kj ij ij v i pi pj þ N P ¼ 0 i; j 2 Q 2 i < j v j ki kj ij ij x 2 Fj ni ; pi P 0 i ¼ 1; . . . ; q; N ij ; P ij P 0 i; j ¼ 1; . . . :qji < j si ; t i binary i ¼ 1; . . . ; q
ð2:1Þ ð2:2Þ ð2:3Þ ð2:4Þ ð2:5Þ ð2:6Þ ð2:7Þ ð2:8Þ ð2:9Þ
where the significance of all variables remains the same as that defined in model (1.0)–(1.9), expect the parameters a, b, c, d and the maximal deviation variable k are scalar due to the fact there are no priority levels in the model. The sets Q1, Q2, Q3 also do not need a priority level index. 4. Example A hypothetical multi-objective transportation model is used to demonstrate the methodology developed in Section 3. The model has two supply points and three demand points where all potential routes between supply points and demand points are feasible. Each route has the following three criteria associated with it: Reliability: Measuring on a normalised range of 0 (worst) to 1 (best). Cost: Measured on a normalised range of 0 (best) to 1 (worst). Transit time: Measured on a normalised range of 0 (best) to 1 (worst).
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
The supply levels are fixed at levels SP1 = 150, SP2 = 200, and SP3 = 300 units respectively. The minimum demand levels for the first two demand points are fixed at DM1 = 100 and DM2 = 300 units respectively. However the decision maker has goals of supplying exactly 300 and 500 units to demand points 1 and 2 respectively. Demand point 3 has a fixed demand of DM3 = 150 units that must be met exactly. This leads to the following five decision maker set goals: Goal 1: Achieve at least 60% of the theoretical maximum reliability. Goal 2: Do not exceed 40% of the theoretical maximum cost. Goal 3: Do not exceed 40% of the theoretical minimum cost. Goal 4: Supply demand point 1 with exactly 300 units. Goal 5: Supply Demand Point 2 with exactly 500 units. Goals 1–3 are one-sided goals (only one deviation to be penalised) whereas goals 4 and 5 are two-sided goals (both deviations to be penalised). Hence there are seven unwanted deviations in total.
The Eigenvalue method (Saaty, 1980) is then used to produce the following set of weights:
w1 ¼ u1 ¼ 0:360;
w2 ¼ v 2 ¼ 0:181;
w3 ¼ v 3 ¼ 0:181;
w4 ¼ u4 ¼ 0:097;
w5 ¼ v 4 ¼ 0:032;
w6 ¼ u5 ¼ 0:104;
w7 ¼ v 5 ¼ 0:045 This extended goal programming model therefore has the following algebraic representation: 3 X Min a ¼ ak þ b Di þ cðu1 s1 þ v 2 t2 þ v 3 t þ u4 s4 þ v 4 t4 i¼1
þ u5 s5 þ v 5 t 5 Þ þ
Subject to :
3 X 3 X
@ X ðNij þ Pij Þ 21 ði;jÞ2Q
ð3:0Þ
Rij xij þ n1 p1 ¼ 420
ð3:1Þ
C ij xij þ n2 p2 ¼ 280
ð3:2Þ
T ij xij þ n3 p3 ¼ 280
ð3:3Þ
i¼1 j¼1 3 X 3 X i¼1 j¼1 3 X 3 X
4.1. Extended non-lexicographic goal programming model
i¼1 j¼1
This section presents the non-lexicographic version of the extended goal programming model with 4 meta-objectives for the example. Introducing the following notation. Let i = 1, . . . , 3 be the index of the supply nodes and j = 1, . . . , 3 be the index of the demand nodes. Xij is the number of items sent between supply node i and demand node j. Rij is the reliability score for the route between supply node i and demand node j. Cij is the cost score for the route between supply node i and demand node j. Tij is the time score for the route between supply node i and demand node j.
3 X xi1 þ n4 p4 ¼ 300
ð3:4Þ
i¼1 3 X xi2 þ n5 p5 ¼ 500
ð3:5Þ
i¼1
Di 6 k i ¼ 1; . . . ; D 3 X 3 X
ð3:6Þ
Rij xij þ s1 P 420
ð3:7Þ
C ij xij t 2 6 280
ð3:8Þ
T ij xij t3 6 280
ð3:9Þ
i¼1 j¼1 3 X 3 X i¼1 j¼1 3 X 3 X i¼1 j¼1
With all other notation corresponding to that introduced in model 2 in Section 3. In particular the following variables are defined in order to simplify the algebraic representation:
u1 n1 v 2 p2 v 3 p3 u4 n4 v 4 p4 D1 ¼ ; D2 ¼ ; D3 ¼ ; D4 ¼ ; D5 ¼ ; 420 280 280 300 300 u5 n5 v 5 p5 ; D7 ¼ D6 ¼ 500 500 w1 ¼ u1 ; w2 ¼ v 2 ; w3 ¼ v 3 ; w4 ¼ u4 ; w5 ¼ v 4 ; w6 ¼ u5 ; w7 ¼ v 5
3 X xi1 þ Ms4 P 300
ð3:10Þ
i¼1 3 X xi1 Mt4 6 300
ð3:11Þ
i¼1 3 X xi2 þ Ms5 P 500
ð3:12Þ
i¼1 3 X xi2 Mt5 6 500
ð3:13Þ
i¼1
A pairwise comparison method (Saaty, 1980) is used to determine the decision maker weights with respect to the unwanted deviations. The pairwise comparison matrix given by Table 1 represents the preferences of the hypothetical decision maker:
Di Dj þ Nij Pij ¼ 0 i; j ¼ 1; . . . ; 7 i < j wj wj 3 X xij 6 SPi i ¼ 1; . . . ; 3
ð3:15Þ
i¼1 3 X xij P DMj
Table 1 Pairwise comparison matrix for the non-lexicographic model.
ð3:14Þ
j ¼ 1; . . . ; 3
ð3:16Þ
i¼1
Deviation
w1 = u1
w2 = v2
w3 = v3
w4 = u4
w5 = v4
w6 = u5
w 7 = v5
w1 = u1 w 2 = v2 w 3 = v3 w4 = u4 w 5 = v4 w6 = u5 w 7 = v5
–
2 –
2 1 –
5 2 2 –
8 5 5 3 –
4 2 2 1 0.25 –
7 4 4 3 0.5 3 –
xij P 0 i ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3; ni ; pi P 0 i ¼ 1; . . . ; 3; N ij ; P ij P 0 i; j ¼ 1 . . . ; 7 j > isi ; t i binary i ¼ 1; . . . ; 5; k P 0; ni pi ¼ 0 i ¼ 1; . . . ; 5 ð3:17Þ where goals (3.1)–(3.5) represent the reliability, cost, transit time, demand point 1 level, and demand point 2 goals respectively. Constraint set (3.6) relates to the L1 meta-objective of minimising the
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
5
D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx Table 2 Distinct solutions produced by the weight sensitivity analysis. Sol
A
b
c
d
N1
P2
P3
N4
P4
N5
P5
L1
L1
L0
PW
A B C D E F G H
0.25 0.001 0.125 0.063 0.997 0.001 0.063 0.495
0.25 0.997 0.625 0.813 0.001 0.001 0.438 0.495
0.25 0.001 0.125 0.063 0.001 0.001 0.063 0.005
0.25 0.001 0.125 0.063 0.001 0.997 0.438 0.005
0 0 0 0 38 0 0 0
67 7.5 64 20 50 47 50 36
0 11 0 0 50 47 38 31
134 200 137 200 100 94 99 129
0 0 0 0 0 0 0 0
200 200 200 200 155 200 200 200
0 0 0 0 0 0 0 0
0.1281 0.1182 0.1276 0.1190 0.1608 0.1327 0.1301 0.1265
0.043 0.065 0.044 0.065 0.032 0.042 0.042 0.042
0.38 0.56 0.38 0.38 0.92 0.56 0.56 0.56
0.3880 0.4135 0.3882 0.4177 0.3764 0.3603 0.3605 0.3689
maximal deviation. Constraints (3.7)–(3.13) relate to the L0 metaobjective of minimising the number of unmet goals. M is an arbitrarily large positive constant. Constraint set (3.14) relates to the pairwise meta-objective of achieving as close as match to the pairwise comparisons given by the decision maker as possible. Constraint sets (3.15) and (3.16) impose hard constraints to ensure supply levels are not exceeded and minimum demand levels are met at each point. The extended goal programming model is solved used LINGO 13.0 (LINDO, 2012) and an initial equal meta-weight solution of a = b = c = d = 0.25 resulting in solution point A in Table 2. In order to investigate the range of solutions available by varying the meta-weights the sensitivity analysis algorithm of Jones (2011) is applied to the meta-weights. This algorithm is designed to give a selection of sufficiently diverse solutions for consideration by the decision maker. The parameter inputs to the algorithm are set at TMax = 2 (indicating that at most two weights are simultaneously raised to give search directions from the initial solution) and Maxlevel = 2 (indicating that most two bisections are allowed in each search direction). No additional preference information is included as the whole meta-weight space is assumed to be of potential interest to the decision maker. A small weight rather than a zero weight has been given to each meta-objective where necessary to avoid potential Pareto inefficiency occurring due to the existence of alternative optimal solutions. The (Jones, 2011) algorithm produces eight distinct solutions which are detailed in Table 2, Columns 2–5 in Table 2 detail the first set of meta-weights for which the solution is found. Additional weight sets that generate the same solution are given beneath the table. Columns 6–12 give the solution in unwanted deviational space. Columns 13–16 give the solution in meta-objective space, where L1, L1, L0, and PW denote the first, second, third, and fourth components (excluding the meta-weight) of the achievement function (3.0) respectively. The number of decimal places shown in Columns 13–16 is sufficient to demonstrate differences in values where they exist. It is noted that all eight solutions are, as expected, Pareto Efficient in both objective and meta-objective spaces.
Table 4 Pairwise comparison matrix for priority level 2 of the lexicographic model. Deviation
w4 = u4
w5 = v4
w6 = u5
w 7 = v5
w4 = u4 w5 = v4 w6 = u5 w7 = v5
–
3 –
1 0.25 –
3 0.5 3 –
Priority Level 2: Minimize unwanted deviations from the supply level goals (goals 4 and 5). The decision maker still wishes to give preferences in a pairwise fashion. Thus, pairwise comparison matrices are elicited for each priority level and the relevant intra priority level weights are found using the Eigenvalue method. The resulting pairwise comparison matrices for the first and second priority levels are given by Tables 3 and 4 respectively. Priority level 1: The Eigenvalue method is then used to produce the following set of weights:
w1 ¼ u1 ¼ 0:5; w2 ¼ v 2 ¼ 0:25; w3 ¼ v 3 ¼ 0:25; Priority Level 2: The Eigenvalue method is then used to produce the following set of weights:
w4 ¼ u4 ¼ 0:367; w5 ¼ v 4 ¼ 0:096; w6 ¼ u5 ¼ 0:391; w7 ¼ v 5 ¼ 0:146 The following pre-emptive extended goal programme is then formulated in accordance with the methodology detailed in Section 3:
" Lex Min a ¼ a1 k1 þ b1
3 X Di þ c1 ðu1 s1 þ v 2 t2 þ v 3 t3 Þ i¼1
7 X d1 X ðN ij þ Pij Þ; a2 k2 þ b2 Di þ 3 ði;jÞ2Q i¼4
4.2. Extended lexicographic goal programming model
1
The unwanted deviations are now assumed to be divided into two priority levels by the decision maker in order to reflect the fact that they regard meeting the three performance related goals as more important than meeting the supply level related goals: Priority Level 1: Minimize unwanted deviations from the reliability, cost, and transit time goals (goals 1–3). Table 3 Pairwise comparison matrix for priority level 1 of the lexicographic model. Deviation
w1 = u1
w2 = v2
w 3 = v3
w1 = u1 w 2 = v2 w 3 = v3
–
2 –
2 1 –
d2 X þ c2 ðu3 s4 þ u5 t4 þ u6 s5 þ u7 t 5 Þ þ ðNij þ Pij Þ 18 ði;jÞ2Q
#
2
ð4:0Þ Subject to: Eqs. (3.1)–(3.5), (3.8)–(3.13), (3.15) and (3.16)
Di 6 k1
i ¼ 1; . . . ; 3
Di 6 k2 i ¼ 4; . . . ; 7 Di Dj þ Nij Pij ¼ 0 ði; jÞ 2 Q 1 [ Q 2 wj wj
ð4:1Þ ð4:2Þ ð4:3Þ
xij P 0 i ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3; ni ; pi P 0 i ¼ 1; . . . ; 3; Nij ; Pij P 0 i; j ¼ 1 . . . ; 7 j > i; si ; ti binary i ¼ 1; . . . ; 5; k1 ; k2 P 0; ni pi ¼ 0 i ¼ 1; . . . ; 5
ð4:4Þ
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
where Q 1 ¼ ði; jÞji ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3; i < j; Q 2 ¼ ði; jÞji ¼ 4; . . . ; 7; j ¼ 4; . . . ; 7; i < j: All other notation remains consistent with that introduced in models 1 and 3. Assuming a parameter setting that gives equal weight to all four meta-objectives in both priority levels (i.e. a1 = a2 = b1 = b2 = c1 = c2 = d1 = d2 = 0.25) and solving a lexicographic goal programming using Lingo 13.0 in accordance with the methodology given in (Jones and Tamiz, 2010, Chapter 5) yields the following solution expressed in terms of non-negative decision and deviational variables rounded to the nearest integer:
x11 ¼ 92; x33 ¼ 111;
x12 ¼ 58; n4 ¼ 200;
x22 ¼ 211; n5 ¼ 200;
x23 ¼ 39;
x31 ¼ 8;
p5 ¼ 20
Which is equivalent to solution point D in Table 2. It is clear that the lexicographic ordering has successfully prioritised the first three goals as required. It is also noted the second priority level is redundant due to a combination of relatively high target levels and the complexity of the multiple distance metric targets being required in the first priority level. 5. Discussion Solutions have been generated in Section 4 for both the nonlexicographic and lexicographic versions proposed in Section 3. It is noted that the larger number of meta-objectives makes finding alternative optimal solutions at each priority level a more difficult task. Therefore if the lexicographic version is going to be used and redundancy is to be avoided then the target values at higher priority levels should be set at fairly relaxed levels. On the other hand, this finding indicates that the non-lexicographic should be the preferred choice when dealing with models with a number of metaobjectives greater than two. The non-lexicographic version has been successfully used alongside the (Jones, 2011) weight sensitivity algorithm to produce a number of distinct solutions relating to different meta-weight combinations. 5.1. Reduction to other goal programming models This paper has worked from a base model of the linear version of the ELGP model of Romero (2001, 2004). Therefore by setting the parameters c and d to zero the linear version of the ELGP is reproduced. By implication, the linear models listed by Romero (2001): conventional mathematical programming, weighted goal programming, lexicographic goal programming, Chebyshev goal programming, the reference point method (although this requires a negative value of b), and the L1 and L1 solutions in compromise programming. Setting d to zero results in a model that has some similarities to the meta goal programming formulation of Rodriguez et al. (2002) with a, b and c relating to the weights of the three meta-goals. Unlike meta goal programming, the reduced model does not require the decision maker to set target values for the meta-objectives but rather allows parametric trade-offs between them. 5.2. Normalisation considerations This paper considers normalisation on two levels: the objective level and the meta-objective level. The need for normalisation at the objective level in goal programming to ensure that deviations are comparable is well known. Not using a (formal or informal) normalisation technique will produce results that are biased toward objectives measured in lower denomination units (Romero, 1991). In the examples in Section 4, the technique of percentage normalisation is used at the objective level.
When using a method with many meta-objectives it is also important to consider whether a formal or informal technique to normalise the meta-objectives is necessary or not. The first observation to make is that the meta-objectives are combined using parameters (a, b, c, d) rather than preferential weights. The idea is generally to vary the parameters to produce differing solutions rather than a single point solution (as demonstrated by the nonlexicographic case in Section 4). Thus any mismatch in units of measurement will be less critical than in the standard objective case. It may, however, make it more difficult to effectively search the parameter search to elicit a range of solutions. Consider the four meta-objectives under consideration in this paper. The first meta-objective is a sum of (normalised) weighted deviations from goals. The second meta-objective is the single largest weighted deviation from a goal. The third objective is the sum of a number weighted (normalised) binary variables equal to the number of unwanted deviations. The fourth objective is equal to the sum of unwanted deviations from the pairs of unwanted deviations. There are kðk1Þ of these, where k is the number of unwanted 2 deviational variables. The informal normalisation applied at the meta-objective level is to divide the fourth meta-objective by kðk1Þ in order to reduce the units of this meta-objectives to a level 2 closer to that of the other three meta-objectives. The results can be seen in the range of meta-objective solutions given in the last four columns of Table 2. In the lexicographic case the fourth metaobjective terms in two priority levels are divided by the number of pairs of unwanted deviational variables associated with that priority level. 5.3. Meta-objective space considerations (Jones, 2011) considers a goal programming as existing in three spaces: decision space, objective space, and weight space. In order to encompass extended goal programming a fourth space: metaweight space has been included. Meta-weight space has dimensions Rm , where is the number of meta-objectives used in the extended goal programming model. Like weight space, all dimensions can by convention be constrained either to the range [0, 1] or to the range [ 1, 1] if Pareto Efficiency restoration techniques or reference point method techniques are required. As the number of potential meta-objectives grows there is a need to provide some formal mechanism for searching meta-weight space in order to ensure the decision maker has access to the full range of possible solutions. Seen in this context, the Jones (2011) algorithm used in Section 4.1 is one means of searching meta-weight space. Other search based methods, heuristics, and meta-heuristics could potentially also be used for this purpose. 6. Conclusions The extended lexicographic goal programming has been shown to provide an effective framework for the inclusion and combination of different underlying philosophies. This is achieved by formulating meta-objectives to represent each of the philosophies and performing a parametric analysis of the resulting meta-objective space. This paper has extended the number of meta-objectives in the framework from two to four. It is likely that further works will be able to incorporate further philosophies by augmenting the number of meta-objectives. Possibilities for augmentation include meta-objectives to reflect the fact that pairwise comparisons are sometimes uncertain or difficult to specify (Deparis et al., 2012) or specialised meta-objectives arising from the field of multi-response optimisation (Kazemzadeh et al., 2008). Of course, not every real-life decision situation necessarily contains all the underlying philosophies and hence the relevant subset
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of meta-objectives should be chosen. For instance, a decision maker who wishes to specify their preferences in a direct rather than a pairwise manner but wishes to consider the efficiency, equality, and number of targets reach should omit the fourth meta-objective and use the first three meta-objectives in this paper. The second meta-objective has been prevalent in most goal programming applications to date (Jones and Tamiz, 2010), although this could be due to historical and computational reasons, as it underlies both the weighted and lexicographic variants. This paper proposes increased usage of the other three meta-objectives described in order to enhance the flexibility and modelling power of goal programming. If any of the meta-objectives is not required for optimisation purposes but is required solely as an information source, then the meta-weight of that meta-objective can be set to zero but the relevant equations left in the formulation. This will allow for calculation of, but not optimisation of, the relevant meta-objective’s value. This paper has focussed on linear goal programming and shown in Section 5.1 to encompass the lexicographic, weighted, Chebyshev, and meta goal programming distance metric based variants. The proposed framework could also in theory be applied to other goal programming decision variable or goal based variants such as integer, non-linear, stochastic, chance-constrained, fractional, interval, and fuzzy goal programming although this would require some further theoretical development in the case of the more advanced variants. Acknowledgements Mariano Jiménez wishes to gratefully acknowledge financial support from the Spanish Ministry of Education, Project ECO2011-26499
7
References Arenas, M., Bilbao, A., Perez, B., Rodriguez, M.V., 2004. Fuzzy extended lexicographic goal programming. In: LopezDiaz, M., Gil, M.A., Grzegorzewski, P., Hryniewicz, O., Lawry, J. (Eds.), Soft Methodology and Random Information Systems, Advances in Soft Computing. Springer-Verlag, Berlin, pp. 543–550. Charnes, A., Cooper, W.W., 1961. Management Models and Industrial Applications of Linear Programming. John Wiley & Sons, New York. De Andres, R., Garcia-Lapresta, J.L., Gonzalez-Pachon, J., 2010. Performance appraisal based on distance function methods. European Journal of Operational Research 207, 1599–1607. Deparis, S., Mousseau, V., Ozturk, M., Pallier, C., Huron, C., 2012. When conflict induces the expression of incomplete preferences. European Journal of Operational Research 221, 593–602. Flavell, R.B., 1976. A new goal programming formulation. Omega 4, 731–732. Gass, S.I., 1986. A process for determining priorities and weights for large scale linear goal programmes. Journal of the Operational Research Society 37, 779– 785. Jones, D.F., Collins, A., Hand, C., 2007. A classification model based on goal programming with non-standard preference functions with application to prediction of cinema-going behaviour. European Journal of Operational Research 177, 515–524. Jones, D.F., Tamiz, M., 2010. Practical Goal Programming. Springer, New York. Jones, D.F., 2011. A practical weight sensitivity algorithm for goal and multiple objective programming. European Journal of Operational Research 213, 238– 245. Kazemzadeh, R.B., Bashiri, M., Atkinson, A.C., Noorossana, R., 2008. A general framework for multiresponse optimization problems based on goal programming. European Journal of Operational Research 189, 421–429. LINDO, 2012. LINGO 13.0 Optimization Modeling Software for Linear, Non-Linear, and Integer Programming. (accessed 20.02.12). Rawls, J., 1973. A Theory of Justice. Oxford University Press, Oxford. Rodriguez, M.V., Caballero, R., Ruiz, F., Romero, C., 2002. Meta-goal programming. European Journal of Operational Research 136, 422–429. Romero, C., 1991. A Handbook of Critical Issues in Goal Programming. Pergamon Press, Oxford. Romero, C., 2001. Extended lexicographic goal programming: a unifying approach. Omega 29, 63–71. Romero, C., 2004. A general structure of achievement function for a goal programming model. European Journal of Operational Research 153, 675–686. Saaty, T.L., 1980. The Analytical Hierarchy Process. McGraw-Hill, New York.
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Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Incorporating additional meta-objectives into the extended lexicographic goal programming framework Dylan Jones a,⇑, Mariano Jimenez b a b
Logistics and Management Mathematics Group, Department of Mathematics, University of Portsmouth, United Kingdom Department of Applied Economics I, University of The Basque Country (UPV-EHU), Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 24 April 2012 Accepted 27 December 2012 Available online xxxx
This paper introduces two new meta-objectives into the extended goal programming framework. The first meta-objective is the number of unmet goals and the second is a measure of closeness to the pairwise comparisons given by the decision maker. These complement the original two meta-objectives of the weighted sum of deviations and the maximal weighted deviation to provide a flexible four metaobjective framework. Lexicographic and non-lexicographic representations of the framework are developed. An example relating to transportation is solved in both the lexicographic and non-lexicographic forms. Weight sensitivity analysis is applied to the meta-weight parameters for the non-lexicographic case in order to find a range of available distinct solutions. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Extended goal programming Pairwise comparisons Multiple objective programming Weight sensitivity analysis
Subject to : uli nli þ v li pli 6 kl
1. Introduction The extended lexicographic goal programming (ELGP) is introduced by Romero (2001) with the purpose of providing a general framework which covers and allows the combination of the most common goal programming variants. It is also encompasses several other distance-based MCDM techniques. This work is further extended by Romero (2004) who provides a more generalised form of the achievement function and by Arenas et al. (2004) who extend the framework to include fuzzy models. The ELGP framework has been recently applied by De Andres et al. (2010) to a performance appraisal model. The general ELGP model is given with the linear form of the achievement function as proposed in (Romero, 2004) is given as:
2
(
X
)!
3
;...;7 þv 6 a1 k1 þ ð1 a1 Þ 7 6 i2H1 7 6 ( ) ! 7 6 7 6 X 6 ;...; 7 al kl þ ð1 al Þ uli nli þ v li pli Min a ¼ 6 7 7 6 i2Hl 6 ( )! 7 7 6 7 6 X 5 4 aL kL þ ð1 aL Þ uLi nLi þ v Li pLi u1i n1i
1 1 i pi
ð0:0Þ
i2HL
⇑ Corresponding author. Address: Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace, Portsmouth PO1 3HF, United Kingdom. Tel.: +44 2392846362; fax: +44 2392846353. E-mail addresses: [email protected] (D. Jones), [email protected] (M. Jimenez).
i 2 Hl ; l ¼ 1; . . . ; L;
f i ðxÞ þ ni pi ¼ bi
i ¼ 1; . . . ; q
ni ; pi P 0 i ¼ 1; . . . ; q
ð0:1Þ ð0:2Þ ð0:3Þ
where the model is defined as having L priority levels and q objectives. fi(x) is the achieved value of the ith objective which has an associated target value of bi. Deviational variables ni and pi denote the negative and positive deviations from the ith target value respectively. The maximal weighted deviation from amongst the set of unwanted deviations is denoted by kl. The sets Hl, l = 1, . . . , L include the objectives with an least one of their deviational variables to be penalised in priority level l. The weights uli and v li are associated with the relative level of importance associated with the minimisation of the negative and positive deviational variables from the ith target value in the lth priority level respectively. Unwanted deviations are given a positive weight and deviations which are not desired to be minimised in that priority level are given a zero weight. al is a parameter which controls the relative importance of efficiency and equity in the lth priority level. The ELGP formulation allows for the inclusion and combination of the optimisation, balancing, ordering, and satisficing underlying philosophies (Jones and Tamiz, 2010, Chapter 1). The lexicographic ordering philosophy is available via the priority structure of the achievement function. The satisfying philosophy is evident in the set of goals. The optimising philosophy can be achieved through the use of Pareto efficiency detection and restoration techniques (Jones and Tamiz, 2010, Chapter 6), or in the case of a single priority level (L = 1) through the setting of sufficiently high target goal values. The balancing philosophy is achieved through the inclusion
0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.12.025
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
of the maximal deviation terms (kl) in each priority level. Furthermore, the balance between optimisation (efficiency) and balance (equity) can be controlled at each priority level through the parameter al which can be varied between complete emphasis on optimisation (a = 0) and complete emphasis on balance (a = 1). The actual value of a chosen depends on the decision maker’s attitude towards balance and optimisation. It is recommended that some form of sensitivity or parametric analysis is carried undertaken on a, such as the algorithm presented in Section 4 of this paper. The ELGP framework is therefore a comprehensive tool for the inclusion of many types of underlying philosophies into the goal programming framework. 2. Meta-objectives This section details each of the proposed meta-objectives and makes linkages to previous work within the field of goal programming. 2.1. Meta objective 1 – Minimisation of the maximal unwanted deviation This is the first of the two meta-objectives taken from the Romero (2001, 2004) extended goal programming model. It is based around the L1 distance metric which links to the Rawlsian theory of social justice (Rawls, 1973). The minimisation of the maximal deviation is first used in a goal programming context by Flavell (1976). It underlies the variant of Chebyshev (or Minmax) goal programming. It ensures a balance between the objectives is maintained. In the context of a multi-stakeholder situation in can be used to ensure that a balance between stakeholders is achieved, thus facilitating the achievement of a consensus. It is hence associated with the concepts of fairness, equality, and social justice. 2.2. Meta objective 2 – Minimisation of average unwanted deviation This is the second of the two meta-objectives taken from the Romero (2001, 2004) extended goal programming model. It is based around the L1 distance metric which underlies the original weighted and lexicographic goal programming models as proposed by Charnes and Cooper (1961). The vast majority of the literature to date has utilised the L1 distance metric. As it is solely concerned with the minimisation of the average without regard for the distribution between individual objectives it is associated with the principles of optimisation (sometimes termed ruthless optimisation due to the above property) and efficiency (Jones and Tamiz, 2010, Chapter 1).
programming in the meta-goal programming formulation of Rodriguez et al. (2002). It is also suggested for pattern recognition using goal programming by Jones et al. (2007) The proposed formulation in Section 3 allows the incorporation of this metric into a parametric goal programming framework where the trade-offs with other meta-objectives can be effectively investigated. 2.4. Meta objective 4 – Minimisation of distance from given pairwise comparisons This paper proposes this novel meta-objective, which can be used for goal programmes where the decision maker preferences are expressed via a pairwise comparison matrix. It minimises the distance between the preferences expressed by the decision maker and those achieved by the goal programming solution. It is thus a measure of consistency with the preferences (this is not the same as consistency of the preferences which is a separate measure). The synergy between goal programming and pairwise comparisons is first noted by Gass (1986) who used the Analytical Hierarchy Process (AHP) to produce a weight set for a weighted goal programming model. Since then the linkage between goal programming and pairwise comparisons has been exploited in two ways. Firstly, goal programming has been used as a substitute to the Eigenvalue method (Saaty, 1980) as a means of generating the weight vectors from the pairwise comparison matrix. Secondly goal programming has been used in the manner of Gass (1986) above, that is the use of goal programming to generate the weights of a goal programme. This is noted by Jones (2011) to produce a valid starting point for a goal programming sensitivity analysis. However, due to the nature of the goal programming optimisation process (especially in the linear weighted case where the solution algorithm is often simplex-based) under current methodologies there is no explicit link between the final solution and the initial preferences. The proposed meta-objective rectifies this situation by allowing the connection to be explicitly measured and optimised. The consistency of the preferences can be measured in the standard fashion and the pairwise comparison matrix returned to the decision maker for re-consideration if the inconsistency level is too high. Thus the four meta-objectives outlined above allow for a comprehensive range of considerations including optimisation, efficiency, equality, fairness, social justice, target achieving, and consistency with preferences to be measured. This paper will use the extended goal programming framework to build a formulation which allows these considerations to be optimised and traded-off in a parametric manner.
3. Formulation of four meta-objective extended goal programming framework
2.3. Meta objective 3 – Minimisation of the number of unmet goals 3.1. Lexicographic version This is the first of the two meta-objectives introduced in this paper. It is based around the distance metric which has the following properties in the context of goal programming:
8 > < 0 if the ith unwanted dev iation v ariable has v alue zero si ¼ 1 if the ith unwanted dev iation v ariable has a positiv e > : v alue The weighted sum of these binary measures of whether the goals have been met or not will give a measure of the level of achievement of goals in the model. It is useful in situations where rewards are given if targets are met or penalties applied for unmet targets. This meta-objective is thus associated with target setting and achieving behaviour. This L0 metric is first seen in the context of goal
This model extends the number of meta-objectives in the extended goal programming model from two to four. The following meta-objectives are used at each priority level: (1) Minimisation of the normalised (L1) (maximum unwanted deviations from the set of goals in that priority level-given a weight al in the lth priority level. (2) Minimisation of the normalised (L1) (weighted sum of unwanted deviations from the set of goals in that priority level-given a weight bl in the lth priority level. (3) Minimisation of the number of unmet goals (L0) (from the set of goals in that priority level-given a weight vl in the lth priority level.
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
expressed level of achievement of the ith and jth unwanted deviational variables respectively.
(4) Minimisation of the discrepancy between the expressed pairwise preferences of the decision maker and the actual preferences indicated by the solution-given a weight d1 in the lth priority level.
3.2. Non-lexicographic version
It is also suggested that for convenience the meta-objective parameter weights sum to one at each priority level (al + bl + cl + dl = 1 l = 1, . . . , L).
8
9 X v 1 pi =
8
The non-lexicographic version is relevant when the decision maker does not have a clear ordering of goals in their preferences but rather wishes to investigate the trade-offs between the
9 =
91 3 = i 6 @a1 k1 þ b þ l1 s þ t1i ti þ d1 ðNij þ Pij Þ A; . . . ; 7 þ v1 1 7 6 ki ; : 1 : 1 i i ; : ; 7 6 1 1 1 i2Q i2Q i2Q i2Q ði;jÞ2Q 7 6 1 2 1 2 3 8 9 8 9 8 91 7 60 7 6 < = < = < = X X X X X l l 7 6 ui ni v p i l l i @al kl þ bl A 7 þ v l s þ t t ðN þ P Þ ; . . . ; þ þ d Min a ¼ 6 ij ij l l i i ki ; 7 6 : l ki : l i i ; : ; 7 6 l l l i2Q i2Q i2Q i2Q ði;jÞ2Q 1 2 1 2 3 7 60 8 9 8 9 8 91 7 6 7 6
u1i ni ki
i2Q 1
Subject to :
i2Q 2
i2Q 1
uli ni v li pi þ 6 kl i ¼ 1; . . . ; q l ¼ 1; . . . ; L ki ki f i ðxÞ þ ni pi ¼ bi i ¼ 1; . . . ; q f i ðxÞ þ Msi P bi
i 2 Q l1 l ¼ 1; . . . ; L Q l2
X
i2Q 2
ð1:1Þ ð1:2Þ ð1:3Þ
l ¼ 1; . . . ; L f i ðxÞ Mt i 6 bi i 2 ui ni nj þ Nij Pij ¼ 0 i; j 2 Q L1 i < j uj ki kj
ð1:4Þ
l ¼ 1; . . . ; L ui ni pj þ Nij Pij ¼ 0 i 2 Q l1 ; j 2 Q l2 ; v j ki kj
ð1:5Þ
l ¼ 1; . . . ; L
v i pi pj þ Nij Pij ¼ 0 v j ki kj
8 < X
ði;jÞ2Q 3
objectives and between the meta-objectives in the model. The algebraic version of the non-lexicographic version is given as:
(
Min a ¼ ak þ b ( þv
i; j 2
Subject to :
i
l ¼ 1; . . . ; L x2F
ð1:7Þ ð1:8Þ
ni ; pi P 0 i ¼ 1; . . . ; q; Nij ; Pij P 0 i; j ¼ 1; . . . qji < j si ; t i binary i ¼ 1; . . . ; q
X ui ni
i2Q 1
X
i2Q 1
ð1:6Þ Q L2
ð1:9Þ
where all existing notation is in common with the ELGP model introduced in Section 1 and the following additional notation is defined: Q l1 is the ordered set of the indices of the unwanted negative deviational variables in priority level l. Q l2 is the ordered set of the indices of the unwanted positive deviational variables in priority level l and Q l3 is the set of pairs of unwanted deviational variables indices in priority level l defined by:
Q l3 ¼ i; j 2 Q l1 ji < j [ i 2 Q l1 ; j 2 Q l2 [ i; j 2 Q l2 ji < j si is a binary variable that takes the value 1 if the achieved value of ith goal is less than the target value and value 0 otherwise. ti is a binary variable that takes the value 1 if the achieved value of ith goal is greater than the target value and value 0 otherwise. The si and ti variables represent whether the goals have been met for the cases of unwanted negative and positive deviations respectively. lli and tli are relevant the relative weights representing the penalty applied in the lth priority level for not meeting the ith goal on the negative and positive direction respectively. M is a large positive constant. The normalisation constant of the ith objective is given by ki. Nij and Pij are the deviations from the decision maker
ð1:0Þ
ki
li si þ
þ
X v 1p i
ki )
i2Q 2
X i2Q 2
mi ti
)
i
(
q 1 X ðN ij þ Pij Þ þd jQ 3 j ði;jÞ2Q
) ð2:0Þ
3
ui ni v i pi þ 6 k i ¼ 1; . . . ; q ki ki f i ðxÞ þ ni pi ¼ bi i ¼ 1; . . . ; q f i ðxÞ þ Msi P bi i 2 Q 1 f i ðxÞ Mt i 6 bi i 2 Q 2 ui ni nj þ Nij Pij ¼ 0 i; j 2 Q 1 i < j uj ki kj ui ni pj þ N P ¼ 0 i 2 Q 1; j 2 Q 2 v j ki kj ij ij v i pi pj þ N P ¼ 0 i; j 2 Q 2 i < j v j ki kj ij ij x 2 Fj ni ; pi P 0 i ¼ 1; . . . ; q; N ij ; P ij P 0 i; j ¼ 1; . . . :qji < j si ; t i binary i ¼ 1; . . . ; q
ð2:1Þ ð2:2Þ ð2:3Þ ð2:4Þ ð2:5Þ ð2:6Þ ð2:7Þ ð2:8Þ ð2:9Þ
where the significance of all variables remains the same as that defined in model (1.0)–(1.9), expect the parameters a, b, c, d and the maximal deviation variable k are scalar due to the fact there are no priority levels in the model. The sets Q1, Q2, Q3 also do not need a priority level index. 4. Example A hypothetical multi-objective transportation model is used to demonstrate the methodology developed in Section 3. The model has two supply points and three demand points where all potential routes between supply points and demand points are feasible. Each route has the following three criteria associated with it: Reliability: Measuring on a normalised range of 0 (worst) to 1 (best). Cost: Measured on a normalised range of 0 (best) to 1 (worst). Transit time: Measured on a normalised range of 0 (best) to 1 (worst).
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
The supply levels are fixed at levels SP1 = 150, SP2 = 200, and SP3 = 300 units respectively. The minimum demand levels for the first two demand points are fixed at DM1 = 100 and DM2 = 300 units respectively. However the decision maker has goals of supplying exactly 300 and 500 units to demand points 1 and 2 respectively. Demand point 3 has a fixed demand of DM3 = 150 units that must be met exactly. This leads to the following five decision maker set goals: Goal 1: Achieve at least 60% of the theoretical maximum reliability. Goal 2: Do not exceed 40% of the theoretical maximum cost. Goal 3: Do not exceed 40% of the theoretical minimum cost. Goal 4: Supply demand point 1 with exactly 300 units. Goal 5: Supply Demand Point 2 with exactly 500 units. Goals 1–3 are one-sided goals (only one deviation to be penalised) whereas goals 4 and 5 are two-sided goals (both deviations to be penalised). Hence there are seven unwanted deviations in total.
The Eigenvalue method (Saaty, 1980) is then used to produce the following set of weights:
w1 ¼ u1 ¼ 0:360;
w2 ¼ v 2 ¼ 0:181;
w3 ¼ v 3 ¼ 0:181;
w4 ¼ u4 ¼ 0:097;
w5 ¼ v 4 ¼ 0:032;
w6 ¼ u5 ¼ 0:104;
w7 ¼ v 5 ¼ 0:045 This extended goal programming model therefore has the following algebraic representation: 3 X Min a ¼ ak þ b Di þ cðu1 s1 þ v 2 t2 þ v 3 t þ u4 s4 þ v 4 t4 i¼1
þ u5 s5 þ v 5 t 5 Þ þ
Subject to :
3 X 3 X
@ X ðNij þ Pij Þ 21 ði;jÞ2Q
ð3:0Þ
Rij xij þ n1 p1 ¼ 420
ð3:1Þ
C ij xij þ n2 p2 ¼ 280
ð3:2Þ
T ij xij þ n3 p3 ¼ 280
ð3:3Þ
i¼1 j¼1 3 X 3 X i¼1 j¼1 3 X 3 X
4.1. Extended non-lexicographic goal programming model
i¼1 j¼1
This section presents the non-lexicographic version of the extended goal programming model with 4 meta-objectives for the example. Introducing the following notation. Let i = 1, . . . , 3 be the index of the supply nodes and j = 1, . . . , 3 be the index of the demand nodes. Xij is the number of items sent between supply node i and demand node j. Rij is the reliability score for the route between supply node i and demand node j. Cij is the cost score for the route between supply node i and demand node j. Tij is the time score for the route between supply node i and demand node j.
3 X xi1 þ n4 p4 ¼ 300
ð3:4Þ
i¼1 3 X xi2 þ n5 p5 ¼ 500
ð3:5Þ
i¼1
Di 6 k i ¼ 1; . . . ; D 3 X 3 X
ð3:6Þ
Rij xij þ s1 P 420
ð3:7Þ
C ij xij t 2 6 280
ð3:8Þ
T ij xij t3 6 280
ð3:9Þ
i¼1 j¼1 3 X 3 X i¼1 j¼1 3 X 3 X i¼1 j¼1
With all other notation corresponding to that introduced in model 2 in Section 3. In particular the following variables are defined in order to simplify the algebraic representation:
u1 n1 v 2 p2 v 3 p3 u4 n4 v 4 p4 D1 ¼ ; D2 ¼ ; D3 ¼ ; D4 ¼ ; D5 ¼ ; 420 280 280 300 300 u5 n5 v 5 p5 ; D7 ¼ D6 ¼ 500 500 w1 ¼ u1 ; w2 ¼ v 2 ; w3 ¼ v 3 ; w4 ¼ u4 ; w5 ¼ v 4 ; w6 ¼ u5 ; w7 ¼ v 5
3 X xi1 þ Ms4 P 300
ð3:10Þ
i¼1 3 X xi1 Mt4 6 300
ð3:11Þ
i¼1 3 X xi2 þ Ms5 P 500
ð3:12Þ
i¼1 3 X xi2 Mt5 6 500
ð3:13Þ
i¼1
A pairwise comparison method (Saaty, 1980) is used to determine the decision maker weights with respect to the unwanted deviations. The pairwise comparison matrix given by Table 1 represents the preferences of the hypothetical decision maker:
Di Dj þ Nij Pij ¼ 0 i; j ¼ 1; . . . ; 7 i < j wj wj 3 X xij 6 SPi i ¼ 1; . . . ; 3
ð3:15Þ
i¼1 3 X xij P DMj
Table 1 Pairwise comparison matrix for the non-lexicographic model.
ð3:14Þ
j ¼ 1; . . . ; 3
ð3:16Þ
i¼1
Deviation
w1 = u1
w2 = v2
w3 = v3
w4 = u4
w5 = v4
w6 = u5
w 7 = v5
w1 = u1 w 2 = v2 w 3 = v3 w4 = u4 w 5 = v4 w6 = u5 w 7 = v5
–
2 –
2 1 –
5 2 2 –
8 5 5 3 –
4 2 2 1 0.25 –
7 4 4 3 0.5 3 –
xij P 0 i ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3; ni ; pi P 0 i ¼ 1; . . . ; 3; N ij ; P ij P 0 i; j ¼ 1 . . . ; 7 j > isi ; t i binary i ¼ 1; . . . ; 5; k P 0; ni pi ¼ 0 i ¼ 1; . . . ; 5 ð3:17Þ where goals (3.1)–(3.5) represent the reliability, cost, transit time, demand point 1 level, and demand point 2 goals respectively. Constraint set (3.6) relates to the L1 meta-objective of minimising the
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx Table 2 Distinct solutions produced by the weight sensitivity analysis. Sol
A
b
c
d
N1
P2
P3
N4
P4
N5
P5
L1
L1
L0
PW
A B C D E F G H
0.25 0.001 0.125 0.063 0.997 0.001 0.063 0.495
0.25 0.997 0.625 0.813 0.001 0.001 0.438 0.495
0.25 0.001 0.125 0.063 0.001 0.001 0.063 0.005
0.25 0.001 0.125 0.063 0.001 0.997 0.438 0.005
0 0 0 0 38 0 0 0
67 7.5 64 20 50 47 50 36
0 11 0 0 50 47 38 31
134 200 137 200 100 94 99 129
0 0 0 0 0 0 0 0
200 200 200 200 155 200 200 200
0 0 0 0 0 0 0 0
0.1281 0.1182 0.1276 0.1190 0.1608 0.1327 0.1301 0.1265
0.043 0.065 0.044 0.065 0.032 0.042 0.042 0.042
0.38 0.56 0.38 0.38 0.92 0.56 0.56 0.56
0.3880 0.4135 0.3882 0.4177 0.3764 0.3603 0.3605 0.3689
maximal deviation. Constraints (3.7)–(3.13) relate to the L0 metaobjective of minimising the number of unmet goals. M is an arbitrarily large positive constant. Constraint set (3.14) relates to the pairwise meta-objective of achieving as close as match to the pairwise comparisons given by the decision maker as possible. Constraint sets (3.15) and (3.16) impose hard constraints to ensure supply levels are not exceeded and minimum demand levels are met at each point. The extended goal programming model is solved used LINGO 13.0 (LINDO, 2012) and an initial equal meta-weight solution of a = b = c = d = 0.25 resulting in solution point A in Table 2. In order to investigate the range of solutions available by varying the meta-weights the sensitivity analysis algorithm of Jones (2011) is applied to the meta-weights. This algorithm is designed to give a selection of sufficiently diverse solutions for consideration by the decision maker. The parameter inputs to the algorithm are set at TMax = 2 (indicating that at most two weights are simultaneously raised to give search directions from the initial solution) and Maxlevel = 2 (indicating that most two bisections are allowed in each search direction). No additional preference information is included as the whole meta-weight space is assumed to be of potential interest to the decision maker. A small weight rather than a zero weight has been given to each meta-objective where necessary to avoid potential Pareto inefficiency occurring due to the existence of alternative optimal solutions. The (Jones, 2011) algorithm produces eight distinct solutions which are detailed in Table 2, Columns 2–5 in Table 2 detail the first set of meta-weights for which the solution is found. Additional weight sets that generate the same solution are given beneath the table. Columns 6–12 give the solution in unwanted deviational space. Columns 13–16 give the solution in meta-objective space, where L1, L1, L0, and PW denote the first, second, third, and fourth components (excluding the meta-weight) of the achievement function (3.0) respectively. The number of decimal places shown in Columns 13–16 is sufficient to demonstrate differences in values where they exist. It is noted that all eight solutions are, as expected, Pareto Efficient in both objective and meta-objective spaces.
Table 4 Pairwise comparison matrix for priority level 2 of the lexicographic model. Deviation
w4 = u4
w5 = v4
w6 = u5
w 7 = v5
w4 = u4 w5 = v4 w6 = u5 w7 = v5
–
3 –
1 0.25 –
3 0.5 3 –
Priority Level 2: Minimize unwanted deviations from the supply level goals (goals 4 and 5). The decision maker still wishes to give preferences in a pairwise fashion. Thus, pairwise comparison matrices are elicited for each priority level and the relevant intra priority level weights are found using the Eigenvalue method. The resulting pairwise comparison matrices for the first and second priority levels are given by Tables 3 and 4 respectively. Priority level 1: The Eigenvalue method is then used to produce the following set of weights:
w1 ¼ u1 ¼ 0:5; w2 ¼ v 2 ¼ 0:25; w3 ¼ v 3 ¼ 0:25; Priority Level 2: The Eigenvalue method is then used to produce the following set of weights:
w4 ¼ u4 ¼ 0:367; w5 ¼ v 4 ¼ 0:096; w6 ¼ u5 ¼ 0:391; w7 ¼ v 5 ¼ 0:146 The following pre-emptive extended goal programme is then formulated in accordance with the methodology detailed in Section 3:
" Lex Min a ¼ a1 k1 þ b1
3 X Di þ c1 ðu1 s1 þ v 2 t2 þ v 3 t3 Þ i¼1
7 X d1 X ðN ij þ Pij Þ; a2 k2 þ b2 Di þ 3 ði;jÞ2Q i¼4
4.2. Extended lexicographic goal programming model
1
The unwanted deviations are now assumed to be divided into two priority levels by the decision maker in order to reflect the fact that they regard meeting the three performance related goals as more important than meeting the supply level related goals: Priority Level 1: Minimize unwanted deviations from the reliability, cost, and transit time goals (goals 1–3). Table 3 Pairwise comparison matrix for priority level 1 of the lexicographic model. Deviation
w1 = u1
w2 = v2
w 3 = v3
w1 = u1 w 2 = v2 w 3 = v3
–
2 –
2 1 –
d2 X þ c2 ðu3 s4 þ u5 t4 þ u6 s5 þ u7 t 5 Þ þ ðNij þ Pij Þ 18 ði;jÞ2Q
#
2
ð4:0Þ Subject to: Eqs. (3.1)–(3.5), (3.8)–(3.13), (3.15) and (3.16)
Di 6 k1
i ¼ 1; . . . ; 3
Di 6 k2 i ¼ 4; . . . ; 7 Di Dj þ Nij Pij ¼ 0 ði; jÞ 2 Q 1 [ Q 2 wj wj
ð4:1Þ ð4:2Þ ð4:3Þ
xij P 0 i ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3; ni ; pi P 0 i ¼ 1; . . . ; 3; Nij ; Pij P 0 i; j ¼ 1 . . . ; 7 j > i; si ; ti binary i ¼ 1; . . . ; 5; k1 ; k2 P 0; ni pi ¼ 0 i ¼ 1; . . . ; 5
ð4:4Þ
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
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D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
where Q 1 ¼ ði; jÞji ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3; i < j; Q 2 ¼ ði; jÞji ¼ 4; . . . ; 7; j ¼ 4; . . . ; 7; i < j: All other notation remains consistent with that introduced in models 1 and 3. Assuming a parameter setting that gives equal weight to all four meta-objectives in both priority levels (i.e. a1 = a2 = b1 = b2 = c1 = c2 = d1 = d2 = 0.25) and solving a lexicographic goal programming using Lingo 13.0 in accordance with the methodology given in (Jones and Tamiz, 2010, Chapter 5) yields the following solution expressed in terms of non-negative decision and deviational variables rounded to the nearest integer:
x11 ¼ 92; x33 ¼ 111;
x12 ¼ 58; n4 ¼ 200;
x22 ¼ 211; n5 ¼ 200;
x23 ¼ 39;
x31 ¼ 8;
p5 ¼ 20
Which is equivalent to solution point D in Table 2. It is clear that the lexicographic ordering has successfully prioritised the first three goals as required. It is also noted the second priority level is redundant due to a combination of relatively high target levels and the complexity of the multiple distance metric targets being required in the first priority level. 5. Discussion Solutions have been generated in Section 4 for both the nonlexicographic and lexicographic versions proposed in Section 3. It is noted that the larger number of meta-objectives makes finding alternative optimal solutions at each priority level a more difficult task. Therefore if the lexicographic version is going to be used and redundancy is to be avoided then the target values at higher priority levels should be set at fairly relaxed levels. On the other hand, this finding indicates that the non-lexicographic should be the preferred choice when dealing with models with a number of metaobjectives greater than two. The non-lexicographic version has been successfully used alongside the (Jones, 2011) weight sensitivity algorithm to produce a number of distinct solutions relating to different meta-weight combinations. 5.1. Reduction to other goal programming models This paper has worked from a base model of the linear version of the ELGP model of Romero (2001, 2004). Therefore by setting the parameters c and d to zero the linear version of the ELGP is reproduced. By implication, the linear models listed by Romero (2001): conventional mathematical programming, weighted goal programming, lexicographic goal programming, Chebyshev goal programming, the reference point method (although this requires a negative value of b), and the L1 and L1 solutions in compromise programming. Setting d to zero results in a model that has some similarities to the meta goal programming formulation of Rodriguez et al. (2002) with a, b and c relating to the weights of the three meta-goals. Unlike meta goal programming, the reduced model does not require the decision maker to set target values for the meta-objectives but rather allows parametric trade-offs between them. 5.2. Normalisation considerations This paper considers normalisation on two levels: the objective level and the meta-objective level. The need for normalisation at the objective level in goal programming to ensure that deviations are comparable is well known. Not using a (formal or informal) normalisation technique will produce results that are biased toward objectives measured in lower denomination units (Romero, 1991). In the examples in Section 4, the technique of percentage normalisation is used at the objective level.
When using a method with many meta-objectives it is also important to consider whether a formal or informal technique to normalise the meta-objectives is necessary or not. The first observation to make is that the meta-objectives are combined using parameters (a, b, c, d) rather than preferential weights. The idea is generally to vary the parameters to produce differing solutions rather than a single point solution (as demonstrated by the nonlexicographic case in Section 4). Thus any mismatch in units of measurement will be less critical than in the standard objective case. It may, however, make it more difficult to effectively search the parameter search to elicit a range of solutions. Consider the four meta-objectives under consideration in this paper. The first meta-objective is a sum of (normalised) weighted deviations from goals. The second meta-objective is the single largest weighted deviation from a goal. The third objective is the sum of a number weighted (normalised) binary variables equal to the number of unwanted deviations. The fourth objective is equal to the sum of unwanted deviations from the pairs of unwanted deviations. There are kðk1Þ of these, where k is the number of unwanted 2 deviational variables. The informal normalisation applied at the meta-objective level is to divide the fourth meta-objective by kðk1Þ in order to reduce the units of this meta-objectives to a level 2 closer to that of the other three meta-objectives. The results can be seen in the range of meta-objective solutions given in the last four columns of Table 2. In the lexicographic case the fourth metaobjective terms in two priority levels are divided by the number of pairs of unwanted deviational variables associated with that priority level. 5.3. Meta-objective space considerations (Jones, 2011) considers a goal programming as existing in three spaces: decision space, objective space, and weight space. In order to encompass extended goal programming a fourth space: metaweight space has been included. Meta-weight space has dimensions Rm , where is the number of meta-objectives used in the extended goal programming model. Like weight space, all dimensions can by convention be constrained either to the range [0, 1] or to the range [ 1, 1] if Pareto Efficiency restoration techniques or reference point method techniques are required. As the number of potential meta-objectives grows there is a need to provide some formal mechanism for searching meta-weight space in order to ensure the decision maker has access to the full range of possible solutions. Seen in this context, the Jones (2011) algorithm used in Section 4.1 is one means of searching meta-weight space. Other search based methods, heuristics, and meta-heuristics could potentially also be used for this purpose. 6. Conclusions The extended lexicographic goal programming has been shown to provide an effective framework for the inclusion and combination of different underlying philosophies. This is achieved by formulating meta-objectives to represent each of the philosophies and performing a parametric analysis of the resulting meta-objective space. This paper has extended the number of meta-objectives in the framework from two to four. It is likely that further works will be able to incorporate further philosophies by augmenting the number of meta-objectives. Possibilities for augmentation include meta-objectives to reflect the fact that pairwise comparisons are sometimes uncertain or difficult to specify (Deparis et al., 2012) or specialised meta-objectives arising from the field of multi-response optimisation (Kazemzadeh et al., 2008). Of course, not every real-life decision situation necessarily contains all the underlying philosophies and hence the relevant subset
Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025
D. Jones, M. Jimenez / European Journal of Operational Research xxx (2013) xxx–xxx
of meta-objectives should be chosen. For instance, a decision maker who wishes to specify their preferences in a direct rather than a pairwise manner but wishes to consider the efficiency, equality, and number of targets reach should omit the fourth meta-objective and use the first three meta-objectives in this paper. The second meta-objective has been prevalent in most goal programming applications to date (Jones and Tamiz, 2010), although this could be due to historical and computational reasons, as it underlies both the weighted and lexicographic variants. This paper proposes increased usage of the other three meta-objectives described in order to enhance the flexibility and modelling power of goal programming. If any of the meta-objectives is not required for optimisation purposes but is required solely as an information source, then the meta-weight of that meta-objective can be set to zero but the relevant equations left in the formulation. This will allow for calculation of, but not optimisation of, the relevant meta-objective’s value. This paper has focussed on linear goal programming and shown in Section 5.1 to encompass the lexicographic, weighted, Chebyshev, and meta goal programming distance metric based variants. The proposed framework could also in theory be applied to other goal programming decision variable or goal based variants such as integer, non-linear, stochastic, chance-constrained, fractional, interval, and fuzzy goal programming although this would require some further theoretical development in the case of the more advanced variants. Acknowledgements Mariano Jiménez wishes to gratefully acknowledge financial support from the Spanish Ministry of Education, Project ECO2011-26499
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Please cite this article in press as: Jones, D., Jimenez, M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.12.025