Compromise Programming Slim Zekri Dept Natural Resource Economics

Compromise Programming Slim Zekri Dept. Natural Resource Economics Slim@squ. edu. om Sultan Qaboos University

• “Investors, of course, are looking for businesses in which management can buy low, sell high, collect early, and pay late. The business plan needs to spell out how close to that ideal the new venture is expected to come” – William A. Sahlman 1997 – Professor of Business Administration at the Harvard Business School.

CP • CP is a complement to MOP • It allows to reduce the set of efficient solutions to a more reasonable size without demanding any information from the DM • Without assumptions on the utility function of the DM • The basic idea in CP is to identify an ideal or utopian solution which is only a point of reference for the DM. • CP assumes that any DM seeks a solution as close as possible to the ideal point, possibly the only assumption made by CP about human preferences

Distance function • To find out the set of solutions closest to the ideal point we need to introduce the concept of distance • The concept of distance is not used in its geometric sense, but as a proxy measure for human preferences.

Introduction to the Concept of Distance Measures 2 + (3 -8)2 ]1/2 = 9. 43 d = [(2 -10) Pythagorean or Euclidean Distance

Extension of the Distance • Pythagorean or Euclidean Distance • Extension to the n-dimension space

Family of Lp Metrics • The notion of distance can be genralized to Lp

Lp distances example

Lp metrics • In a strict two-dimensional geometric sense the use of Lp metrics for values of the parameter p greater than two is meaningless – It would mean the existence of distances shorter than the straight line! – The use of Lp metrics is very useful, if not used in a geometric sense as a measure for human preferences. • Observe that – L 1 = 13 – L 2 = 9. 43 • As P increases more weight is given to the largest deviation – L∞ = 8 or │2 -10│ • Thus when P →∞ the distance is given exclusively by the largest deviation • All possible distances are bounded by the 'longest', the L 1 metric, and the 'shortest', the L∞ metric, distances. • It is also called the Chebyshev distance

Discrete approximation of the best Compromise Solution • Zeleney’s (1972) Axiom of Choice: The best-compromise solution is the nearest solution to the ideal • Calculate distances between the set of solutions and the ideal – Let’s call the ideal Z*j • The degree of closeness between the objective Zj(x) and it’s ideal is given by the distance dj dj = Z*j - Zj(x) dj = Zj(x) - Z*j if Zj(x) is to maximize or if Zj(x) is to minimize or • Deviations should be normalized to avoid biases towards objectives with large numericals •

Normalize using Ideal and antiideal • Z, j is the anti-ideal or nadir point for the jth objective • The normalised degrees of closeness are bounded between 0 and 1; • When an objective achieves its ideal solution the degree of closeness is 0 • when an objective achieves its anti-ideal solution the degree of closeness is 1. • The normalised degrees of closeness measure the percentage of achievement of one objective with respect to its ideal value.

Generalizing to family of Metrics • The best-compromise solution changes according to p and Wj • Wj are the weights representing the relative importance of the jth objective to the DM • p acts as a weight attached to the deviations according to their magnitudes

B’ is the closest solution to the ideal regardless of the values of W or P

CP a continuous setting • The discrete method has two weaknesses – Requires the determination of the efficient set – The best-compromise point should always be an extreme efficient point. In many instances the best-compromises could be Pareto-interior points • See next figure

Only A’; B’ or C’ can be best compromise in the discrete method!!!

CP continuous setting… • Solve LP problems to determine the set of solutions closest to the ideal – For p = 1 – For p → ∞

When p → ∞ Min-Max

Compromise set • Yu (1973) proved that for problems with two objectives L 1 and L∞ metrics define a subset of the efficient set, • Zeleny (1974) calls the L 1 -L∞ the compromise set. • All the other best-compromise solutions fall between the solutions corresponding to L 1 and L∞ metrics

Solving for L 1 • For W 1 = W 2 (equal importance) B‘ is the solution -B' is the best compromise when the metric L 1 is used

Solving for p→∞ • For the L∞ metric (p→∞) the maximum deviation from among the individual deviations is minimized • When p→∞ only the largest deviation counts • The best-compromise solution is obtained by solving the following LP problem

Solution for L∞ • For W 1 = W 2 solution is the point Z‘ – NPV = ₤ 119, 000 and – Hiring = 7, 616 hours of casual labor. – X 1 = 19. 04 and X 2 = 0 • Nonlinear algorithms are needed to obtain the best-compromise solutions for other metrics

CP and displaced ideal • CP reduced the set of efficient solution of interest from A’B’C’ to Z’B’ to help the DM • It’s inconvenient to work with large CP sets – it is not easy for the DM to choose the optimum solutions from the efficient compromises • How can we further reduce the compromise set without heavy assumptions? ? ? • Zeleny (1974, 1976) has suggested a method called the displaced ideal which helps reducing the compromise set

Method of Displaced ideal • Requires some information from the DM • CP : we cannot have a reliable mathematical representation of the DM's actual utility function in practice • CP does not even attempt to determine it • CP identifies a compromise set interpreted as the portion of the efficient set where the tangency between the iso-utility functions and the efficient set will likely take place – The compromise set is a “landing area” for the utility curve • Area Z’B’ in our example

Displaced ideal • Assume the compromise set Z'B' is too large • Interact with the DM and Ask him : – “which portions of the efficient set are not of interest for you at all? ” – Discard these portions • Assume the DM discards solutions belonging to: – The segment B'C' because they require too many hours of casual labour – The solutions along A'Z' as they provide too little NPV • We now have a new ideal: I' (163, 625; 7, 616)

New Ideal I’ • Solve 2 new LP problems to determine the set of solutions closest to the new ideal I’ – For p = 1 – For p → ∞ B’ Z’’

Pros and cons of GP, MOP and CP • Information required from the DM – GP is the most difficult approach. The DM has to provide precise target values, weights attached to each deviational variable, pre-emptive ordering of preferences, • Much of this is very difficult to obtain in some cases, as with the target values, it could be argued that this is the kind of information that the model should provide to the DM rather than it being an input requirement. – Building a MOP model does not require any information about the DM's preferences. • The mathematical expression of the objectives being considered is sufficient. – For CP we need to know only the relative preferences of the DM for each objective

Pros and cons… • Information produced by the model for use by the DM – GP variants provide only a single solution – Sensitivity analysis may help generating more info to the DM • GP provides rather too little information compared to MOP or CP

Info to the DM… • The efficient set generated by MOP provides the DM with extremely valuable information – Trade-offs for the objectives involved – Evaluating different alternatives before makeing a choice. – An advantage of MOP when the number of objectives is limited to 2 or 3 to display the efficient set and the corresponding trade-offs graphically; • CP provides the same information as MOP by identifying the bounds of that portion of the efficient set which is closest to the ideal point.

Conclusion • No definite superiority of one MCDM approach relative to others • As Ignizio (1983) says: “there is not now, and probably never shall be, one single "best“ approach to all types of multi-objective mathematical programming problems” • The choice of the technique is left to the analyst and depends on several criteria 1 - number of attributes: the higher the # the more difficult to use MOP • DM’s ability to provide information about targets and priorities • Analysis detail required by the DM

Relationships between different MCDM approaches • A WGP model in which all the targets have been set as the ideal values is a CP model with the L 1 metric without normalized objectives • Let’s consider the general WGP with minimization of ni

Assume now that all targets are set to tj* the ideal values • F(x) + nj – pj = tj* • nj = tj* - F(x) • The structure then turns into the following model which is a non normalized (p=1) CP model • Or Min tj* - F(x) is equivalent to Max F(x)

• when L∞ metric is used in CP, and the objectives are not normalized, the model is equivalent to a MINIMAX GP with targets set as the ideal values of the goals. • For tj = tj* and pj = 0 the following CP model for the L∞ = metric when the objectives are not normalized is obtained • In models where the targets have been set as ideal values the analysts are not actually utilising GP but the MOP technique without being aware of the situation