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Maximin Fairness in Project Budget Allocation
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Electronic Notes in Discrete Mathematics 55 (2016) 65–68
www.elsevier.com/locate/endm
Maximin Fairness in Project Budget Allocation Maurizio Naldi a,1 Gaia Nicosia b,2 Andrea Pacifici a,3 Ulrich Pferschy c,4 a
b
Dipartimento di Ingegneria Civile e Ingegneria Informatica, Universit` a di Roma Tor Vergata, Roma, Italy
Dipartimento di Ingegneria, Universit` a Roma Tre, Roma, Italy c
Department of Statistics and Operations Research, University of Graz, Graz, Austria
Abstract This work addresses a multi agent allocation problem in which multiple departments compete for shares of a company budget. Each department has its own portfolio of projects with given expected profits and costs and selects an optimal subset of its projects consuming its assigned budget share. Besides considering the total profit of the company a central decision maker should also take fairness issues into account. Thus, we introduce an equity criterion based on maximin fairness. The resulting trade-off between total profit and fairness indicators is studied in this contribution. To this purpose a bicriteria ILP model is presented where one of the objectives is the maximization of the overall profit and the other is the maximization of the minimum budget allocated to one of the departments. We perform an experimental analysis showing a nearly perfect linear anticorrelation between profit and fairness index values. Keywords: Knapsack problem, project management, multi agent system, fair allocation, decision support system.
http://dx.doi.org/10.1016/j.endm.2016.10.017 1571-0653/© 2016 Elsevier B.V. All rights reserved.
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M. Naldi et al. / Electronic Notes in Discrete Mathematics 55 (2016) 65–68
Introduction
We consider an allocation problem faced by the general manager of a company with a set D of d departments. A total budget b has be allocated to several projects, each belonging to one of the departments. We indicate by Ji the set of projects of department i ∈ D and w.l.o.g. assume that |Ji | = n, for i ∈ D. The j-th project of the i-th department requires a budget sij , which must be obtained in full for the project to be undertaken, and yields an estimated return on investment (ROI) rij (i = 1 . . . d, j = 1 . . . n). Since the overall required budget i∈D, j∈Ji sij is usually larger than the available budget b, the company must select a subset of the projects submitted for funding. Naturally, the company’s objective is the maximization of the total profit obtained from the budget investment. However, this may correspond to selecting projects in a way that can be perceived as unfair by one or more departments, since the budget may well be allocated in an unbalanced way. To avoid such potentially biased solutions the company should take into account some idea of equity or fairness in the allocation decision. In economic analysis, an axiomatic characterization of what might be a fair resource allocation has been the subject of several studies in the last decades. In the context of optimization, even though some studies date back to the Nineties ([4]), only recently “fairness” concepts received considerable attention (see e.g. [1], [6] and [7]). Here, we formulate the allocation decision as a bicriteria problem in which both profit and fairness maximization are considered. In this paper we follow the idea of Rawlsian justice and adopt a maximin fairness approach, i.e., we aim at maximizing, together with the overall profit, the minimum value of the budget allocated to a department. A similar problem has been addressed in [3], where the author uses range as a fairness indicator, i.e. the maximum difference between the budget allocated to any two departments. The author presents an ILP model and a two-phase algorithm for determining Pareto optimal solutions. In the future we will consider also other fairness measures, e.g. based on HHI or the Gini-coefficient. Note that our problem is also strongly related to variants of the binary knapsack problems with multiple agents as, for instance, the knapsack sharing problem [2]. 1
Email: [email protected] Email: [email protected] 3 Email: [email protected] (Corresponding author.) 4 Email: [email protected] Supported by the Austrian Science Fund (FWF): P 23829-N13. 2
M. Naldi et al. / Electronic Notes in Discrete Mathematics 55 (2016) 65–68
2
67
Bicriteria model
We use a set of decision variables as in the standard 0-1 knapsack model: Variable xij ∈ {0, 1} equals 1 iff the j-th project of department i is accepted. Then the budget bi (x) allocated to the i-th department and the overall expected profit π(x) for the company may be expressed respectively as bi (x) = s x and π(x) = ij ij j∈Ji i∈D j∈Ji rij sij xij . The total investment by the company is obviously limited through a knapsack constraint by the available budget b. We indicate by X the set of feasible allocations, i.e. d×n X = {x ∈ {0, 1} : i∈D bi (x) ≤ b}. Under any allocation x ∈ X , the maximin fairness index is then expressed by F (x) = mini∈D bi (x) and the obvious resulting bicriteria knapsack model for the overall profit and fairness maximization can be written as follows: max {(π(x), F (x)) : x ∈ X }
(1)
In this work we study P (λ) as a surrogate model of (1) for any fairness bound λ: P (λ) : max{π(x) : F (x) ≥ λ, x ∈ X }. (2)
3
Computational Experiments
We performed a large number of computational tests using Gurobi as ILP solver on randomly generated instances: individual project budgets are drawn from a lognormal distribution and the corresponding ROI values from a uniform distribution (whose lower bound reflects the assumption that only projects with a positive estimated return larger than the given interest rate are considered). The lognormal model has been validated in [5] for datasets spanning ten years of data. In particular, the following values have been set for the parameters of the lognormal model: μ = 5.2 and σ = 1.35. These values are quite central in the range observed in the analyzed datasets and give a mean project size of 451 M$. We generated 5 different classes of 1000 instances each. The instances in each class share the same budget value b, the number of departments d ∈ {2, 5, 10} and the number of projects for each department n ∈ {10, 50, 100}. We performed a statistical analysis on the test instances set to measure the distribution of fairness, normalized by the maximum fairness that can be achieved, i.e., the fairness that would result if all the departments got the same share of the budget (max F = b/d). The resulting empirical probability
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M. Naldi et al. / Electronic Notes in Discrete Mathematics 55 (2016) 65–68
density function (obtained through a Gaussian kernel approach) gets more and more slanted towards 1 and exhibits a diminishing dispersion as λ grows. Most importantly, we investigate how the value of λ in (2) impacts the average profit and fairness in the solution of P (λ). Since the optimization procedure aims at maximizing the profit, and the quest for fairness is considered as a constraint, we expect fairness to be achieved at the expense of profit, the more so the higher the λ-threshold is set. Such a behavior is convincingly verified by the data for all instance classes. In fact, the correlation between fairness and profit turns out to be a startling -0.9975, i.e., profit and fairness exhibit a nearly perfect linear anticorrelation. However, taking a different point of view, it can be shown that the relationship of profit and fairness to λ is not linear. Indeed, if we progressively raise the fairness bound λ, the average fairness grows more rapidly than its minimum guaranteed value λ especially as λ is increased over 50% of the budget value b. Moreover, we can observe that the simultaneous decrease in profits is also non linear. At the time being, additional experiments are in progress aiming at better characterizing the trade-off between profit and fairness criteria.
References [1] D. Bertsimas, V.F. Farias, N. Trichakis, The price of fairness, Operations Research 59, 17–31, 2011. [2] M. Fujimoto, T. Yamada, An exact algorithm for the knapsack sharing problem with common items, European J. of Operational Research 171, 693–707, 2006. [3] G. Kozanidis, Solving the linear multiple choice knapsack problem with two objectives: profit and equity, Computational Optimization and Applications 43, 261–294, 2009. [4] M. Mandell, Modeling effectiveness-equity tradeoffs in public service delivery systems, Management Science 37, 467–482, 1991. [5] M. Naldi, A Probability Model for the Size of Investment Projects, in: UKSimAMSS 9th IEEE European Modelling Symposium on Mathematical Modelling and Computer Simulation, Madrid, 169–173, 2015. [6] G. Nicosia, A. Pacifici, U. Pferschy, Competitive subset selection with two agents, Discrete Applied Mathematics 159, 1865–1877, 2011. [7] G. Nicosia, A. Pacifici, U. Pferschy, Price of Fairness for Allocating Bounded Resources, arXiv Preprint Series, no. 1508.05253, 2015.
Electronic Notes in Discrete Mathematics 55 (2016) 65–68
www.elsevier.com/locate/endm
Maximin Fairness in Project Budget Allocation Maurizio Naldi a,1 Gaia Nicosia b,2 Andrea Pacifici a,3 Ulrich Pferschy c,4 a
b
Dipartimento di Ingegneria Civile e Ingegneria Informatica, Universit` a di Roma Tor Vergata, Roma, Italy
Dipartimento di Ingegneria, Universit` a Roma Tre, Roma, Italy c
Department of Statistics and Operations Research, University of Graz, Graz, Austria
Abstract This work addresses a multi agent allocation problem in which multiple departments compete for shares of a company budget. Each department has its own portfolio of projects with given expected profits and costs and selects an optimal subset of its projects consuming its assigned budget share. Besides considering the total profit of the company a central decision maker should also take fairness issues into account. Thus, we introduce an equity criterion based on maximin fairness. The resulting trade-off between total profit and fairness indicators is studied in this contribution. To this purpose a bicriteria ILP model is presented where one of the objectives is the maximization of the overall profit and the other is the maximization of the minimum budget allocated to one of the departments. We perform an experimental analysis showing a nearly perfect linear anticorrelation between profit and fairness index values. Keywords: Knapsack problem, project management, multi agent system, fair allocation, decision support system.
http://dx.doi.org/10.1016/j.endm.2016.10.017 1571-0653/© 2016 Elsevier B.V. All rights reserved.
66
1
M. Naldi et al. / Electronic Notes in Discrete Mathematics 55 (2016) 65–68
Introduction
We consider an allocation problem faced by the general manager of a company with a set D of d departments. A total budget b has be allocated to several projects, each belonging to one of the departments. We indicate by Ji the set of projects of department i ∈ D and w.l.o.g. assume that |Ji | = n, for i ∈ D. The j-th project of the i-th department requires a budget sij , which must be obtained in full for the project to be undertaken, and yields an estimated return on investment (ROI) rij (i = 1 . . . d, j = 1 . . . n). Since the overall required budget i∈D, j∈Ji sij is usually larger than the available budget b, the company must select a subset of the projects submitted for funding. Naturally, the company’s objective is the maximization of the total profit obtained from the budget investment. However, this may correspond to selecting projects in a way that can be perceived as unfair by one or more departments, since the budget may well be allocated in an unbalanced way. To avoid such potentially biased solutions the company should take into account some idea of equity or fairness in the allocation decision. In economic analysis, an axiomatic characterization of what might be a fair resource allocation has been the subject of several studies in the last decades. In the context of optimization, even though some studies date back to the Nineties ([4]), only recently “fairness” concepts received considerable attention (see e.g. [1], [6] and [7]). Here, we formulate the allocation decision as a bicriteria problem in which both profit and fairness maximization are considered. In this paper we follow the idea of Rawlsian justice and adopt a maximin fairness approach, i.e., we aim at maximizing, together with the overall profit, the minimum value of the budget allocated to a department. A similar problem has been addressed in [3], where the author uses range as a fairness indicator, i.e. the maximum difference between the budget allocated to any two departments. The author presents an ILP model and a two-phase algorithm for determining Pareto optimal solutions. In the future we will consider also other fairness measures, e.g. based on HHI or the Gini-coefficient. Note that our problem is also strongly related to variants of the binary knapsack problems with multiple agents as, for instance, the knapsack sharing problem [2]. 1
Email: [email protected] Email: [email protected] 3 Email: [email protected] (Corresponding author.) 4 Email: [email protected] Supported by the Austrian Science Fund (FWF): P 23829-N13. 2
M. Naldi et al. / Electronic Notes in Discrete Mathematics 55 (2016) 65–68
2
67
Bicriteria model
We use a set of decision variables as in the standard 0-1 knapsack model: Variable xij ∈ {0, 1} equals 1 iff the j-th project of department i is accepted. Then the budget bi (x) allocated to the i-th department and the overall expected profit π(x) for the company may be expressed respectively as bi (x) = s x and π(x) = ij ij j∈Ji i∈D j∈Ji rij sij xij . The total investment by the company is obviously limited through a knapsack constraint by the available budget b. We indicate by X the set of feasible allocations, i.e. d×n X = {x ∈ {0, 1} : i∈D bi (x) ≤ b}. Under any allocation x ∈ X , the maximin fairness index is then expressed by F (x) = mini∈D bi (x) and the obvious resulting bicriteria knapsack model for the overall profit and fairness maximization can be written as follows: max {(π(x), F (x)) : x ∈ X }
(1)
In this work we study P (λ) as a surrogate model of (1) for any fairness bound λ: P (λ) : max{π(x) : F (x) ≥ λ, x ∈ X }. (2)
3
Computational Experiments
We performed a large number of computational tests using Gurobi as ILP solver on randomly generated instances: individual project budgets are drawn from a lognormal distribution and the corresponding ROI values from a uniform distribution (whose lower bound reflects the assumption that only projects with a positive estimated return larger than the given interest rate are considered). The lognormal model has been validated in [5] for datasets spanning ten years of data. In particular, the following values have been set for the parameters of the lognormal model: μ = 5.2 and σ = 1.35. These values are quite central in the range observed in the analyzed datasets and give a mean project size of 451 M$. We generated 5 different classes of 1000 instances each. The instances in each class share the same budget value b, the number of departments d ∈ {2, 5, 10} and the number of projects for each department n ∈ {10, 50, 100}. We performed a statistical analysis on the test instances set to measure the distribution of fairness, normalized by the maximum fairness that can be achieved, i.e., the fairness that would result if all the departments got the same share of the budget (max F = b/d). The resulting empirical probability
68
M. Naldi et al. / Electronic Notes in Discrete Mathematics 55 (2016) 65–68
density function (obtained through a Gaussian kernel approach) gets more and more slanted towards 1 and exhibits a diminishing dispersion as λ grows. Most importantly, we investigate how the value of λ in (2) impacts the average profit and fairness in the solution of P (λ). Since the optimization procedure aims at maximizing the profit, and the quest for fairness is considered as a constraint, we expect fairness to be achieved at the expense of profit, the more so the higher the λ-threshold is set. Such a behavior is convincingly verified by the data for all instance classes. In fact, the correlation between fairness and profit turns out to be a startling -0.9975, i.e., profit and fairness exhibit a nearly perfect linear anticorrelation. However, taking a different point of view, it can be shown that the relationship of profit and fairness to λ is not linear. Indeed, if we progressively raise the fairness bound λ, the average fairness grows more rapidly than its minimum guaranteed value λ especially as λ is increased over 50% of the budget value b. Moreover, we can observe that the simultaneous decrease in profits is also non linear. At the time being, additional experiments are in progress aiming at better characterizing the trade-off between profit and fairness criteria.
References [1] D. Bertsimas, V.F. Farias, N. Trichakis, The price of fairness, Operations Research 59, 17–31, 2011. [2] M. Fujimoto, T. Yamada, An exact algorithm for the knapsack sharing problem with common items, European J. of Operational Research 171, 693–707, 2006. [3] G. Kozanidis, Solving the linear multiple choice knapsack problem with two objectives: profit and equity, Computational Optimization and Applications 43, 261–294, 2009. [4] M. Mandell, Modeling effectiveness-equity tradeoffs in public service delivery systems, Management Science 37, 467–482, 1991. [5] M. Naldi, A Probability Model for the Size of Investment Projects, in: UKSimAMSS 9th IEEE European Modelling Symposium on Mathematical Modelling and Computer Simulation, Madrid, 169–173, 2015. [6] G. Nicosia, A. Pacifici, U. Pferschy, Competitive subset selection with two agents, Discrete Applied Mathematics 159, 1865–1877, 2011. [7] G. Nicosia, A. Pacifici, U. Pferschy, Price of Fairness for Allocating Bounded Resources, arXiv Preprint Series, no. 1508.05253, 2015.