2012 International Conference on Information and Computer Applications (ICICA 2012)
IPCSIT vol. 24 (2012) © (2012) IACSIT Press, Singapore
R&D Multi-Project Optimization Problem
Xingwen Zhang
+
Google Inc.
Abstract. In this paper we formulate the multi-project optimization problem for a R&D firm. The firm is
facing a set of potential projects and needs to select the projects to work on. If a project is selected, the firm
should also simultaneously determine the num- ber of employees assigned to the project. This multi-project
optimization problem is NP-hard, for which both heuristic and metaheuristic methods are developed. Computational results demonstrate that simulated annealing method generates superior project plans.
Keywords: R&D, Project selection, Manpower assignment, Heuristic, Metaheuristics, Simulated
annealing
1. Introduction
In this paper we consider multi-project selection and optimization problem for an R&D firm (see [6], [4],
[5] and [2] for works on R&D management). The firm is considering multiple projects, and has to decide
which of the projects to work on and how much eport to spend on them. We assume that each project has a
fixed setup cost, and that the the expected revenue of the project is ai xi if xi employees are assigned to
project i. For example, by spending more efort on the project we can speed up its development and thus
realize its gain more quickly, which would lead to a larger net present value (NPV). Furthermore, the
revenues of the projects are correlated. For example, the actual realized revenues of the projects could be
a ected by macro-economic variables such as GDP and interest rate. In addition, the R&D firm has limited
manpower, and thus may not able to assign as much manpower as needed to each project. We assume that
the firm pays no extra cost to assign its employees to the projects since the employees’ salaries are
considered as sunk cost for the firm. However, the supply of the manpower is not unlimited, and is capped
above by C .
In the following sections, we will formulate the multi-project optimization problem with limited
manpower. A hill climbing method is used to construct a project plan, and a metaheuristic method is
developed to improve the solution quality. We will present computational results, and conclude the paper
with potential future work.
2. Problem Formulation
We consider the multi-project optimization problem for an R&D firm that has multiple potential projects.
Let N be the number of projects. In each period the firm produces a forecast on the revenue of each project.
It also has forecasts, derived from its understanding of factors like macroeconomic statistics or simply
purchased from an external provider, on the correlation of the revenues. Further, if the firm decides to carry
out a project, it incurs a fixed setup cost for that project. The firm has limited manpower, C , and is not
allowed to use more than C manpower (since, for example, extra hirings would lead to prohibitively large
cost).
+
Corresponding author. : Tel.: + 1 (650-796-6463).
E-mail address: zhangxingwen@gmail.com.
103
Mathematically, let ai denote the expected revenue the firm will gain for assigning a single employee to
project i, and let sij denote the covariance of the unit revenues for project i and j. ki denotes the fixed setup
cost for project i. Then, the firm maximizes its risk-adjusted utility function, f (x), as follows.
N
N
N
x1 ,... xN
i =1
i =1 j =1
s.t.
∑a x
N
max ∑ ai xi − r ∑∑ xi x j sij − ∑ ki1{ xi ≠ 0}
N
i =1
i i
(1)
i =1
≤C
X i ≥ 0 for i = 1,...N ,
(2)
xi is an integer for
i = 1,...N ,
(3)
where 1{·} is the indicator function and r is the risk-adjustment coefficient. Constraint (1) limits the amount
of manpower the firm can utilize.The problem is clearly NP-hard, since the quadratic knapsack problem [10]
is a specialcase of the model presented above. To solve the problem, we will first construct a solutionusing a
hill-climbing based approach. A meta-heuristic approach is then developed toimprove the solution quality.
3. Construction Method
Constructing a good solution for the multi-project optimization with correlated revenues is not a simple
task due to the correlation between the revenues. For instance, the well- known dynamic programming
approach for knapsack problems [9] does not work here, due to lack of sub-problem optimality (i.e.
Bellman’s principle of optimality [3]). Instead, we develop and implement a hill-climbing based construction
method as follows.
We start with the naive solution x = (0, . . . , 0), that is, the firm doesn’t carry out any project. We then
iteratively improve the solution quality by changing xi by δi , with δi ∈ {−1, 1}; we call this operator 1-Opt.
More specifically, in each iteration we go through i ∈ {1, . . . , N } and check whether changing xi to xi + δi
will improve current solution or not. If it does, we modify current solution by setting xi = xi + δi , and repeat
the process. We stop until no further improvement can be made.
To search a larger neighborhood, we also implement a variant of the construction method with 2-Opt
that allows simultaneously changing xi and xj by δi and δj respectively, where i = j and δi , δj ∈ {−2, −1, 1,
2}. In each iteration of this construction method, we modify the allocated manpower of two projects aiming
to improve the risk-adjusted utility for the firm while maintaining the feasibility of the solution. Again, we
repeat the step until there is no further improvement.
4. Simulated Annealing
Simulated annealing (SA) algorithm is used to further improve the solution quality. SA is an extension to
Hill-climbing algorithm. Given a new solution xnew that is better than or equal to the current solution xcurrent ,
SA always accepts xnew . In addition, a move that leads to a solution worse than the current solution is
accepted with probability Paccept = e
+
, where ∆ = f (xnew ) − f (xcurrent ) is the change in the firm’s utility
T
function f (x), and T is a control parameter. Simulated annealing algorithm for our multi-project optimization
problem is described in Algorithm 1 (see [7], [8] and [1] for more references on SA).
In our computational experiment, the initial solution is either x = (0, . . . , 0) or the solution produced by
the construction heuristic, and the initial temperature T is set as 60. The new temperature is calculated by
the formula T ′ = λ T , witse T is the new temperature and λ = 0.995 is the cooling rate. The stop criterion
in Algorithm 1 is to stop the iterations when xcurrent is not updated in the inner loop. The inner loop
criterion is to perform the loop L times. The value of L is set as 30000.
5. Computational Results
104
We implemented the construction method and SA algorithm in Java, and tested the meth- ods with 100
randomly generated problem instances with N = 100 projects. The results in Table 1 show that with 1-Opt
only SA produces much better solutions than the construc- tion heuristic. The average improvement is about
69.04%, at the cost of more computation time (however, on average the SA method took about 80 seconds
on a Intel Core2Duo com- puter with 2 2.26GHz CPUs and 2G RAM). Lastly, when both 1-Opt and 2-Opt
were used the gain of SA decreased to 2.39%, which demonstrated the power of a larger neighborhood move
for the construction heuristic.
Table 1: Statistics of Solutions by Construction Heuristic and Simulation Annealing with 1-Opt
mean
standard deviation
signal to noise ratio
Construction
7018.88
1306.15
5.37
SA
11864.77
2015.54
5.89
6. Conclusion and Future Work
We have formulated the multi-project optimization problem and solved it using construc- tion heuristic
and simulated annealing, and the computational results demonstrated that simulated annealing produced
better results. For future work, we will explore the prob- lems with variable cost for assigning employees to a
project; another possible extension is to consider employee skill types, and only allows employees with
matched skills to work on projects that require those skills. In addition, other popular metaheuristic methods,
such as Tabu Search and Genetic Algorithm, may produce solutions of better quality. Lastly, neighborhood
moves that involve more than two projects are worth exploring as well.
7. References
[1] E. Aarts and J. Korst. Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial
Optimization and Neural Computing. Wiley, Chichester, 1989.
[2] P. Aghion and J. Tirole. The Management of Innovation. Quarterly Journal of Economics, 109(4):1185–1209,
1994.
[3] R. E. Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, 1957.
[4] J. A. DiMasi. New Drug Development in the United States from 1963 to 1999. Clinical Pharmacology &
Therapeutics, 69(5):286–296, 2001.
[5] J. A. DiMasi. Risks in New Drug Development: Approval Success Rates for Investi- gational Drugs. Clinical
Pharmacology & Therapeutics, 69(5):297–307, 2001.
105
[6] J. A. DiMasi, R. W. Hansen, H. G. Grabowski, and L. Lasagna. Research and Development Costs for New
Drugs by Therapeutic Category. A Study of the US Pharmaceutical Industry. Pharmacoeconomics, 7(2):152–169,
1995.
[7]
S. Kirkpatrick and C. Gelatt amnd M. Vecchi. Optimization by Simulated Annealing. Science, 220(4598):671–
680, 1983.
[8] P. J. M. Laarhoven and E. Aarts. Simulated Annealing: Theory and Applications. Reidel, Dordrecht, 1987.
[9] S. Martello. Knapsack Problems: Algorithms and Computer Implementations. John Wiley & Sons, Inc., New
York, NY, 1990.
[10] D. Pisinger. The Quadratic Knapsack Problem-a Survey. Discrete Applied Mathe- matics, 155(5):623–648,
2007.
106