Multi-period resource allocation for estimating project costs in competitive bidding

Abstract

In competitive bidding for project contracts, contractors estimate the cost of completing a project and then determine the bid price. Accordingly, the bid price is markedly affected by the inaccuracies in the estimated cost. To establish a profit-making strategy in competitive bidding, it is crucial for contractors to estimate project costs accurately. Although allocating a large amount of resources to cost estimates allows contractors to prepare more accurate estimates, there is usually a limit to available resources in practice. To the best of our knowledge, however, none of the existing studies have addressed the resource allocation problem for estimating project costs in competitive bidding. To maximize a contractor’s expected profit, this paper develops a multi-period resource allocation method for estimating project costs in a sequential competitive bidding situation. Our resource allocation model is posed as a mixed integer linear programming problem by making piecewise linear approximations of the expected profit functions. Numerical experiments examine the characteristics of the optimal resource allocation and demonstrate the effectiveness of our resource allocation method.

Appendix: Probability of winning

The function ${\mathcal {P}}_i$, i.e., the probability of winning a contract i is necessary to calculate the expected profit (1). By following Takano et al. (2014), this “Appendix” describes the formula for computing the probability of winning derived by Friedman (1956), who assumes that the number of competitors follows a Poisson distribution and their bid prices follow identical gamma distributions.

Let us suppose that the bid prices of the competitors for contract i have the same probability density function:

$$\begin{aligned} {\mathcal {F}}_i(y) = y^{\kappa _i - 1}\frac{\exp (-y/\theta _i)}{(\kappa _i - 1)!\,\theta _i^{\kappa _i}},~y \ge 0, \end{aligned}$$

(12)

where y is the bid price, $\kappa _i\,(\ge$1) is a shape parameter, and $\theta _i\,(>$0) is a scale parameter. The mean and variance of the gamma distribution (12) are $\kappa _i \theta _i$ and $\kappa _i \theta _i^2$, respectively. Moreover, suppose that the number of competitors who bid for contract i has the following probability mass function:

$$\begin{aligned} {\mathcal {G}}_i(k) = \frac{\lambda _i^k}{k !}\exp (-\lambda _i),~k=0,1,2,\ldots , \end{aligned}$$

(13)

where k is the number of competitors who bid for contract i, and $\lambda _i\,(>$0) is a parameter which represents both the mean and variance of the Poisson distribution (13). It then follows from Friedman (1956) that

$$\begin{aligned} {\mathcal {P}}_i [b]= & {} \sum \limits _{k=0}^{\infty } {\mathcal {G}}_i(k) \left( \int _{b}^{\infty } {\mathcal {F}}_i(y)\,\text{ d }y \right) ^{k}\nonumber \\= & {} \exp \left( -\lambda _i \left( 1 - \exp \left( - \frac{b}{\theta _i} \right) \sum \limits _{\ell =0}^{\kappa _i - 1} \frac{1}{\ell !} \left( \frac{b}{\theta _i} \right) ^{\ell } \right) \right) , \end{aligned}$$

(14)

where b is a contractor’s bid price.