Resource leveling in projects with flexible structures

Abstract

Project resource leveling, which is usually performed in the project planning phase, aims to minimize fluctuations in resource usage such that the project success probability is enhanced. The existing studies on resource leveling mainly assume a fixed project structure, i.e., the activities and precedence relationships are given in advance. However, in real-world projects, the project structure is usually flexible, i.e., not all activities are implemented, and the precedence relationships exist only between the activities that are implemented. Aiming at providing support for project managers to make effective resource leveling decisions when facing flexible project structures, we study the resource leveling problem with flexible structures (RLP-PS). We present a nonlinear integer programming model for the RLP-PS and a linearization method to transform the nonlinear model into a linear model. To efficiently solve the RLP-PS, we devise two problem-specific algorithms: a two-stage heuristic algorithm and a customized genetic algorithm. Based on the PSPLIB benchmark dataset, extensive computational experiments are performed to analyze the performances of our algorithms. The experimental results reveal the effectiveness and competitiveness of our algorithms.

Appendix: Calculation of the lower bounds

Lower bound ${\varvec{l}}{{\varvec{b}}}_{{\varvec{i}}}\_{\varvec{M}}$. $l{b}_{i}\_M$ is the lower bound of the objective function value of instance $i$ in the most ideal situation. Specifically, for each instance $i$, we consider only the mandatory activities, then the average usage of resource $k$ at each time period is ${\overline{u}}_{k}= ({\sum }_{i\in M}{r}_{ik}\times {d}_{i})/\overline{d}$. This corresponds to the most ideal situation in resource leveling. In this case, the objective function value can be used as the lower bound $l{b}_{i}\_\text{M}={\sum }_{k=1}^{K}{\sum }_{t=1}^{\overline{d}}{({\overline{u}}_{k})}^{2}$.

Lower bound ${\varvec{l}}{{\varvec{b}}}_{{\varvec{i}}}\_{\varvec{L}}{\varvec{a}}{\varvec{g}}{\varvec{r}}$. $l{b}_{i}\_Lagr$ is obtained by applying Lagrangian relaxation to model M1. Specifically, we first relax M1 by moving its constraint (11) to the objective function (10) and the resulting new objective function is

$$\left\{{\sum }_{k=1}^{K}{c}_{k}\sum_{t=1}^{\overline{d} }\sum_{h=1}^{{H}_{kt}}(2h-1){y}_{kth}-{\sum }_{k=1}^{K}{\sum }_{t=1}^{\overline{d}}{\lambda }_{kt}({\sum }_{i\in N}{r}_{ik}{\sum }_{\tau =\mathrm{max}\left\{{es}_{i},t-{d}_{i}+1\right\}}^{\mathrm{min}\left\{t,{ls}_{i}\right\}}{x}_{i\tau }-{\sum }_{h=1}^{{H}_{kt}}{y}_{kth})\right\}$$

(19)

where the multiplier ${\lambda }_{kt}\ge 0$, $k=1,\dots ,K, t=1,\dots ,\overline{d }$. We further rewrite (A1) as:

$$\left\{{\sum }_{k=1}^{K}\sum_{t=1}^{\overline{d} }\sum_{h=1}^{{H}_{kt}}\left[{c}_{k}\left(2h-1\right)-{\lambda }_{kt}\right]{y}_{kth}-{\sum }_{k=1}^{K}{\sum }_{t=1}^{\overline{d}}{\sum }_{i\in N}{\sum }_{\tau =\mathrm{max}\left\{{es}_{i},t-{d}_{i}+1\right\}}^{\mathrm{min}\left\{t,{ls}_{i}\right\}}{\lambda }_{kt}{r}_{ik}{x}_{i\tau }\right\}$$

(20)

Now we obtain the relaxed model LR($\lambda$):

Minimize (20).

Subject to: (2)-(6), (8), (12).

Then $l{b}_{i}\_Lagr$ is obtained by solving the Lagrangian dual of LR($\lambda$) using the subgradient method (Bianco & Caramia, 2012). In each iteration, the Lagrangian multipliers are updated according to the following steps:

• (a) The subgradient values for each relaxed constraint related to resource $k$ and time period $t$ is determined as ${SG}_{kt}={\sum }_{i\in N}{r}_{ik}{\sum }_{\tau =\mathrm{max}\left\{{es}_{i},t-{d}_{i}+1\right\}}^{\mathrm{min}\left\{t,{ls}_{i}\right\}}{x}_{i\tau }-{\sum }_{h=1}^{{H}_{kt}}{y}_{kth}$.

• (b) The step $ST=\beta ({Z}_{UB}-{Z}_{LB}^{*})/{\sum }_{k=1}^{K}\sum_{t=1}^{\overline{d}}S{G }_{kt}^{2}$, where the upper bound ${Z}_{UB}$ is obtained by using the GA to solve the same instance,${Z}_{LB}^{*}$ is the current best lower bound that is obtained by solving LR(λ) with CPLEX, and $\beta$ is a scaling factor with an initial value of 2. If ${Z}_{LB}^{*}$ is not improved after five consecutive iterations, the value of $\beta$ is reduced to half of its current value.

• (c) The Lagrange multiplier ${\lambda }_{kt}=\mathrm{max}(0, {\lambda }_{kt}+ST\bullet {SG}_{kt})$.

  The Lagrangian relaxation algorithm stops when any of the following conditions are met: 300 iterations, or 1800s, or $\beta <0.0001$.