In this section, the models proposed in the last few sections are illustrated with numerical values. The proposed models can be illustrated with the data from any project supply chain. However, in this research, the models are mostly illustrated to be fit in with the data from the construction sector. This is because of the enormous importance of the construction sector to the economy. As per the UK Contractor Group (2012), every £1 spent in the construction sector, generates £2.84 in output. Moreover, the construction sector value chain is worth 14% of UK’s overall GDP (UKCG, 2012).
4.5.1
For polynomial discounted cash-flows
It is assumed that the project value follows the following equation q(T ) = 30 − 1.5T where ψ = 0.05 and q0 = £ 30,000. In practice, the value of A in equation (4.1) would usually be
less than 1 with a maximum value of 1, but for simplicity, it is assumed that A = 1 in this numerical analysis. The resource cost per unit time has been considered as kλn. Following Savaskan & Van Wassenhove (2006) and Swami & Shah (2013b), we assume that n = 2. k can be considered as the resource price per unit per unit time. Assuming an average wage of £10 per hour, with 20 hours. per week, the approximate value of k is assigned as £ 200 per week. In the existing literature by Kwon et al. (2010), the authors did not consider the costs incurred by the project manager which are independent of λ. According to Potts & Ankrah (2014), these costs can vary between 50-70 percent of the overall cost. The Co value
contract fails, is assumed πout = £1,800. The value of λ was derived as resource unit per
week.
For polynomial discounting, if m=1, then the project reward is discounted linearly. From equations (4.18) and (4.19), we get the unique set of solutions for the first best profit and first best resource consumption rate for the uniform, gamma, beta and Weibull distributed completion time as π0 = q0− q0.ψ h µ1 λA 0 i − kµ1λN0 − Co and λ0 = Aq0ψ kN A+N1 respectively. The optimal values of h* and g* from equations (4.20) and (4.21) are: h∗ = q0ψ and g∗ =
q0− (π0− πout) − Co. The profits are presented in figure 4.1 and in table D.1 (in appendix
D). An increase in the value of h while keeping g constant reduces(increases) the profit of
Figure 4.1: Profit values and Efficiency for polynomial discounted cash-flows with m=1
the contractor (project manager) and vice-versa. The total profit increases at first. Then, it reaches the maximum first best solution for profit value of 6.24 at g = 10.56 and h = 1.5. After that, it starts to decline. This can be explained by setting the value of h=0. When h attains a value of zero, the contract becomes a fixed price contract equivalent. It yields a value of selected resource consumption rate (λ) as zero and the profit functions become undefined (- ∞). This fails to coordinate the supply chain. This supports the findings from the proposition 2. Thus, any positive entry of h above zero would increase the profit of both the members of the supply chain and eventually total profit at first. However, further increasing h reduced the contractor’s profit and increased the project manager’s profit. In the beginning, the increase in project manager’s profit overcomes the decrease in contractor’s profit. As a result, the total profit increased slowly until it reached the first best solution. After that, the reduction of contractor’s profit was found to be substantial. As a result, the total profit starts to drop from
the maximum value with every positive increment of h.
On the contrary, keeping h constant at the optimal value of 1.5, changing the g values was found to have no impact on the total profit of the decentralized setting. However, this has an impact on the individual profit share of the members in the decentralized setting. Increasing g while keeping h constant increases (reduces) the profit of the contractor (project manager) and vice-versa. Thus, setting h constant at the optimal solution, multiple values of g can yield the total decentralized profit equal to the first best solution. However, if the members of the supply chain can earn a minimum profit outside the present contract, then there exists an individual rationality constraint. This makes some of the solutions to be invalid in the given context.
Furthermore, the values of the exponent m are changed. λ0 was found to be increasing
with the increase in increase in m. This caused a decrease in first best profit π0. In fact, after
m started attaining some higher values of 3 or 4 or above, the optimal π0 values, started to
become negative. On those, cases the proposed model would become invalid for the supply chain under consideration. The results are presented in table D.2 in Appendix D.
4.5.2
For exponential discounted cash-flows
For the numerical example, the effective value of project reward is assumed as q0= £4 million.
α is the continuous discounting rate. According to the Govt. of UK., Cabinet Office (2015), the present discount rate in practice is 3.5%. This research assumed the value as 0.04. Co is
assumed as £2 million. It is also assumed that µ1 is 10 years and πout= £0.25 million.
Furthermore, the per hour wage for the work is assumed as £15. The worker will be employed for 150 hours each month of the year. This approximates to a value of k = 0.03 per year. n is assumed to be 2 as before (to be consistent with the previous literature) and again A is assumed as 1
For recoverable product life
In this case, it was assumed that the product life is recoverable upon completion of the projects. Thus, any project reward reduction due to anticipated revenue loss is not considered. The results for the profits are presented in figures 4.2 and 4.3.
Figure 4.2: Profit values for Uniform Distribution with exponential cash-flows and time- based contracts
Figure 4.3: Profit values for Gamma Distribution with exponential cash-flows and time-based contracts
Once again in the decentralized setting, keeping g as constant at the optimal value (1.55 for uniform, and 1.54 for gamma), increasing h was found to be reducing (increasing) the profit of the contractor (project manager) and vice-versa. Again the total profit increased at first. Then, it reached the maximum first best solution of the profit value (0.77 for uniform and 0.78 for gamma) for the optimal values of h (0.13 for bothe the distributions). After that, these values started to decline.
Similar results are found as found in section 4.5.1 while keeping h constant at the optimal values and changing g. Changing g was not found to have any considerable impact on the overall profit.
For irrecoverable product life
Similar results were derived for this case as for the case with recoverable product life. The product life of the outcome of the project cannot be recovered if there is any delay in com- pletion. Thus, a polynomial reduction of project reward was considered with the discounting factor ψ = 0.05 times the project value per unit time. The results are presented in the figures 4.4 and 4.5.
Figure 4.4: Profit values for uniform distributed time with exponential cash-flows and time- based contracts for irrecoverable product life
Figure 4.5: Profit values for gamma distributed time with exponential cash-flows and time- based contracts for irrecoverable product life
Once again in the decentralized setting, keeping g as constant at the optimal value (1.72 for uniform, and 1.71 for gamma), increasing h was found to be reducing (increasing) the profit of the contractor (project manager) and vice-versa. Again the total profit increased at
first. Then, it reached the maximum first best solution of profit value (0.61 for uniform and 0.62 for gamma) for optimal values of h (0.17 for both the distributions). After that, these values started to decline.
Comparing the results of the two scenarios of the long-term projects in sections 4.5.2 and 4.5.2, there are similar results for both the distributions. Both the contract parameters have increased in the case of the polynomial reduction in comparison to the nonreduction. However, the first best profit has decreased in the case of the polynomial reduction of project reward in comparison to the case with no reduction of reward.