Read the article below and answer ALL the questions that follow.

A robust scheduling for the multi$-$mode project scheduling problem with a given deadline under uncertainty of

activity duration.

Important issues within project scheduling are deadline and cost. Quite often, a specific project must be finished by a

certain deadline else penalty costs may incur or customers and or profit may be lost. Therefore, the most important goal

of the project manager is to finish the project on the predefined deadline and with minimum cost. This problem is known

as a time constrained scheduling problem. When these problems occur in many projects from practice and deadlines

must be met, different ways to speed up the project are employed, e$.$g$.,$ by working in overtime, hiring extra resource

capacity, changing the mode of activities if it is possible or even outsourcing $($parts of$)$ the project. All these options are

costly but often unavoidable. With multi$-$mode projects, two options are considered simultaneously to resolve scheduling

problems with predefined deadlines and uncertainty in the duration of activities where deadlines must be met and with the

least cost. These are to change the mode of activities with a less costly mode and or hiring extra resource capacity. Thus,

the questions arise as to how much, when, and what kind of extra capacity in which mode of activity should be used to

meet the deadline against the minimum cost of applying additional resources.

Uncertainties may lead to considerable schedule interruptions during the project progress. In fact, the uncertainty may

arise from several possible sources such as the activities may take more or less time than originally estimated $($activity

disruptions$),$ resources may become unavailable $($resource disruptions$),$ materials may arrive behind the schedule, ready

times and due dates may change, new activities may be added to the project or some existing activities may be removed

due to changes in the scope of the project, weather conditions may cause several delays, etc. Some reasons for

uncertainty in the duration of activities include incorrect estimation of the task duration and the need to rework according

to the inspector$$s analysis. The proposed problem in this research is unresolved in the literature, especially in the cases

where there is uncertainty in the project environment and a robust scheduling is needed. In addition, for projects in

practice, it is important to obtain a baseline schedule that responds to changes and uncertainties. This provides the

motivation to investigate a solution for the multi$-$mode project scheduling problem with the given deadline under

uncertainty of activity duration. Therefore, the main objective is to obtain a baseline schedule for the proposed problem,

which while meeting the existing limitations and having the lowest cost, will be robust against uncertainties. QUESTION $1(20$ Marks$)$

The article makes reference to Time$$cost trade$-$off problems which is one of the most frequent and critical decisions that

project managers usually make. Using an appropriate diagram, demonstrate, with reference to the article above, how you

would effect the time$-$cost trade$-$off on projects with the various uncertainties $($or variability$),$ as mentioned in the article,

that may impact the project.