Network Analysis - Resource Restrictions

8/14/2019 Network Analysis - Resource Restrictions

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Network analysis - resource restrictions

One important extensions to the basic network analysis technique relates to resource

restrictions.

Typically, in real-world network analysis, each activity has associated with it someresources (such as men, machinery, materials, etc). We mentioned before that, in

calculating the minimum overall project completion time, we took no account of any

resource restrictions. To illustrate how network analysis can be extended to deal with

resource restrictions consider the activity on node network as shown below.

The Gantt chart below illustrates when each activity can take place. To meet the

minimum project completion time of 24 weeks for the above project critical activities

must take place at fixed points in time. For non-critical activities however we have

flexibility (within limits) as to precisely when these activities take place.

The interpretation of the Gantt chart below the package displays both the earliest and

latest times for each activity and the critical path for the project (using suitable

colours). Each activity has two horizontal bars. The first bar represents when it willtake place if it is started at its earliest start time. The second bar represents when it

will take place if it is started at its latest start time. Obviously for critical activities the

two bars are identical.


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Hence for this project the critical activities are 1,3,5,7,8,9 and 11 and the non-criticalactivities are 2,4,6 and 10.

We consider two different types of resource restricted problem:

resource smoothing (also known as resource levelling) problems

resource constrained (also known as resource allocation) problems

Note here that dealing effectively with resource restrictions is hard (both

computationally and theoretically). Resource constrained problems are very hard to

solve and resource smoothing problems, although easier, are also hard to solve.

The word resource here is used to denote any item (e.g. men, machinery, materials,

cash) needed to ensure activities take place. Resources can essentially be classified as:

renewable per time period (e.g. we have a certain number of men that are

available each and every time period); or

non-renewable over the lifetime of the project (e.g. we might have a certain

budget for the project and each sum of money that is spent eats into this

budget).


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Resource smoothing

Suppose now that we have just one resource (men) associated with each activity and

that the number of men required are:

2 men for activity 1 1 man all the other activities (activities 2 to 11 inclusive)

The spreadsheet is shown (in part) below. Column B in that spreadsheet gives the

resource usage for each activity, column C a suggested start time (if we wish to

impose our own suggested start time for each activity). Columns D and E show the

start and finish time for each activity. The values for the cells in these columns are set

using the underlying precedence relationships in the network seen above. Cell D13

shows the end time (when all activities have been completed, i.e. the project

completion time).

Suppose now that we decide:

we wish to meet the minimum overall project completion time of 24 weeks

(hence implying that the times at which critical activities occur are fixed)

we wish to start all non-critical activities at their earliest possible start times

then in the light of these decisions what does the plot (profile) of resource usage(number of men used) against time look like?

Using our spreadsheet we can see that the plot of resource usage against time is as

below.


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The peak of the resource profile is associated with the start of the project, when

activity 1 requires 2 men and the other activities, which are being performedsimultaneously with activity 1, require one man.

A key question is:

What resource usage profile would we have most liked to have seen here?

Clearly what we would have most liked to have seen here is a constant profile of

resource against time, i.e. a constant usage of resource over time. This is because

variations from a constant (straight line) profile most likely cost us extra money -

either in terms of hiring extra resource to cover peaks in the resource profile, or interms of unutilized resources when we have troughs in the resource profile. Whilst it

may not be possible to achieve an ideal resource profile we should keep this ideal in

mind.

Now another key question is:

Is the resource profile that we see fixed or can it be altered?

The simple answer is that the resource profile is not fixed. It can be altered. Simply

put changing the start time for an activity changes the resource profile. This isillustrated below where we have changed the start time for activity 2 to time 2. It can

be seen that the project still completes in 24 weeks, but the resource profile is

different.


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So is our current usage of resource ideal? Plainly not, but what flexibility do we have?

If we still wish to complete in 24 weeks we can do NOTHING with regard to the

critical activities.

However we have some choice for the non-critical activities. Recall that such

activities have an associated float (slack) time. We could artificially delay starting

some of these activities. If we do so the resource profile will change, maybe for the

better. Indeed this is what we did implicitly above. There, in delaying the start of

activity 2 until time 2, we still completed the project on time, but had a different

resource profile. In fact for this particular example it is easy to see that delaying

starting activity 2 until time 6 leads to a better resource profile, and is still feasible in

terms of the 24 weeks completion time. Delay starting activity 2 until time 6, when

activity 1 will have finished we have:


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Here we clearly have a resource profile more in line with our ideal of a constant

resource usage.

It is important to note however that artificially delaying the start of non-critical

activities in order to improve a resource profile is not free. Rather it costs us. Simply

put time lost by delaying the start of an activity cannot later be regained (if things do

not turn out as planned) given the desired fixed completion time for the overall

project.

This has illustrated the resource smoothingorresource levellingproblem, which canbe stated as:

Given afixedoverall project completion time (which we know is feasible with

respect to the resource constraints) "smooth" the usage of resources over the

timescale of the project (so that, for example, we do not get large changes

between one week and the next in the number of men we need).


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This smoothing process makes creative use of floatto artificially delay activities inorder to smooth resource usage.

There are two disadvantages to smoothing:

if we have multiple resources then we may well find that smoothing oneresource makes another resource less smooth, hence we have to make tradeoffs

between resource profiles. We will consider a problem with multiple resources

later below.

time lost cannot be regained, once we have delayed an activity to smooth a

resource profile then, if things go wrong later (e.g. some activities take longer

than we planned) we cannot regain the time we have lost

Extending the example

You may have noticed that in the above example the resource used per time period

never went above the desired resource level of 2. This, in fact, was purely

coincidental, nothing in the Solver model for resource smoothing prevented the

resource used in any time period exceeding this level.

To illustrate this, and to give an example that is less easy to smooth by eye consider

the same network as before but now with a different resource usage for each activity,

as seen in the spreadsheet below.


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Note that for this example we do exceed the desired resource level of 7 at times, and

also (unlike our previous example) there is a "gap" between the end of activity 2 and

the start of activity 4.


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Now it may be that we consider the above resource profile is not smooth enough. We

could however investigate the effect of increasing the project duration. This may

enable us to produce a better resource profile. It certainly has one important effect -

whereas before critical activities were fixed in time to meet the overall minimum

project completion time, now if we are prepared to see an increase in project duration

these are not fixed in time but their start times can be varied, with consequently

potential beneficial effects upon the resource profile.

Considering, for example, a completion time of 25 weeks the changes necessary to

Solver are shown below.

It can be seen that the only change relates to the constraint for the project completion

time, which is now constrained to be exactly equal to 25. Note in passing that now wecan genuinely adjust the start times for all activities (cells C2 to C12 inclusive). This

is because some critical activities can perhaps be delayed in meeting the completion

time of 25 weeks.

After a number of random starts with Solver we have the solution below


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with the resource profile and Gantt chart being


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Here the Gantt chart shows that activity 6 has been inserted between the previously

successive critical activities of 5 and 7. Visually the resource profile seems smoother

for this 25 week solution than for the 24 week solution - yet the objective value of 171

is larger than the previous objective value of 168. How then can we explain this?

The answer lies in the way we defined our objective - as the sum of the squared

deviations from the desired resource level over the time period. Hence the 171 relates

to a 25 week period, and the 168 to a 24 week period. Comparing them on a weekly

basis (mean squared deviation from desired level per week) the 25 week solution has a

weekly value of 171/25 = 6.84 and the 24 week solution has a weekly value of 168/24

= 7.00. Hence on a weekly basis the 25 week solution is indeed "more smooth" than

the 24 week solution (as we would expect from a visual comparison).

Continuing in this vein the best solution I can manage to produce for a 26 week

completion using Solver is shown below.


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On a weekly evaluation this has a smoothness of 196/26 = 7.54, so less smooth than

either the 24 week or the 25 week solution.

Note here because of the inherent mathematical difficulties in solving a nonlinear

optimisation problem alluded to before we can never be sure whether ANY

solution is the best solution (i.e. optimal) that can EVER be found (logically there

must be a set of start times that yield the minimum value for the smoothing

objective we have adopted) or not.

However, the solutions we have are the best we can get (optimal or not!) and we now

have a decision problem - given possible completion times:

24 weeks, smoothness value 7.00



which completion time would you choose?

The situation here is exactly analogous to the one that occurred when we

considered cost/time tradeoff. The package gives you information as to the effect of

trading off completion time against smoothness. The actual decision as to the

completion time to choose is one that must be taken by a human decision-maker.
http://people.brunel.ac.uk/~mastjjb/jeb/or/netcpm.htmlhttp://people.brunel.ac.uk/~mastjjb/jeb/or/netcpm.html


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Two resource smoothing

To expand our example suppose that we now have two resources associated with each

activity. If you clickhere you will be able to download an Excel 97 spreadsheet called

res2.xls with which you can explore using two resources in the network above. The

examples given below are produced using this spreadsheet.

In this spreadsheet the data for Resource 1 can be found under theResource 1 data taband the data for resource 2 can be found under theResource 2 data tab. The data for

resource 1 is

and the data for resource 2 is
http://people.brunel.ac.uk/~mastjjb/jeb/or/res2.xlshttp://people.brunel.ac.uk/~mastjjb/jeb/or/res2.xls


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Now if we wish to smooth this profile, for the desired resource level of 2 for resource

1 and the desired resource level of 7 for resource 2 we can simply add together the

squared deviations from desired resource level for each resource in order to generate a

single objective to be minimised. Note here that this assumes that each resource is

equally important. Whilst we could weight our objective to place differing importance

on each resource this is an unnecessary complication here.

In the spreadsheet the objective value shown in theResource 2 data tab sheet is the

total squared deviations from desired level for resource 2. The objective value shown

in theResource 1 data tab sheet is the total squared deviations from desired level forresources 1 and 2 combined. We can now carry out the same procedure as before. The

Solver parameters are set in theResource 1 data tab sheet and are as before (seebelow) for a completion time of 24 weeks.

After a number of tries using Solver the best solution I can obtain is shown below.


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The corresponding resource profile and Gantt chart are shown below.


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It can be seen that in this case activity 6 is delayed until some time after the end of

activity 4.

On a weekly evaluation this solution has a smoothness of 204/24 = 8.5. Continuing as

we did for the single resource example above, the best solution I can find for a

completion time of 25 weeks is:


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which has a weekly smoothness value of 185/25 =7.4, and the best solution I can find

for a completion time of 26 weeks is:

which has a weekly smoothness value of 212/26 = 8.15.


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Hence, as we did before for a single resource, we have the information to make an

informed decision as to which completion time to have:

24 weeks with a (weekly) smoothness value of 8.5,

25 weeks with a smoothness value of 7.4, or

26 weeks with a smoothness value of 8.15.

For interest the resource profile for the lowest smoothness value of 7.4 found is shown

below.

Resource constrained problems

So far in smoothing a resource profile we have been prepared to accept resource usage

above our desired resource level. In some circumstances however we have an absolute

constraint upon the maximum level of resource that can be used in any time period.

To illustrate this if you clickhere you will be able to download an Excel 97

spreadsheet called res3.xls with which you can explore resource constrained problemsin the network we are considering. The examples given below are produced using this

spreadsheet.

To remind you of it the network for the project we are considering is


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The spreadsheet is shown below.

The resource profile for the start times shown above is as below.


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Here we have assumed that we wish to impose an absolute limit of 8 on the usage of

our single resource in any time period. The current resource profile violates this

constraint and this is indicated by cell C13 in the spreadsheet saying "Infeasible".

Whilst we know that from the time point of view the minimum project completion

time is 24 weeks a key question is: can we still achieve the same minimum project

completion time when a resource constraint is added?

The answer to this question is that sometimes it is impossible to achieve the same

completion time. For this problem, for example, activities 9 and 10 are performed

simultaneously in achieving the minimum overall project completion time of 24

weeks (e.g. consider the network diagram shown above). Activity 9 requires 6

resource units, activity 10 requires 6 resource units. It is clear therefore that, in the

presence of a resource constraint of 8 units these two activities cannot be performed

simultaneously. Hence for this network we immediately know that the original

completion time of 24 weeks is no longer feasible.

The question therefore arises as to what is the minimum overall project completiontime for this project when resource constraints are taken into account.

This has illustrated the resource constrainedorresource allocationproblem, which

can be stated as:

Given the activities, their associated resource usage, precedence relationships

and resource restrictions (typically expressed as the amount of each resource


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available per time period) find the minimum overall project completion time

(which satisfies both the precedence relationships and the resource restrictions).

Note here that resource constrained problems are much harder (both theoretically and

computationally) to solve than resource smoothing problems. This will become

apparent computationally below if you use the spreadsheet to attempt to solve aresource constrained problem. Whilst for our small example we might well be able to

see using our eyes/brain what is the minimum overall project completion time (which

satisfies both the precedence relationships and the resource restrictions) for

larger/more complex examples this is impossible.

Estimating the minimum project completion time

In order to gain some insight into how we might go about estimating the minimum

project completion time when we have a resource constraint of 8 we can start from theresource unconstrained solution. This is shown below together with the resource

profile and Gantt chart.


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Examining the resource profile the first point at which we exceed the resource

constraint of 8 is at time 4-5. Examining the Gantt chart this is associated with activity6 (requiring 5 resource units) being performed simultaneously with activity 1 (which

requires 4 resource units).

This illustrates a general truth: increases in the resource profile can only be caused

by one (or more) activities starting


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Examining the resource profile above the first point at which we now exceed the

resource constraint of 8 is at time 20-21. Examining the Gantt chart above this is

associated with activity 9 (requiring 6 resource units) being performed simultaneously

with activity 10 (which requires 6 resource units). Plainly one (or other) of these

activities needs to be delayed. Here, we choose to delay the activity with the smallest

completion time (activity 10) until activity 9 has finished. (More logically we

typically choose to delay activities which are non-critical in the unconstrained case,

and if there are more than one such activity choose the one with the smallest

completion time). Hence we have a suggested start time for activity 10 of 23. Enteringthis into the spreadsheet we get:

which we can see is declared to be feasible (cell C13 above, i.e. it satisfies the

resource constraint), with a resource profile of:


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Hence we have a feasible solution to the problem of determining the start time for

each activity so as to respect the resource constraints and the network precedence

relationships. This solution involves completing the project in 25 weeks. This may, or

may not, be the optimal (minimum) project completion time when we have our

resource constraint of 8 present.

Hence we can suggest a general procedure (algorithm) for estimating the minimum

possible project completion time in the presence of resource constraints as follow:

take the start times associated with the resource unconstrained project

Then repeat the following until the resource constraints are satisfied:

examine the resource profile

take the first point at which a resource constraint is violated and examine the

activities associated with that violation. This violation must be associated with

one (or more) activities starting.

from the activities which were starting choose the activity (which is non-criticalin the resource unconstrained case) with the smallest completion time

attempt to remove the violation by preventing the chosen activity from starting

until the next time at which an activity ends


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Although we did not phrase our approach above in these terms you should be able to

see that if we had set out this general approach first, and then applied it to the above

example, we would have ended up with exactly the same solution.

Minimum resource constrained project completion time

Now we might expect that instead of estimating the minimum possible project

completion time we could estimate it directly using our Excel spreadsheet and Solver.

Even though we cannot guarantee to find the minimum possible project completion

time we might perhaps expect (certainly for this small example) Solver to perform

well. The Solver input parameters for this example, in order to estimate the minimum

resource constrained project completion time, are as below.

The target cell to be minimised is cell D13, which contains the project completion

time. The constraint forcing D13 (the project completion time) to be


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indicating that Solver could not find a solution which satisfied the constraints of the

problem. The difficulty here is in ensuring that the resource constraints for each and

every period are satisfied.

Hence again we need to provide Solver with a starting solution. My experience here

has been that Solver works better if it is provided with a starting solution which isalready feasible rather than a starting solution which is infeasible. One way of doing

this, for this particular example, is to set the suggested start time for the first activity

to 0 and the suggested start time for all other activities i equal to the finish time of

activity (i-1). Doing this gives the feasible starting solution:

Unfortunately Solver fails to improve on this solution - which is disappointing given

that we know that we have estimated the minimum project completion time as 25.Indeed it is especially disappointing since changing the suggested start time for

activity 2 in the spreadsheet above immediately gives:

which is a better feasible solution, which again Solver fails to improve on.


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Consider now the Solver input parameters shown below. Here we have a similarmodel as before, but now we are minimising cell D14 (which contains the project

completion time plus a resource based term, as discussed above). Also we no longer

need a constraint relating to resource in Solver - if when we solve the solution has

value less than 100 it is automatically feasible, if the solution has value more than 100

it is infeasible.

Technically this incorporation into the Solver objective function of the resourceconstraint is known as a penalty based (penalty function) approach.

Starting from the solution below (see the discussion previously)


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and using Solver we get

which is equal to the (feasible) solution we found before. This example illustrates that

thinking laterally when faced with a system like Solver (which prefers fewer

constraints) can pay substantial benefits.

Combining resource smoothing and resource constraints

Clearly it is possible to combine resource smoothing and resource constraints. To

illustrate this suppose we take the resource constrained problem considered above and

seek to smooth the project within our 25 week deadline with a desired resource level

of 7 (note here that the desired resource level need not be equal to the maximum

resource level, 8 in this case). Using Solver to achieve this we will use the

spreadsheet res4.xls we used above. The spreadsheet (starting from the starting

solution discussed above) and Solver parameters will be:


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where the constraint D14=25 ensures that we stay feasible with a project completion

time of 25. The solution obtained is shown below.


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with the resource profile and Gantt chart being


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It can be seen from the start times given above that the smoothing process has made

no difference to the times we had derived previously. However, we did not know this

beforehand and consequently attempting smoothing has told us that the resource

profile is already "as smooth" for a 25 week completion time as we can make it.

Network Analysis - Resource Restrictions

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