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Dynamic programming method is yet another constrained optimization method of project selection. In this method, you break a complex problem into a sequence of simpler problems. This method provides a general framework of analyzing many problem types. In this framework, you use various optimization techniques to solve a specific aspect of the problem. This method requires your creativity before you can decide if the problem needs to use dynamic programming for its solution.

This method of project selection is a mathematical technique to make a sequence of correlated decisions. The method consists of systematic procedure to determine the best combination of decisions. In the linear programming method of project selection, you have standard mathematical formula. However, in case of the dynamic programming method of project selection, you do not have any standard mathematical formula.

In the dynamic programming method, you use a general method to solve a problem. You must develop the equations used to solve the problem to the requirement of the problem.

Features of Dynamic Programming Method

The dynamic programming method of projects selection has the following features:

• Stages
• States
• Recursive Optimization

Stages

In the dynamic programming method, you structure the problem in multiple stages. You solve each of these stages sequentially, one at a time. Even though you solve one stage at a time, the solution of the problem in one stage defines the characteristics of the problem in the next stage. On a time line of the project planning, you represent each stage at different time frames. However, at times you might realize that some stages do not have any time association. When you formulate a dynamic program for a problem that has such stages, it becomes difficult to recognize the problems.

States

When you define a stage for a problem, you also associate it with a state of the process. The state of a process is the information you need to assess the effect of the decision has on the future action. You do not have to follow any set rules to specify a state. However, it is a critical parameter for dynamic programming method. Specifying a state is more of an art, and requires creativity and deep understanding of the problem.

You must consider the following properties of a state when specifying it:

• You should keep the number of state variables to the minimal because of the cost involved in the computation efforts.
• You must provide sufficient information to enable future decision without considering process used to reach the state.

Recursive Optimization

After you have structured stages and states associated to the stages, you need to develop a recursive optimization procedure. The recursive optimization procedure builds a solution of up to the n number of stages, with one stage at a time in the specified sequence. The recursive procedure solves each stage until an overall optimum solution is available.

You can base the recursive procedure on any of the following induction processes:

• Forward induction process: The recursive procedure starts with the first stage and concludes with the last stage by including each stage sequentially, one stage at a time.
• Backward induction process: The recursive procedure starts with the last stage and concludes with the first stage by including each stage sequentially, one stage at a time, in the reverse order.

Mathematical Formula

Consider that you have a multistage decision process where the return of the specific stage is represented by the following function (f):

Here:

• d_n: is the decision that you can chose form the set D_n.
• s_n: is the state of the process with n stages remaining in the N number of stages in the procedure.

The next state of the process completely depends on the current state and decision of the process. Therefore, you can define the transition function (t) with n-1 stages to be remaining in the procedure as:

Now that the dynamic programming method consists of a recursive optimization procedure, you can use the following optimal value function, which represents the maximum return possible over the n stages remaining:

Subject to:

The function  uses only the decision variable  and not the decision variable . Therefore, you can first maximize the latter part of the equation and then chose  to maximize the entire equation.

You can now rewrite the equation as follows:

Multi objective programming is another type of constrained optimization method of project selection. In this method, you make decision for multiple problems with mathematical optimization. In case, in a multi objective programming, a single solution cannot optimize each of the problems, then the problems are said to be in conflict and there is a probability of multiple optimal solutions. A solution is called as non dominated if values of none of the problem can be optimized without degrading values of another problem.

There are multiple terms used to define multi objective programming, such as multi objective optimization, vector optimization, multi criteria optimization, multi attribute optimization, or Pareto optimization. This method is an area of making decisions based on multiple criteria. You make decision mainly based on mathematical optimization of problems that requires you to simultaneously optimize more than one objective function. In this method of project selection, you make an optimal decision in presence of a trade off among multiple conflicting objectives. For example, minimizing cost, maintaining quality control, and adhering to deadlines are common multi objective optimization problems you face in most of the projects.

You can use the following methods of multi objective programming methods for selecting a project:

• No Preference Methods
• A Priori Methods
• A Posteriori Methods
• Interactive Methods
• Hybrid Methods

No Preference Methods

In a no preference method of multi objective programming, you do not require any preference information from the decision maker. One of the most common no preference methods you can use is the method of global criterion. In the global criterion method, you solve the scalar problem in the following form:

Subject to 

The global criterion method is sensitive to objective function scaling. Therefore, you must normalize the objectives into a uniform and dimensionless scale.

A Priori Methods

In contrast to the no preference methods, in the a priori methods, you need to specify sufficient preference before the solution process. The most common a priori methods that you can use include utility function method, and lexicographic method.

For example, in a utility function method, you assume that the utility function is available. A utility function, such as a mapping u: Y→R, specifies an ordering of the decision vectors. After you have got the value for u, you solve the problem in the following form:

Subject to  

Similarly, in a lexicographic method, you assume that you can rank objectives by the importance. Therefore, you assume that objective functions are ranked by importance such that f₁ is most important and f_k is least important. In the lexicographic method, you solve a sequence of single objective optimization problems in the following form:

Here, y_j is the optimal value of the problem with n = m.

A Posteriori Methods

In the a posteriori methods, your goal is to prepare all the Pareto optimal solutions or subset of the Pareto optimal solutions. You can categories most of the a posteriori methods into the following classes:

• Mathematical programming based a posteriori methods: In these methods, you repeat the algorithm and with each run of the algorithm, you produce a Pareto optimal solution. Examples of this class include Normal Boundary Intersection (NBI), Modified Normal Boundary Intersection (NBIm), Normal Constraint (NC), Successive Pareto Optimization (SPO), and Direct Search Domain (DSD).
• Evolutional algorithms: In these methods, you produce a set of Pareto optimal solutions when you run the algorithm. Examples of this class include Non dominated Sorting Genetic Algorithm II (NSGA II) and Strength Pareto Evolutionary Algorithm 2 (SPEA 2).

Interactive Methods

In the interactive methods, you work on an interactive solution process and keep interacting with it when searching for a most favorable solution. In these methods, the process expects you to provide you preferences for each iteration of the process to get Pareto optimal solutions.

The following is a common algorithm of the procedure followed in the interactive methods:

1. Start the process with initial values.
2. Fix a starting point for the Pareto optimal solution.
3. Provide your preferences.
4. Generate a Pareto optimal solution.
5. Select the best solution so far, if multiple solutions are generated.
6. If it is an optimal solution, then stop. Else, repeat step 3 through step 6.

Hybrid Methods

In the hybrid methods, the algorithm consists of a combination of algorithms from multi criteria decision making (MCDM) and evolutionary multi objective optimization (EMO). The hybrid methods are used to overcome the shortcomings and utilizing the strengths of these methods.

Multi Objective Programming Software

Multi objective programming involves complex mathematical computations. Therefore, either you need help from an expert or use any of the multi objective programming software available in the market for this purpose. The following is a list of some of the software available in the market:

• BENSOLVE
• Distributed Evolutionary Algorithms in Python
• MOEA Framework
• Decisionarium
• GUIMOO
• IDSS Software
• IND – NIMBUS
• jMetal
• MakeItRational
• Midacomo
• Multiple Goal Optimization (MGO)
• WWW – NIMBUS

Integer programming is a yet another type of constrained optimization method of project selection. In this method, you look towards a decision that works on integer values and not on fractional values. For example, producing a number of cars can never be fractional.

In contrast to the linear programming method, where you work on a continuous model that enables you to define decision variables to be fractional, in the integer programming model, you must consider only integer values for the decision variables. For example, when you can produce 189.86 tones of sugar in a plant, you cannot produce 23.6 airplanes. While you can use linear programming method to select a project in the former case, you must use the linear programming method to select a project in the latter case because fractional solution is not at all realistic for it.

In the integer programming method, you optimize a solution by using the following formula:

Subject to:

 

You define an integer programming method as mixed integer program when you restrict only some of the decision variables as integers. However, when you make sure that the value of all the decision variables must be integers, then it is a pure integer program.

As in case of the linear programming method, where if the decision constraints are of a network type, you can get an integer solution by ignoring integrality restriction. However, in general the decision variables are fractional. For such a solution to be an integer solution, you must consider further steps to arrive at a solution.

Formulation of an Integer Program

 

The integer programming method has extensive programming capabilities that are better than the linear programming capabilities. The integrality restriction of this method reflects the natural sense of non possibility of dividing a problem. For example, when you decide to add a room to a building, it does not make any sense to have a fractional solution. In such problems, you must consider a solution that is integral in nature.

When formulating an integer program, you must consider the following components:

• Binary Variables
• Logical Constraints

Binary Variables

At times, you might have just two values for making a decision, which are Yes and No. In such cases, the variables in the program are restricted to only two values – 0 or 1. The variables for which you restrict values to 0 or 1 are known as binary, logical, bivalent, or 0 – 1 variables. In formulating an integer program, you use such variables quite frequently.

To formulate a problem solution using a binary variable in an integer program, you use the following formula:

Logical Constraints

When you use decision variables in an integer program, you impose logical constraints. You can consider the following constraints at the time of formulating a solution by using an integer program:

• Constraint feasibility: For a decision variable, you ask a question whether the choice available for the decision variable satisfies the constraint. If the constraint is satisfied, you assign value 0 to the variable and 1 if it does not.
• Alternative constraints: For a decision variable, you might have more than one constraint. The decision variable must satisfy at least one constraint and not both.
• Conditional constraints: In contrast to the alternative constraints, you might have multiple constraints, the decision variable must satisfy both the constraints.
• Compound alternatives: Compound alternatives include sets of alternative constraints. These sets of constraints can lie in disjointed regions when plotted on a graph or can be on overlapping regions.

Linear programming method is a type of constrained optimization method of project selection. In this method, you look towards reducing the project cost by efficiently reducing the duration of the project. You look for running an activity in its normal time or the crash time. The crash time of the activity enables you to reduce the activity time or the project as a whole.

When you complete only a specific activity for a duration that incurs the smallest cost, you term it as the normal time. You crash an activity by spending more efforts to make sure that the activity takes lesser time to complete. Adding efforts to the activity definitely increases the cost. However, when you include overhead costs, you might realize that running multiple activities at crash level is financially advantageous, if it reduces the overall duration and overhead costs.

However, one of the most important aspects of the project you need to consider is if crashing an activity reduces the overall time of the project. For example, if the activity does not lie in the critical path of the project, then you might not want to consider crashing the activity. Critical path of the project is the sequence of the activities that you need to perform from start to end, considering that you can perform other activities, which can be another sequence of activities, in parallel. The alternate sequence of activities that you perform in parallel is known as an alternate path.

At times, you might realize that crashing activities in the critical path might result in making an alternate path a critical path because the time required to complete such path is now more than the former critical path. In such a case, if you justify the costs involved, you can consider further crashing the activities of the new critical path too.

Crashing an activity means that you reduce the time required to complete the activity by adding additional resources, including man power and machinery. This not only adds efforts and reduces the time required to complete the activity faster, but also increases the cost for completing the activity.

Consider the following table of activities for a simple project that indicates the normal and crash time along with the costs of various activities of a project. The table also indicates the dependencies of various activities:

Activity	Predecessor	Normal		Crash
		Week	Cost	Week	Cost
A	–	2	80	1	150
B	A	3	90	2	120
C	A	4	120	2	180
D	B, C	2	100	2	100
E	D	1	50	1	50
F	E	1.5	200	0.5	300

Consider the activity F. This activity requires four weeks to complete. However, you can crash this activity to two weeks with an additional cost of $ 150. However, there is always proportionality between the time of reduction and additional cost. Therefore, if you want to crash the activity F by two weeks, you can calculate the cost of activity by calculating the slope cost and then the cost of activity as follows:

Slope Cost = (Crash Cost – Normal Cost) / (Normal time - Crash time)

Cost of Activity = Normal Cost + (Slope Cost * Crash time)

Applying the preceding formulas, we can calculate the activity cost of activity F as:

Slope Cost = (300 – 200) / 1
           = 100

Cost of activity = 200 + (100 * 0.5)
                 = 250

You can calculate the cost of activity for each activity and sum all of the costs to arrive at the final cost of project. You can also represent the activities from the table with a network diagram to understand the dependencies of the activities. The following is the network diagram for the sample project:

One of the types methods you use to select a project is Benefit Measurement Methods of Project Selection. In these methods, you calculate or estimate the benefits you expect from the projects and then depending on the highest benefits, you select a project. However, these methods are more suitable to select projects that are simple and easy to calculate benefits from such projects. In big organizations, where projects are of complicated and are of a large scale, these methods might not be appropriate to select a project.

Considering the magnitude of the project, the organization has a lot on stake. Therefore, you cannot take chances to select a project based on just judgments or simple calculation. Such projects need complicated mathematical calculation before you decide on considering a project. For large and complicated projects, you can use constrained optimization methods to select a project.

In a constrained optimization method, you make complex mathematical calculations to select a project. These mathematical calculations are based on various best and worst case scenarios, and probability of the project outcome. Depending on the outcome of these calculations, you compare the candidate projects and the select a project with the best outcome.

You can use any of the following constrained optimization methods to select a project:

• Linear Programming Method of Project Selection: In this method, you look towards reducing the project cost by efficiently reducing the duration of the project. You look for running an activity in its normal time or the crash time. The crash time of the activity enables you to reduce the activity time or the project as a whole. To know more about this project selection method, refer Linear Programming Method of Project Selection.
• Integer Programming Method of Project Selection: In this method, you look towards a decision that works on integer values and not on fractional values. For example, producing a number of cars can never be fractional. To know more about this project selection method, refer Integer Programming Method of Project Selection.
• Dynamic Programming Method of Project Selection: In this method, you break a complex problem into a sequence of simpler problems. This method provides a general framework of analyzing many problem types. In this framework, you use various optimization techniques to solve a specific aspect of the problem. This method requires your creativity before you can decide if the problem needs to use dynamic programming for its solution. To know more about this project selection method, refer Dynamic Programming Method of Project Selection.
• Multi Objective Programming Method of Project Selection: In this method, you make decision for multiple problems with mathematical optimization. In case, in a multi objective programming, a single solution cannot optimize each of the problems, then the problems are said to be in conflict and there is a probability of multiple optimal solutions. A solution is called as non dominated if values of none of the problem can be optimized without degrading values of another problem. To know more about this project selection method, refer Multi Objective Programming Method of Project Selection.

As discussed in other tutorials, at any given moment of time, an organization has multiple ideas waiting to be considered as a project. However, it is practically as well as financially not possible to consider each and every idea as a project and work on it. This does not means that the ideas that are not taken up as a project do not have potential of generating any revenue.

Due to limited resources, it is not practical for you to convert each idea into a project and pursue it. As a result you lose on the capitalizing an opportunity to realize the potential revenue from all such ideas. The revenue that you lose by giving up an idea and converting another idea to a project is known as opportunity cost.

Opportunity cost is just a value and not a benefit. Additionally, you do not add opportunity costs of all candidate projects to arrive at a value. Opportunity cost is just a value of the project you lose for not selecting it. When you consider the opportunity cost of a project, it is always the cost of the value of the next best alternative project and not the cumulative value of all candidate projects.

Therefore, you can also define the opportunity cost as the relative cost because it is a cost of one project relative to the other project.

Calculating an Opportunity Cost

Calculating an opportunity cost for a project, you do not need any special or complicated formula. You just need to calculate the net present value of the projects you are considering and compare the values. You select the project that has a higher net present value. The net present value of the project you do not select is the opportunity cost. To know more about calculating net present value, refer Economic Model for Project Selection – Net Present Value.

Consider a scenario where you have two projects IOS Test App and Android Test App. The net present value for the project IOS Test App is calculated as $ 80,000 and the same for the project Android Test App is calculated as $ 102,000. In this scenario you select the Android Test App project. As a result, the opportunity cost for the Android Test App is $ 80,000. This means that the opportunity cost for selecting the Android Test App project is $ 80,000.

Notice that when calling the opportunity cost, you do not refer to the losing project but the one you have selected, though the value is that of the losing. For example, in the preceding scenario, you have selected the Android Test App project. Therefor, you refer to the opportunity cost of $ 80,000 for the Android Test App and not the IOS Test App project.

The name Murder Board method of project selection sounds quite unconventional and somewhat like some anti-social activity group or more so sounds like a title of a thriller movie. However, the name does not have anything to do with any anti-social activity. The name is more of derived from the fact that this method completely works on negative and constructive methodology.

In this method of project selection, you create a murder board. A murder board is also known as a scrub-down. You constitute a murder board, which is a committee that comprises of senior managers and subject matter experts from different areas. The murder board scrutinizes the project to find reasons why the project should not be selected. You must defend the project and counter all the queries of the board members.

The main responsibility of the murder board is to critically review the proposed project. As the name suggests, the members of the board are supposed to review the project aggressively. During this review, there is no scope of pleasantries and is done without any constraints.

The reviews are more like grilling sessions where every possible effort is made to prove that the project is not worth considering. It is the sole responsibility of the proposer to satisfy each and every query of the board members and prove the worth of the project. The murder board meets with a sole objective of killing a well prepared project proposal based on the technical merits. The name of the method itself is derived from the fact that the objective of board members is to murder the whole idea of the project.

The murder board makes sure that every aspect of the project is looked into and reviewed in depth. The review includes the following aspects of the project:

• Problem statement
• Assumptions
• Risks
• Mitigation
• Proposed solution

As a part of its responsibility, the murder board is not supposed to tolerate on holding back even a least suspicion of a problem. The murder board session is supposed to be very aggressive and argumentative mostly because it consists of subject matter experts from various areas. These subject matter experts grill the presentation in their respective areas of expertise.

Notice that as per the nature and goals of the murder board, you can expect that all your skills be tested when you make a presentation. You should be prepared to face new challenges with a very high intensity.

In an organization that has a wide range of project to select from, the murder board plays a vital role. The board reviews each and every project proposal before these are given a final go ahead.

Advantages

A murder board method of project selection has the following advantages:

• It enables the organization to select best of the ideas and convert the same into projects.
• You must be very well prepared before you present the proposal to the board.
• With expertise from a wide range of fields, chances of committing mistakes are reduced considerably.
• You can take an advantage of the knowledge of the board members to further enhance the project.

Disadvantages

A murder board method of project selection has the following disadvantages:

• The aggressive review can destroy the confidence of the presenter.
• A really good idea with potential of generating good revenue might unnecessarily get shot down.
• There are chances of board members tempted to prove their supremacy and compete to kill the project first.