Posted: September 13th, 2017

A Combined AHP- transportation for Project Management

Paper, Order, or Assignment Requirements

 

 

this is Operation research class. you need to read this article throughly and understand the method. Then we need to re write the paper and per teacher’s request we will be combining the AHP method with the transportation method to find the best Contractor. The paper that I have is not very clear about the problem statement. In fact after seraching for the author’s name online, I found another portion of this paper that has the problem statement.
Odiriginal problem is evaluating between 5 Contractors I think but the teacher okeyed us to choose the best out of 3 of them only (which is what has been done in my friend’s paper). We need to rework this and I think if everyone spends about 2-3 hours we can get this out of the way.
MSE 606A Spring 2014

A combined AHP-GP model for project management

Mahsa Sadighi, Celina Bedrosian, Mehrnaz Taherkhani

Department of Manufacturing Systems Engineering and Management, California State University Northridge, California, USA

Submitted 17 May 2014

Abstract

Making the best decision is always a challenging process in project management. Although the Analytic Hierarchy Process (AHP) can help the project managers to prioritize the alternatives based on their criteria and make a descending-order list, their limitation and constrains are not considered within the confines of this model. To address this issue, this paper proposes the combined AHP-GP (Analytical Hierarchy Process and Goal programming) method in order to help decision makers to include their real world restrictions. The contractor selection problem is used as an example in this paper. First, six main factors were selected using the previous studies for prequalification of contractors. Then these factors, through the Analytic Hierarchy Process (AHP), were weighted accurately and consistently. Subsequently, the priority weights along with resource constrains were incorporated with a goal-programming model to help select the most appropriate contractor. Finally a sensitivity analysis has been applied to test the model and depict the impact of changes in resource limitation on the final solution of combined AHP-GP model.

Keywords: Goal Programming, Analytical Hierarchy Process, Project Management, Contractor Selection

 

 

Introduction

“Project management is the art of making the right decisions” ‎[1]. In managing a project, decision-making could be very complicated due to some reasons. First of all, there are multiple objectives associated with   problems. Project managers deal with customers, project team members and sponsors and should attempt to satisfy their different objectives and preferences as much as possible. Secondly, project managers are faced with different alternatives and making choices between them could be very complex process. Uncertainties can even make to process more complex. In the modern world, complexity and cost of projects has increased significantly ‎[1]. As a consequence wrong decisions can have expensive results for the decision makers. Therefore, project managers typically need a proper analysis to select the best among the alternatives.

Contractor selection problem is used in this paper as an example of project management decision. It has always been a key factor in the success of any construction project ‎[2]. In order to select the most appropriate contractor, decision makers need a proper and accurate method ‎[3]. Considerable part of decision analysis methods is multiple criteria decision-making (MCDM) methods ‎[4]. These methods help decision-makers to better understand the problems they face, their personal values as well as organizational objectives and aim them in identifying a preferred course of action ‎[5].

Several MCDM methods have been used in contractor selection decision-making. Hatush, Zedan and Skitmore, Martin R. (1998) by using the multi-attribute analysis (MAA) method generated a quantitative method in order to choose the best contractor ‎[6]. This method is claimed to be extremely data intensive, as it requires incredible amount of measures at every step to be able to record the decision maker’s preferences ‎[7]. The other group of researchers (2014) came up with Fuzzy Theory for this purpose ‎[8]. Fuzzy systems is also claimed to be difficult to develop. In many cases, the decision maker should do many simulations to be able to use it in the real world ‎[7]. Y-K. Juan (2009) ‎[8] approached to contractor selection by case-based reasoning (CBR) and data envelopment analysis (DEA). The disadvantages of CBR method is its sensitivity to inconsistent data and dependency to many cases. Additionally, DEA assumes that all the data are exactly known and does not deal with imprecise information ‎[7]. Analytic hierarchy process (AHP) method was used by the other group ‎[9]. AHP method is easy to use, scalable and not data intensive ‎[7]. It also provides some level of consistency in order to reduce decision biases. However, AHP method is only able to solve problems with a hierarchy and does not consider the real world resource limitations ‎[11].

The aim of this paper is to extend the AHP model presented by Kamal M. A. AL-Harabi in contractor selection process ‎[4] by adding Goal Programming (GP) model to it. GP is another MCDM method that provides an optimum solution for decision makers considering all the objectives, constrains and limitations associated with the problem. GP model is capable of handling large-scale problems but its major drawback is its inability to prioritize the decision-making criteria quantifiably ‎[7]. Thus, this method is not useful when decision-makers only are able to express their objectives in a manner of preferences and cannot quantify their objectives. Therefore, many studies find it necessary to combine the GP with other techniques. According to William Ho ‎[11] the combined AHP-GP is the most commonly used integrated AHPs that can be applied to vast array of problems.

In a combined AHP-GP model, the AHP can be used to prioritize the alternatives based on decision makers’ preference. Then GP model can be formulated based on resource and system limitations as well as AHP priorities constrains. In this way the AHP-GP model can overcome the disadvantages of both AHP and GP and can be beneficial is selecting best alternative without violating the limited available resources. ‎[11]

Methodology

The AHP Approach

The analytic hierarchy process (AHP) is an approach and compound technique to arrange and analyze complex and intricate decisions. It was established by Thomas L. Saaty in the 1970s ‎[17] and has been studied and regenerated after that. AHP modeling is a scoring and organized method that appoints the priority from decision alternatives for multiple criterion problems, based on mathematics and psychology. Generally, AHP is useful for multi criteria decision-making in many fields such as education, business, pharmaceutical, and games etc.

AHP has a remarkable characteristic, which is allocating numeral values to judgments of decision makers and prioritizing criteria for decision-making. For this purpose, decision-maker should perform pair wise comparison among criteria and alternatives. Based on these comparisons, the weight of criteria and ranking for alternatives based on these criteria will be determined.

Figure 1, shows AHP basic structure of hierarchy. As can be seen, it has three primary levels: the first level is the goal of the decision, objectives are in the second level and in the third level are alternatives. The preferences assigned to each level effect the level above and so on, until the final goal is met. Since many real world decision-making situations are hierarchical, decision maker can use this arrangement to analyses each part of the complex problem.

Another advantage of AHP method is that it provides a technique for checking the consistency of the evaluator’s judgments. In the analytical hierarchy process, consistency of pair wise comparisons should be found in the final step to assure unbiased decisions.

 

The GP Approach

 

Decision makers are often faced by different objectives and goals that could have a conflict with each other. The Goal programming is a mathematical programming model that helps decision makers to find an optimum solution to solve multiple and often conflicting problem. Charnes, Cooper and Ferguson proposed this method in 1955 ‎[18]. The purpose of using GP model is to attain goals according to their priorities by applying constrains to each objective. There are two kinds of variables in the GP model; decision variables and deviational variables. Deviational variables indicate differences between the values of the decision variables and the goals. In the zero-one model decision variables can take only two values. If the decision variable is selected its value is 1 and if it is not selected it takes 0 value.

Resource constraints can be stated in equations consisting of linear combinations of deviation and decision variables, with the target level of resource utilization or benefit on the right hand side. The objective function is to minimize the deviational variables by finding the value of the decision variables in the problem constrains.

The model first aims to achieve the first goal. As it is met, the values of deviational variables are added on constraint and the process is repeated until all goals are met.

The Combined AHP-GP Model

As explained in section 1, the goal programming does not provide enough information about decision maker preferences in objective function. In order to overcome this problem, several researchers have combined the AHP and GP models [12-16].

In the combined model, the deviational variables associated with AHP model are included in the objective function. In comparison to GP model, combined model has two more set of constrains and deviational variables. First set of constrains and deviational variables are based on AHP overall ranking. Second sets are based on each of the criteria in AHP model. The overall AHP-GP process is shown in Figure 2.

 

Formulation of the problem

In this section, an example of contractor selection problem will be formulated. It illustrates how the combined AHP-GP model can be applied to a real world cases.

AHP Formulation

The first step in AHP is synthesizing the pair-wise comparison matrix. For this purpose, each project alternative should be compared against the other alternatives based on each selection criterion separately.

Table 1 shows an example of the preference matrix for N^th selection criterion. X_ij indicates the rank of alternative i with respect to alternative j, based on N^th selection criterion and X_ji is the reciprocal of X_ij.

Criterion N	1	2	3	4	5
1	X11	X12	X13	X14	X15
2	X21	X22	X23	X24	X25
3	X31	X32	X33	X34	X35
4	X41	X42	X43	X44	X45
5	X51	X52	X53	X54	X55

After conducting the pair-wise comparison for each criterion, matrixes should be normalized by diving each score by sum of the scores in its column. The relative priority of each alternative will be the average of normalized score within each row.

These procedures should also be employed for the selection criteria themselves to find the relative preference of them. Table 2 shows preference matrix for selection criteria. Here Y_ij indicates the rank of criterion i with respect to criterion j and Y_ji is the reciprocal of Y_ij.

	Criterion 1	Criterion 2	Criterion 3	Criterion 4
Criterion 1	Y11	Y12	Y13	Y14
Criterion 2	Y21	Y22	Y23	Y24
Criterion 3	Y31	Y32	Y33	Y34
Criterion 4	Y41	Y42	Y43	Y44

The procedure described above gives us three set of information including overall scores of each alternatives with respect to each criterion, overall ranking of alternatives based on all criteria and scoring of the criteria amongst themselves. This information will be then incorporated in the combined model.

Goal Programming Formulation

In contractor selection process, the first goal is not to violate limitations and constraints associated with the problem. A study was done on 2009 ‎[19] comparing selection attributes and modeling approach used by different authors in the process of selecting the best contractor. Based on this study we decided to define four factors to be considered in our decision. We defined our limitations as total available budget and total amount of time taken for completion of the project. We also decided to consider two expectations including satisfactory warranty period and number of services during warranty period. Each of these limitations and constraints will have deviational variables associated with them. By adding deviational variables to the overall objective function, we attempt to minimize the over-utilization (positive deviations) of cost and time and the under-utilization (negative deviations) of warranty and services.

The GP model constrains can be expressed as shown in equations 1-4, with the desired amount of budget, completion time, warranty period and number of services on the right hand side of each equation.

: The bid price issued by contractor i.

: The completion time by contractor i.

: The warranty issued by contractor i.

: The services issued by contractor i.

In these equations and are deviational variables representing how much under and over the target will be achieved respectively. As the first goal we seek to minimize the over-spending of available budget () and time () besides minimizing under-contribution of warranty () and services (). The objective function can be stated as:

(+)

(5)

Combined model Formulation

In the combined AHP-GP methodology, all the scores derived in the section 3.1 from AHP model, should be added in the original goal-programming model as constraint equations. Additionally, deviation variables associated with AHP model should be added to objective function of GP model. The combined model seeks to minimize deviations from desired level of limitations as well as AHP ranking.

Equation 6 shows the constraint equation for the overall score of alternatives with respect to criteria preference.

Where is overall priority vector of contractor i based on all criteria.

A set of constrains based on each criteria should be added to the combined model which is shown in equation 7.

Here is overall priority vector of i^th contractor based on k^th criteria. Also, and are deviation variables form target of.

The deviational variables of equation 6 and 7 should be included in the original objective function just like the deviational variables for the resource constraint (equation 5). However, there will be two major differences. First, minimizing AHP deviational variables will be a lower priority goal in comparison to goal presented in equation 5. Secondly, this goal should be weighted according to the relative preference scores of the criteria. The overall objective including the AHP constraints can be stated as:

Z = ( + )

+ () +(                 (8)

Where is overall priority vector of criteria k. This objective function can be solved using commercial linear programming solvers, such as TORA or Excel Solver, in order to obtain the best alternative.

Case study

In this section, we use the combined AHP-GP model discussed in section 3 for contractor selection process. For the sake of simplicity, among five contractors provided in the reference publication ‎[4], we decided to consider only three and select the best among them. We choose the first three contractors to be considered. Table 3 presents the data for these three alternatives ‎[4].

 

 

	Contractor A	Contractor B	Contractor C
Experience	5 year experience Two Similar Projects	7 year experience One similar project Special procurement experience	8 year experience No similar project 1 international project
Financial stability	$7 M assets High Growth rate No liability	$10 M assets $5.5 M liabilities Part of a group of companies	$14 M assets $6 M liabilities
Quality performance	Good organization C.M. personnel Good reputation Many certificates Safety program	Average organization C.M. personnel Two delayed projects Safety program	good organization C.M. team Government award Good reputation QA/QC program
Manpower resources	150 laborers 10 special skilled laborers	100 laborers 200 by subcontract Availability in peaks	Good skilled labors 25 special skilled laborers
Equipment resources	4 mixer machines 1 excavator 15 others	6 mixer machines 1 excavator 1 bulldozer 20 others 15,000 sf steel formwork	1 batching plant 2 concrete transferring trucks 2 mixer machines 1 excavator 1 bulldozer 16 others 17,000 sf steel formwork
Current works load	1 big project ending 2 projects in mid (1 medium + 1 small)	2 projects ending (1 big + 1 medium)	1 medium project started 2 projects ending (1 big + 1 medium)

 

AHP Formulation

In this section we use AHP method proposed in section 3.1, for prequalification of three contractors. In prequalification process, project managers screen the contractors based on their criteria, in order to determine their ability for participation in the project. ‎[4] This process could be very beneficial for project managers since it ensures them a qualified contractor will be selected to construct the project. Several studies have focused on contractor prequalification based on different criteria. In our paper, we focused on 6 criteria of experience, financial stability, quality performance, manpower resources, equipment resources and current works load proposed by the reference paper ‎[4]. But we changed the priorities of prequalification criteria as well as priorities of each contractor based on each criterion. Table 4-9 show the pair-wise comparison matrix and priority vector for three alternatives of A, B and C based on our six criteria. It can be seen our judgments are acceptable since all the CR values are less than 0.1.

Exp.	A	B	C	Priority vector
A	1	2	4	0.557
B	1/2	1	3	0.320
C	1/4	1/3	1	0.123

FS	A	B	C	Priority vector
A	1	6	7	0.755
B	1/6	1	2	0.154
C	1/7	1/2	1	0.092

QP	A	B	C	Priority vector
A	1	8	4	0.702
B	1/8	1	1/4	0.072
C	1/4	4	1	0.227

MPR	A	B	C	Priority vector
A	1	1/4	3	0.231
B	4	1	5	0.665
C	1/3	1/5	1	0.104

ER	A	B	C	Priority vector
A	1	1/6	1/3	0.093
B	6	1	4	0.685
C	3	1/4	1	0.221

CWL	A	B	C	Priority vector
A	1	5	1/2	0.366
B	1/5	1	1/4	0.102
C	2	4	1	0.532

Similarly we weighed the criteria themselves against each other in pair wise comparisons. The Table 10 reflects the normalized pair-wise comparison matrix and priority vector for six criteria.

 

	Exp.	FS	QP	MPR	ER	CWL	Priority Vector
EXP.	0.432	0.375	0.480	0.294	0.294	0.367	0.374
FS	0.108	0.094	0.080	0.176	0.176	0.061	0.116
QP	0.144	0.281	0.240	0.235	0.235	0.367	0.251
MPR	0.086	0.031	0.060	0.059	0.059	0.041	0.056
ER	0.086	0.031	0.060	0.059	0.059	0.041	0.056
CWL	0.144	0.188	0.080	0.176	0.176	0.122	0.148

 

 

	Exp. (0.374)	FS (0.116)	QP (0.251)	MPR (0.056)	ER (0.056)	CWL (0.148)	Overall priority vector
A	0.557	0.755	0.702	0.231	0.093	0.366	0.544
B	0.320	0.154	0.072	0.665	0.685	0.102	0.246
C	0.123	0.092	0.227	0.104	0.221	0.532	0.210

 

Table 11 shows the priority values of three alternatives based on each criteria and their overall priority vector. Looking at the AHP scored reveals that contractor A is the best alternative among these three options for prequalification purposes. If prequalification was the only restriction, we could simply choose contractor A. Also contractor B and C would be our second and third preference, respectively. However, in the real world we will face many other limitations such as cost and time that should be considered in our decision making process. These factors will be discussed in the next section in a GP model.

Goal Programing Formulation

As discussed in section 3.2, we decided to define four factors to be considered in our decision. Table 12 shows our estimations about the desirable amount of our restrictions and quotations submitted by the contractors. Based on our estimations, the total available budget for the project is $3 million. Additionally, we want to complete the whole project within one year and are willing to receive at least three year warranty and 6 services during the warranty period. Looking at Table 12, contractor A which was selected by AHP model is the most expensive decision alternative with the bid price of $2.726 million but this contractor can do the project within 6 months. On the other hand contractor C requires 10 months to complete the project with the lowest bid price of $2.193 million. Table 12 also shows that contractor B is the best alternative based on provided warranty and services. These numbers should be incorporated in the combined model as system constraints.

 

	Bid Price ($million)	Completion Time (month)	Warranty (year)	Number of Services during Warranty Period
A	2.726	6	3	7
B	2.561	8	5	8
C	2.193	10	4	6
Goal	3.000	12	3	6

 

These constrains can be stated as:

2.726 * + 2.56 * + 2.193 * + – = 3,      (9)

6 * +8 * + 10 * + – =12,

(10)

3 * + 5 * + 4 * + – = 3,                                    (11)

7 * + 8 * + 6 * + – = 6                         (12)

The first goal of the combined model is not to violate any of these constrains.

Formulation of the combined model

Our goal in this problem is to find best one and only one of the three contractors. So different alternatives are mutually exclusive and we should add one constrain as blew to our combined model:

.                  (13)

Constrains for the first goal of the problem has been formulated in section 4.2. The second goal in this problem is to ensure that contractor with the highest AHP weights will be selected. This goal can be achieved by minimizing the under-utilization () of the AHP overall weights:

0.544 * + 0.246 * + 0.210 * + – = 1.        (14)

The development of the third goal of the problem originates from the AHP data derived in section ‎4.1. These data can be stated in equations with deviational variables like as resource constraint equations written earlier, with two differences. First, we place the goal of minimizing these AHP deviational variables at a lower priority in the combined objective function; and second, we weight the deviational variables according to the relative preference scores of the criteria.

The right-hand-side values of these equations seek to pressure a selection of a contractor with the highest scores and can be set to any value depending on the nature of the problem. For our case, each of the right-hand-side was simply derived by taking the highest score within each measures and using it as the maximum possible value. This is valid since in our model only one of the three alternatives can be selected. Therefore it’s impossible to have any value higher than the largest score of the three contractors. Constrains can be stated as:

0.557 * + 0.320 * + 0.123 * + – = 0.557,             (15)

0.755 * + 0.154 * + 0.092 * + – = 0.755,            (16)

0.702 * + 0.072 * + 0.227 * + – = 0.702,             (17)

0.231 * + 0.665 * + 0.104 * + – = 0.665,            (18)

0.093* + 0.685 * + 0.221 * + – = 0.685,             (19)

0.366 * + 0.102 * + 0.532 * + – = 0.532,                        (20)

These equations seek to minimize the under-utilization of the AHP score by minimizing negative deviations (dX^–, dF^–, dQ^–, dM^–, dE^–, dC^–) based on overall weight of each criteria.

Our objective function would be minimizing the overall deviations in each of the goal constraints including resource limitations as well as desired AHP scores. In Summary the objective function for complete AHP-GP model can be stated as:

Minimize:

Z = ( + )

+ () +(

0.056, 0.056, 0.148(21)

Subject to following constrains:

2.726* + 2.56 * + 2.193 * x₃ + – = 3

6 * + 8* + 10* + – = 12,

3 * + 5 * + 4 * + – = 3,

7 * + 8 * + 6 * + – = 6,

0.544 * + 0.246 * + 0.210 *+ – = 1,

0.557 * + 0.320 * + 0.123 * + – = 0.557,

0.755 * + 0.154 * + 0.092 * + – = 0.755,

0.702 * + 0.072 * + 0.227 * + – = 0.702,

0.231 * + 0.665 * + 0.104 * + – = 0.665,

0.093* + 0.685 * + 0.221 * + – = 0.685,

0.366 * + 0.102 * + 0.532 * + – = 0.532,

= 0 if it’s not chosen or 1 if it’s chosen.

To solve this problem, first we should consider the first goal and set aside all other objectives. If deviational variables associated with the first goal could be minimized to zero, it means that we achieved our first goal and can we move on to the next goal. In this step the first deviational variable should be set to zero and be added as additional constraints for the second goal. We continue this process until all the deviational variables have been minimized.

Notice that by selecting one of the contractors, we will also select the corresponding AHP scores. It means that we will have some under-utilization of prequalification measures, since no contractor was better than the others in every single criterion. For example, if contractor A is selected, it would receive the highest values of 0.557 in experience, 0.755 in financial stability and 0.702 in quality performance. However, if contractor B is selected, it would receive the highest values of 0.665 in manpower resources and 0.685 in equipment resources. Finally, if contractor C is selected, it would receive the highest values of 0.532 in desirable current work load.

Results and discussion

For solving the combined AHP-GP model we used Excel Solver in Microsoft Excel ‎[20] shows the optimal selection of contractors. The optimal choice turned out to be contractor B. As can be seen this solution is not consistent with AHP solution because of the restrictions we put in place.

 

Decision Variable	Value	Remarks
A (x1)	0	Do not select contractor A
B (x2)	1	Select Contractor B
C (x3)	0	Do not select contractor C

 

	Goal 1				Goal 2	Goal 3
Deviational variables	dg1	dg2	dg3	dg4	dA	dE	dF	dQ	dM	dE	dC
Upward Deviations (+)	0	0	2	2	0	0	0	0	0	0	0
Downward Deviations (-)	0.44	4	0	0	0.75	0.24	0.60	0.63	0.00	0.00	0.43

 

The associated slack in the goal constraints are shown is Table 14. Choosing contractor B contributes to $440 k leftover in the available budget, 4 months earlier finishing the project, 2 year more warranty and having 2 more services during the warranty period.

The amount of unused contribution to AHP scores is 0.753, which is summation of AHP scores of contractors that were not selected (0.544 for contractor A and 0.210 for contractor B).

Additionally, contractor B was ranked highest in 2 out of 6 contractor prequalification criteria, there are 4 under-utilization slacks associated with its selection.

After getting the satisfactory results from our model, we applied sensitivity analysis to see the impact of changes in resource limitation on the final solution of combined AHP-GP model. When we decreased the available budget to $2.2 million, the model changed the optimal selection to contractor C. it was reasonable since contractor C was the only alternative that did not violate this constraint regardless of its undesirable scores. Similarly, completion time was reduced to 6 months, when the contractor A was selected by our model. Finally, when completion time was reduced to 9 months, contractor B was selected and multiple contractors were not chosen, indicating that the model selects one and only one contractor. These results confirmed that our model is stable, accurate and flexible.

Concluding remarks

The combined AHP-GP model presented in this paper extends previous research in the use of AHP in contractor selection process by incorporating resource allocation in the GP model. The combined AHP-GP model has prevented over-use of resources and under-contribution of decision maker’s preference of criteria.

The factors used in AHP part of the model for prequalification of contractors were experience, financial stability, quality performance, manpower resources, equipment resources and current works load. Pairwise comparisons of different contractors based on these factors provided an unbiased preference order of alternatives in a consistent manner. In addition, GP part of the model was used to do the trade-offs between cost, time, warranty, number of services during the warranty period and select the best alternative.

The combined AHP-GP model used in this paper is a stable, accurate and flexible method. This was confirmed by testing the model with different constrains. Thus, project managers can update and refine the model based on their preferences and restrictions and use it to aid them in their contractor selection process.

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Excel Solver. http://www.solver.com/