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Integrated resource management for simultaneous project selection and scheduling
Accepted Manuscript Integrated Resource Management for Simultaneous Project Selection and Scheduling Mohammad Shariatmadari, Nasim Nahavandi, Seyed Hessameddin Zegordi, Mohammad Hossein Sobhiyah PII: DOI: Reference:
S0360-8352(17)30133-X http://dx.doi.org/10.1016/j.cie.2017.04.003 CAIE 4690
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
25 November 2016 29 March 2017 3 April 2017
Please cite this article as: Shariatmadari, M., Nahavandi, N., Zegordi, S.H., Sobhiyah, M.H., Integrated Resource Management for Simultaneous Project Selection and Scheduling, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie.2017.04.003
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Integrated Resource Management for Simultaneous Project Selection and Scheduling
Mohammad Shariatmadaria Nasim Nahavandia,* Seyed Hessameddin Zegordia Mohammad Hossein Sobhiyahb
a
: Faculty of Industrial & Systems Engineering, Tarbiat Modares University, P.O. Box 14115 143, Tehran, Iran. b
: Project & Construction Management Group, Faculty of Arts & Architecture, Tarbiat Modares University (TMU), P.O. Box 14155 4838, Tehran, Iran. *, Corresponding Author
1
Integrated Resource Management for Simultaneous Project Selection and Scheduling
Abstract Nowadays, resource management is one of the most critical decision making processes in projectoriented companies. To reach the highest level of profitability, these companies should consider profiting from available projects and optimal management of their (renewable and nonrenewable) resources in terms of selection and scheduling of the most optimal project portfolio. In this regard, this paper presents the Integrated Resource Management (IRM) approach for simultaneous project selection and scheduling. A mixed-integer programming model is also proposed for the approach. To solve the research problem, first a heuristic algorithm is developed for generating feasible initial solutions; then a method for improving the resulted solutions is presented based on Gravitational Search Algorithm (GSA). Finally, the proposed method is compared with Lingo on a set of 270 test problems. The comparison demonstrates that the proposed method outperforms Lingo in most cases and has less time complexity. Also, it will be shown that using IRM approach in test problems improves average profitability by 13.2 %.
Keywords: Integrated Resource Management (IRM), Project Selection, Project Scheduling, Resource Investment Problem (RIP), Gravitational Search Algorithm (GSA).
1. Introduction Project-oriented companies need to answer the following questions to achieve the best performance in competitive markets: 1. Regarding the company’s renewable and nonrenewable resources and the profits of project, which one should be chosen? 2. How should the selected projects be scheduled during the planning horizon considering the existing constraints? 3. How should company’s renewable and nonrenewable resources be managed during the planning horizon? Many attempts have been made to provide separate responses to each of the above questions. The first question was answered through studies on the Project Portfolio Selection (PPS) problem (Tavana, KhaliliDamghani, & Sadi-Nezhad, 2013); (Li, Fang, Tian, & Guo, 2015); (Schaeffer & Cruz-Reyes, 2016). In general, the aim of the PPS is to select a project portfolio from several candidate projects considering the existing limitations and facilities so as to achieve an optimum objective function (Tofighian & Naderi, 2015). One of the most important and commonly used objective functions in this field is the maximization of projects profit (Liu & Wang, 2011).
2
To answer the second question, numerous studies have been conducted in the past five decades in the form of the Resource-Constrained Project Scheduling Problem (RCPSP). The objective of the RCPSP is to allocate the resources (which are considered to be constant in the standard model) to perform activities. Therefore, the optimum objective function value is obtained while all of the constraints are considered. Various objective functions can be used for this problem; for examples, time-related objectives (Khoshjahan, Najafi, & Afshar-Nadjafi, 2013), cost-related objectives (Khoshjahan et al., 2013) and cash flow-related objectives (Leyman & Vanhoucke, 2017). The answer to the third question involves a special type of RCPSP known as the Resource Investment Problem (RIP) in which renewable resources are variable. In spite of development of this problem in 1984 (Möhring, 1984), fewer studies have been carried out on this problem than the standard RCPSP. This problem is referred to as the Resource Availability Cost Problem (RACP) in most articles (Zhu, Ruiz, Li, & Li, 2016). Although several articles aimed at presenting exact methods for solving this problem (Rodrigues & Yamashita, 2010), most of them adopted heuristic and meta-heuristic approaches (AfsharNadjafi, 2014); (Zhu et al., 2016). Previously, it was tried to answer the first and second questions simultaneously by studying the project selection and scheduling problem. (Chen & Askin, 2009) proposed a mixed-integer programming (MIP) model with an NPV maximization objective function and used an implicit enumeration procedure to solve the problem. (Liu & Wang, 2011) developed an optimization model for the project selection and scheduling problem based on the Constraint Programing (CP) method using time-dependent resource constraints. (Huang & Zhao, 2014) studied this problem in the absence of historical data for the problem parameters. They also developed a Genetic Algorithm (GA) to solve the suggested model. As well as for answering the second and third questions simultaneously, combination of the project scheduling and material ordering have been discussed in some researches (Aquilano & Smith, 1980); (Smith-Daniels & Aquilano, 1984). In this regard, (Smith-Daniels & Smith-Daniels, 1987) presented a MIP model where activity durations are fixed in the problem. In contrast to previous research, (Dodin & Elimam, 2001) developed a model for the problem with variable activity durations. In addition, Several papers developed heuristic approaches (Schmitt & Faaland, 2004) and meta-heuristic approaches (Zoraghia, Najafib, & Niaki, 2014) to solve the problem. (Zoraghi, Shahsavar, Abbasi, & Van Peteghem, 2016) studied the multi-mode resource constrained project scheduling combined with the material ordering. They also proposed three meta-heuristic approaches to obtain near-optimal solutions. For minimizing the renewable and nonrenewable resource costs, (Shahsavar, Zoraghi, & Abbasi) presented the integration of the RIP with the quantity discount problem in material ordering. Given the efforts made in the project selection and scheduling area and also the project scheduling and material ordering area, the objective of the present research is to answer all of the three questions simultaneously. In other words, based on the IRM and considering the time-dependent revenues of the projects, the project portfolio is selected and scheduled in the planning horizon simultaneously. Since the problem is an NP-hard problem, a Gravitational Search Algorithm (GSA) is developed in this research to solve it as a set of test problems. The effect of the IRM on the profitability of Project Selection and Scheduling Process (PPSP) is also examined. The remainder of the paper is outlined as follows: section 2 proposes a mathematical model and approach to the research problem. Section 3 describes different steps of the GSA proposed for this 3
problem. Section 4 presents the calculation results of applying the proposed algorithm to a set of test problems. Finally, section 5 covers the research conclusions and future research.
2. Problem Statement and Mathematical Formulation 2.1. The Proposed IRM Approach Resources are elements undertaking project activities. Hence, resource management, which involves assessment of resource levels during a planning horizon, plays a vital role in determining the profits of project-oriented companies.
Figure (1): Different resource management approaches to the project selection and scheduling problems
Figure (1) depicts three different approaches for resource management in the performance of companies. In the first approach, the company assesses the resource levels against criteria that are beyond the scope of Project Selection and Scheduling Process (PSSP), and the results are used as input parameters for these processes. In the second approach, resource management covers project scheduling, but project selection is still beyond the scope of the process. The Resource Investment Problem (RIP) is an example of the second approach, but in the standard RIP, resource management is limited to management of renewable resources. Finally, the third approach, which is proposed and used in this research, includes resource management during simultaneous project selection and scheduling. This approach (IRM) includes the following features: -
The possibility of simultaneous management of renewable and nonrenewable resources to achieve optimum resource productivity during the planning horizon. Considering different costs (included running cost, decreasing cost and increasing cost) of renewable resources for using them to accomplish activities of the selected project portfolio. Considering the company’s cash flow during the planning horizon as a parameter determining resource preparation and accomplishing activities of the selected project portfolio. Implicitly considering the Resource Leveling Problem wherever resource leveling improves the profitability of the company.
2.2. Problem Statement The problem is defined as follows in accordance with the decision-making environment of many projectoriented companies: There are a few projects to select at the beginning of planning horizon and completion of each project involves time-dependent revenues, which change the company’s cash flow during the planning horizon. The initial level of renewable and nonrenewable resources is known in the beginning of the planning horizon. Without loss of generality, it is assumed that the only nonrenewable resource is money. In practical problems it is possible to calculate the monetary equivalent of other nonrenewable resources by applying the suitable conversion factor and reducing the revenue of 4
activities proportionally. During the planning horizon, if the company’s cash is adequate it is possible to change the level of renewable resources. Costs of renewable resources are defined in three forms: A) Cost of running a resource unit per time unit (which cannot be included in direct costs of an activity because the resource may be unemployed in a period of time) such as staff salaries; B) Cost of increasing a resource unit, like the hiring and training cost of human resources; C) Cost of reducing a resource unit, such as the releasing cost of human resources. Also, the amount of the renewable resources required for completion of any activity has been considered as predetermined constant. According to non-inflationary conditions in real world, in this problem it is assumed that if negative costs are associated with reducing a resource unit (in the case of resources the sales or dismissal of which generates revenues), the sales price will be lower than the cost of increasing a resource unit. In other words, it is presumed that the company cannot gain profit by buying and selling a resource unit in different periods. The resources required for accomplishing the activities are known and invariant. In addition, all the precedence relations are considered to be finish-start with zero lag. The early start time for the first activity of all project is equal to zero. Then the early start time for the activity of project is the maximum of the early finish times of its immediate predecessors ( ), i.e. : The late finish time for the last activity of all project is equal to the planning horizon. Therefore early finish time for the activity of project is the minimum of the late start times of its immediate successors ( ), i.e. . Any activity of project must be done inside the time window . The goal is to create project portfolios, schedule them in the planning horizon, and determine the level of each resource in each period so that the company’s assets are maximized at the end of the planning horizon.
2.3. Notations and Mathematical Model Indices : Project’s indices
.
: Activity’s indices
.
: Renewable Resource’s indices : Period’s indices
.
.
Parameters : Resource Units
required by activity of project (nonnegative).
: Cost of increasing resource
per-unit. 5
: Cost of decreasing resource : Cost of running resource
per-unit.
per-unit per-period.
: Duration of activity of project . : Expected profit (expected revenue minus the cost of non-renewable resources) if activity project is completed in period . : Set of immediate successor activities of project . : Set of immediate predecessor activities of project . : Company’s current assets at the beginning of the planning horizon. : Resource
available in the beginning of the planning horizon.
: Early start time for activity of project . : Late start time for activity of project . : Early finish time for activity of project . : Late finish time for activity of project . : Number of activities of project .
Decision Variables : 1 if project is selected, 0 otherwise. : 1 if activity in project is completed at period ; and 0 otherwise. : Resource
available in period .
Intermediate Variables : Increased amount of resource : Decreased amount of resource
in period compared to period in period compared to period
: Company’s current assets at the beginning of period .
6
. .
in
Mathematical Model (1) St. (2)
,
(3)
(4)
(5)
(6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
Objective function (1) shows maximization of cash available at the end of the planning horizon assuming that the level of renewable resources at the end of the planning horizon equals the renewable resources level at the beginning of the planning horizon. Constraint (2) ensures that no activity of non-selected projects is accomplished and all of the activities of the selected projects are accomplished. The constraints (3) and (4) are multi-project version of the discrete-time (DT) formulation for RCPSP (Pritsker, Waiters, & Wolfe, 1969). Constraint (3) includes the precedence relations for each project. The level of renewable resources used at each period should not exceed the level of available resources at that particular period, which is presented by constraint (4). Upper and lower bands of the innermost sigma presented in the left side of the equation 4, seek to detect in-progress activities at time . An 7
activity is in progress at time if it ends at time know that activity of project cannot end beyond
(Pritsker et al., 1969). Also, we . Therefore, using the lower band of
and the upper band of
in
,
the amount of type resource, which is required for all in-progress activities at time , is calculated. It should be noted that whenever at , the lower band in the innermost sigma presented in the left side of the equation 4 is more than the upper band ,that is, , Sigma will equal zero based on the summation properties. Also, note that being nonnegative of has been implicitly considered in this constraint. Constraints (5), (6), and (7) follow cash flow conditions in each period. In constraint (5), since represents the running (operating) cost of -type renewable resources at each and must be paid even if it is not used, must be calculated for all available resources during period . Constraints (8) to (11) calculate the increase or decrease of each renewable resource during the planning horizon. Constraints (12) and (13) determine the level of renewable resources at the beginning and end of the planning horizon, respectively. Constraint (14) ensures that the project activities are not started beyond the earliest and latest onset of activities. Finally, constraints (15) and (16) reflect the problem variables. In the above model, if the number of available projects equals one, the problem will be transformed into a special RIP. Since the NP-hard nature of the RIP has been proved (Möhring, 1984), the aforementioned model is also an NP-hard model and a method is proposed in the following to solve this model using the GSA.
3. The Proposed GSA The GSA, which is inspired by Newton's second law of gravity and motion, is one of the newest evolutionary optimization algorithms (Rashedi, Nezamabadi-Pour, & Saryazdi, 2009). This algorithm has been used in various fields such as image processing (Rashedi & Nezamabadi-Pour, 2013), network (Packiasudha, Suja, & Jerome, 2017), supply chain (Pei et al., 2014) and marketing (Kumar, Kumar, & Edukondalu, 2013). In the GSA, the optimization path is drawn using the gravitational laws of physics. In this algorithm, each agent (solution) has a mass depending on its position in the solution space and the related fitness function and exerts forces on other agents (solutions) in proportion to its mass. The sum of forces acting on each agent reveals the acceleration that affects the initial velocity of that agent and specifies the future position of the agent. The process of using the GSA method to solve the described Integrated Resource Management (IRM) problem consists of two major phases. In the first phase, given the limitations of the problem, a heuristic method is proposed to generate an initial population of solutions. In the second phase, with regard to the GSA updating process, changing the position of agents continues to improve the fitness function until termination conditions are met.
8
3.1. Creating the Initial Population 3.1.1. Representation Based on the framework of every meta-heuristic method, the first step in using the GSA method is to determine the representation method. As explained in describing the mathematical model, any solution to this problem should include three variables: 1) selected projects; 2) scheduling of selected projects; 3) level of each renewable resource at each period during the planning horizon. The representation method is defined as a pair of matrices. The first matrix, i.e. the start time matrix (S), represents the first two variables and the second matrix, i.e. the renewable resource level matrix (R), represents the third variable. The first matrix is a matrix in which . The entry in this matrix denotes the beginning of activity of project (
). For projects with activities fewer than
,
virtual activities are assumed with zero resources and time. The second matrix is a matrix in which the entry denotes the level of resource in period . The relationship between these two matrices is described in the following.
3.1.2. Project Selection and Scheduling Process (PSSP) To increase the probability of selecting more profitable projects, an initial fitness value is calculated for each project using the following equation: (17)
In each phase, probability of selection of each non-selected project is as follows. (18)
Where, is the set of projects not selected so far. Following the selection, each project is carried out with probability of based on a schedule. Each project is not carried out with probability of with an all-zero row in the start time matrix. (19)
If the above process leads to accomplishment of the project, its activities should be scheduled. In this regard, a random sequence of activities of the selected project is created such that the precedence relations defined for the project are considered. The resulting sequence is called . Afterwards, each activity can start in the interval that activity, where
considering the precedence relations defined for shows the set of direct predecessors of activity of project , and
the finish time of activity of project , and project .
are
represents modified earliest start time of activity of (20)
Moreover, 9
(21)
In the above relation, denotes the problem planning horizon, and and are the earliest start and finish times of activity of project , respectively. The length of the aforementioned period is shown by Γ. To improve the quality of initial solutions and increase the probability of satisfying the problem constraints, each time Γ is selected with a probability, which is calculated by combining two probability functions. The first probability function is used to improve resource leveling, whereas the second probability function is employed to postpone the start of an activity. Evidently, the aforementioned probability functions should be calculated times. is defined as the matrix of the maximum resources available during the planning horizon until step x and is defined as the matrix of the minimum resources available during the planning horizon until step x. These two matrices have dimensions, and to calculate them relations (22) and (23) are used for the first activity of the first selected project. (22) (23)
Afterwards, the first probability function is calculated as follows: (24)
The second probability function is also calculated using relation (25) (Tofighian & Naderi, 2015): (25)
Finally, the probability of the start of activity of project at each time during the interval calculated using relation (25):
Γ is
(26)
In each step, the start time of activity of project is determined by generating a uniformly distributed random number and considering the above probability, and then the and matrices are updated with recurrence relations (27) to (29). (27) (28) (29)
Next, the scheduled activity is omitted from the
sequence and the process continues with the next
activity in this sequence. The aforementioned process continues until the
sequence and the
set
become empty. If sequence becomes empty, it indicates that all the projects which we have decided to do, have been scheduled. And if set becomes empty, it suggests that reviewing all the projects in the project selection and scheduling process has been completed. 10
Figure (2) presents the pseudocode of the PSSP.
Figure (2): Pseudocode of Project Selection and Scheduling Process (PSSP)
3.1.3. Available Resources Finalization Process (ARFP) As explained in the previous section, during the PSSP, a minimum available resources matrix ( ) corresponding to each S matrix is obtained. However, in order to improve the objective function, the available resource can be increased in terms of some entries. This increase can be attributed to a condition in which the use of resource leveling leads to a decrease in the cost of using resources during the planning horizon. The matrix of final available resources with the same dimensions as matri R , is called R. The entry shows the final level of the type resource in period . As explained in the problem statement section, in this problem it is assumed that no profit is gained by buying and selling the resources. In such conditions, the level of each resource can escalate in a period provided that the minimum available resource in future periods exceeds the final available resources in the previous period. It is assumed that the goal is to determine the final level of resource in period and for , the inequality (30) occurs for the first time in period . (30)
Therefore, the final available resource
Ζ
in period
. The number of possible
(
) should be an integer in the interval
values equals: (31)
Hence, the Z range can be re-written as follows. Ζ
(32)
As increases in relation to , larger values should be selected from this set. On the other hand, as η escalates, smaller values are selected from this set. Hence, first the approximate cost of selecting each member of is calculated using relation (33), and then the probability of selecting each value is obtained using relation (34). 𝜂
(33) (34)
If relation (30) is not satisfied for any
value, then: (35)
Using relations (33) to (35) and starting with , the probability of selecting possible values of is calculated and by generating a uniformly distributed random number and considering the aforementioned probability, a member is selected from the set and then the next period will be initiated. This process continues until values for are obtained. 11
After finalizing the available resources matrix, using relation (5), which was described in the mathematical model section, it is possible to calculate the company’s cash at each period during the planning horizon ( ). Evidently, the only feasible solution is the one in which the resulting cash at all periods is positive. On the other hand, the company’s cash at the end of the last period, , is known as the objective function of that solution. (36)
The pseudocode of the process of finalization of the available resources matrix and checking feasibility of the solution (which was explained above) is depicted in Figure (3).
Figure (3): Pseudocode of Available Resources Finalization Process (ARFP)
3.2. Changing Position of Agents Using GSA 3.2.1. Calculating Agent Fitness In the GSA, the searcher agents, which represent the solutions, move in the solution space in a way that their fitness improves over time. To avoid mistaking this “time” for the time defined in previous sections, the timeline or solution stages are shown by τ . The gravitational mass of agents matches their fitness. Assume that the number of searcher agents is shown by . Therefore, the gravitational mass of agent is obtained through relations (37) and (38) (Rashedi et al., 2009). 𝜏
𝜏
𝜏
𝜏
(37)
𝜏
(38)
𝜏
Where,
denotes the gravitational mass of agent
at time 𝜏, and
a at time 𝜏, which is equal to the final cash calculated for that agent
𝜏 shows fitness of agent . In addition, for 𝜏
𝜏 in the current maximization problem we have,
and 𝜏
𝜏 𝜏
(39) 𝜏
(40)
3.2.2. Updating Agent Velocity and Position Since the original GSA is designed for the continuous solution space, hypotheses and modifications should be used to apply this algorithm to the discrete space of this problem. Firstly, in the standard GSA, calculations of changes of agent positions are separately presented for dimensions of the solution space 12
depending on the continuous n-dimension space under study. In the discrete problem under study, based on the selected representation method, it is possible to consider each of the entries of the start time matrix (s) a problem dimension, and then use the algorithm presented in the 3.1.3 section to create the related R matrix. Therefore, the number of problem dimensions is set to . In addition, as stated in section 3.1.2, each of the defined dimensions can vary in the interval . Fortunately, this characteristic can also be considered by making changes to the velocity and position relations. On the other hand, since selection or non-selection of each project (like project j) causes intense changes in the solution, the velocity and position of agents are calculated in two steps. In step one, considering the relative fitness of agents in the set (for which project is selected), decisions are made about selection or non-selection of project in agent in a future time (𝜏 ). Hence, it is assumed that is a set of factors in which project is selected. In this state, first the probability is calculated as follows: 𝜏
(41)
𝜏
By generating a uniformly distributed random number and considering the aforementioned probability, if a decision is made to avoid project , the row in the start times matrix of agent is set to zero for
𝜏
; otherwise, if a decision is made to carry out project , the process of updating positions in each of the dimensions (entries) of this row continues as follows. As stated in section 3.2.1, a random sequence of selected project activities is selected such that the precedence relations defined for that project are considered. This sequence is used for updating purposes as the sequence of dimensions in that project. Based on (Rashedi et al., 2009) and dimensions defined for this problem at the beginning of this section, the total force exerted on the dimension of agent by factors is obtained as follows.
𝜏
𝜏
𝜏
𝜏
𝜏
𝜏
𝜏
(42)
Afterwards, to calculate the acceleration, the aforementioned total force is divided by the gravitational mass of agent : 𝜏
𝜏 𝜏
𝜏
𝜏
𝜏
𝜏
(43)
Where, -
is a uniformly distributed random number in the interval [0,1].
𝜏 stands for the Manhattan distance of the matrices corresponding to agents and at time 𝜏. is an insignificant value used to avoid a zero denominator. 𝜏 is the gravitational coefficient which is set to (the initial value of 𝜏 ) in the beginning (with a value in the interval [0, 1]) and decreases over time. In the end, it equals (the final value of 𝜏 with a value in the interval [0, 1]) as shown in relation (44): 13
𝜏
𝜏
Velocity of agent
along the
𝜏
Where,
𝜏
However, if follows. 𝜏
dimension at time 𝜏
is calculated via the following relation:
𝜏
(45)
is a uniformly distributed random number in the interval [0, 1]. If dimension at time 𝜏
position along the 𝜏
(44)
𝜏
𝜏
(46)
is negative, position along the
𝜏
𝜏
is positive,
is calculated as follows:
𝜏
𝜏
𝜏
dimension at time 𝜏
𝜏
is calculated as
(47)
Figure (4) depicts the pseudocode of the mass position variation process.
Figure (4): Pseudocode of Agent’s Position Changing
4. Experimental Results 4.1. Test problems Since the research problem has not been discussed so far, through modifications and application of some parameters for the test problems defined for the RIP/max problem (which is available in PSPLIB), it was tried to use it within this framework. Table (1) presents the parameters and modifications. In this table, Moreover,
denotes the cost of resource , which is provided in the RIP/max test problems. shows a uniformly distributed random number in the interval (A, B).
Using the mentioned constructive relations, 27 groups of test problems are defined based on the number of available projects, , the maximum number of activities in each project, , and the number of renewable resources required for activities, . Each category consists of 10 test problems, and details of each category are presented in Table (2).
Table (1): Used Relations for Creating Parameters
Parameters
Constructive Relations for Test Problems
14
4.2. Results of the Proposed GSA The following relations are used to use the proposed GSA with the K and G parameters during solution steps (𝜏). 𝜏
𝜏
𝜏 𝜏
𝜏
In relation (48),
(48) 𝜏
(49)
𝜏 decreases linearly over time by starting with one and ending with zero. In relation
(49), 𝜏 which represents the number of elements exponentially decreases from to five percent of the elements. In this relation and represented the initial and final number of elements, respectively. This process allows the algorithm to properly explore the entire solution space over time or in the early steps, and search the solution space and exploit optimal solutions after finding the suitable spaces. Since there is no exact or heuristic solution to obtain the optimum solution or an approximately optimum solution to the research problem, the solutions resulted from Lingo are used in this section to assess the quality of the proposed GSA. Although the method used in Lingo does not deterministically yield the global optimum solution, it could be used as a suitable approach to assess the quality of results obtained from the proposed GSA. The experiments were performed on a personal computer with an Intel Core i7, 2.5 GHz processor and 8 GB memory. The main stopping criteria in this algorithm is the value considered for T which has been assumed to be 300. Also, if in each time (repetition) in the 30 final repetitions no improvement is achieved, the algorithm will be stopped. The number of searching elements (s) has been assumed to be 20. Therefore, considering function 𝜏 introduced above, . With the passage of time and increase of , the number of decreases and eventually, as explained above, it will reach 5% of searching elements, that is, 1. Thus, Also, considering function 𝜏 introduced above, and Table (3) illustrates the performance of the proposed GSA as compared to the performance of the Lingo. The comparative parameters used in this table are described in the following. : The number of problems solved by Lingo in 3600 CPU seconds.
15
: The number of problems solved by the proposed GSA in 3600 CPU seconds. : The average CPU time consumed to solve problems in Lingo. : The average CPU time consumed to solve problems using the proposed GSA. : The average relative deviation. : The maximum relative deviation. Relative deviation in the last two parameters (D1) is calculated as follows. (50)
As seen in Table (3), Lingo managed to solve all the problems in only 14 of the 27 categories during the predetermined time. In addition, less than 50% of problems in 8 categories and none of the problems in 3 categories were solved in Lingo. This is while, the proposed GSA yielded solutions to all of the 270 problems in all of the 27 categories. In addition, the average and maximum relative deviations in all of the categories were negligible by varying in the 2.02% to 3.07% range. However, the average CPU time spent on solving the problems in all of the categories with the proposed GSA was much shorter than Lingo. In addition, categories (21), (24), and (27) in which no problem was solved in Lingo had 5 renewable resources. On the other hand, the highest average time spent with the proposed GSA in these three categories was 125, 129, and 141 CPU seconds. 4.3. The Effect of IRM on PSSP Analysis This section analyzes the effect of the IRM on the Project Selection and Scheduling Process (PSSP). Considering results of the previous section which proved efficiency of the proposed GSA, the 27 problem categories defined using this and assuming constant available resource ( ) were solved once again. To this end, values of and were replaced with a very large number for all resources. Table (3) presents results obtained with constant values for . The comparison parameters used in this table are as follows assuming that denotes the problem objective function with variable values for , and denotes the objective function with constant values for . The comparative parameters used in this table are described in the following. : The average relative deviation. : The maximum relative deviation. Relative deviation ( ) is calculated as follows. (51)
16
Table (2): Parameter Settings for Test Problems Category’s Number
Number of Test Problem
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
5
5
5
5
5
5
5
5
5
10
10
10
10
10
10
10
10
10
15
15
15
15
15
15
15
15
15
10
10
10
20
20
20
30
30
30
10
10
10
20
20
20
30
30
30
10
10
10
20
20
20
30
30
30
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Table (3): Computational Results
Category’s Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
No. Lingo No. GSA AT Lingo AT GSA ARD1 % MRD1 % ARD2 % MRD2 %
10 10 85 5 0.00 0.00 12.31 13.37
10 10 91 5 0.00 0.00 13.27 15.99
10 10 241 8 0.00 0.00 13.32 14.39
10 10 104 6 0.00 0.00 14.71 18.03
10 10 163 6 0.00 0.00 12.19 13.59
10 10 472 9 0.00 0.00 12.64 13.87
10 10 146 6 0.00 0.00 14.39 18.26
10 10 351 8 0.12 0.21 13.78 16.26
10 10 861 33 0.47 0.88 12.62 14.05
10 10 219 7 0.09 0.18 12.63 14.29
10 10 717 14 0.39 0.65 12.44 13.49
9 10 1233 33 0.97 1.60 14.02 17.34
10 10 632 17 0.14 0.20 12.50 13.92
8 10 3029 42 0.69 1.00 15.09 18.79
Category’s Number
15
16
17
18
19
20
21
22
23
24
25
26
27
-
No. Lingo No. GSA AT Lingo AT GSA ARD1 % MRD1 % ARD2 % MRD2 %
4 10 3280 46 1.51 2.21 12.39 14.09
10 10 994 18 0.15 0.19 12.81 14.77
9 10 3117 51 1.37 1.99 14.07 16.16
4 10 3415 92 1.63 2.81 12.05 13.58
10 10 1414 35 0.37 0.54 13.10 15.63
5 10 3167 85 2.02 3.07 12.11 13.87
0 10 125 12.08 13.46
7 10 2354 59 0.55 0.58 14.19 16.96
3 10 3029 57 2.32 3.4 12.26 13.01
0 10 129 14.84 17.53
8 10 2942 69 0.81 1.27 12.20 13.64
1 10 3111 118 1.91 1.91 15.31 18.78
0 10 141 13.64 15.94
-
17
As shown in Table (3), the average relative deviation obtained for all of the categories is larger than 12% and the average improvement of profitability is 13.2%. There were problems in some categories with relative deviations of approximately 18%. Although these deviations partly depend on the values of and , given the rational and random process used to determine these values in the used test problems, it is possible to use these results as proof of the considerable effect of the IRM on the growth of profitability and efficiency of project-oriented companies.
5. Conclusions In this paper, the issue of Integrated Resource Management (IRM), which has not been discussed previously, was studied. The previous approaches to resource management were described and advantages of the new proposed approach were enumerated. Also, a mixed-integer programming model was developed for the proposed approach. The objective function of the model was maximization of the company’s cash at the end of the planning horizon. Since the problem was NP-hard, first a heuristic method was developed for generating feasible initial solutions, then a combinatorial algorithm was developed based on GSA to solve it. The Results of the proposed method with 27 groups of test problems of different sizes were assessed against the results of Lingo. It was found that the method achieve acceptable results similar to the results of Lingo, in much shorter time; also, the proposed method was capable of solving many problems which Lingo couldn't solve them in a rational time. The effect of the IRM on the project selection and scheduling process was also assessed, and it was revealed that the average improvement of profitability on test problems was 13.2%. Inclusion of time-dependent costs for increasing or decreasing renewable resources can be the subject of future studies. Considering this inclusion in inflationary condition makes the problem similar to the condition of markets in which inflation causes considerable changes to resource supply costs and projects proceeds. It is also useful to use actual data to measure and analyze the effect of the IRM on profitability of project-oriented companies. References Afshar-Nadjafi, B. (2014). Multi-mode resource availability cost problem with recruitment and release dates for resources. Applied Mathematical Modelling, 38(21), 5347-5355. Aquilano, N. J., & Smith, D. E. (1980). A formal set of algorithms for project scheduling with critical path scheduling/material requirements planning. Journal of Operations Management, 1(2), 57-67. Chen, J., & Askin, R. G. (2009). Project selection, scheduling and resource allocation with time dependent returns. European Journal of Operational Research, 193(1), 23-34. Dodin, B., & Elimam, A. (2001). Integrated project scheduling and material planning with variable activity duration and rewards. IIE transactions, 33(11), 1005-1018. Huang, X., & Zhao, T. (2014). Project selection and scheduling with uncertain net income and investment cost. Applied Mathematics and Computation, 247, 61-71. Khoshjahan, Y., Najafi, A. A., & Afshar-Nadjafi, B. (2013). Resource constrained project scheduling problem with discounted earliness–tardiness penalties: Mathematical modeling and solving procedure. Computers & Industrial Engineering, 66(2), 293-300. Kumar, J. V., Kumar, D. V., & Edukondalu, K. (2013). Strategic bidding using fuzzy adaptive gravitational search algorithm in a pool based electricity market. Applied Soft Computing, 13(5), 2445-2455. 18
Leyman, P., & Vanhoucke, M. (2017). Capital-and resource-constrained project scheduling with net present value optimization. European Journal of Operational Research, 256(3), 757-776. Li, X., Fang, S.-C., Tian, Y., & Guo, X. (2015). Expanded model of the project portfolio selection problem with divisibility, time profile factors and cardinality constraints. Journal of the Operational Research Society, 66(7), 1132-1139. Liu, S.-S., & Wang, C.-J. (2011). Optimizing project selection and scheduling problems with timedependent resource constraints. Automation in Construction, 20(8), 1110-1119. Möhring, R. H. (1984). Minimizing costs of resource requirements in project networks subject to a fixed completion time. Operations Research, 32(1), 89-120. Packiasudha, M., Suja, S., & Jerome, J. (2017). A new Cumulative Gravitational Search algorithm for optimal placement of FACT device to minimize system loss in the deregulated electrical power environment. International Journal of Electrical Power & Energy Systems, 84, 34-46. Pei, J., Liu, X., Pardalos, P. M., Fan, W., Yang, S., & Wang, L. (2014). Application of an effective modified gravitational search algorithm for the coordinated scheduling problem in a two-stage supply chain. The International Journal of Advanced Manufacturing Technology, 70(1-4), 335-348. Pritsker, A. A. B., Waiters, L. J., & Wolfe, P. M. (1969). Multiproject scheduling with limited resources: A zero-one programming approach. Management Science, 16(1), 93-108. Rashedi, E., & Nezamabadi-Pour, H. (2013). A stochastic gravitational approach to feature based color image segmentation. Engineering Applications of Artificial Intelligence, 26(4), 1322-1332. Rashedi, E., Nezamabadi-Pour, H., & Saryazdi, S. (2009). GSA: a gravitational search algorithm. Information sciences, 179(13), 2232-2248. Rodrigues, S. B., & Yamashita, D. S. (2010). An exact algorithm for minimizing resource availability costs in project scheduling. European Journal of Operational Research, 206(3), 562-568. Schaeffer, S., & Cruz-Reyes, L. (2016). Static R&D project portfolio selection in public organizations. Decision Support Systems, 84, 53-63. Schmitt, T., & Faaland, B. (2004). Scheduling recurrent construction. Naval Research Logistics (NRL), 51(8), 1102-1128. Shahsavar, A., Zoraghi, N., & Abbasi, B. Integration of resource investment problem with quantity discount problem in material ordering for minimizing resource costs of projects. Operational Research, 1-28. Smith-Daniels, D. E., & Aquilano, N. J. (1984). Constrained resource project scheduling subject to material constraints. Journal of Operations Management, 4(4), 369-387. Smith-Daniels, D. E., & Smith-Daniels, V. L. (1987). Optimal project scheduling with materials ordering. IIE transactions, 19(2), 122-129. Tavana, M., Khalili-Damghani, K., & Sadi-Nezhad, S. (2013). A fuzzy group data envelopment analysis model for high-technology project selection: A case study at NASA. Computers & Industrial Engineering, 66(1), 10-23. Tofighian, A. A., & Naderi, B. (2015). Modeling and solving the project selection and scheduling. Computers & Industrial Engineering, 83, 30-38. Zhu, X., Ruiz, R., Li, S., & Li, X. (2016). An effective heuristic for project scheduling with resource availability cost. European Journal of Operational Research. Zoraghi, N., Shahsavar, A., Abbasi, B., & Van Peteghem, V. (2016). Multi-mode resource-constrained project scheduling problem with material ordering under bonus–penalty policies. TOP, 1-31. Zoraghia, N., Najafib, A., & Niaki, S. (2014). Resource Constrained Project Scheduling with Material Ordering: Two Hybridized Meta-Heuristic Approaches. International Journal of Engineering.
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S0360-8352(17)30133-X http://dx.doi.org/10.1016/j.cie.2017.04.003 CAIE 4690
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
25 November 2016 29 March 2017 3 April 2017
Please cite this article as: Shariatmadari, M., Nahavandi, N., Zegordi, S.H., Sobhiyah, M.H., Integrated Resource Management for Simultaneous Project Selection and Scheduling, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie.2017.04.003
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Integrated Resource Management for Simultaneous Project Selection and Scheduling
Mohammad Shariatmadaria Nasim Nahavandia,* Seyed Hessameddin Zegordia Mohammad Hossein Sobhiyahb
a
: Faculty of Industrial & Systems Engineering, Tarbiat Modares University, P.O. Box 14115 143, Tehran, Iran. b
: Project & Construction Management Group, Faculty of Arts & Architecture, Tarbiat Modares University (TMU), P.O. Box 14155 4838, Tehran, Iran. *, Corresponding Author
1
Integrated Resource Management for Simultaneous Project Selection and Scheduling
Abstract Nowadays, resource management is one of the most critical decision making processes in projectoriented companies. To reach the highest level of profitability, these companies should consider profiting from available projects and optimal management of their (renewable and nonrenewable) resources in terms of selection and scheduling of the most optimal project portfolio. In this regard, this paper presents the Integrated Resource Management (IRM) approach for simultaneous project selection and scheduling. A mixed-integer programming model is also proposed for the approach. To solve the research problem, first a heuristic algorithm is developed for generating feasible initial solutions; then a method for improving the resulted solutions is presented based on Gravitational Search Algorithm (GSA). Finally, the proposed method is compared with Lingo on a set of 270 test problems. The comparison demonstrates that the proposed method outperforms Lingo in most cases and has less time complexity. Also, it will be shown that using IRM approach in test problems improves average profitability by 13.2 %.
Keywords: Integrated Resource Management (IRM), Project Selection, Project Scheduling, Resource Investment Problem (RIP), Gravitational Search Algorithm (GSA).
1. Introduction Project-oriented companies need to answer the following questions to achieve the best performance in competitive markets: 1. Regarding the company’s renewable and nonrenewable resources and the profits of project, which one should be chosen? 2. How should the selected projects be scheduled during the planning horizon considering the existing constraints? 3. How should company’s renewable and nonrenewable resources be managed during the planning horizon? Many attempts have been made to provide separate responses to each of the above questions. The first question was answered through studies on the Project Portfolio Selection (PPS) problem (Tavana, KhaliliDamghani, & Sadi-Nezhad, 2013); (Li, Fang, Tian, & Guo, 2015); (Schaeffer & Cruz-Reyes, 2016). In general, the aim of the PPS is to select a project portfolio from several candidate projects considering the existing limitations and facilities so as to achieve an optimum objective function (Tofighian & Naderi, 2015). One of the most important and commonly used objective functions in this field is the maximization of projects profit (Liu & Wang, 2011).
2
To answer the second question, numerous studies have been conducted in the past five decades in the form of the Resource-Constrained Project Scheduling Problem (RCPSP). The objective of the RCPSP is to allocate the resources (which are considered to be constant in the standard model) to perform activities. Therefore, the optimum objective function value is obtained while all of the constraints are considered. Various objective functions can be used for this problem; for examples, time-related objectives (Khoshjahan, Najafi, & Afshar-Nadjafi, 2013), cost-related objectives (Khoshjahan et al., 2013) and cash flow-related objectives (Leyman & Vanhoucke, 2017). The answer to the third question involves a special type of RCPSP known as the Resource Investment Problem (RIP) in which renewable resources are variable. In spite of development of this problem in 1984 (Möhring, 1984), fewer studies have been carried out on this problem than the standard RCPSP. This problem is referred to as the Resource Availability Cost Problem (RACP) in most articles (Zhu, Ruiz, Li, & Li, 2016). Although several articles aimed at presenting exact methods for solving this problem (Rodrigues & Yamashita, 2010), most of them adopted heuristic and meta-heuristic approaches (AfsharNadjafi, 2014); (Zhu et al., 2016). Previously, it was tried to answer the first and second questions simultaneously by studying the project selection and scheduling problem. (Chen & Askin, 2009) proposed a mixed-integer programming (MIP) model with an NPV maximization objective function and used an implicit enumeration procedure to solve the problem. (Liu & Wang, 2011) developed an optimization model for the project selection and scheduling problem based on the Constraint Programing (CP) method using time-dependent resource constraints. (Huang & Zhao, 2014) studied this problem in the absence of historical data for the problem parameters. They also developed a Genetic Algorithm (GA) to solve the suggested model. As well as for answering the second and third questions simultaneously, combination of the project scheduling and material ordering have been discussed in some researches (Aquilano & Smith, 1980); (Smith-Daniels & Aquilano, 1984). In this regard, (Smith-Daniels & Smith-Daniels, 1987) presented a MIP model where activity durations are fixed in the problem. In contrast to previous research, (Dodin & Elimam, 2001) developed a model for the problem with variable activity durations. In addition, Several papers developed heuristic approaches (Schmitt & Faaland, 2004) and meta-heuristic approaches (Zoraghia, Najafib, & Niaki, 2014) to solve the problem. (Zoraghi, Shahsavar, Abbasi, & Van Peteghem, 2016) studied the multi-mode resource constrained project scheduling combined with the material ordering. They also proposed three meta-heuristic approaches to obtain near-optimal solutions. For minimizing the renewable and nonrenewable resource costs, (Shahsavar, Zoraghi, & Abbasi) presented the integration of the RIP with the quantity discount problem in material ordering. Given the efforts made in the project selection and scheduling area and also the project scheduling and material ordering area, the objective of the present research is to answer all of the three questions simultaneously. In other words, based on the IRM and considering the time-dependent revenues of the projects, the project portfolio is selected and scheduled in the planning horizon simultaneously. Since the problem is an NP-hard problem, a Gravitational Search Algorithm (GSA) is developed in this research to solve it as a set of test problems. The effect of the IRM on the profitability of Project Selection and Scheduling Process (PPSP) is also examined. The remainder of the paper is outlined as follows: section 2 proposes a mathematical model and approach to the research problem. Section 3 describes different steps of the GSA proposed for this 3
problem. Section 4 presents the calculation results of applying the proposed algorithm to a set of test problems. Finally, section 5 covers the research conclusions and future research.
2. Problem Statement and Mathematical Formulation 2.1. The Proposed IRM Approach Resources are elements undertaking project activities. Hence, resource management, which involves assessment of resource levels during a planning horizon, plays a vital role in determining the profits of project-oriented companies.
Figure (1): Different resource management approaches to the project selection and scheduling problems
Figure (1) depicts three different approaches for resource management in the performance of companies. In the first approach, the company assesses the resource levels against criteria that are beyond the scope of Project Selection and Scheduling Process (PSSP), and the results are used as input parameters for these processes. In the second approach, resource management covers project scheduling, but project selection is still beyond the scope of the process. The Resource Investment Problem (RIP) is an example of the second approach, but in the standard RIP, resource management is limited to management of renewable resources. Finally, the third approach, which is proposed and used in this research, includes resource management during simultaneous project selection and scheduling. This approach (IRM) includes the following features: -
The possibility of simultaneous management of renewable and nonrenewable resources to achieve optimum resource productivity during the planning horizon. Considering different costs (included running cost, decreasing cost and increasing cost) of renewable resources for using them to accomplish activities of the selected project portfolio. Considering the company’s cash flow during the planning horizon as a parameter determining resource preparation and accomplishing activities of the selected project portfolio. Implicitly considering the Resource Leveling Problem wherever resource leveling improves the profitability of the company.
2.2. Problem Statement The problem is defined as follows in accordance with the decision-making environment of many projectoriented companies: There are a few projects to select at the beginning of planning horizon and completion of each project involves time-dependent revenues, which change the company’s cash flow during the planning horizon. The initial level of renewable and nonrenewable resources is known in the beginning of the planning horizon. Without loss of generality, it is assumed that the only nonrenewable resource is money. In practical problems it is possible to calculate the monetary equivalent of other nonrenewable resources by applying the suitable conversion factor and reducing the revenue of 4
activities proportionally. During the planning horizon, if the company’s cash is adequate it is possible to change the level of renewable resources. Costs of renewable resources are defined in three forms: A) Cost of running a resource unit per time unit (which cannot be included in direct costs of an activity because the resource may be unemployed in a period of time) such as staff salaries; B) Cost of increasing a resource unit, like the hiring and training cost of human resources; C) Cost of reducing a resource unit, such as the releasing cost of human resources. Also, the amount of the renewable resources required for completion of any activity has been considered as predetermined constant. According to non-inflationary conditions in real world, in this problem it is assumed that if negative costs are associated with reducing a resource unit (in the case of resources the sales or dismissal of which generates revenues), the sales price will be lower than the cost of increasing a resource unit. In other words, it is presumed that the company cannot gain profit by buying and selling a resource unit in different periods. The resources required for accomplishing the activities are known and invariant. In addition, all the precedence relations are considered to be finish-start with zero lag. The early start time for the first activity of all project is equal to zero. Then the early start time for the activity of project is the maximum of the early finish times of its immediate predecessors ( ), i.e. : The late finish time for the last activity of all project is equal to the planning horizon. Therefore early finish time for the activity of project is the minimum of the late start times of its immediate successors ( ), i.e. . Any activity of project must be done inside the time window . The goal is to create project portfolios, schedule them in the planning horizon, and determine the level of each resource in each period so that the company’s assets are maximized at the end of the planning horizon.
2.3. Notations and Mathematical Model Indices : Project’s indices
.
: Activity’s indices
.
: Renewable Resource’s indices : Period’s indices
.
.
Parameters : Resource Units
required by activity of project (nonnegative).
: Cost of increasing resource
per-unit. 5
: Cost of decreasing resource : Cost of running resource
per-unit.
per-unit per-period.
: Duration of activity of project . : Expected profit (expected revenue minus the cost of non-renewable resources) if activity project is completed in period . : Set of immediate successor activities of project . : Set of immediate predecessor activities of project . : Company’s current assets at the beginning of the planning horizon. : Resource
available in the beginning of the planning horizon.
: Early start time for activity of project . : Late start time for activity of project . : Early finish time for activity of project . : Late finish time for activity of project . : Number of activities of project .
Decision Variables : 1 if project is selected, 0 otherwise. : 1 if activity in project is completed at period ; and 0 otherwise. : Resource
available in period .
Intermediate Variables : Increased amount of resource : Decreased amount of resource
in period compared to period in period compared to period
: Company’s current assets at the beginning of period .
6
. .
in
Mathematical Model (1) St. (2)
,
(3)
(4)
(5)
(6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
Objective function (1) shows maximization of cash available at the end of the planning horizon assuming that the level of renewable resources at the end of the planning horizon equals the renewable resources level at the beginning of the planning horizon. Constraint (2) ensures that no activity of non-selected projects is accomplished and all of the activities of the selected projects are accomplished. The constraints (3) and (4) are multi-project version of the discrete-time (DT) formulation for RCPSP (Pritsker, Waiters, & Wolfe, 1969). Constraint (3) includes the precedence relations for each project. The level of renewable resources used at each period should not exceed the level of available resources at that particular period, which is presented by constraint (4). Upper and lower bands of the innermost sigma presented in the left side of the equation 4, seek to detect in-progress activities at time . An 7
activity is in progress at time if it ends at time know that activity of project cannot end beyond
(Pritsker et al., 1969). Also, we . Therefore, using the lower band of
and the upper band of
in
,
the amount of type resource, which is required for all in-progress activities at time , is calculated. It should be noted that whenever at , the lower band in the innermost sigma presented in the left side of the equation 4 is more than the upper band ,that is, , Sigma will equal zero based on the summation properties. Also, note that being nonnegative of has been implicitly considered in this constraint. Constraints (5), (6), and (7) follow cash flow conditions in each period. In constraint (5), since represents the running (operating) cost of -type renewable resources at each and must be paid even if it is not used, must be calculated for all available resources during period . Constraints (8) to (11) calculate the increase or decrease of each renewable resource during the planning horizon. Constraints (12) and (13) determine the level of renewable resources at the beginning and end of the planning horizon, respectively. Constraint (14) ensures that the project activities are not started beyond the earliest and latest onset of activities. Finally, constraints (15) and (16) reflect the problem variables. In the above model, if the number of available projects equals one, the problem will be transformed into a special RIP. Since the NP-hard nature of the RIP has been proved (Möhring, 1984), the aforementioned model is also an NP-hard model and a method is proposed in the following to solve this model using the GSA.
3. The Proposed GSA The GSA, which is inspired by Newton's second law of gravity and motion, is one of the newest evolutionary optimization algorithms (Rashedi, Nezamabadi-Pour, & Saryazdi, 2009). This algorithm has been used in various fields such as image processing (Rashedi & Nezamabadi-Pour, 2013), network (Packiasudha, Suja, & Jerome, 2017), supply chain (Pei et al., 2014) and marketing (Kumar, Kumar, & Edukondalu, 2013). In the GSA, the optimization path is drawn using the gravitational laws of physics. In this algorithm, each agent (solution) has a mass depending on its position in the solution space and the related fitness function and exerts forces on other agents (solutions) in proportion to its mass. The sum of forces acting on each agent reveals the acceleration that affects the initial velocity of that agent and specifies the future position of the agent. The process of using the GSA method to solve the described Integrated Resource Management (IRM) problem consists of two major phases. In the first phase, given the limitations of the problem, a heuristic method is proposed to generate an initial population of solutions. In the second phase, with regard to the GSA updating process, changing the position of agents continues to improve the fitness function until termination conditions are met.
8
3.1. Creating the Initial Population 3.1.1. Representation Based on the framework of every meta-heuristic method, the first step in using the GSA method is to determine the representation method. As explained in describing the mathematical model, any solution to this problem should include three variables: 1) selected projects; 2) scheduling of selected projects; 3) level of each renewable resource at each period during the planning horizon. The representation method is defined as a pair of matrices. The first matrix, i.e. the start time matrix (S), represents the first two variables and the second matrix, i.e. the renewable resource level matrix (R), represents the third variable. The first matrix is a matrix in which . The entry in this matrix denotes the beginning of activity of project (
). For projects with activities fewer than
,
virtual activities are assumed with zero resources and time. The second matrix is a matrix in which the entry denotes the level of resource in period . The relationship between these two matrices is described in the following.
3.1.2. Project Selection and Scheduling Process (PSSP) To increase the probability of selecting more profitable projects, an initial fitness value is calculated for each project using the following equation: (17)
In each phase, probability of selection of each non-selected project is as follows. (18)
Where, is the set of projects not selected so far. Following the selection, each project is carried out with probability of based on a schedule. Each project is not carried out with probability of with an all-zero row in the start time matrix. (19)
If the above process leads to accomplishment of the project, its activities should be scheduled. In this regard, a random sequence of activities of the selected project is created such that the precedence relations defined for the project are considered. The resulting sequence is called . Afterwards, each activity can start in the interval that activity, where
considering the precedence relations defined for shows the set of direct predecessors of activity of project , and
the finish time of activity of project , and project .
are
represents modified earliest start time of activity of (20)
Moreover, 9
(21)
In the above relation, denotes the problem planning horizon, and and are the earliest start and finish times of activity of project , respectively. The length of the aforementioned period is shown by Γ. To improve the quality of initial solutions and increase the probability of satisfying the problem constraints, each time Γ is selected with a probability, which is calculated by combining two probability functions. The first probability function is used to improve resource leveling, whereas the second probability function is employed to postpone the start of an activity. Evidently, the aforementioned probability functions should be calculated times. is defined as the matrix of the maximum resources available during the planning horizon until step x and is defined as the matrix of the minimum resources available during the planning horizon until step x. These two matrices have dimensions, and to calculate them relations (22) and (23) are used for the first activity of the first selected project. (22) (23)
Afterwards, the first probability function is calculated as follows: (24)
The second probability function is also calculated using relation (25) (Tofighian & Naderi, 2015): (25)
Finally, the probability of the start of activity of project at each time during the interval calculated using relation (25):
Γ is
(26)
In each step, the start time of activity of project is determined by generating a uniformly distributed random number and considering the above probability, and then the and matrices are updated with recurrence relations (27) to (29). (27) (28) (29)
Next, the scheduled activity is omitted from the
sequence and the process continues with the next
activity in this sequence. The aforementioned process continues until the
sequence and the
set
become empty. If sequence becomes empty, it indicates that all the projects which we have decided to do, have been scheduled. And if set becomes empty, it suggests that reviewing all the projects in the project selection and scheduling process has been completed. 10
Figure (2) presents the pseudocode of the PSSP.
Figure (2): Pseudocode of Project Selection and Scheduling Process (PSSP)
3.1.3. Available Resources Finalization Process (ARFP) As explained in the previous section, during the PSSP, a minimum available resources matrix ( ) corresponding to each S matrix is obtained. However, in order to improve the objective function, the available resource can be increased in terms of some entries. This increase can be attributed to a condition in which the use of resource leveling leads to a decrease in the cost of using resources during the planning horizon. The matrix of final available resources with the same dimensions as matri R , is called R. The entry shows the final level of the type resource in period . As explained in the problem statement section, in this problem it is assumed that no profit is gained by buying and selling the resources. In such conditions, the level of each resource can escalate in a period provided that the minimum available resource in future periods exceeds the final available resources in the previous period. It is assumed that the goal is to determine the final level of resource in period and for , the inequality (30) occurs for the first time in period . (30)
Therefore, the final available resource
Ζ
in period
. The number of possible
(
) should be an integer in the interval
values equals: (31)
Hence, the Z range can be re-written as follows. Ζ
(32)
As increases in relation to , larger values should be selected from this set. On the other hand, as η escalates, smaller values are selected from this set. Hence, first the approximate cost of selecting each member of is calculated using relation (33), and then the probability of selecting each value is obtained using relation (34). 𝜂
(33) (34)
If relation (30) is not satisfied for any
value, then: (35)
Using relations (33) to (35) and starting with , the probability of selecting possible values of is calculated and by generating a uniformly distributed random number and considering the aforementioned probability, a member is selected from the set and then the next period will be initiated. This process continues until values for are obtained. 11
After finalizing the available resources matrix, using relation (5), which was described in the mathematical model section, it is possible to calculate the company’s cash at each period during the planning horizon ( ). Evidently, the only feasible solution is the one in which the resulting cash at all periods is positive. On the other hand, the company’s cash at the end of the last period, , is known as the objective function of that solution. (36)
The pseudocode of the process of finalization of the available resources matrix and checking feasibility of the solution (which was explained above) is depicted in Figure (3).
Figure (3): Pseudocode of Available Resources Finalization Process (ARFP)
3.2. Changing Position of Agents Using GSA 3.2.1. Calculating Agent Fitness In the GSA, the searcher agents, which represent the solutions, move in the solution space in a way that their fitness improves over time. To avoid mistaking this “time” for the time defined in previous sections, the timeline or solution stages are shown by τ . The gravitational mass of agents matches their fitness. Assume that the number of searcher agents is shown by . Therefore, the gravitational mass of agent is obtained through relations (37) and (38) (Rashedi et al., 2009). 𝜏
𝜏
𝜏
𝜏
(37)
𝜏
(38)
𝜏
Where,
denotes the gravitational mass of agent
at time 𝜏, and
a at time 𝜏, which is equal to the final cash calculated for that agent
𝜏 shows fitness of agent . In addition, for 𝜏
𝜏 in the current maximization problem we have,
and 𝜏
𝜏 𝜏
(39) 𝜏
(40)
3.2.2. Updating Agent Velocity and Position Since the original GSA is designed for the continuous solution space, hypotheses and modifications should be used to apply this algorithm to the discrete space of this problem. Firstly, in the standard GSA, calculations of changes of agent positions are separately presented for dimensions of the solution space 12
depending on the continuous n-dimension space under study. In the discrete problem under study, based on the selected representation method, it is possible to consider each of the entries of the start time matrix (s) a problem dimension, and then use the algorithm presented in the 3.1.3 section to create the related R matrix. Therefore, the number of problem dimensions is set to . In addition, as stated in section 3.1.2, each of the defined dimensions can vary in the interval . Fortunately, this characteristic can also be considered by making changes to the velocity and position relations. On the other hand, since selection or non-selection of each project (like project j) causes intense changes in the solution, the velocity and position of agents are calculated in two steps. In step one, considering the relative fitness of agents in the set (for which project is selected), decisions are made about selection or non-selection of project in agent in a future time (𝜏 ). Hence, it is assumed that is a set of factors in which project is selected. In this state, first the probability is calculated as follows: 𝜏
(41)
𝜏
By generating a uniformly distributed random number and considering the aforementioned probability, if a decision is made to avoid project , the row in the start times matrix of agent is set to zero for
𝜏
; otherwise, if a decision is made to carry out project , the process of updating positions in each of the dimensions (entries) of this row continues as follows. As stated in section 3.2.1, a random sequence of selected project activities is selected such that the precedence relations defined for that project are considered. This sequence is used for updating purposes as the sequence of dimensions in that project. Based on (Rashedi et al., 2009) and dimensions defined for this problem at the beginning of this section, the total force exerted on the dimension of agent by factors is obtained as follows.
𝜏
𝜏
𝜏
𝜏
𝜏
𝜏
𝜏
(42)
Afterwards, to calculate the acceleration, the aforementioned total force is divided by the gravitational mass of agent : 𝜏
𝜏 𝜏
𝜏
𝜏
𝜏
𝜏
(43)
Where, -
is a uniformly distributed random number in the interval [0,1].
𝜏 stands for the Manhattan distance of the matrices corresponding to agents and at time 𝜏. is an insignificant value used to avoid a zero denominator. 𝜏 is the gravitational coefficient which is set to (the initial value of 𝜏 ) in the beginning (with a value in the interval [0, 1]) and decreases over time. In the end, it equals (the final value of 𝜏 with a value in the interval [0, 1]) as shown in relation (44): 13
𝜏
𝜏
Velocity of agent
along the
𝜏
Where,
𝜏
However, if follows. 𝜏
dimension at time 𝜏
is calculated via the following relation:
𝜏
(45)
is a uniformly distributed random number in the interval [0, 1]. If dimension at time 𝜏
position along the 𝜏
(44)
𝜏
𝜏
(46)
is negative, position along the
𝜏
𝜏
is positive,
is calculated as follows:
𝜏
𝜏
𝜏
dimension at time 𝜏
𝜏
is calculated as
(47)
Figure (4) depicts the pseudocode of the mass position variation process.
Figure (4): Pseudocode of Agent’s Position Changing
4. Experimental Results 4.1. Test problems Since the research problem has not been discussed so far, through modifications and application of some parameters for the test problems defined for the RIP/max problem (which is available in PSPLIB), it was tried to use it within this framework. Table (1) presents the parameters and modifications. In this table, Moreover,
denotes the cost of resource , which is provided in the RIP/max test problems. shows a uniformly distributed random number in the interval (A, B).
Using the mentioned constructive relations, 27 groups of test problems are defined based on the number of available projects, , the maximum number of activities in each project, , and the number of renewable resources required for activities, . Each category consists of 10 test problems, and details of each category are presented in Table (2).
Table (1): Used Relations for Creating Parameters
Parameters
Constructive Relations for Test Problems
14
4.2. Results of the Proposed GSA The following relations are used to use the proposed GSA with the K and G parameters during solution steps (𝜏). 𝜏
𝜏
𝜏 𝜏
𝜏
In relation (48),
(48) 𝜏
(49)
𝜏 decreases linearly over time by starting with one and ending with zero. In relation
(49), 𝜏 which represents the number of elements exponentially decreases from to five percent of the elements. In this relation and represented the initial and final number of elements, respectively. This process allows the algorithm to properly explore the entire solution space over time or in the early steps, and search the solution space and exploit optimal solutions after finding the suitable spaces. Since there is no exact or heuristic solution to obtain the optimum solution or an approximately optimum solution to the research problem, the solutions resulted from Lingo are used in this section to assess the quality of the proposed GSA. Although the method used in Lingo does not deterministically yield the global optimum solution, it could be used as a suitable approach to assess the quality of results obtained from the proposed GSA. The experiments were performed on a personal computer with an Intel Core i7, 2.5 GHz processor and 8 GB memory. The main stopping criteria in this algorithm is the value considered for T which has been assumed to be 300. Also, if in each time (repetition) in the 30 final repetitions no improvement is achieved, the algorithm will be stopped. The number of searching elements (s) has been assumed to be 20. Therefore, considering function 𝜏 introduced above, . With the passage of time and increase of , the number of decreases and eventually, as explained above, it will reach 5% of searching elements, that is, 1. Thus, Also, considering function 𝜏 introduced above, and Table (3) illustrates the performance of the proposed GSA as compared to the performance of the Lingo. The comparative parameters used in this table are described in the following. : The number of problems solved by Lingo in 3600 CPU seconds.
15
: The number of problems solved by the proposed GSA in 3600 CPU seconds. : The average CPU time consumed to solve problems in Lingo. : The average CPU time consumed to solve problems using the proposed GSA. : The average relative deviation. : The maximum relative deviation. Relative deviation in the last two parameters (D1) is calculated as follows. (50)
As seen in Table (3), Lingo managed to solve all the problems in only 14 of the 27 categories during the predetermined time. In addition, less than 50% of problems in 8 categories and none of the problems in 3 categories were solved in Lingo. This is while, the proposed GSA yielded solutions to all of the 270 problems in all of the 27 categories. In addition, the average and maximum relative deviations in all of the categories were negligible by varying in the 2.02% to 3.07% range. However, the average CPU time spent on solving the problems in all of the categories with the proposed GSA was much shorter than Lingo. In addition, categories (21), (24), and (27) in which no problem was solved in Lingo had 5 renewable resources. On the other hand, the highest average time spent with the proposed GSA in these three categories was 125, 129, and 141 CPU seconds. 4.3. The Effect of IRM on PSSP Analysis This section analyzes the effect of the IRM on the Project Selection and Scheduling Process (PSSP). Considering results of the previous section which proved efficiency of the proposed GSA, the 27 problem categories defined using this and assuming constant available resource ( ) were solved once again. To this end, values of and were replaced with a very large number for all resources. Table (3) presents results obtained with constant values for . The comparison parameters used in this table are as follows assuming that denotes the problem objective function with variable values for , and denotes the objective function with constant values for . The comparative parameters used in this table are described in the following. : The average relative deviation. : The maximum relative deviation. Relative deviation ( ) is calculated as follows. (51)
16
Table (2): Parameter Settings for Test Problems Category’s Number
Number of Test Problem
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
5
5
5
5
5
5
5
5
5
10
10
10
10
10
10
10
10
10
15
15
15
15
15
15
15
15
15
10
10
10
20
20
20
30
30
30
10
10
10
20
20
20
30
30
30
10
10
10
20
20
20
30
30
30
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Table (3): Computational Results
Category’s Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
No. Lingo No. GSA AT Lingo AT GSA ARD1 % MRD1 % ARD2 % MRD2 %
10 10 85 5 0.00 0.00 12.31 13.37
10 10 91 5 0.00 0.00 13.27 15.99
10 10 241 8 0.00 0.00 13.32 14.39
10 10 104 6 0.00 0.00 14.71 18.03
10 10 163 6 0.00 0.00 12.19 13.59
10 10 472 9 0.00 0.00 12.64 13.87
10 10 146 6 0.00 0.00 14.39 18.26
10 10 351 8 0.12 0.21 13.78 16.26
10 10 861 33 0.47 0.88 12.62 14.05
10 10 219 7 0.09 0.18 12.63 14.29
10 10 717 14 0.39 0.65 12.44 13.49
9 10 1233 33 0.97 1.60 14.02 17.34
10 10 632 17 0.14 0.20 12.50 13.92
8 10 3029 42 0.69 1.00 15.09 18.79
Category’s Number
15
16
17
18
19
20
21
22
23
24
25
26
27
-
No. Lingo No. GSA AT Lingo AT GSA ARD1 % MRD1 % ARD2 % MRD2 %
4 10 3280 46 1.51 2.21 12.39 14.09
10 10 994 18 0.15 0.19 12.81 14.77
9 10 3117 51 1.37 1.99 14.07 16.16
4 10 3415 92 1.63 2.81 12.05 13.58
10 10 1414 35 0.37 0.54 13.10 15.63
5 10 3167 85 2.02 3.07 12.11 13.87
0 10 125 12.08 13.46
7 10 2354 59 0.55 0.58 14.19 16.96
3 10 3029 57 2.32 3.4 12.26 13.01
0 10 129 14.84 17.53
8 10 2942 69 0.81 1.27 12.20 13.64
1 10 3111 118 1.91 1.91 15.31 18.78
0 10 141 13.64 15.94
-
17
As shown in Table (3), the average relative deviation obtained for all of the categories is larger than 12% and the average improvement of profitability is 13.2%. There were problems in some categories with relative deviations of approximately 18%. Although these deviations partly depend on the values of and , given the rational and random process used to determine these values in the used test problems, it is possible to use these results as proof of the considerable effect of the IRM on the growth of profitability and efficiency of project-oriented companies.
5. Conclusions In this paper, the issue of Integrated Resource Management (IRM), which has not been discussed previously, was studied. The previous approaches to resource management were described and advantages of the new proposed approach were enumerated. Also, a mixed-integer programming model was developed for the proposed approach. The objective function of the model was maximization of the company’s cash at the end of the planning horizon. Since the problem was NP-hard, first a heuristic method was developed for generating feasible initial solutions, then a combinatorial algorithm was developed based on GSA to solve it. The Results of the proposed method with 27 groups of test problems of different sizes were assessed against the results of Lingo. It was found that the method achieve acceptable results similar to the results of Lingo, in much shorter time; also, the proposed method was capable of solving many problems which Lingo couldn't solve them in a rational time. The effect of the IRM on the project selection and scheduling process was also assessed, and it was revealed that the average improvement of profitability on test problems was 13.2%. Inclusion of time-dependent costs for increasing or decreasing renewable resources can be the subject of future studies. Considering this inclusion in inflationary condition makes the problem similar to the condition of markets in which inflation causes considerable changes to resource supply costs and projects proceeds. It is also useful to use actual data to measure and analyze the effect of the IRM on profitability of project-oriented companies. References Afshar-Nadjafi, B. (2014). Multi-mode resource availability cost problem with recruitment and release dates for resources. Applied Mathematical Modelling, 38(21), 5347-5355. Aquilano, N. J., & Smith, D. E. (1980). A formal set of algorithms for project scheduling with critical path scheduling/material requirements planning. Journal of Operations Management, 1(2), 57-67. Chen, J., & Askin, R. G. (2009). Project selection, scheduling and resource allocation with time dependent returns. European Journal of Operational Research, 193(1), 23-34. Dodin, B., & Elimam, A. (2001). Integrated project scheduling and material planning with variable activity duration and rewards. IIE transactions, 33(11), 1005-1018. Huang, X., & Zhao, T. (2014). Project selection and scheduling with uncertain net income and investment cost. Applied Mathematics and Computation, 247, 61-71. Khoshjahan, Y., Najafi, A. A., & Afshar-Nadjafi, B. (2013). Resource constrained project scheduling problem with discounted earliness–tardiness penalties: Mathematical modeling and solving procedure. Computers & Industrial Engineering, 66(2), 293-300. Kumar, J. V., Kumar, D. V., & Edukondalu, K. (2013). Strategic bidding using fuzzy adaptive gravitational search algorithm in a pool based electricity market. Applied Soft Computing, 13(5), 2445-2455. 18
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