Controllable Processing Times in Project and Production Management: Analysing the trade-off between processing times and the amount of resources

Review Article Controllable Processing Times in Project and Production Management: Analysing the Trade-Off between Processing Times and the Amount of Resources Victor Fernandez-Viagas and Jose M. Framinan IndustrialManagement,SchoolofEngineering,UniversityofSeville,CaminodelosDescubrimientos,s/n,41092Seville,Spain Correspondence should be addressed to Victor Fernandez-Viagas; [email protected] Received 4 December 2014; Revised 10 February 2015; Accepted 17 February 2015 Academic Editor: Hang Xu Copyright © 2015 V. Fernandez-Viagas and J. M. Framinan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The amount of resources assigned to a task highly influences its processing time. Traditionally, different functions have been used in the literature in order to map the processing time of the task with the amount of resources assigned to the task. Obviously, this relation depends on several factors such as the type of resource and/or decision problem under study. Although in the literature there are hundreds of papers using these relations in their models or methods, most of them do not justify the motivation for choosing a specific relation over another one. In some cases, even wrong justifications are given and, hence, infeasible or nonappropriated relations have been applied for the different problems, as we will show in this paper. Thus, our paper intends to fill this gap establishing the conditions where each relation can be applied by analysing the relations between the processing time of a task and the amount of resources assigned to that task commonly employed in the production and project management literature. 1. Introduction The completion time of a project (project lead time) has turned out to be one of the main sources of competitive advantage for companies (see, e.g., [1,2]). This lead time depends on the processing time of each task in the project and, traditionally, these processing times have been assumed to be fixed data. However, in real life, they are usually dependent on the amount of resources assigned to the tasks [3]. In this case, they are denoted as controllable processing times, and it is then assumed that jobs can be accomplished in shorter or longer durations by increasing or decreasing the available resources, such as manpower and equipment [4]. The assumption of controllable processing times acquires great importance when looking at the completion time of the project. A good choice of the amount of resources to be assigned to a task before it starts becomes essential since changing the amount of resources during the execution of the task is detrimental both for the makespan of the task [5]andforitscost[6]. Furthermore, the amount of allocated resources becomes important as a factor which affects productivity (see, e.g., [7]). The current economic crisis emphasizes this fact [8], which is being used by companies to reduce expenses, to optimize their processes, and to adjust their production resources. Note that, properly speaking, resource-dependent processing times have been usually classified depending on the level of skill of the assigned resources and on the amount of resources. In the former case, a task is performed by a resource (typically an employee) with a given level of skill or experience, and the processing times of that task are different depending on such level (examples can be found in [9– 13]). In the latter case, the processing time of a task changes with the amount of resources assigned to the task. The term “controllable processing times” has traditionally been used in the literature to reflect this case. This paper focuses on this second type. Regarding the type of resources, the classical classification of resources proposed by [14,15]isadopted here whereas resources are classified from the viewpoint of renewability and divisibility. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 826318, 19 pages http://dx.doi.org/10.1155/2015/826318 2Mathematical Problems in Engineering With respect to renewability, we note the following. (i) A resource is denoted as renewable if only its total usage is constrained at every moment; that is, once a resource has been used by a task, it may be assigned to another task. (ii) A resource is denoted as nonrenewable if its total consumption is constrained; that is, once it is consumed, it cannot be allocated to another task. (iii) A resource is denoted as doubly constrained if both previous aspects are considered. With respect to the divisibility, we consider the following. (i) A resource is called discrete if it can be allocated to the tasks in discrete amounts; that is, we have a finite setofresources. (ii) A resource is denoted to be continuous if it can be assigned to the tasks in a continuous amount within an interval. Controllable processing times have been widely analyzed in the scheduling literature (see reviews in [4,16]). Additionally,therearecontributionsinrelateddecisionproblemssuch as resource allocation or software development size team. Processing times depending on the amount of resources have been used both in production and in project management, for example, for single-machine/flow shop scheduling, resource allocation problems, and multimode resource-constrained project scheduling problem (MRCPSP), and both for discrete and for continuous resources. However, to the best of our knowledge, there is no analysis or detailed discussion regarding the different types of relations between processing times andtheamountofresourcesusedincontrollableprocessing times based scheduling problems. In this paper, the main contributions regarding processing times and the amount of resources used in both production and project management are analyzed. Several properties are defined with the aim of analyzing the relations between processing times and the amount of resources. Some inconsistencies and difficulties to implement the relations, due to infeasible configurations, complexities of the relations, nonlinearities, and so forth, are identified and discussed. More specifically, the remainder of the paper is as follows. A general notation for the controllable processing times is defined in Section 2, along with the theoretical concepts and the basic properties for the relations. The main relations between processing times and the amount of resources found in the literature are analyzed in Section 3.Finally,inSection 4theconclusionsaredescribed. 2. Notation and Basic Properties Since this paper presents results from different research fields, itisnecessarytofirstclarifythenotationtobeusedhere.We have tried to accommodate the terms while maintaining the original notation as far as possible. Let us assume a project (in production management this term is usually denoted as job) composed of several tasks (operations in production management) that have to be performed using an amount of resources 𝑢. Depending on 𝑢,𝑝the processing time of the task may change; that is, 𝑝=𝑝(𝑢).Notethattheterm“project”canbealsoused instead of “task” when the project is not divided in tasks (e.g., [17,19,90,92]). Additionally, let us define the effort, 𝑒=𝑒(𝑢), as the amount of man-hours or man-months (depending on the unit of 𝑝) that a task needs to be carried out. Let us now define 𝑆as the size of the task or workload, which indicates the amount of work that has to be performed to complete the task. 𝑆is measured in the unit of the work, for example, source lines of codes or function points (FP) forasoftwaretask.Itisanattributeofeachtaskthatwillbe assumed constant in this paper. Thus, the productivity of the task, 𝑃𝑟,canbewrittenasafunctionofthesizeofthetask and the effort (see, e.g., [93–96]): 𝑃𝑟= 𝑆 𝑒(𝑢)=𝑃𝑟(𝑢).(1) An increase in the productivity indicates an increase in the task size if the effort is constant or a decrease in the effort when the size of the task remains the same. As it can be seen in (1), the productivity only depends on the amount of resources, 𝑢. The amount of resources for which the maximum productivity, max(𝑃𝑟), is achieved is denoted as 𝑢∗and can be obtained by maximizing the productivity or minimizing the effort. More specifically, we intend to find 𝑢∗for which 𝑃𝑟(𝑢∗) ≥ 𝑃𝑟(𝑢)∀𝑢 =𝑢 ∗.Since𝑆is assumed constant, it is clear that 𝑒(𝑢∗)≤𝑒(𝑢)∀𝑢 =𝑢∗. The tuple (𝑢∗,𝑝(𝑢∗)), with maximum productivity, is denoted as the productive configuration or productive points of the problem. This approach is implicitly used in several papers related to software projects management. An example canbefoundin[17] where the minimal effort is sought in order to determine the most productive relation between processing times and amount of resources. Next, we present two basic properties which must hold. By doing so, two regions are distinguished to establish the limits of the area where each configuration (𝑢,𝑝) can take place. Second, a general law for productive processes is introduced in order to analyze the relations. Property 1. Assuming that the same amount of resources is availableforeachperiod,theprocessingtimeofthetaskmust fulfill 𝑝≥constant/𝑢; that is, the processing time must be over an ideal boundary which corresponds to an inverse proportional relation between 𝑝and 𝑢as defined in Figure 1. Proof. If the same amount of resources is employed throughout the duration of the task, the effort can be written as the amount of resources times the processing time of the task (2): 𝑒=𝑝(𝑢)⋅𝑢. (2) Substituting in the expression (1): 𝑃𝑟(𝑢)=𝑆 𝑝(𝑢)⋅𝑢.(3) Mathematical Problems in Engineering 3 Processing time Ideal boundary Number of resources Figure 1: Ideal boundary. Processing time Number of resources Infeasible region Figure 2: Infeasible region. As shown in (1), maximizing the productivity is equivalent to minimizing the effort; therefore, 𝜕(𝑝(𝑢)⋅𝑢) 𝜕𝑢 =0=𝑢⋅𝜕𝑝(𝑢) 𝜕𝑢 +𝑝(𝑢)󳨀→ 𝜕𝑝(𝑢) 𝜕𝑢 =−𝑝(𝑢) 𝑢. (4) Thus, solving the differential equation, the point (𝑢,𝑝) reaching the maximal productivity is 𝑝(𝑢)=𝑘 𝑢.(5) Corollary 1. Each tuple (𝑢,𝑝) under the ideal boundary, 𝑝(𝑢)≤𝑘/𝑢,isinfeasible(seeFigure 2); otherwise, 𝑝(𝑢)≥𝑘/𝑢 is feasible. Proof. The proof is trivial in view of Property 1. Corollary 1 has been implicitly applied by [46,47]and implies that each tuple (𝑢,𝑝) in the region over the curve Dominated region Infeasible region Number of resources Processing time p1 u1 Figure 3: Dominated region. Dominated region Number of resources Processing time Pareto frontier p∗ u∗ Infeasible region Figure 4: Example for many configurations 𝑝-𝑢. is feasible. Nevertheless, for a tuple (𝑢1,𝑝1)in the ideal boundary, any other tuple (𝑢,𝑝)placed in the region “dominated region” (in Figure 3)isdominatedby(𝑢1,𝑝1),since the latter achieves less processing times with less amount of resources. Note that points that are nondominated by others are labelled as “efficient” according to the discussion shown in, for example, [46,97]. Hence, any tuple (𝑢,𝑝)candidate to be chosen as configuration of the problem must be located outside both the infeasible region and the dominated region, as we can see in Figure 4. In general, at least one tuple with minimal 𝑢⋅𝑝must exist representing the most productive configuration to perform the task, that is, the productive configuration (𝑢∗,𝑝(𝑢∗)). In the following sections, it will be assumed that productive points exist (even if they may be unknown) to have a reference so the different approaches can be compared. Since the goal pursued by companies is to minimize both the processing times and the amount of resources, different trade-offs can be established, which leads to a number of nondominated solutions forming a Pareto frontier. Note that the points over this frontier are dominated 4Mathematical Problems in Engineering and cannot be considered as possible configurations for the task; that is, nondominated processing times must be a nonincreasing function of the amount of resources assigned to the task, 𝑑𝑝(𝑢)/𝑑𝑢≤0for each 𝑢∈[𝑢,𝑢]. Property 2. Given some amount of resources 𝑢󸀠,thefulfillment of the law of diminishing marginal returns is the same as the fulfillment of (𝑑(1/𝑝(𝑢))/𝑑𝑢)(𝑢2)< (𝑑(1/𝑝(𝑢))/ 𝑑𝑢)(𝑢1)∀𝑢 2>𝑢1>𝑢󸀠. Proof. The law of diminishing marginal returns establishes that, given some amount of resources denoted as 𝑢󸀠,the output of a productive process increases at a decreasing rate when the amount of recourse increases (see [98–100]). Considering the output 𝑌=𝑆/𝑝(𝑢)as the amount of the task size performed in each time period and the input 𝑋=𝑢,the law of diminishing marginal returns can be written as 𝑌(𝑋2)>𝑌(𝑋1), 𝑑𝑌 𝑑𝑋(𝑋2)<𝑑𝑌 𝑑𝑋(𝑋1), ∀𝑋2>𝑋1>𝑋󸀠.(6) Substituting 𝑌and 𝑋in expression (6), 𝑑(𝑆/𝑝(𝑢)) 𝑑𝑢 (𝑢2)<𝑑(𝑆/𝑝(𝑢)) 𝑑𝑢 (𝑢1) 󳨀→ 𝑑(1/𝑝(𝑢)) 𝑑𝑢 (𝑢2)<𝑑(1/𝑝(𝑢)) 𝑑𝑢 (𝑢1) ∀𝑢2>𝑢1>𝑢󸀠. (7) Corollary 2. Given some amount of resources, 𝑢󸀠,theprocessing times of the task must satisfy 𝑑2(1/𝑝(𝑢))/𝑑𝑢2<0∀𝑢 󸀠< 𝑢<𝑢. Proof. The proof is trivial in view of Property 2. This property, together with the previous one, is used in this paper to analyze the different relations used in production management. Although in production management the output must satisfy the law of diminishing marginal returns, there is no such condition in project management. However, there are several results based on the experimentation for renewable discrete resource (manpower) in the literature. Among them, [101] established that the 𝑢-productivity graphics must be similar to an inverted U-shape where it is assumed that there is only a single productive configuration with maximum productivity defined by the tuple (𝑢∗,𝑝(𝑢∗)) [102]. The productivity decreases for 𝑢>𝑢 ∗;thisscenario isdenotedasECinthefollowing,duetothefactthat there is too much coordination and communication if more employees are assigned [92,103] and that these difficulties in communication increase with the size of the team [7,104]. It is also assumed that there is a decrease in productivity due to lack of specialization, denoted as LS, in small teams if fewer employees are assigned (i.e., for 𝑢<𝑢 ∗). This fact is confirmed by [105], who also cite other difficulties such as making trade-off decisions or managing error backlog. As a summary, a number of properties have been presented in this section. These will be discussed in the next sections. 3. Analysis of the Main Relations in the Literature The main relations between processing times and amount of resources are presented and classified in this section. This section takes into account relations used in production and project management since most of them (except 3) have been used interchangeably. The environment where each relation is usedisindicatedwithinbracketsafterthetitleoftherelation. The goal is to analyze how processing times depending on the amount of resources have been treated in the literature and to discuss their use. A detailed summary of the papers using processing times as a function of the amount of resources presented here can be seen in Table 1,andasummaryofthe section is shown in Section 3.6. 3.1. Linear Relation (Production and Project Management). Although the majority of scheduling problems in the literature consider fixed processing times [3,65], different relations between 𝑝and 𝑢have been also assumed for scheduling problems and for decision problems. According to [16], most papers using controllable processing times assume a linear relation between processing times and the amount of resources. For instance [8,25,65–80], consider for the singlemachine scheduling ([81] in a two-agent scheduling problem) that processing times can increase or decrease within a lower and upper bound (𝑝and 𝑝, resp.) depending linearly on the nonrenewable resources (cost) assigned. This single-machine problem is also solved considering discrete processing times by [78]. The linear approach for renewable and nonrenewable continuous resources is considered for a single-machine by [85]andby[86] for the hybrid flow shop problem. For discrete resources, the linear approach is assumed by [83,84] for the single-machine and by [82]forflowshopscheduling. Both discrete and continuous resources are considered in single-machine problems by [87] for renewable resources and for renewable and nonrenewable resources by [26,58–62]and by [60] for parallel-machine scheduling. Learning effect is addedtothelinearrelationsin[33–35,63]withoutchanging the linearity of the relation between 𝑝and 𝑢. In the area of project management, there are few papers using this relation. See, for example, [19] which includes this assumption in a multiple project scheduling using renewable resources. Traditionally, most problems considering controllable processingtimeshaveusedthefollowingrelation(e.g.,[16, 34,84]): 𝑝(𝑢)=𝑝−𝑏⋅𝑢 with 𝑢≤𝑢≤𝑢, (8) where 𝑢and 𝑢are, respectively, the lower (typically 0) and upper bound for 𝑢.𝑝and 𝑏are constants referring to the basic processing times (also denoted normal or nominal processing time) and to the compression rate of the task, which identifies the slope of the line. The linear expression could be also Mathematical Problems in Engineering 5 Table 1: Summary of several p-urelations used in the literature. 𝑝(𝑢) Domain of application Decision problem Renewable or nonrenewable resource Discrete or continuous resource References Hyperbola Software project Software development team size Renewable Discrete (manpower) See, for example, [17] Project management Resource allocation problem Nonrenewable and renewable Continuous (money) and discrete (manpower) See, for example, [18] Convex (𝑘=1) Multiple project management Multiple project scheduling Renewable Continuous and discrete See, for example, [19] Software project Software development team size Renewable Discrete (manpower) See, for example, [20–22] Government and industrial projects Resource allocation problem Renewable Discrete (manpower) See, for example, [18] Convex (𝑘=0.5) VLSI circuit design Resource allocation problem Nonrenewable Continuous (silicon area) See, for example, [18] Convex (𝑘=0.398) Software project Software development team size Renewable Discrete (manpower) See, for example, [20] Production-scheduling Single-machine scheduling Nonrenewable and renewable Continuous See, for example, [23,24] Convex (𝑘<1) Production-scheduling Single-machine; two-machine flow shop; parallel-machine; and open-shop scheduling Nonrenewable Continuous See, for example, [25–32] Production-scheduling Single-machine scheduling Nonrenewable and renewable Continuous and discrete See, for example, [33–36] Production-scheduling Single-machine scheduling Nonrenewable Continuous and discrete See, for example, [37] Convex (𝑘=0.5,1,2)+constant Production-scheduling Single-machine scheduling Nonrenewable Continuous See, for example, [38] Data (multimode) Project management MRCPSP Nonrenewable and renewable Discrete See, for example, [39–45] Project management; production DTRTP; project task and manpower scheduling; parallel-machine scheduling; and MRCSP Renewable Discrete See, for example, [46–57] 6Mathematical Problems in Engineering Table 1: Continued. 𝑝(𝑢) Domain of application Decision problem Renewable or nonrenewable resource Discrete or continuous resource References Multiple project management Multiple project scheduling Nonrenewable and renewable Continuous and discrete See, for example, [19] Production-scheduling Single-machine, parallel-machine scheduling, and flow shop scheduling Nonrenewable and renewable Continuous and discrete See, for example, [26,33–35,58–63] Production-scheduling Single-machine scheduling Nonrenewable Continuous and discrete See, for example, [64] Linear Production-scheduling Single-machine, two-agent scheduling Nonrenewable Continuous (money) See, for example, [8,65–81] Production-scheduling Hybrid flow shop Nonrenewable and renewable Discrete See, for example, [82] Production-scheduling Single-machine Nonrenewable and renewable Discrete See, for example, [83,84] Production-scheduling Single-machine and hybrid flow shop scheduling Nonrenewable and renewable Continuous See, for example, [85,86] Production-scheduling Single-machine scheduling Renewable Continuous and discrete See, for example, [87] Piecewise linear Project management RCPSP + staff assignment with CPT Nonrenewable Continuous and discrete See, for example, [88] Production-scheduling Time-cost trade-off Nonrenewable Continuous See, for example, [89] Convex 𝑘= 1 + polynomial Project management Task estimating Renewable Discrete (manpower) See, for example, [5,90,91] Mathematical Problems in Engineering 7 expressed in function of some tuple (𝑢1,𝑝1)insteadof𝑏;that is, 𝑝1=𝑝−𝑏⋅𝑢1󳨀→ 𝑏 = 𝑝−𝑝1 𝑢1, 𝑝(𝑢)=𝑝+𝑝−𝑝1 𝑢1⋅𝑢. (9) The expression of the productivity as a function of the team size is, according to Section 2,thefollowing: 𝑃𝑟= 𝑆 [𝑝−𝑏⋅𝑢]⋅𝑢.(10) If 𝑝and 𝑏are known and fixed, the productivity only depends on the number of resources assigned to the task. Obviously, the tuple with maximum productivity (𝑢∗,𝑝(𝑢∗)) must be in one of the two endpoints (see Figure 5). Regarding the fulfillment of the law of diminishing marginal returns, this can be easily checked by Property 2 deriving the inverse of the processing time with 𝑢1<𝑢2: 𝑑(1/𝑝(𝑢)) 𝑑𝑢 (𝑢2)<𝑑(1/𝑝(𝑢)) 𝑑𝑢 (𝑢1) 󳨀→ 𝑑(1/(𝑝−𝑏⋅𝑢)) 𝑑𝑢 (𝑢2)<𝑑(1/(𝑝−𝑏⋅𝑢)) 𝑑𝑢 (𝑢1) 󳨀→ 𝑏 (𝑝−𝑏⋅𝑢2)2<𝑏 (𝑝−𝑏⋅𝑢1)2 󳨀→ (𝑝−𝑏⋅𝑢1)2<(𝑝−𝑏⋅𝑢2)2. (11) Since processing times must be positive (𝑝−𝑏⋅𝑢>0), the expression can be written as 𝑝−𝑏⋅𝑢1<𝑝−𝑏⋅𝑢2󳨀→ 𝑢2<𝑢1(12) which is not possible. Hence, the linear relation does not satisfy the law of diminishing marginal returns for any amount of resources and it can be only employed as an approximation of 𝑝(𝑢). Note that, when using this configuration, only a productive point typically exists, and it is placed in the most productive endpoint, 𝑢or 𝑢. If both endpoints are equally productive, two productive points exist. Additionally, the linear relation does not realistically model the processing time of a task since the task could be performed even if no resource is assigned and in a zero time if a large amount of resources is allocated. The widespread use of the linear relation is probably due to its simplicity and as an approximation of the convex relation with 𝑘<1and 𝑘=1(see Sections 3.3.1 and 3.3.2), which is validonlyforsmallintervals. As mentioned before, this relation has been mostly used for production management. Nevertheless there are also references in project management (see, e.g., [19]). For this relation, the u-productivity graphicisnotsimilartoaninverted U-shape (see Figure 5). In fact, it is more similar to the opposite figure, a U-shape. Thereby, in project management, thisrelationcouldbeonlyusedtomodelanapproximationof the lack of specialization or of the excess of communication but never to model both effects since the productive points arealwaysintheendpoints. 3.2. Piecewise Linear (Production and Project Management). A piecewise linear decreasing relation between processing times and the amount of nonrenewable continuous resources assigned to the task is proposed by [89]inproductionand by [88] in project management. In the former, this relation isadjustedtotheconvexcurvewith𝑘=1by setting some points of the piecewise linear relation over the convex curve (see an example in Figure 6). The goal of this relation is to linearize the convex expression and to facilitate the construction of an integer linear programming model to solve the decision problem (the same idea could be extended to represent other behaviors of the task, such as a convexrelation with 𝑘=0.5, by changing the chosen points for other points in the new curve). As with the convex relation (see Section 3.3.1), using this approach, the piecewise linear relation would not satisfy the law of the diminishing returns since it is formed by several lines that do not satisfy this law (see Section 3.1). In the latter, a piecewise linear relationship with two sections is proposed to approximate the behavior of the uproductivity to an inverted U-shape where the productive configuration is the point between both sections. 3.3. Convex Relation (Production and Project Management). The general expression used for this case is 𝑝=(𝑑 𝑢)𝑘,(13) where 𝑑is a constant usually denoted as workload and 𝑘is a positive constant which holds 1 for many actual government and industrial projects [18], distributed communication network, time-sharing computing system, chemical plant, or commercial construction projects [19], while 𝑘=0.5has been used in many large scale integration (VLSI) circuit designs where the resource is silicon area [18]. In production management, a convex relation between processing times and amount of nonrenewable continuous resources with a generic 𝑘has been used, for example, by [25–31] in single-machine, two-machine, parallel-machine, and open-shop scheduling problems. Renewable resources are assumed by several authors (see, e.g., [23,24]) also for the single-machine scheduling problem. Continuous and discrete nonrenewable resources are considered, for example, by [37] for the single-machine scheduling problem with deteriorating jobs. Examples of papers using the convex relation in a single-machine scheduling problem with learning effect taking into account any type of resource can be found in [33– 36]. Both learning effect and deteriorating jobs do not change theconvexrelationbetween𝑝and 𝑢. In contrast with the linear relation, this convex relation seems to fit better into the aforementioned real problems. Additionally, only one parameter has to be calculated to implement it as compared to the 2 parameters needed by the 8Mathematical Problems in Engineering Infeasible region Number of resources Processing time uu Convex k=1 Linear Infeasible region Number of resources Processing time uu Number of resources u u Number of resources uu Convex k=1 Linear Productivity Max (Pr) Productivity Max (Pr) Figure 5: Processing times and productivities for linear relation. Infeasible region Number of resources Processing time u uu∗ Convex k=1 Piecewise linear Number of resources u uu∗ Piecewise linear Max (Pr) Productivity Figure 6: Piecewise linear relation. Mathematical Problems in Engineering 9 linear relation. However, its nonlinearity makes modeling the decision problems using linear programming difficult. Despite the aforementioned shortcomings, many papers in the literature justify this relation with the argument that the linear relation does not obey the law of diminishing marginal returns, whereas the convex relation does (see, e.g., [3,18,23, 27]). In our opinion, this argument must not be further used as justification of the use of the convex relation, according to the findings in this section. 3.3.1. Convex Relation with 𝑘=1(Production and Project Management). Here, there is an inversely proportional relation between the processing times and the amount of resources assigned [27]: 𝑝=𝑑 𝑢󳨀→𝑝⋅𝑢=𝑒=𝑑, (14) where 𝑑, usually denoted as workload, is a constant and, consequently, the effort is also a constant for this relation. In Section 2, it was shown that any point in this relation is within the ideal boundary and the constant 𝑑can therefore be calculated considering that the curve cuts one of these points (𝑢1,𝑝1): 𝑝1=𝑑 𝑢1󳨀→ 𝑑 = 𝑢1⋅𝑝1.(15) Substituting in (14),theconvexrelationwith𝑘=1can be obtained: 𝑝=𝑢1⋅𝑝1 𝑢.(16) When no relation between 𝑝and 𝑢is known, the constant 𝑑 has been also calculated in the literature using the workload, 𝑆, or a function of it (see [20–22]), 𝑓(𝑆).Asmentionedin Section 2,therelationbetween𝑝and 𝑢corresponds to the ideal boundary whose points have maximum productivity. Accordingly, the convex relation with 𝑘=1must be used in the case where each configuration between 𝑝and 𝑢has the same value of the productivity and, therefore, each point is a productive point. That is, when the amount of resources increases, the processing time decreases in the same proportion. This corresponds with the ideal case both inproductionandinprojectmanagementwherethereare no penalties for employing additional resources. In fact, this relation does not fulfill the law of diminishing returns as the derivate of the output is constant, so it does not decrease: 𝑌(𝑋)=𝑌(𝑢)=𝑆 𝑝(𝑢)=𝑆⋅𝑢, 𝑑𝑌 𝑑𝑋(𝑋)=𝑑(𝑆⋅𝑢) 𝑑𝑢 (𝑢)=𝑆󳨀→𝑑𝑌 𝑑𝑋(𝑋1)=𝑑𝑌 𝑑𝑋(𝑋2). (17) Regarding project management, this relation does not consider penalties due to excess of communication (excess of the number of employees) or lack of specification for manpower since the productivity is constant for any amount of resources. 3.3.2. Convex Relation with 𝑘<1(Production and Project Management). The convex relation with 𝑘<1is analysed here: 𝑝=(𝑑 𝑢)𝑘,with 𝑢≤𝑢≤𝑢, (18) where 𝑑isaconstantwhichcanbedefinedwhenapointof the curve is known. Hence, assuming that the curve includes the point (𝑢1,𝑝1), the constant 𝑑can be calculated; that is, 𝑝1=(𝑑 𝑢1)𝑘󳨀→ 𝑑 = 𝑢1⋅(𝑝1)1/𝑘 .(19) As with the convex relation with 𝑘=1,onepointisonly necessary to completely define the relation, in contrast to relations as the hyperbola (see Section 3.4) that needs to define up to 4 parameters. Thereby,therelationcanbewrittenagaininexpression (20) using a point in the curve (expression (19))(fora representation see Figure 7): 𝑝=𝑝1⋅(𝑢1 𝑢)𝑘.(20) Substituting in the expression of the productivity: 𝑃𝑟= 𝑆 𝑝⋅𝑢=𝑆 𝑝1⋅(𝑢1/𝑢)𝑘⋅𝑢=𝑆 𝑝1⋅𝑢1𝑘⋅𝑢1−𝑘 =constant 𝑢1−𝑘 . (21) As we can see in expression (21) or in Figure 7,the productivity always increases by assigning fewer resources to the task for 𝑘<1. This can be easily seen using expression (21): 𝑢1<𝑢2󳨀→ 1 𝑢1>1 𝑢2󳨀→ ( 1 𝑢1)1−𝑘 >(1 𝑢2)1−𝑘 󳨀→ 𝑃𝑟1>𝑃𝑟2. (22) Hence, this relation within 𝑝-𝑢corresponds to tasks whose maximum productivity (productive configuration) is in 𝑢resources (𝑢∗=𝑢), and this productivity decreases with the increase of resources assigned. The law of diminishing marginalreturnsisfulfilledasitcanbeseenin(23) using Property 2 for the point 𝑢󸀠=0: 𝑑(1/𝑝(𝑢)) 𝑑𝑢 (𝑢2)<𝑑(1/𝑝(𝑢)) 𝑑𝑢 (𝑢1) 󳨀→ 𝑑((𝑢/𝑑)𝑘) 𝑑𝑢 (𝑢2)<𝑑((𝑢/𝑑)𝑘) 𝑑𝑢 (𝑢1) 󳨀→ 𝑘⋅𝑢2𝑘−1 𝑑𝑘<𝑘⋅𝑢1𝑘−1 𝑑𝑘 󳨀→ 𝑢2𝑘−1 <𝑢1𝑘−1 𝑘<1 󳨀→ 𝑢1<𝑢2. (23) This expression is satisfied for any amount of resources from the left endpoint (𝑢󸀠=0). However, the law of the 16 Mathematical Problems in Engineering and production management, several relations between processing times and amount of resources have been applied. However, to the best of our knowledge, the advantages, disadvantages, and properties of these relations have not been analyzed.Inthispaper,wehaveanalyzedpapersassuming controllable processing times in project and production management, and the different relations between processing times and the amount of resources have been classified. Due to the uncertainty in the processing times of the tasks [39], it is impossible to know exactly the relation between the processing time and the amount of resources assigned to the task and, analogously, to choose a relation over another. Approximations must be taken into account in order to represent the relations and to adapt them to solve the problems. Thereby, a total of 10 relations have been analyzed in this paper. Papers in the production management literature havemainlyfocusedonlinearandconvexrelationsbothfor discrete and for continuous resources. In project management, the research has widely been focused on multimode approach. We have defined the feasible and unfeasible regions for 𝑝=𝑓(𝑢), as well as several definitions and properties, to be able to compare the main functions of the literature. For each one, the productivity for each amount of resources has been measured. Additionally, the law of diminishing marginal returns in the field of production management has been analyzed, together with several hypotheses for each relation in the area of the project management. As a result, it has been shown that the most employed relation in the literature, that is, the linear one, does not fulfill the requirements of both production and project environments. Nevertheless it has been extensively used as an approximation of the reality, probably due to its simplicity and the possibility to maintain the linearity of the models. On the other hand, the assumption, widely used in the literature (see, e.g., [3,18,23,27]), of applying the convex relation instead of the linear one to fulfill the law of diminishing marginal returns has been shown not to be correct and must not be further used as a justification of its consideration. Only aconvexrelationwith𝑘<1couldbeconsidered,butonly to model the decreasing part of the law, that is, from 𝑢󸀠on. When a constant is added to the expression, the law can be fulfilled by the configuration 𝑘=2+ constant where the point 𝑢󸀠is equal to 𝑑/√3⋅𝑏. Something similar happens regarding project management, since the u-productivity graphicisonly similartoaninvertedU-shapefortheconfiguration𝑘= 2+ constant. However, to the best of our knowledge this configuration has not been used yet. In this regard, the best relation related to u-productivity graphic is achieved by the hyperbola. Nevertheless, the high number of constants (four) needed for its configuration makes it difficult to use since at least four p-upoints should be known in advance. This is similar to the multimode and piecewise linear relations, which obviously have the most complete configuration in order to model the p-urelation since the decision maker can adapt these relations in function of the studied problem. This last aspect also remarks the great importance of the previous knowledge of the problem (i.e., which p-upoints are known) whenchoosingtherelationtobeapplied. Finally, regarding the future research lines of this paper, new relations may be considered for discrete resources since there are almost no papers using them. In addition, to the best of our knowledge, only the hyperbola has been used to represent the inverted U-shape of the productivity in project management. 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