Algorithms for Investment Project Distribution on Regions

This paper proposes an optimization system for solving an NP-hard problem by using several new algorithms and application programs. This study aims to identify a suitable distribution of investment projects across several developed industrial regions. It is assumed that all industrial regions involved have the same economic and strategic characteristics. The problem involves a set of projects that are to be assigned across regions. Each project creates an estimated number of new jobs, and the distribution of projects can be guided by minimizing the maximum total number of newly created jobs. The problem is NP-hard one, and it is difficult to determine the most appropriate distribution. We apply scheduling algorithms in order to solve the analyzed problem. Severalheuristics are developedto obtain the appropriate distribution of newly created jobs across all regions. A branch-and-bound method is employed in order to obtain the exact solution. The performance of the algorithm is demonstrated by the experimental results for a total number of 1850 instances.


Introduction
It is important to distribute investments throughout a given country. A "good" distribution of investment resources is very important both from an economic and a social perspective since this may have a direct effect on the behavior of cities in terms of business and economic cycles. e context of this paper is the regional allocation of industrial projects. e problem is as follows: when there are several projects to be assigned across different industrial regions, how should these projects be allocated? e solution to this problem may be based either on the principle of equity distribution or on a distribution process which is meant to minimize the maximum total number of projects to be assigned. We start with a set of projects to be undertaken in several regions, each of which represents an amount of newly created jobs and will be entirely undertaken in the chosen region. For the sake of equity, certain heuristics are applied to guide the allocation of industrial projects to different regions in order to increase the minimum total number of new jobs created [1]. To answer the research question, it is necessary to choose between minimizing the maximum total number of new jobs created and maximizing the minimum total number of new jobs created. In this study, we assume that the industrial regions involved are identical and that they share the same characteristics that affect both investment allocation and investment opportunities. For example, if the investment project consists in the creation of a marine port, then it should be allocated to a coastal region since this is an investment with specific economic characteristics. All types of investments with such specific characteristics will be excluded from this work. Heuristics and exact solutions will be used to ensure a good distribution of all investment projects to industrial regions. e solutions to the aforementioned problem affect individuals and their everyday activities and have significant impacts on the public and private sectors and on society at large. A wide variety of resource allocation methods with different mathematical approaches to medialization have previously been used in various application areas to compute optimal solutions to the problem of the distribution of projects to several regions. e present work aims to determine the distribution of investment projects across regions by using a new objective function and newly proposed mathematical model. New solutions were developed based on the utilization of the heuristics and the optimal solutions for P||C max (minimizing the maximum total completion time for parallel machines) and P||C min (maximizing the minimum total completion time for parallel machines). For the first problem, which is P||C max , we use the heuristics and optimal solution cited by Haouari and Jemmali [2]. e respective derived bounds cited in latter work were embedded in an exact procedure which yielded an exact and effective algorithm based on a symmetry-breaking branching strategy. For the second problem, P||C min , a branch-and-bound (B & B) algorithm was developed by Haouari and Jemmali [1] based on enhanced lower and upper bounds. e P||C min problem is also referred to as "machine covering." A study of Walter et al. [3] also proposed an exact B & Bmethod. e rest of this paper is structured as follows. Section 2 discusses previous work related to the aforementioned problem, while Section 3 sets forth the analyzed problem. Section 4 develops the lower bound for the analyzed problem, and Section 5 introduces and describes several heuristics. Section 6 presents the proposed solution, and Section 7 describes the experimental tests applied and their results. Section 8 concludes this paper.

Related Work
e theory of optimal control is used to tackle the unequal distribution of income between regions associated with high rates of aggregate growth for the regional allocation of investments [4]. Other applications for optimizing the allocation of investments include linear dynamic programming problems. e objective of a study by Ohtsuki [5] was to search for the optimum solution to the regional allocation problem. Bagchi et al. [6] provided a formulation for the problem of investment allocation across cities in the framework of a planning model, and a case study examining the impact of investment on the economies of Japan was carried out [7]. e study by Castells and Solé-Ollé [8] was undertaken in order to help managers make decisions, and in order to address the need for a system that provided data and information, two metaheuristic algorithms were compared, the first of which was based on a genetic algorithm and the second on the mean variance technique. e work of Salo et al. [9] focused on improved methods of resource allocation compared to the previous work that would facilitate decision analytics by helping public organizations and firms allocate resources for "lumpy" investment opportunities. Resource allocation problems are ubiquitous in governmental organizations and business. In addition to the aforementioned cited works, Luss [10] specifically addressed equitable resource allocation problems with applications in manufacturing, networks, communication, and other areas.
Ranjbar et al. [11] investigated the problem of investment project scheduling with the purpose of reducing the related costs and the amount of necessary resources and of avoiding any delay in the implementation of those projects. e project in this case consisted in a set of interrelated activities, and the authors suggested two methods for resolving the project scheduling problem, which are a genetic algorithm and relinking, whereby a schedule is created with a list of priorities. Shadrokh and Kianfar [12] suggested a genetic algorithm to solve the resource investment problem. e performance of this algorithm has been noticed to be satisfactory when compared with that of other algorithms.
Castells and Solé-Ollé [8] analyzed the main determinants of investment distribution across different regions. e expected objectives of the investment were set based on a general description of the objectives of the respective country, which focused on effective and fair trade and on the rules arising from political factors. is study employed a dynamic equation that allowed for gradual adjustment and readjustment, and the results indicated that criteria of effective trade played an important part in the geographical distribution of investment within the government infrastructure. Hartmann and Briskorn [13] provided an overview of the aforementioned extensions, which were designed based on the structure of the resource-constrained project scheduling problem (RCPSP).
is paper gave a general summary related to the concepts of activity, precedence relationships, and resource constraints and also gave a summary of many lesser-known concepts. Singh [14] developed an algorithm that incorporated the priority of a project with the development of a programmed schedule, with the objective of reducing the project makespan and the penalty cost when some projects have higher priority. e proposed algorithm was shown to be an improvement of the current priority rules.
Rodrigues and Yamashita [15] developed a new precise algorithm that solved the cost of the resource availability (RACP) problem in scheduling projects. is new algorithm relied on a hybrid method in which an initial feasible solution was determined heuristically. e branching scheme resolved an RCPSP at each node, where the resources of the RACP were fixed. e new algorithm can be employed for reducing the cost of making resources available.
Sanaei et al. [16] studied the firefly algorithm, which was originally developed by Yang to optimize numerical tasks.
is method was employed for solving an RCPSP. e difficulty in solving this type of problem arises from a lack of resources and priority of constraints. e algorithm begins by configuring a set of random scheduling solutions, and the primitive scheduling is then repeatedly improved using the firefly algorithm. Qi et al. [17] developed a model for multimode resources that reduced the cost of making resources available for all activities, with the aim of completing all activities of a project on schedule. When the deadline for a project is confirmed, the contractor will attempt to reduce the project cost, and this study proposes an optimization method and a useful heuristic algorithm for identifying the optimal solution. Rahmani et al. [18] proposed a new method for solving an RCPSP based on a differential search. 2 Computational Intelligence and Neuroscience eir approach could provide a set of preliminary schedules. Rabbani et al. [19] suggested a new approach to the multiobjective time-cost constrained RACP.
More recently, several works have proposed solutions for the fair distribution problem. For example, Jemmali [25] gave different approximate solutions for the project revenue assignment problem. Another study [26] focused on budgetbalancing algorithms for solving the aforementioned problem and gave several algorithms. An application of load balancing of the files on different storage supports was treated in [27].

Description of the Problem
e problem analyzed in this paper can be described as follows. Given a set of n p industrial investment projects P 1 to be undertaken in n r different industrial regions Re, each project j can be assigned to only one industrial region i with i � 1, . . . , n r . Each project j with j � 1, . . . , n p can be evaluated by applying three indicators. e number of newly created jobs is denoted by NJ j . e objective of this study is to ensure a suitable distribution of projects, and it is essential to define a variable that can be evaluated in order to measure the performance of the project allocation process. In the first approach, the number of new jobs created was set as the objective variable, and the overall objective was to maximize the minimum total number of new jobs created. e question then arises as to whether minimizing the maximum total number of new jobs created yields the same result.
When assigning a project j to an industrial region i, the total cumulative number of new jobs created is denoted by t j . e total number of new jobs created in each region i after the project allocation is completed is denoted by C i . Example 1 illustrates the analyzed problem. e maximum number of new jobs created for all regions is denoted by C max , and the minimum number of new jobs created for all regions is denoted by C min . For each schedule, the values of C i are indexed as follows: C 1 ≤ C 1 ≤ · · · ≤ C n r . Example 1. We assume that, in the case of industrial investment projects, there are five new projects, i.e., n p � 5. In addition, there are two regions, i.e., n r � 2. Table 1 displays the number of new jobs created for each project. Now, let us suppose that the assignment of projects to regions will be as follows. Projects 1, 4, and 5 are allocated to region 1, while projects 2 and 3 are allocated to region 2. Figure 1 illustrates this allocation.
From Figure 1, it can be noticed that there is a large gap between regions 1 and 2, which is equal to C 2 − C 1 � 17. is means that more new jobs were created in region 2 than in region 1. e objective is to decrease this gap in order to ensure a better distribution between the regions involved.
ere is also another arrangement that can be used in order to ameliorate the manner of project assignment with the purpose of achieving a good project distribution. Figure 2 illustrates the new arrangement.
It can be seen from Figure 2 that the gap between regions is lower compared with the one in the first arrangement shown in Figure 1. After rearrangement, the gap is reduced to three.

Remark 1.
e gap for each region i is given by Proof. e maximum number of new jobs created is C max . us, for each region, C max − C i gives the difference between the number of new jobs created in region i and that in the region with the maximum number of new jobs. e indicator that can be measured to minimize the gap between regions is represented by the following formula:

Proposition 1. e objective described in equation (1) can be expressed as
Proof. From equation (1), we have By applying the minimum function to the latter result, we obtain equation (2).

Computational Intelligence and Neuroscience
Hereafter, the gap between regions that was obtained by using a given heuristic is denoted by g NJ max � n r C max − n r i�1 C i . e objective of this study is to minimize g NJ max . It is worth mentioning that the constraint applied in this work consists in the fact that the project cannot be divided (i.e., the number of new jobs created by a project cannot be divided across two or more industrial regions). e optimal solution is denoted by g NJ * max , and LB max denotes the lower bound and UB max the heuristic value for the P||C max problem.

Lower Bounds
Two lower bounds are proposed for the analyzed problem. e first is obtained from the optimal solution to P||C max and the optimal solution to P||C min . Hence, the balance between P||C max and P||C min will be considered. e second lower bound is obtained from the lower bound on P||C max . C * max denotes the optimal solution for P||C max , and C * min denotes the optimal solution for P||C min .

Min-Max-Based Lower Bound (MML).
is lower bound is determined based on the following proposition: Proof. In any schedule σ, the value n r C max − n r i�1 C i is calculated after C max is evaluated. is value is the difference between C max and C i for each region. is also indicates that the difference between C max and C i for all the remaining regions (excluding the region that has C i � C max ) will be zero in this case.
In addition, σ(max) is the schedule obtained by applying the optimal solution to P||C max . In this schedule, it is obvious that no C i is greater than C max , and hence, this schedule is the optimal one, with C max being minimized. Let us suppose that the aforementioned problem may be re-envisaged as follows. Let us assume that, in the ideal case, when the two regions with C max and C min are excluded, the remaining regions have the same number of new jobs, which is equal to C max . In this case, the comparison is only between C max and C min . Now, if this relaxation is applied while applying the schedule σ(max), the minimum value can be obtained when the value of C min is maximized. us, the difference will be C * max − C * min . Accordingly, σ(max) is then selected, which is the schedule obtained by applying the optimal solution to P||C min and the relaxation; the ideal case has a difference of C * max − C * min . e proposition can also be written mathematically. For any schedule, C min ≤ C * min can be written as Furthermore, in any schedule, we have Combining equations (5) and (4) yields From equation (6), it can be deduced that C * max − C * min is a valid lower bound.

C max -Based Lower Bound (CML).
is lower bound is based on the P||C max one. e C max -based lower bound, denoted by CML, is based on the following proposition: e obtained value for CML is a valid lower bound.
Proof. From the objective function described in equation (2), we have Furthermore, for any schedule, we have When considering the lower bound on the P||C max problem, we have From equation (8), after multiplication by a positive integer, we obtain en, n p j�1 NJ j is subtracted from equation (9) to give From equations (7) and (10), it can be deduced that n r LB max − n p j�1 NJ j is the lower bound for the problem. In this study, two lower bounds from the literature for the P||C max problem are used for developing two versions of CML. Denoted by LB CML FS and LB CML TV , these two lower bounds are based on L FS and L TV , respectively, as described by Haouari and Jemmali [2].

Approximate Solutions
In this section, certain heuristics developed for the problem under study are presented, and these are compared to demonstrate their performance. e first heuristic is based on the nonincreasing dispatching rule, while the second is based on the nondecreasing dispatching rule. e multifit method is employed for deriving the third heuristic. In the fourth approach, we apply several times the resolution of the subset-sum problem in order to calculate the value of the heuristic. In a similar way, the fifth heuristic will solve a wellknown problem iteratively; difference lies in choosing the knapsack problem which is used rather than the subset-sum problem. e last heuristic is based on rescheduling of the most charged and least charged regions to find a better result. 4 Computational Intelligence and Neuroscience (NIJ). For this schedule, the total number of new jobs created for all projects is placed in a nonincreasing order. Highest NJ j is then allocated to the first region on the list, i.e., the one with the lowest number of new jobs created.

Heuristic 1: Nonincreasing Number of New Jobs Created
Example 2. In this example, the same instance cited in Example 1 is considered. Figure 3 shows the schedule after applying NIJ.

Heuristic 2: Nondecreasing Number of New Jobs Created (NDJ).
In this schedule, the total number of new jobs created for all projects is placed in a nondecreasing order. Lowest NJ j is then allocated to the first region on the list, i.e., the one with the lowest number of new jobs created. e schedule obtained after applying NDJ is as follows: region 1: 5, 1, 2 and region 2: 4, 3. us, g NJ max � 50 − 33 � 17.

Heuristic 3: Multifit Heuristic Based on the Number of New Jobs Created (MFJ).
is approximate solution is inspired by the machine job scheme presented by LEE and David Massey [28]. is method uses the bin-packing method for determining the minimum capacity that will allow all n new jobs created to fit into the m regions. For each bin capacity, the first-fit decreasing (FFD) method is employed for fitting the number of newly created jobs to the bin. Let us assume that the number of new jobs was ordered such that NJ 1 ≥ · · · ≥ NJ n p . e FFD method successively assigns the project to the lowest indexed region with the necessary capacity for that respective number of new jobs. Hence, LB max � max(NJ 1 , NJ n r + NJ n r +1 , [ n p j�1 NJ j /n r ]), and UB max is the value achieved by applying the LPT heuristic for P||C max . Here, bin denotes the number of bins used after applying FFD, and iter is a predetermined, fixed number of iterations for FFD. In this study, iter � 35.
e MFJ algorithm is illustrated in Algorithm 1.
Example 3. Assume that, in case of industrial investment projects, the number of projects is equal to 10, i.e., n p � 10.
In addition, there are two regions, i.e., n r � 2. Table 2 presents the number of new jobs created for each project.
For the instance given in Comparing this value with the results obtained using NIJ, we see that the associated g NJ max is equal to 19. is example shows that the results obtained using MFJ are better than those obtained using NIJ. However, the difference in the number of new jobs created in regions 1 and 2 after applying MFJ is only 5.

Heuristic 4: Subset-Sum Problem-Based (CSS) Approach.
is heuristic is based on a greedy algorithm. It is denoted by (P) k (k � 1, . . . , n r − 1 ) as follows: (i) S 1 � P I and S k�1 � S k /P I k , where P I k is an optimal subset sum for (P) k (k � 1, . . . , n r − 1). (ii) L(S, K) denotes a valid lower bound for P||C max on the makespan of a reduced instance, defined by k ≤ n r regions and a subset of the number of new jobs created S⊆P I . Hence, the first region will be the number of new jobs created until L on (P) 1 is reached. e remaining number of new jobs created in the remaining regions will constitute the second problem (P) 2 , which will consist in solving the new SSP until L is reached, and so on [29]. e subset-sum problem will be solved by using the dynamic programming algorithm developed by Pisinger [30] to solve the subset-sum problem.

Heuristic 5: Knapsack Problem-Based (CKS) Approach.
is heuristic is based on the division of the main problem into m smaller problems, each of which involves scheduling of the projects to regions. In the first region, a knapsack problem is used to assign a project to region 1, and the remaining projects are assigned to the remaining regions using a further knapsack problem, and so on. For each region, a knapsack problem must be solved as follows: Computational Intelligence and Neuroscience

Heuristic 6: Min-Max Iteration Heuristic (MMI)
. is heuristic is based on the reformulation of P 2 ||C max as a subset-sum problem [29]. In each iteration, the most loaded region r max and the least loaded region r min were fixed, and a subset-sum problem was solved with the projects assigned to r max and r min . is procedure was repeated several times, and the best solution was identified. e number of iterations was set to 100 based on an experimental study.

Lemma 1.
e optimal solution to the approached problem is Proof. Based on the Corollary 1 above, Based on the definition of P||C max , the problem is

Experimental Results
In this section, we present the results given after the implementation of all algorithms coded in C++ using an Intel(R) Xeon(R) CPU E5-2687W v4 @ 3.00 GHz and 64 GB RAM. Several instances and statistical analyses are presented in order to illustrate the results obtained by applying the algorithms described in the previous sections. Different instances were generated which were inspired by Dell'Amico and Martello [31]. e overall instances were based on the choice of n, m, and the type of new jobs created, which is denoted by Class. e selection of the pair (n p , n r ) was as follows: n p � 10, 15, 50, { 100, 200, 300} and n r � 2, 3, 5, 10, 20, 25, 50 { }. Table 3 shows the fixed distribution of (n p , n r ). Step 0. Set I � 0, C u � UB max , and C l � LB max .
Step 2. Apply FFD with capacity C.
Step 3. If all projects n p can be assigned to n r regions, then set C u � C and go to Step 4; else, C l � C and go to Step 4.
Step 4. If I � iter then STOP; otherwise, go to Step 1. End. If bin > n r or bin < n r , then the result given by LPT is accepted; otherwise, the result given by FFD is accepted. ALGORITHM 1: MFJ algorithm.  is result was obtained based on the performance of the heuristic and lower bounds. Another notable result is the number of times that the lower bound was equal to the upper bound: 1684 from a total of 1850 instances (representing 91.03% of the total number of instances). is indicates that the optimal solution is reached when the lower and upper bounds are calculated. An exact solution is required for only 8.97% of the overall instances. It is worth mentioning that, for the lower bound MML which is based on the solution provided by the B & B method, if the instance is not solved by the B & B method, then the value of C * max is not determined. In this case, the best lower bound for P||C max is chosen. e same procedure is followed for P||C min .
In Table 4, it is obvious that the gap increases when m increases for MML. is phenomenon is not applicable for LB CML TV and LB CML FS . For LB CML TV , when n r � 2, GAP l � 0.06, and when n r � 5, GAP l � 1.71, but when n r � 10, the value of the gap decreases to 0.02. e values of Time, as illustrated by Table 4, do not vary along with other values. e maximum time was 467.640 s and was reached when n r � 50 for the MML bound. is is because this last lower bound is a consequence of solving two different problems in an exact way. e shortest time was achieved for LB CML TV and was less than 0.001 s. e total minimum gap was 0.04 and was obtained for n r � 2.
Other analyses are illustrated in Table 5, which shows the variation in the gap and time according to the number of new jobs created.
Time increases when n p increases for LB CML TV and LB CML FS . e smallest gap of zero is obtained for LB CML FS when n p � 100, n p � 200, and n p � 300. e maximum gap value, which is 8.413, was obtained for MML when n p � 300. e total maximum time was 198.545 s for n r � 50. is explains the nonresolved exact solution provided by employing the B & B method for calculation of MML. Table 6 presents the variation in gap and time according to Class. Table 6 shows that all classes are approximately characterized by the same magnitude of difficulty.
In Table 7, Perc indicates that the percentage of the analyzed lower bound is equal to the best value of all lower bounds for all total number of instances.
In Table 7, it can be noticed that the best lower bound, which reaches a maximum percentage of 95.89%, is LB CML Further information on the results obtained is included in Table 8.
To show how the analyzed heuristics perform, several tables of results are presented and discussed. e analyses are based on the variations in GAP u and Time with n p , n r , and Class. Table 9 presents the variations in gap and time based on the number of regions.
It is obvious in Table 9 that the MMI heuristic has a constant gap of zero for all values of n r . e heuristics NDJ and NIJ are much faster than the others, and the necessary execution time for these heuristics is lower than 0.001 s. e maximum gap of 491.61 is achieved for NDJ when n r � 50. Furthermore, the highest time consumption of 0.018 s is seen for MMI for n r � 50. e results show that, in almost all cases, MMI provides better results than other heuristics, but there is no dominance amongst the heuristics. Table 10 compares GAP and Time based on n p .
As it can be noticed in Table 10, the gap does not vary along with the variation of n p values. e time increases according to n p for CSS. e results of all executed heuristics are shown in Figure 4, where an increasing curve can be noticed. Computational Intelligence and Neuroscience From Table 11, it is vivid that there is a difference between the classes in terms of difficulty.
For the CSS heuristic, the greatest values for the gap can be noticed for Classes 2 and 3, meaning that these classes are more difficult to solve than the others for CSS. e same behavior was noticed for the CKS, MFJ, NDJ, and NIJ heuristics. It can be concluded that this problem is more difficult to solve for Classes 2 and 3. Furthermore, this heuristic is more timeconsuming, reaching 0.585 s for Class 2.
In Table 12, it is vivid that the best heuristic is MMI, with the highest percentage of 98.2%, an average gap of zero, and an average execution time of 0.502 s. e lowest percentage that is 14.5% was obtained by NDJ, with a gap of 251.67 and a time of less than 0.001 s. Further information on the results obtained is included in Table 13. Now, the experimental study focuses on the gap between the values of the minimum heuristic and the maximum lower bound. is requires a comparison between the best lower bound (L) and the best heuristic (U), denoted by GAP u l � (U − L)/L. In Table 14, the maximum gap is 0.74, which is obtained when n p � 50, and the average time is 98.194 s. For n p � 200, n p � 300, and n p � 500, the gap is zero, indicating that the optimal solution is obtained. Figure 5 shows the variation in GAP u l with the number of new jobs created, which peaks when n p � 50 and tends asymptotically to zero when n p is higher. Table 15 illustrates the impact of the number of regions on the analyzed problem and shows that the maximum gap of 0.86 is reached when n r � 20. However, a minimum gap of zero is reached for n r � 2 with a reasonable time of 0.048 s and for n r � 10 with a time of 1.262 s. Table 16 shows the variation in GAP u l and Time with Class. is table confirms once again that Classes 2 and 3 are       more difficult to solve and require more time for that by comparison with Classes 1, 4, and 5. e maximum gap of 0.23 is obtained for Class 2, and a value of 0.28 is seen for Class 3. e performance of the exact solution was determined by applying the B & B method, and the time limit settled at 1200 s. If the time required for solving an instance and obtaining the exact solution exceeded this time limit, the     resolution process was stopped, and it was noted that the respective instance remained unresolved. Table 17 Figure 5: Variation of the GAP u l value according to n p . which represents 3.78% of the total number of instances that is 1850. Table 18 shows that the instances that remained unresolved when using the exact method are concentrated for n r � 20, n r � 25, and n r � 50. e maximum value of Time was observed when n r � 20 and was equal to 179.407 s. Table 19 shows the differences between classes for the exact solution g NJ * max .

Computational Intelligence and Neuroscience
In Table 19, it obvious that the maximum number of unsolved instances was obtained for Class 3 with NS � 20. Further results obtained by employing the B & B method is presented in Table 20.
e results included in Tables 17 and 18 motivated an investigation of n p and n r in order to determine the instances which are the most difficult to solve. A study based on the ratio n p /n r was carried out. For any ratio n p /n r for the given instances amongst the total number of 1850 instances, the unsolved instances NS, the average number of nodes NN, and the average value of Time were calculated.
For the total number of instances given at the beginning of this section, there were 22 different values of the ratio n p /n r . e results are illustrated in Table 21. Indeed, Table 21 shows the performance indicators used for measuring the results obtained by the B & B method for variations in the ratio n p /n r . e maximum average time of 870.017 s was obtained for (n p /n r ) � 2.5. It is obvious that the unsolved instances were concentrated for (n p /n r ) � 2 and (n p /n r ) � 2.5. It can therefore be concluded that these types of instances are the most difficult to solve. Figure 6 shows the concentrated unsolved instances for index 2 and index 3, which are represented in Table 21 and referred to as (n p /n r ) � 2 and (n p /n r ) � 2.5, respectively. e ratio n p /n r represents the economic scale that can define the hardness of the problem to give a fair distribution of the projects to create new jobs on the regions. e experimental results give an appreciated gap between regions using heuristics. On the contrary, the exact method gives the optimal solution of the problem analyzed. Reducing the gap related to the new job created between regions decreases the unemployment rate in the internal cities of the country and strengthens regional development.

Conclusion
e problem analyzed above is of important practical interest as it models real-life situations in the field of economics and finance, such as equal distribution between different regions, and attempts to minimize the differences between them in terms of levels of investment. e solution presented in this paper will therefore be useful from a practical point of view. From a theoretical perspective, the approached problem is a challenging one as it is difficult to solve; thus, its resolution will enhance the theoretical knowledge in this field. e analyses and resolution of this problem may therefore have an important impact on the economy and the industrial sector. In this paper, a comprehensive literature review on resource allocation and optimization studies was presented, and the theory of optimization was applied to a real-life case study to illustrate the importance of this practice. e theoretical and practical analyses are closely related. is study was applied to the industrial sector, and the results achieved by applying heuristics and an exact solution to the analyzed problem demonstrate the importance of optimization applied on the project distribution field. e results contribute to reducing the difference between various industrial regions and to achieving equity among them. More other branching methods can be developed to ameliorate the execution time of the exact solution. e proposed solutions in this paper can be applied to several fields.

Data Availability
All the data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.