Multi-Objective Optimization Concept Based on Periodical and Permanent Objective within a Process

Multi-objective optimization (MOO) is an optimization involving minimization of several objective functions more than the conventional one objective optimization which have useful applications in Engineering. Many of the current methodologies addresses challenges and solutions to multi-objective optimization problem, which attempts to solve simultaneously several objectives with multiple constraints, subjoined to each objective. Most challenges in MOO are generally subjected to linear inequality constraints that prevent all objectives from being optimized simultaneously. This paper takes short survey and deep analysis of Random and Uniform EntryExit time of objectives. It then breaks down process into sub-process and then presents some new concepts by introducing methods in solving problem in MOO, which comes due to periodical objectives that do not stay for the entire duration of process lifetime unlike permanent objectives, which are optimized once for the entire process lifetime. A methodology based on partial optimization that optimizes each objective iteratively and weight convergence method that optimizes sub-group of objectives is given. Furthermore, another method is introduced which involve objective classification, ranking, estimation and prediction where objectives are classified base on their properties, and ranked using a given criteria and in addition estimated for an optimal weight point (pareto optimal point) if it certifies a coveted optimal weight point. Then finally predicted to find how much it deviates from the estimated optimal weight point. Although this paper presents concepts work only, it’s practical application are beyond the scope of this paper, however base on analysis presented, the concept is worthy of igniting further research and application.


INTRODUCTION
Optimization of process is of paramount eminence in science, engineering, finance and social issue. Its application ranges from biological process control, chemical industrial control to physical process control and many more. Optimal control succor in abridging cost and wastage of resources and minimizes time for process execution within given constraints to meet desire objectives as illustrated in many method in Optimal Control[1][2] [3][4] and adaptive control [5]. Take for example a case in a flight or marine vessel, which is required to maximize the amount of load (goods) it, carries meanwhile minimizing it weight and size. Some common approach is to minimize the amount of fuel it uses, as reduction in fuel use is directly proportional to decrease in overall weight of a vessel and ultimately the size of the fuel tank. Nevertheless, how does one reduce fuel without affecting the distance to be covered by the vessel? Some engineers would use the shortest path/rout possible for navigation, others might use the approach of redesigning an energy efficient engine to ensure minimum consumption and minimum wastage of fuel [6][7] during navigation. All these are targeted at minimizing cost and wastage of resources such as fuel.
Another example, in chemical industrial setting, where the objective is to maximize the product of reaction. Some engineers would device a methodology by maximizing rate of reaction but some reaction are endothermic which requires too much input energy to run the reaction process fully and efficiently. So how does one minimize energy inputs required to run such endothermic reaction meanwhile maximizing product of the reaction? All these are some of the notable situation that requires good optimization strategy.
A simple System often consists of a single objective and perhaps a single constraint and they are linear normally. However a system start getting complicated when it's non-linear and in addition, when there are more than one or many constraints and many or multiple objectives to be solve simultaneously. Such systems are quite not unexacting to optimize due to their Complexity.
Most challenges in multi-objective optimization often referred to as multi-criteria programming are generally subjected to linear inequality constraints that prevent all objectives from being simultaneously solve for instance a case where number of objectives are more than that of controllable variable or, perturbation that generates uncertainties. Many scholars have put several Approaches and model for finding optimal weight point of multiobjective optimization in place. The methods addresses several challenges in multi-objective optimization and their application in many field such as science, finance, engineering and many more for instance in field of electricity in electrical power balance where trade-off between voltage and electrical grid requirement [8] where demand of electricity and electricity generation needs to be balance with considering constraints bonded to it.
The challenges involve in multi-objective optimization is finding an optimal weight point solution thought there may exist multiple solution for a given multi-objective optimization problem, however the problem is finding such solution which isn't straight forward such as that of a single objective optimization.

LITERATURE REVIEW.
Below are some of the notable multi-objective optimization methods and their application in many problem domains. In addition, to currently and often use standard approach by setting a fix optimal weight point among multiple objectives. This paper initially appeared in preprint [9], which presented some of survey in the field. The multi-objective optimization approaches are categories into four major categories [10][11] as below: No preferences, Priori method (lexicographical programming, Goal programming, Utility programming), Posteriori Method and Interactive Method.
A paper by Stefan ET el [12], introduces an approach based on transformation of multi-stage optimal control model (OCM) with random switching time. They separated the problem into two sub-domain i.e. optimal control with random time horizon and Multi-stage Optimal Control Model (OCM). In their model and reformulation as deterministic (OCM), they assume that switching time divides the time horizon that is define as stage 1 and stage 2 or even more by random variable out of sample space Ω = [0, ∞]. They further applied reformulation problem introduced in [13] by Boukas ET el about deterministic Optimal Control Model with infinite time horizon and they got the results (2.1) and (2.2) and there are probability at some point that the switch has not occurred as the pointed out. A case study by Bialaszewski, Tomasz ET el [14] showed their method base on genetic gender approach for solving multi-objective optimization challenges of detection observers. In their method the previous knowledge about a single gender of all included solutions is use for the purpose of making difference among groups of objective. The knowledge is from fitness of a single person and use during a current parental crossover in evolutionary multi-objective optimization process.
An approach by Fazlollah, Samira ET el [16] presented a work on multi-objective optimization model for sizing and operation optimization district heating system with heat storage tanks. The model includes process design and energy integration method for optimizing the temperature interval, the volume and the operation strategy of thermal storage tanks.
The application of multi-objective optimization in water distribution system by Shokoohi, Meisam [16], they use Ant-colony-Optimization for the optimization algorithm which concern with water quality base objectives in Water Distribution System design alongside other common objectives. A multi-objectives decision support system developed for rehabilitation planning of public infrastructure by Farran Mazan [17]. Their method provides decision makers a collection of optimal rehabilitation tradeoff over a preferred analysis period. They handled two main objectives function cost and performance at once, together with the collection of attached constraints. The mechanism is based on a fitness-oriented method where challenges information is taken into account. To further analyze, cost and performance all together, a normalization methods of all objectives is attained through time-value concept for both cost and condition states. Their proposed methodology is based on life-cycle costing approach using a dynamic markov chain to constitute the degeneration methodology and optimal rehabilitation profile is found using algorithm. Several applications of multi-objective optimization problem has been extensively use in different and many field for example in large scale cluster [18] [19] and in some of the field like the one presented on the paper [20].
In conclusion, many scholars propose several methods; however, some method might be feasible for one or more situations while in another situation it might not be perfectly feasible.

CONCEPTS IN MULTI-OBJECTIVE OPTIMIZATION.
This portion discusses analysis and presents some of the challenges and solution involve in multi-objective optimization which do not appear in single objectives optimization and then it gives ways forward in solving those problems. First is to classify objectives base on how long they take or are actively needed in a process lifetime during process execution. They are classified into two categories as describe here below.

PERIODICAL/SHORT-TERM OBJECTIVES
In this context, a temporal objective is the one that is needed only for a particular period or for a short while, less than the time for process execution and it's not throughout the entire process. Periodical objectives can pop in or out at any moment during process execution time, these could be due to perturbation or any other factor which may cause it, for example consider the objectives function given below (3.1) in MOO min( 1 ( ), 2 ( ) … ( )) (3.1) Where integers, ≥ 2 are the number of objectives. Out of total objective ( ), one or more might be very crucial from the start of the process or in the middle or even towards the end of process execution but not throughout the entire process execution and somewhere somehow, the periodical objective will be no longer be relevant once it is not needed.
This periodical objectives which switch on and off, results in the entire multi-objectives being solve again and again, with and without the periodical objective in and out of the process execution as the optimal weight point of multi-objectives varies and are not the same when one include or remove a given objective from many objectives. For example, the optimal weight 1 ≠ 2 of the equation in multi-objective (3.2) and (3.3) are not the same because of the absent of the objective in other word, 1 is not an optimal weight point of equation (3.3) whose optimal weight point is 2 2) The MOO (3.2) contains five objectives function that need to be minimize.
2 ≔ min( 1 ( ), 2 ( ), 3 ( ), 4 ( ), 5 ( ), 6 ( )) (3.3) In addition, the MOO (3.3) contains six objectives functions not the same as equation (3.2) however all the five objectives are the same with additional objective. These gives challenge of redesigning another new optimal weight point, which is optimal for the remaining objectives, which are only relevant during execution excluding the objective that is no longer needed, or including the additional objectives as in equation (3.3).

PERMANENT/LONG-TERM OBJECTIVES
Unlike in periodical or temporary objective in multi-objective optimization, permanent or long-term objective is defined as the one that stay or is needed from the starts of a process execution up to the end of process execution which does not result in redesigning the optimal weight point since the optimal weight point remains the same. For instance, if a given process starts at time ( 0 ) and ends at time ( ) , permanent objectives also starts at time ( 0 ) and ends at time ( ) as the process.

RANDOM ENTRY-EXIT TIME
Indexing and or ranking objectives based on time in multi-objective optimization when dealing with shortterm/periodical, objectives and long-term objectives/permanent objectives mixed altogether within a process lifetime. It is very crucial to index objectives based on time they either enter ( ) or exit ( ′ ) a process for all the objectives involve for easy ranking, classification, and solving using partial optimization technique presented here below. Given time series for entry time = { 0 , 1 , 2 , 3 , … , } and exit time as ′ = { 0 ′ , 1 ′ , 2 ′ , … , ′ }. Several works have been done on time base optimization such as those presented by Boukas ET el [13] 3.3.1 RANDOM ENTRY TIME In this case, all objectives are assume to ends at the same time a process terminates however, their entry time into the process varies randomly. Given example of objectives function from (3.3) are index based on time objectives enter in a process but first their time of entry is estimated. Supposed the following estimated time for objectives in (3.3) are recorded, such that 1 ( ) enter at 0 , 2 ( ) enter at 6 , 3 ( ) enter at 3 , 4 ( ) enter at the same time as 2 ( ), and 5 ( ) enter at ( 7 ) Indexing the five objectives as below: ( 1 ( )) 0 , ( 2 ( )) 6 , ( 3 ( )) 3 , ( 4 ( )) 6 , ( 5 ( )) 7 (3.4) Rearranging (3.4) in ascending order of entry time such that objective that enter first is rank first and given high priority, and those that enter last are given last priority or ranked last. However for two or more objectives with same time of entry are rank using any other criteria such as dependency on other conditions, to give them high priority. If no such or any other condition exist, those objective with same entry time can be put at any order see condition (3.5) after rearranging.

RANDOM EXIT TIME
Here, all objectives are assumed to enter at the same time a process execution begins however, their exit time out of the process varies. Given example of objectives function from (3.3) are index based on time objectives exit a process but first their time of exit out of the process is estimated. Supposed the following estimated exit time for objectives in (3.3) are noted, such that 1 ( ) exit at ′ 2 , 2 ( ) exit at 6 ′ , 3 ( ) exit at ′ 3 , 4 ( ) exit at the same time as 2 ( ), and 5 ( ) exit at ′ 7 Indexing the five objectives as below: ( 1 ( )) ′ 2 , ( 2 ( )) ′ 6 , ( 3 ( )) ′ 3 , ( 4 ( )) ′ 6 , ( 5 ( )) ′ 7 (3.6) Rearranging (3.6) in descending order of exit time such that objective that exit first is rank last and given low priority, and those which exit last are given high priority or ranked first. However for two or more objective with same time of exit are ranked using any other criteria such as dependency on other conditions, to give them high priority. If no such or any other condition exist, those objectives with same exit time can be put at any order see condition (3.7) after rearranging.

MIXED RANDOM ENTRY-EXIT TIME
In this scenario, it is based on the idea that in multi-objective optimization some objectives have random entry and exit time in that an objective can enter at any time and exit at any time before the process terminate and for all objectives in a process their entry and exit times are scatter randomly within the process. Also considering this process also contains permanent objectives. It is like a combination of objective with varying entry-exit time. Take for instance the case below: 3 ) 4 ( ) ⟹ enter( 6 ) → exit( ′ 6 ) 5 ( ) ⟹ enter( 7 ) → exit( ′ 7 ) For case above with different entry-exit time, there are two scenario prior to ranking using time of entry or exit. We consider Exit time( ′ ) and Entry time( ) so we rank them by taking which is more important, Entry time or Exit time. If we priorities Entry time ( ) , then we can rank them base on entry time only and ignore exit time( ′ ), See (3.4) where it is ranked base on entry time so exit time is of less or no priority. However for the case when exit time are of high priority, entry time is ignored. See (3.7). Many approach can be use more efficiently for this special case.

UNIFORM EXIT-ENTRY TIME
In this case, all objectives, exit ( ′ ) and another enter at ( ) a process at the same time one or more time in a process ( ) noting that ′ = . The question is how to handle this scenario. One of the way presented here is to split the process into multi-Process referred to as sub-process ( ) within a process. Given that { 1 ( ), 2 ( ), 3 ( ), … , ( )} ∈ ( ), ∈ and parameter ( ) of the process represents the objectives to execute within the process. So i.e. = { 1 ( ), 2 ( ), 3 ( ) 4 ( ), 5 ( ) … . ( )}. Where the objective ends, a new one starts. And each sub-process is solve separately by using the presented methods just normally like a full objectives within a process. However splitting a process into a sub-process does not means terminating the entire process. The previous subprocess terminates and the new sub-process continuous instantaneously as the previous sub-process exits. For example of two sub-process 1 ( ) , 2 ( ) see (3.8) where objectives 1 ( ), 2 ( ), 3 ( ) all terminates at time 6 ′ and objectives 4 ( ), 5 ( ) begins at time 6 but as you know 6 = 6 Given that = +1 Now for the first sub-process can be group, index using random Entry-Exit time method for ranking and solve by either partial optimization or Objective Classification, ranking, Estimation and Predictive measurement presented here below. The second sub-process also follows the procedure for the first one and it continues for all the sub-process up to the last one.

PARTIAL OPTIMIZATION CONCEPT IN MULTI_OBJECTIVE OPTIMISATION
In response to the challenges introduced by the presents of periodical objective in multi-objective optimization, I present this concepts which deals with partial optimization of many objectives in multi-objective optimization, two initial idea related to multi-objective optimization is presented. 3.5.1. ITERATIVE MULTI-LEVEL APPROACH.
The approach is iterative in that, it involves taking two or more solvable set from multiple objectives set and their optimal weight for the sub-set solve. This solution becomes or is set as constraints of the next solvable set or objectives from the multiple set. This is done iteratively until all objectives are finished and the final optimal weight point is assumed to be the most optimal weight among multi-objectives. Consider objective function below: min( 1 ( ), 2 ( ), 3 ( ), … , ( )) (3.9) . ∈ Where integers ≥ 2 are the number of objectives and the set are number of feasible set of decision vectors. However, element * ∈ is further defined as a feasible solution or feasible decision of an objectives vector. * ≔ ( * ) ∈ ℝ (3.10) Initially, we first take the second objective function 2 ( ) and set the previous (first) objective function 1 ( ) as a constraint and find the optimal weight 0 as their optimal weight between the two objectives just like the one presented in a paper [21] which attempt to solve two objective by setting one objective as a constraint and another objective to be minimize.
Next iteration is to take objective function 3 ( ) and set the previous optimal weigh 0 for objectives function 1 ( ), 2 ( ) as a constraint to find optimal weight 1 . 1 ≔ min( 3 ( )) (3.12) . 0 ≤ 1 The process continues until all the functions in multi-objectives ( )are finished and the final weight is assumed to be the optimal weight for all objectives. An overall formulation is as below: ≔ min( +2 ( )) (3.13) .
The second approach, which is based on the followings, objective classification, Ranking, Estimation, and Predictive measurement to find how far a system will deviates from a preferred optimal weight. This in the present of a decision maker (DM), where there could be preference. The following steps are how the process can be executed in order to find an optimal weight point.
Step 1: Objectives Classification Several criteria can be used to classify objectives. For instance, each objective can be categories as independence and dependence objective. Independence is objectives which does not rely on other objectives as oppose to dependence objective, which depend on other objectives in other word solving one objective, affect the other which is dependent on it also known as Pareto Optimal solution which means a solutions that cannot be improve without degradation of at least some of the solutions. Another way of classifying objectives is by categorizing them into either temporary or permanent objective as discussed above. This is mainly to priorities and takes care of objectives such that when dynamically choosing weight, such objectives are taken care of.
Step 2; Objective Ranking In this stage 2 of objective ranking, in a set of multi-objectives, objectives is first rank in order of preference or merit base. Second, each individual participating objective its minimum and maximum optimal weight point is determined such that when finding weight such points are taken care of. Furthermore, objective can be rank in order with which they exit process execution, with the one that exit earliest being the last to be optimize and the one that exit last being the first to be optimize in partial optimization. For instance, − and + are defined as lower and upper bound limits for both minimum and maximum optimal weight point for a given single objective see condition (11) − ≤ * ≤ + ∶ . { − , … … . + } ≤ (3. 19) Step 3: Objective Estimation In estimation stage, this requires prior knowledge of objectives ranking and classification then an optimal weight point is first estimated that certifies the ranking and classification in step 1 and 2.
This step is based on the principle of predictive measurement in that control variable can be predicted base on prior and current state, and hence a control strategy is laid such that a controller parameter is adjusted accordingly to meet the estimated weight point of multi-objective optimization base on step 1 and 2. Suppose is deviation between estimated /desired optimal weight and predicted optimal weight and in addition and are quantifiable i.e. can be express numerically where: = | − | (3.20) As → 0, the better as the condition can be express as 0 ≤ ≤ ∆ where ∆ is deviation limits within which it is unacceptable and an appropriate strategy needs to be put in place to drive the system towards zero i.e. ≤ ∆. The best predicted optimal weight occurs when = 0 or when | − | = 0 . Although theoretically it's convincing and achievable, in practice it could be very difficult to reach = 0, however ≤ ∆ is fine. 3.7 OBJECTIVE ALIGNMENT IN PARTIAL OPTIMIZATION.
In the above method presented under partial optimization, Objectives has to first be align in order to make it easily possible to remove an objective or many objective without needs to redesign the entire equation. This purposely is done to give temporary objective less priority and permanent objective very high priority.
a. Classification. First before beginning to optimize the entire multi-objective using method of either iterative or weight convergence, it is first classified into either periodical or permanent objectives. This is basically to give temporary objective less priority and permanent objective very high priority b. Ranking After classifying the objective, it further rank based on the time the objectives takes to exit the process before the process execution ends, then optimization is done in a way that objective which are classified as permanent are optimized first without any preference. In addition, the one that are classified as temporary or periodical objective are then optimized later in the order with which they exit the process and the one that exit earliest will be optimized last such that removing will not affect or need to redesign the entire equation. See equation below (3.21) and (3.22) where 5 ( ) is removed and the previous weight 2 automatical become the weight of the current equation without the present of the objective function 5 ( ). 3 ≔ min( 1 ( ), 2 ( ), 3 ( ), 4 ( ), 5 ( )) (3.21) The MOO (3.23) contains five objectives function which need to be minimized.
2 ≔ min( 1 ( ), 2 ( ), 3 ( ), 4 ( )) (3.22) The same applies to weight convergence method where permanent objectives are classified and optimized separately and the one, which are temporary or periodical, are optimized separately. Then later they are finally optimized such that removing portion of the periodical objectives does not affect the entire solution weight (optimal weight point).

ANALYSIS OF THE CONCEPT.
The following analysis tries to shows the drawback and some advantages of this propose methodology compared to the currently existing methods 4.1. Disadvantage of Existing methods Compared to Presented Concepts In most of the current methodologies presented by many scholars tries to solve all multi-objective problem simultaneously which when periodical/short-term objectives is added or remove will result in destabilizing the optimal weight point hence need to redesign the entire solution again. This isn't that much problem, however during runtime it could be tiresome and the hardship to resolve the same thing time and again for example in equation given in (3.2) and (3.3) where objective is removed and the entire weight are not the same and also (3.21) and (3.22).

Advantage of Presented Concepts
The advantage of this solution presented here is that unlike simultaneous solution, partial objective optimization an objective can be remove safely without need to redesign the entire equation as the previous solution become the current solution without need to solve again. Also addition of an objective does affect much, all is needed is to optimize that objective together with existing solution as explained above. However, objective classification and ranking should be carefully done to ensure alignment before optimization.

CONCLUSION
Due to ultimatum in seeking the best optimal weight point among multi-objective optimization, this paper then discusses some notable published paper by scholars, which presents the novel technique in tackling problems in multi-criteria optimization such as evolutionary algorithm methods, flower pollination algorithm and many more. However, the author further notices some dare in the area of multi-objective optimization where there are situation when one or more objectives are needed either in the beginning or towards the ends or even in the middle of the process execution. The author called this as periodical or temporary objectives unlike usual objectives that are needed from the start up to the end of process execution. This is called permanent or long-term objectives. The problem with this is that every time an objective is added or removed from the running process with multiple objectives will results in destabilization of existing optimal weight point as shown in the above example above in (3.2) and (3.3).
In respond to the challenges, the author presented a theoretical concept which uses partial optimization iteratively and weight convergence including some methods which use objective classification, ranking, estimation and predictive measurement. These concepts are very convincing theoretically however their feasibility is far beyond the scope of this paper. Further in-depth studies will still be conducted to check the practical application and its feasibility in real world scenario including a numerical simulation or any analytical solution when conducted.