Optimisation problems of networking manufacturing processes

The globalization of economy and market led to increased networking in the field of manufacturing. The processes of these manufacturing including logistics became more and more complex. This paper aims to report design aspects of networking manufacturing focusing on NP-hard optimisation problems. The authors describe general models of logistic processes of networking manufacturing systems and present heuristic algorithms to solve the optimisation problems focusing on the solution of knapsack and traveling salesman problems with harmony search and firefly algorithms.


Introduction
The increased diversity of customer's needs has led to the increase of the complexity of manufacturing processes.Complex, networking manufacturing processes required complex logistics and supply chain solutions both in-plant and outside of the manufacturing plant.
Heuristic algorithms offer suitable tools to support the design and control of these networking systems.A wide range of networking problems can be defined as knapsack and traveling salesman problems.Some special and generalized algorithms were created and published in different research works: the first exact solution methods focused on either bounded or unbounded problems [1,2]; the new solution methods are based on heuristic: assessing solution quality of biobjective 0-1 knapsack problem using evolutionary and heuristic algorithms [3], bounded set-up knapsack problem [4], image knapsack problems [5].Much literature has been published in the field of optimization of large scale networks and complex engineering systems by the aid of Harmony Search Algorithm (HAS).There are several applications for harmony search algorithms from the sizing optimization of truss structures [6] to the transport energy modeling [7] or nuclear energy production [8].
The harmony search algorithm can be used to create hybrid algorithms to develop more effective solution methods for large systems [9][10][11].Harmony search algorithms make it possible to realize global harmony search [12,13] and local-best harmony search [14].
This paper is organized as follows.Section 1 presents a short literature review.Section 2 describes two manufacturing related NP-hard optimization problem with heuristic solutions.Conclusions and future research directions are discussed in Section 3.

Manufacturing optimization in the case of networking companies
There are many problems in the field of logistics.Most of these problems are so complicated that a simple model can't specify them.When this occurs several model and methods needed to get solutions, and we have to decide it which we will choose.The appropriate model and approach, can significantly reduce the computational time, human resources and save money.

Knapsack problems in manufacturing
The optimization of logistic systems and processes is the key factor of the economical operation of a company.Capacity problems occur every day in every field of manufacturing and logistics.It has the biggest effect on purchasing, distribution and production, but the problem can appear in services and recycling too.Just to mention some activities the problem affects scheduling of transportation; transportation; loading and unloading of products; storage in the warehouses or internal storages (buffering); building of homogeneous and inhomogeneous loading units; handling of production resources.
There are lots of ways to improve the capacity of a company.The easiest way is to buy more space, building, machine or transport vehicle.But the companies don't want to spend millions of dollars on huge investments, which they don't necessary need, so the thing to do is to overview the capacity management.Most of the time capacity problem can be represented as a knapsack problem (Fig. 1.).The one-dimensional knapsack problem focuses on one question: which products should be chosen to maximize the profit within a limited space (or another parameter)?The one-dimensional knapsack problem can be solved by basic combinatory or linear programming tools, like brute-force search, but unfortunately most of the cases one parameter is not enough to describe a problem in a company, and certainly won't do in a supply chain.Lots of factors, limits, parameter connections and constraints affect the solution and this gives us a NP-hard problem.From this point of view, linear programming can't be used effectively, but heuristic algorithms can give us answers very quickly if we use the good ones with proper settings.
There are several predefined knapsack problems, and all of them have some kinds of solution in the literature: unbounded knapsack problems; bounded set-up knapsack problems; image knapsack problems.We only create a little application to solve the basic knapsack problem, but with some work, it can be used for the most difficult ones too.
One of our previous works was a modification of a meta-heuristic algorithm, that can easily solve the knapsack problem [15].The base of the method called Harmony Search Algorithm, and it's a music-based swarm type algorithm.It has 3 types of variable modification; copying, slightly changes or total randomization; which chosen randomly affected by weights.These 3 modes with the proper settings can be so robust that it can get out of a local optimal point, but it also can refine the solution greatly if it has time.
where is the cost, is the weight, is decision variable, is the weight or capacity limit.The basic knapsack problem doesn't need a complicated model as we can see above.It only contains an objective function, that maximizes the profit and a limit of the space we have, which cannot be exceed.We can make it more complicated, if we add another limit, like weight; or we need to give a formation or sequence for the things inside.
HSA is a hybrid algorithm which means it contains several separated parts from other algorithms, so we made some modifications too.First a memory function was added, because when the search space is too flat or chaotic (noisy) it's easily lose the optimal point.Then a solution refining mechanism took place inside the method, that gives us more punctual solution if the running has more time.

Traveling salesman problem
This problem always occurs when somebody or something must reach at least two places and we must decide in which order.Every company that does some sort of transporting meets this problem on daily basis.There are lots of solutions in the literature and on the market too, they called route planning applications.We are working on a basic one too, but we focused on some extra functions, that can save money.
The basic TSP is based on distances between the reachable points.We try to find the shortest summarized distance as we reach all the points we require.The points and distances are fixed, so the only thing that can be modified is the sequence.In this case the basic TSP is just a permutational problem, and it's not hard to handle.But most of the time minimize the traveled distant is just not enough.There are some parameters, and requests that must be fulfilled: energy consumption, fastest route, emission, deadlines, driving and resting hours, occupancy, empty space, utilization, and we are only speaking about solo transportation.If we place parallel transporting and multiple warehousing in the equation, they just make the computing even harder.With these parameters, the problem gets too hard to be solved by standard linear programming tools, we must use heuristics again.
We create a basic route planning application with HSA, and it worked perfectly, but we only counted with the distances.For the next step, we leave the Matlab language and create a basic route planning application with C# for better visualizing the results.After the language, the base of the application is also changed.A new algorithm has been introduced and we want to try it.It's called Firefly Algorithm (FA).It's also a nature based method, but it simulates the movement of a group of fireflies.
The next update for the application was a specific function, where we not only measure the distance, but to examine the stock levels, and create multiple routes for solutions.A new visual and mathematical model was created.For these new parameters had to be added: costs, efficiency, volumes, weights, etc. Summarized route length: where: is the number of cycles of the investigation period, is the route ID, is the number of routes in the t th investigation period, , is the central storage ID in the t th investigation period visited by the α th route, is the objects number in a given route, , , is the objects ID number in the t th investigation period visited by the α th route in the th place, , is the number of objects in the t th investigation period visited by the α th route.
The following objective function is determining the routes occupancy: where is the maximum storage capacity of the th object, , , actual filled capacity of the th object, is the α th route assigned vehicle capacity.To minimize the investments costs it's important to optimize the routes number.The central storage plays an important role in the system, therefore during the optimization of the supply chain transport and storage costs should be minimized.
where is the central storage replenishment cycle time, is the order-processing cost related to the central storages refill process.The storage costs in the full time horizon can also be calculated.
Based on these two costs, the objective function for the central storage can be written in the following form.
For the routes we can also create an important limitation, which expresses the fact that the volume served by the route can't be more than the vehicles capacity.
The application was created for ATM refilling, but it can be easily modified for other route planning purposes with inventory management: milkrun, vending machine management, smart recycling bins, smart container refilling or emptying, etc.The biggest problem was that the algorithm and the model serve different purpose.The algorithm works with floating points, and tries to find the global minimum by changing the distance between points, but the model needs a simple integer and permutational solution.We have hard time to discretize the algorithm, but in the end it worked greatly.
For our last two applications we change the algorithm again, because the Firefly algorithm was too specific for these tasks, too many modifications were required to work properly.We find a novelty algorithm, that called Black hole, and it works brilliantly.The method it's use is quite simple, and easy to discretize.( 1 0 ) where is the original transportation cost, which is proportional with the distance, is the transportation cost of a new transport vehicle, is the cost of the s th order processing and assigning a new vehicle, profit from fulfilling the t th demand, is the distance between the current position of the i th vehicle and the i th final destination, is the distance between k th vehicle parking site and the j th sub-pickup point, is the distance between the j th sub-pickup point and j th sub-destination, isthe distance between the current position of the i th vehicle and the j th sub-pickup point, is the distance between the j th sub-destination and the i th final destination, is the number of original transports and vehicles that are on road, is the number of incoming transport demands, is the number of transport vehicles parking sites, is the number of new vehicles needed to fulfil the IManE&E 2017 6 transport demands, is the number of vehicles will be transporting including the new demands, is the set of elements of j, that the i th vehicle can transport due to capacity restriction, / is the volume/weight capacity of the i th vehicle, / is the volume/weight of the transported goods in the i th vehicle, / is the volume/weight of the j th transport demand, is the time needed to go from A to B, is the current time of the i th vehicle at the CP point, is the requested arrival time of the i th vehicle to its original destination.
The above mentioned models make it possible to support the design and operation processes of manufacturing related supply chain problems, while a huge number of in-plant supply chain problems can be defined as a special type of knapsack or TSP problems.

Conclusions and future research directions
Manufacturing systems are more and more complex.Heuristic optimization offers powerful tools and methods to design networking manufacturing systems.Within the frame of this paper authors described two possible optimization methods of knapsack and traveling salesman problem with harmony search and firefly algorithm to support the optimization of various manufacturing related logistic problems.As a result of the demonstrated research it can be expected that in the future years more and more optimization application will be developed in the field of supply chain.The results of this study can be used to improve the whole logistics process of companies from the purchasing through production logistics to distribution.
This research work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 691942.This research was partially carried out in the framework of the Center of Excellence of Mechatronics and Logistics at the University of Miskolc.

Fig. 2 .
Fig. 2. TSP flowchart.The first application with the new algorithm analyses the running transport vehicles if they can fulfill another request, within their original goal.The objective in this case is the money that can be saved by running extra errands. ) Fig. 2. TSP flowchart.