Simulating operational control of production in lumber house building businesses

Substantial reserves for higher operation efficiency in wooden house building businesses exist in improvements of production control system through methods of simulation and optimization. This report presents analysis of key factors of lumber house building processes that influence approaches to production control simulation. The need to factor in interconnection and dependence of lumber and house part production processes is highly difficult for operational control of the production system in question. The optimization mathematical models that we developed will help to solve the tasks of operational scheduling at all stages of wooden house manufacturing, from log cutting to production of finished assembly sets.


Introduction
The current economic situation in the wooden housebuilding sector in Russia is experiencing a shortage of funds for radical upgrade of production equipment, as well as persistent deficit of working capital. The market forces businesses to expand their product ranges and move on to small-scale production and to customer-built houses. All this calls for improved control methods, primarily ensuring flexible and optimal production processes in real time [1].
Under the circumstances, businesses are confronted with difficult problems of ensuring timely deliveries to users, well-paced manufacturing, optimized equipment load distribution, creating regulated reserves, balanced production plans and resources, ensuring strict synchronization of warehouses, motor vehicles and equipment, production facilities and operation sites [2].
Solution of the above tasks should be addressed out by a production control system built with optimization mathematical models.
There have been numerous studies to develop efficient operation control systems, and such research goes well back in time [3][4][5][6][7][8][9]. However, studies of operation control in woodworking business are rather few, and those available fail to exhaustively cover the capability of today's information technologies and production control system simulation methods. Besides, most of such studies mainly focus on operation management of sawmills [10][11][12][13][14], pulp-and-paper mills [15,16] or furniture manufacturing businesses [17].
The main purpose of this paper is to design mathematical optimization models addressing issues of time operation planning for multi-stage processes of wooden house building.

Problem description
As its operation control object, this study examines production of wooden houses that uses processes to transform lumber and other materials and power to semi-finished and finished products within a complex business. As the key feature, each such business must include sawmills and woodworking units to manufacture joinery products, wooden structures, and wooden houses assembly sets. Such production system is a complex one because both a sawmill and a woodworking shop use highly complicated processes, and when they are integrated into a united interrelated production process the system's complexity rises to a whole new level. Wooden house manufacturing is a multi-stage process that includes lumber shaping to finished-product set delivery or on-site assembly. The stages of processing are handled by a central logistical system with many warehouses to accumulate stock between steps (saw logs, sawn lumber, blanks, parts, and assembly units).
Unlike in other industries, material is a major extra randomization factor for the production environment of wooden house building, because as a product of living nature, wood is highly diverse and unpredictable in terms of dimensional and qualitative characteristics.
One of the main objectives for production control systems is to achieve coordinated control of sawmilling and woodworking in the context of supply of lumber, blanks and parts within the premises, and thus organizing a comprehensive production process.

Modelling
To work out some rational managerial solutions, a wooden house building business needs to prepare mathematical models addressing the problems of optimized time scheduling for each stage of house manufacturing, from lumber cutting to specific finished parts.
The task of operation time scheduling for log sawing is stated as follows.
The sawmill uses L parallel production flows to process logs. Let l = 1……L be the numbers of the process flows.
Timber logs are sorted to form M dimension groups. Let m = 1, …, Мbe the numbers of the log size groups.
A variety of cutting patterns can be used to mill the timber. Total number of usable rational cutting patterns is P. Let then p = 1,…, P be the serial number of any pattern in the list of cutting patterns.
Specification for lumber to be produced during the planned time period contains K of various type-sizes of boards. Let k = 1, ..., K be the number of lumber type-sizes in the specification.
The entire scheduled operation time period is divided into T time intervals of equal length, such as half-shift. Let t = 1, …, T is the serial number of the respective planning interval (control step).
For the planning period, we set the deadlines τ k and volumes V k, of lumber of k-th typesize, to be produced by the deadlines. We also assume permissible above target k V and below target k V for production of k-th type-size lumber during the planning period. We know number Q m of logs in each m-th group, stored at the stock warehouse during the planning period.
During the same interval of the planning period, each flow may only use one mill (cutting shape) to process material.
Total quantity of type-size lumber produced during each interval of the planning period is limited by number N of pallets in the sorting bay (or recesses in the sorting line). In addition, some pairs of lumber type-sizes exist that are illegal for simultaneous production, because they are hard to tell apart during sorting. We assume that we have a total S of such pairs, while s = 1 ,…, S is the serial number of the pair in the list. Let us take that Ω s is a set of numbers of lumber type-sizes, which stands for a pair illegal for simultaneous production.
Now we need to generate a calendar time schedule for stock milling: we therefore need to know which group of logs will be milled during each interval of the planning period, using which cutting pattern and as part of which flow. The following constraints of technology and planning apply.
The quality of any time schedule is decided by two criteria. First, economic criterion F 1 requires maximized total cost of all production output milled during the planning period. Second, technological criterion F 2 represents the requirement of least changes in the product range during the planning period, helping to rapidly accumulate batches of boards in the same type-size and raise equipment load efficiency during lumber drying and finishing.
Let us adopt the following notation: x t lmp -the problem's variable that is 1, if during the t-th planning period in l-th flow to mill the material of m-th group using p-th cutting pattern; else the value is 0.
a kmpquantity of k-th type-size lumber, made of logs of m-th group using p-th cutting pattern.
b lmpoutput capacity of l-th flow while milling logs of m-th group using p-th cutting pattern during the planning interval.
The production process to make assembly sets of wooden houses from lumber is treated as a multi-stage line production system. The system includes the stages of lumber patterns cutting to make blanks, blank processing to make parts, and preparing sets of parts as per respective specification. Certain cases, depending on the wooden house design, may include the stage that assembles wooden panels. Each such stage can include multiple steps of blank and part processing. The number of steps and the sequence for the blanks and parts to follow through depend on engineering and art design of the housing product, and the technology adopted by the business. Manipulations of the production process can be performed on several machines working in parallel. To coordinate operation of various stages and steps of the production process, the business must establish in-process stocks to store blanks. Such stock enables control of all blanks passing each processing stage, because blanks can enter a specific stage from both the previous processing stage and from the warehouse. The blank/part processing stage is simulated as an in-process stock control system. For ongoing control of the production process, the business must draw a time schedule of operations to plan for changes in sizes of in-process stock with time.
Let us examine the problem of operation time schedule planning for the woodworking stages. We adopt the following notation: Ntotal number of wooden house projects /products made by the business; j -number of project/product in the business's product range, j =1,…, N; Mtotal number of type-sizes of parts/blanks made by the business; i -number of part/blank as per the manufacturer's general specification, i = 1,…, М; а ijnumber of parts/blanks of i-th type-size in j-th product; 3. capacity of stock warehouse before processing of l-th stage 4. capacity of stock warehouse after processing of l-th stage 6. readjustment cycles for machines used to process blanks during l-th stage per shift

Conclusions
Mathematical analysis of the resulting models demonstrates that they can be reduced to the problem of unconditional pseudo-Boolean optimization with target functions preset algorithmically as a problem of conditional optimization with continuous or discrete variables. Coefficients for Boolean variables are not constant values, but they represent functions of the same Boolean variables. To solve this problem, we use a hybrid adaptive heuristic search algorithm. The proposed set of interrelated mathematical models is basis for the designed system to support operational control of wooden house manufacturing.
Future studies are expected to consider the uncertainty of the production environment of the businesses of wooden house building using the methods of fuzzy mathematical programming.