PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV,
HOLDER OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER OF THE RED BANNER OF LABOR

Volume 269　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1976

MANAGING THE PROCESS OF PREPARING NEW PRODUCTION

A. I. GUDZENKO, L. V. PERFILIEV

(Presented by the scientific-technical seminar of the Department of Automated Control Systems and the Control Laboratory)

Managing the process of preparing new production occupies a special place among the numerous tasks of production management. The nature of the course of this process determines not only the principal economic indicators of an industrial enterprise, but also its mobility in the sense of the speed of its response to the pace of scientific and technological progress. These issues acquire particular importance at enterprises with a unit and small-batch character of production, where the task of mastering new products has a large share in the total volume of industrial output.

Managing the process of preparing new production includes calendar planning, analysis of the process on the basis of operational monitoring of plan fulfillment, and the formation of control actions in order to ensure the planned course of the process. The basis of calendar planning is a model of the process of preparing new production, by means of which the optimal sequence of servicing operations is determined in accordance with the vector $\vec{A}_{\mathrm{opt}}$ and the deadlines for performing each operation are calculated in accordance with the vector $\vec{t}_{\mathrm{opt}}$. The action of a set of random factors, as a rule, leads to the occurrence of certain deviations from the plan obtained as a result of statistical modeling. The deviation of the real process $(\vec{A}_{\phi}, \vec{t}_{\phi})$ from the planned one $(\vec{A}_{\mathrm{opt}}, \vec{t}_{\mathrm{opt}})$ is precisely the feedback signal on the basis of which control actions are formed to eliminate the discrepancy that has arisen. Control actions of this kind have a substantial influence on reducing the limits of fluctuations in the duration of individual operations and on reducing the degree of uncertainty in the deadlines for plan fulfillment. Consequently, their action is equivalent to reducing the probabilistic properties of the process and bringing it closer to the properties of a deterministic one.

The formation of rational control actions cannot be carried out without reliable information reflecting the state of all essential parameters of the process at each stage of its development. The need to obtain the indicated information requires solving issues of organizing operational control. One of the first tasks that must be solved in this connection is the selection of well-founded

control times. It is known that increasing the frequency of control surveys is always accompanied by an increase in the volume of information processed and in the associated costs. On the other hand, long intervals of time between control moments lead to a decrease in management effectiveness because substantial deviations of the process from the optimum arise as a result of the action of random factors. It is expedient to link the determination of control times with the results of plan fulfillment in the preceding period and with forecasts for the future. A substantiation of rational control times on the basis of such an approach was carried out in [1].

The approach described in this work makes it possible to determine rational times \(t_l\) for monitoring the state of the process, using optimistic and pessimistic dependences of the volume of plan fulfillment on time, as well as information on the actual course of the process.

Management of the course of preparing new production should be regarded as a process consisting of a sequence of decision-making acts at the times \(t_l\). In the intervals between these moments (on the intervals \([t_l, t_{l+1}]\), \(l=0, 1, 2\ldots\)) the development of the process proceeds under constant control actions, which are the consequence of decisions made at the time \(t_l\). The time intervals \([t_l, t_{l+1}]\), \(l=0, 1, 2\ldots\), will hereafter be called operational periods of management.

At the stage of making decisions on managing the process of preparing new production, the following alternative actions on the process are available:

\(u_1\)—redistribution of human and material resources among subdivisions;

\(u_2\)—intensification of the performance of individual operations within subdivisions;

\(u_3\)—reduction of the volumes of work of lagging overloaded subdivisions by means of an external load;

\(u_4\)—re-optimization of the plan, taking into account information on the state of the process at time \(t_l\);

\(u_5\)—attraction of additional resources.

The elements \(u_1 \div u_5\) form the set \(u\) of alternative actions.

Let us dwell briefly on the essence and expediency of using each of the listed actions. Redistribution of resources among subdivisions is carried out in each operational management period in order to determine the optimal variant of using the resources of the subdivisions for performing the entire set of operations of that period. Both in the normal course of the process and when it lags substantially behind the plan, situations may arise in certain periods in which some subdivisions will be significantly overloaded and, as a result, will restrain the entire course of the process, while others are underloaded and idle while awaiting subsequent operations. In such situations it is natural to accelerate the performance of the limiting operations through the use of the resources of lightly loaded subdivisions. This can be achieved by redistributing operations among subdivisions. In searching for the best variant of such redistribution, both the very possibility of a subdivision \(i\) performing operations of type \(j\), and the effectiveness of such use, must be taken into account. Below, the problem of redistributing operations among subdivisions will be formulated and a method for solving it indicated. Intensification of the performance of individual opera-

...which is carried out mainly through the use of all forms of increasing the labor productivity of performers (including by means of moral and material incentives) and improving the organization of work within subdivisions. These measures are applied, above all, to operations lying on the critical path and, first of all, to those among them whose start-of-service date has shifted relative to the calculated one. In doing so, however, it is necessary to take into account the possibility of such a reduction. The smaller the difference between the planned and optimistic time for performing a given operation, the lower the probability of shortening the duration of that operation through intensification.

Influencing the process by reducing the volumes of work of lagging overloaded subdivisions amounts to a short-term prohibition, for such subdivisions, on performing external load. Thereby, for the duration of the next operational period, the subdivision has the opportunity to allocate additional resources to carrying out the production-preparation plan.

Re-optimization of the calendar schedule consists in periodically forming, with the aid of a mathematical model, a new plan on the basis of information about the state of the process, taking into account the random events that have occurred. Since information on the process is collected in periods \(t_l\), \(l=0, 1, 2\ldots\), re-optimization of the plans, carried out in these same periods, completes the operations of monitoring and analyzing the course of the process.

The attraction of additional resources may be expressed in assigning part of the work on production preparation to subcontracting organizations, and also in obtaining additional staffing, material, and financial opportunities. Both the former and the latter, as a rule, lie beyond the authority of the persons responsible for making decisions on managing the process of preparing new production. The possibility and necessity of using the listed control actions are determined by the specifics of the process of preparing new production at a particular industrial enterprise, as well as by the features of situations arising in the course of plan execution. Obviously, primary attention should be given to the first four management alternatives, which ensure fulfillment of the plan within the established time limits through internal reserves.

As already noted, in order to make well-founded decisions in managing the process of preparing new production, reliable information is needed that makes it possible to analyze the possible consequences of these decisions. The parameters and characteristics of the process under study are used as such information. In each specific case, the characteristics and monitored parameters of the process must correspond to the goals, tasks, and principles of management, since only in this case will they concentrate the information necessary for management purposes. At the same time, naturally, one should take into account the capabilities of the mathematical model of the process in terms of obtaining, with its aid, one or another characteristic.* For the process under consideration, taking into account the described management alternatives, the following characteristics and parameters are used, according to which monitoring is organized at the boundaries of operational management periods:

* The results of the study of managing the process of preparing new production considered in this work are applicable to models similar to the one proposed in [2].

1) the optimistic \(V(t)_{\mathrm{opt}}\) and pessimistic \(V(t)_{\mathrm{pess}}\) dependences of the volume of operations performed over time;

2) \(M_0^{l-1}\) is the set of operations that must be performed during the preceding \((l-1)\)-th operational control period;

3) \(M_1^{l-1}\) is the set of operations that were serviced during the preceding \((l-1)\)-th operational control period;

4) \(\tau_i^l\) is the time fund of the \(i\)-th subdivision that may be allocated to performing operations during the next \(l\)-th operational control period;

5) \(b_i^l\) is the time fund required by the \(i\)-th subdivision to perform the operations that must be serviced during the next \(l\)-th operational control period;

6) \(t_i^p\) is the amount of time during which the lag in meeting the plan deadlines of the \(i\)-th subdivision can be eliminated;

7) \(P_i\) is the probability of failure to meet the plan through the fault of the \(i\)-th subdivision.

Let us briefly consider each of the listed parameters.

The dependences \(V(t)_{\mathrm{opt}}\) and \(V(t)_{\mathrm{pess}}\), necessary for calculating the duration of the next operational control period, are determined on the basis of information about the value of the pessimistic and optimistic times for performing operations, and also the sequence in which they are serviced. In any of the operational control periods, additional optimization of the schedule may be carried out (the control action \(u_4\) is used). The plan obtained as a result of this additional optimization corresponds to the sequence vector \(\vec A_{\mathrm{opt}}^{\,l}\) and the duration vector \(\vec t_{\mathrm{opt}}^{\,l}\). With a fixed vector \(\vec A_{\mathrm{opt}}^{\,l}\) and a known value of the service time of each operation, equal to the optimistic (pessimistic) value, it is not difficult to construct, as a function of time, the curve of the optimistic (pessimistic) volume of work performed, expressed in man-hours or in the number of serviced operations.

The set of operations that must be performed during some operational control period is determined by the plan accepted for implementation. The result of an element-by-element comparison of the sets \(M_0^{l-1}\) and \(M_1^{l-1}\) is a feedback signal establishing the need to use control actions.

Important monitored parameters at the stage of operational control are the time fund required by a subdivision to perform operations, as well as the time fund that may be allocated to servicing these operations, since the magnitude of the ratio of these parameters substantially affects the deadlines for fulfilling the plan and, consequently, the decision made on control of the process. It is necessary to note, however, that the actual completion times of work depend not only on the ratio of the indicated parameters, but also on the structure of the schedule, determined by the specific “technological” sequence of servicing operations. This dependence is manifested in the fact that downtime of subdivisions, caused by the need to take into account the interrelation of operations, postpones the final completion dates of work. The final management decision can be made only after a new schedule has been formed, taking into account the control actions introduced into the control system.

The values of the quantities \(\tau_i^l\) and \(b_i^l\) are calculated from the expressions:

\[ b_i^l=\sum_{j\in M_0^l} b_{ij}; \tag{1} \]

\[ b_{ij}=\frac{t_j-r_i}{c_{ij}}. \tag{2} \]

\[ \tau_i^l = \begin{cases} t_0^l r_i(1-p_i^{v}), & \text{if } \tau_i^l \ge b_i^l,\\ t_0^l r_i, & \text{if } \tau_i^l < b_i^l, \end{cases} \tag{3} \]

where

\(b_{ij}\) is the time fund required for the \(j\)-th operation to be performed by the \(i\)-th subdivision;

\(t_j\) is the planned value of the time for performing the \(j\)-th operation;

\(r_i\) is the size of the \(i\)-th subdivision (persons or units of equipment);

\(c_{ij}\) is the efficiency of using the \(i\)-th subdivision in performing the \(j\)-th operation (the value \(c_{ij}\) lies in the interval \([0,1]\));

\(t_0^l\) is the duration of the \(l\)-th operational control period;

\(p_i^{v}\) is the probability that external-load requirements will arise for the \(i\)-th subdivision.

The next, no less important parameter, used in the case when it is not possible to eliminate schedule slippage within a single operational control period, is the quantity \(t_i^0\). This quantity is calculated under the assumption that equipment downtime is excluded and that the entire time fund of the subdivision is used to perform only the planned operations of preparing new production. Another essential assumption is the restriction on the operation service time, which is taken to be equal to the optimistic value. The quantity \(t_i^0\) is determined by the expression

\[ t_i^0=\frac{b_i+r_i\sum_{j\in M_i} t_{j\text{opt}}}{r_i}, \tag{4} \]

where

\(t_{j\text{opt}}\) is the optimistic time for performing the \(j\)-th operation;

\(M_i\) is the set of operations that the \(i\)-th subdivision still has to perform in preparing new production. The algorithm for calculating the quantity \(P_i\) is given in [3].

The Problem of Redistributing Operations among Subdivisions

There are \(M^0\) operations that must be performed by the subdivisions during the operational control period. A matrix \(C=\|c_{ij}\|\) is given, whose elements characterize the efficiency of using the \(i\)-th subdivision in performing the \(j\)-th operation. The following quantities are also considered known:

\(\tau_i\) is the time fund of the \(i\)-th subdivision that can be allocated during the operational control period to performing operations;

\(b_{ij}\) is the time fund required for the \(j\)-th operation to be performed by the \(i\)-th subdivision.

As noted above, it is required to find the most effective admissible variant of distributing operations among the subdivisions, taking into account the actual resource constraints. An admissible variant of the distribution of operations is understood to be one in which each operation is assigned to only one subdivision, and the total volume required to perform the operations assigned to a subdivision does not exceed the value \(\tau_i\).

To each admissible variant of the distribution of operations there may be put into one-to-one correspondence a matrix \(X=\|x_{ij}\|\), whose elements satisfy the following constraints:

\[ x_{ij}= \begin{cases} 1, & \text{if operation } j \text{ is assigned to subdivision } i;\\ 0, & \text{otherwise.} \end{cases} \tag{5} \]

\[ \sum_{i=1}^{N} x_{ij}=1,\qquad j=\overline{1,M^{0}}, \tag{6} \]

\[ \sum_{j=1}^{M^{0}} b_{ij}\cdot x_{ij}\leq \tau_i,\qquad i=\overline{1,N}. \tag{7} \]

Constraint (6) formalizes the first admissibility condition for the desired distribution (each operation is assigned to only one subdivision); constraint (7), the second condition (the amount of time required to perform all operations assigned to the \(i\)-th subdivision must not exceed the time resources available to it). In accordance with the above, the problem of determining the optimal variant of distributing operations among subdivisions consists in determining such a matrix \(X\), satisfying constraints (5), (6), and (7), for which the objective function

\[ Z=\sum_{i=1}^{N}\sum_{j=1}^{M^{0}} c_{ij}\cdot x_{ij}, \tag{8} \]

is maximized, that is, for which maximization of the total efficiency of carrying out the set of operations \(M^{0}\) is ensured.

Relations (5), (6), (7), and (8) show that the problem described belongs to the class of distribution problems of discrete programming with Boolean variables. An algorithm for solving such a problem is proposed in [4] and is, in essence, an implementation of the well-known “branch and bound” method. A detailed description is also given there of the procedure for constructing the decision tree, the method for computing the estimate (the upper bound of the objective function (8) on a set of solutions), and the strategy for moving through the vertices of the tree.

It should be noted that the problem under consideration may have no solution, owing to the fact that, with a substantial lag in meeting the schedule, constraint (7) may fail to be satisfied. This circumstance requires the use of other control actions that make it possible to eliminate the lag and complete the entire scope of work by the planned or directive deadline. Below is given a substantive description of an algorithm for controlling the process, establishing the interrelation between the values of the monitored parameters and the decision to use one or another control action, and thereby effecting control over the course of the process of preparing new production.

Algorithm for Controlling the Process

Control of the production-preparation process is a multistage system of actions carried out on the basis of information contained in the calendar plan, information on the course of the process arriving at control moments, and also on the basis of the use of available alternatives for influencing the process.

In the initial state (time instant \(t_0\)), the setting control action for the first operational control period is the plan \((\vec A_{\mathrm{opt}}, \vec t_{\mathrm{opt}})\), which regulates the development of the process over the time \(0 — T_{\mathrm{opt}}\), where \(T_{\mathrm{opt}}\) is the completion time for preparation of new production. At the instant \(t_1\) of the end of the first operational control period, the controlled parameters of the process are measured and compared with the planned values. If discrepancies arise, a decision is made to use a control action \(u_\varphi \in U,\ \varphi=1 \div 5\). On the basis of the information about the process obtained at time \(t_1\) and the decisions made, the setting control action is adjusted, i.e., the plan \((\vec A_{\mathrm{opt}}, \vec t_{\mathrm{opt}})\) is replaced by the plan \((\vec A_{\mathrm{opt}}^{\,1}, \vec t_{\mathrm{opt}}^{\,1})\).

The formation of the optimal calendar plan \((\vec A_{\mathrm{opt}}^{\,1}, \vec t_{\mathrm{opt}}^{\,1})\) for the time \((t_1 — T_{\mathrm{opt}}^{\,1})\) is carried out by means of a modeling algorithm analogous to that used to form the plan \((\vec A_{\mathrm{opt}}, \vec t_{\mathrm{opt}})\). Similar actions are carried out at subsequent time instants \(t_2, t_3,\ldots\). Consequently, in the \(l\)-th operational period \([t_l \div t_{l+1}]\) the process develops under the influence of the setting action \((\vec A_{\mathrm{opt}}^{\,l}, \vec t_{\mathrm{opt}}^{\,l})\), which remains unchanged until the end of the operational period. The basic element of such a multistep system of actions is the decision to use one or another element of the set \(U\) of control actions in the \(l\)-th operational control period.

This decision is determined by the result of monitoring the process parameters. As already noted, the signal establishing the need to use control actions is the result of elementwise comparison of the sets \(M_0^{\,l-1}\) and \(M_1^{\,l-1}\). The presence in \(M_0^{\,l-1}\) of at least one such element that is not contained in \(M_1^{\,l-1}\) means that there is a lag in deadlines. The main control action used to eliminate the lag during one operational control period is the redistribution of work among subdivisions. As already noted, with a significant lag of the plan under conditions of a limited time fund of the subdivisions, a case is possible in which redistribution cannot be carried out. Then the lag can be eliminated over several control periods. Questions related to the use of control actions in this case will be considered below.

If in the \(l\)-th control period it is possible to carry out redistribution, this means that it is potentially possible to eliminate the lag. However, the effectiveness of using the control action \(u_1\) is determined only after the formation of a new calendar plan taking into account the redistribution performed. In this case, the plan is formed as a reoptimization for the set of works that must be performed in the \(l\)-th control period. Consequently, the plan obtained in this way differs from the original only over the interval of the \(l\)-th operational control period.

If the use of the described control actions leads to the elimination of the lag for all works, then the resulting plan \([\vec A_{\mathrm{opt}}^{\,l}, \vec t_{\mathrm{opt}}^{\,l}]\) is accepted for implementation. In the case where, during the \(l\)-th operational control period, it is not possible to eliminate the lag of the plan, the potential possibility of doing this over a time interval exceeding the duration of the period under consideration is checked. In this case the value of the time \(t_i^0\) is calculated, which

is compared with \(T_{\mathrm{opt}}^{\,l-1}\)—the optimal planned completion time for all work, and also with \(T_g\)—the directive deadline for fulfillment of the plan. The result of the comparison is determined by one of the following inequalities:

\[ \begin{aligned} &1)\quad t_i^0 \leq T_{\mathrm{opt}}^{\,l-1},\\ &2)\quad T_{\mathrm{opt}}^{\,l-1} < t_i^0 \leq T_g,\\ &3)\quad t_i^0 > T_g. \end{aligned} \]

If inequality (3) holds, it is necessary to appeal to higher management with a request to provide additional reserves in order to prevent a failure to meet the directive deadlines.

The case in which inequality (1) or (2) holds means that there is a potential possibility of fulfilling the plan within the time \(T_{\mathrm{opt}}^{\,l-1}\) or \(T_g\), respectively, provided that the service times of individual operations, as well as the volumes of work of subdivisions due to external load, are reduced. In this case the following directed iterative procedure for further optimization of the plan is used. At each iterative step the subdivision \(n\) is determined for which the probability of plan failure through its fault has the maximum value. Then, taking into account the random events that have occurred, a calendar plan is formed by means of the process model, with the sole difference that the external load does not take part in constructing the plan and, when the next operation is inserted for servicing by subdivision \(n\), the execution time of this operation is taken to be equal to the optimistic one. For the plan thus obtained, \(T^l\) is computed—the time of its execution, which is compared with \(T_{\mathrm{opt}}^{\,l-1}\), the optimal planned completion time for all work.* If \(T^l \leq T_{\mathrm{opt}}^{\,l-1}\), the computation process is terminated. If \(T^l > T_{\mathrm{opt}}^{\,l-1}\), the transition to the next iteration is carried out. This process continues until either, in the newly formed plan, the service time of each operation assumes the optimistic value, or it becomes evident that each subsequent iteration leads to an insignificant decrease in the value \(T^l\). Then, in the case \(T^l \leq T_g\), the finally formed plan is accepted for implementation, while in the case \(T^l > T_g\) it is necessary to mobilize additional reserves in order to prevent a failure to meet the directive deadlines.

REFERENCES

1. M. K. Babunashvili. On an algorithm for determining the quantization step in network planning and control systems. Moscow, Proceedings of the Moscow City Pedagogical Institute named after V. I. Lenin, 1968.

2. A. I. Gudzenko, L. V. Kochnev, V. Z. Yampolsky. An algorithm for solving the calendar-planning problem. Reports of the Fourth Scientific and Technical Conference on Problems of Production Automation, vol. IV, Tomsk, 1969.

3. A. I. Gudzenko, L. V. Kochnev, L. V. Perfiliev. On the question of the analysis of calendar plans. Article in the present collection.

4. T. D. Vinogradova. The integer distribution problem. Moscow, “Nauka,” Izvestiya of the USSR Academy of Sciences, “Technical Cybernetics,” 1969, No. 4.

* Here and below, in describing the iterative procedure, the case in which inequality (1) holds is considered. The procedure described is also retained for the case in which inequality (2) holds, with the sole difference that the value \(T^l\) is compared not with \(T_{\mathrm{opt}}^{\,l-1}\), but with the quantity \(T_g\).