The modern methods of scenario analysis and forecasting, represented by tree-like graphs, are investigated in this paper. The basic theoretical positions of influence diagrams are considered. Its mathematical apparatus is based on Bayesian networks and supported a combination of graphical representation of analyzed process and its probabilistic behavior. The influence diagrams are an effective tool for modeling uncertain events in a complex decision-making process, which reflect decision-making situations in a more compact form. Influence diagrams also reflect all possible factors of the problem and their relationship in a clear and understandable way; provide all relevant information and implementation of the most appropriate decision-making procedure; allow to perform realtime analysis of possible scenarios. The numerical calculations that illustrate the possibilities of practical application of influence diagrams in complex decisionmaking problems under uncertainty are provided. The results obtained are aimed at improving the quality and effectiveness of decision-making in scenario analysis process with a large number of interrelated factors.
The initial situation in decision-making tasks is often characterized by complexity and
uncertainty, and various combinations of uncertain events can influence the outcomes
of decisions made. These events are known as scenarios [
The probability trees and decision trees are some of the most widely known methods
for modeling uncertain situations, which are successfully used for a small number of
uncertain events (n≤4-5). With the increasing number of uncertain events, probability
trees are characterized by a large dimension, which makes it extremely difficult to
display the information that they contain (a set of alternative solutions, systems of
uncertain events and outcomes of alternative solutions) [
In this regard, a new “tool” called the “influence diagram” (ID) appeared to model
uncertain events [
The purpose of the paper is to present the basic theoretical principles of the influence diagrams and propose an example of their practical application in scenarios analysis tasks with a large number of interrelated factors. 2
Let us consider the basics of IDs that reveal the essence and features of them in
accordance with the works [
The following types of nodes are distinguished on any ID. The decision node displays a variety of alternative solutions, which are graphically depicted in the form of rectangles or squares on the ID graph. An event node displays a set (complete system) of uncertain events that directly or indirectly determine the outcome of alternative decisions. The event nodes are depicted in the form of circles or ovals in ID graph. The chance node displays a function that evaluates the outcomes of alternative solutions. The chance nodes shown as a diamond in ID graph.
For these types of nodes, the following notation is accepted: 1. D used for decision nodes; 2. C used for chance nodes; 3. V used for value nodes.
An influence diagram, which is presented in the form of a graph and does not contain any other information, is called partially defined. To perform a decision analysis, each node in the graph must be provided with all relevant information. Lists of alternative solutions are associated with each decision node. A list of events and unconditional or conditional probabilities of their implementation are associated with each chance node. A value node is associated with a list of outcomes of all alternative solutions, the probability of their implementation and the value of the assessed function. ID with all information related to the analysis is called completely defined.
Let us consider some general approaches to ID construction [
Next, the situation is analyzed to determine whether there are groups of undefined events that affect events in nodes C1 and C2.
Suppose the existence of two such systems of uncertain events C3 and C4 was
established (Fig. 3 [
The analysis continues until all the vague (uncertain) factors that can influence the outcome of the alternative actions are established. The process of analyzing the task and constructing the influence diagram ends when all factors are taken into account. After building the ID, the necessary calculations are performed. The calculation of probabilities of the outcomes of alternative decisions is the most difficult task in ID construction. The outcomes of alternative decisions are determined by combinations of uncertain events in event nodes that are direct predecessors of the value node.
Let us consider the ID construction algorithm, which is based on a sequential
calculation of the total probability of events in nodes over the entire set of events in nodes
that are direct predecessors of these nodes. Fig. 4 [
Since decision nodes are not related to outcomes probabilities, they are not shown in Fig. 4.
The following standard types of connection between chance nodes are distinguished in
the ID:
1. linear connection (nodes C1, C2);
2. divergent connection (node C4 and nodes C3 and C6);
3. convergent connection (nodes C2, C4 and node C3; nodes C4, C5 and node C6).
Let us consider the procedure for calculation of total probability for various types of
connection between nodes [
1. Linear connection (Fig. 5 [
n p(Cqj ) p(Cpi ) p(Cqj | Cpi ) , j 1, m .
i1 (1) (2) If necessary, the calculated values of p(Cqj) are normalized so that the sum of the probability values is equal to 1.
The calculated values p(Crl) are taken as the a priori probability of the events in node Cr and, if necessary, are used in further calculations.
2. Divergent connection (Fig. 6 [
m p(Crl ) p(Cqj ) p(Crl | Cqj ) .
j1 node Cq and the conditional probability p(Crl|Cpi), i 1, n ; l 1, k of events in the node Cr are given.
Since the events in nodes Cq and Cr are conditionally independent of each other, until a certain event occurs in node Cp (here we are talking about a priori event probabilities), the total event probability in nodes Cq and Cr is calculated as follows: m p(Crl ) p(C pi ) p(Crl | C pi ) , l 1, k ; j1 m p(Cqj ) p(C pi ) p(Cqj | C pi ) , j 1, m .
j1
(3)
(4)
In the case when the node Cp has more than two proper descendants, the total
probability calculation is carried out in exactly the same way.
3. Convergent connection (Fig. 7 [
The total event probability in nodes Cp is calculated as follows:
m k p(C pi ) p(Cqj ) p(Crl ) p(C pi | Cqj , Crl ) , i 1, n .
j1 l1 (5)
Such calculations can be performed for the entire ID, starting with the initial nodes and ending with nodes that are proper ancestors of the chance node. The sequence of procedures for the total probability calculation for the nodes of the diagram is determined by the structure of the relations (connections) between the nodes of the events.
So, for the ID fragment showed in Fig. 4, the first procedure is a calculation of the total probability of events in node C2 for the entire set of events in node C1. There is a linear type of connection between nodes. Next, the total probability of events in node C3 can be calculated over the entire set of events in nodes C2 and C4, since there is a converging type of connection. The next procedure is a calculation of the total probability of events in node C6 for the entire set of combinations of events in nodes C4 and C5. The calculated total probability of events in node C6 are used as a priori probability for calculating the total probability of events in node C7. The sequence of procedures is shown by arrows and numbers in Fig. 4. At the same time, the calculation procedures order is generally arbitrary. So, the procedure 3 could be performed first, then procedure 4, and only after this the procedures 1 and 2. The following calculation procedures order is also acceptable: 1–3–4–2 or 1–3–2–4. Thus, the execution order for these calculation procedures is determined only by the structure of the influence diagram and the convenience and simplicity of calculations 3
Application of Influence Diagrams for Decision Support Let us consider an example of a scenario analysis of the ship construction costs coordination in the pre-contract stage using an influence diagram.
Suppose it is planning to purchase a combined tanker with the following
characteristics: the deadweight (DWT) is 40 000 tons; the cost of one DWT ton is 1250 cu; the
vessel contract price is 50.0 million cu. The initial information for the analysis is the
cost structure, presented in Table 1, based on which the wholesale price was 50.4
million cu. In order for the contract price to be 50.0 million cu, it is necessary to establish,
with VAT (20%), the required value of the wholesale price equal to 50.0 / 1.2 = 41.7
million cu. To reduce the wholesale price by 50.4 – 41.7 = 8.7 million cu, it is necessary
to reduce the cost of a number of articles costing.
It can be done by reducing the cost of two groups of indicators: articles costing that
depend on market conditions (materials, ship equipment, counterparty service) and
articles costing that can be managed by a shipbuilding company (factory-wide and factory
workshop-wide overhead rates, equipment maintenance and operating costs). With all
that said, let us construct an influence diagram for the task of ship construction cost
reduction (Fig. 8 [
ID nodes carry the following information:
Decision node D to enter into a contract for the purchase of a ship; not enter into a contract for the purchase of a ship.
Chance nodes are:
С1 characterizes the probability of cost reduction of indicators that depends on market conditions.
С2 characterizes the probability of reducing costs which are managed by a shipbuilding company.
С3, С4, С5 characterize the probability of cost reduction of materials, ship equipment and counterparty service, respectively.
С6, С7, С8 characterize the probability of reducing costs associated with factory-wide and factory workshop-wide overhead rates, equipment maintenance and operating costs.
Node V is represented by the following evaluation expression:
V = V1+V2 = p(C1) (S1+S2+S3)+p(C2) (S6+S7+S11), where p is the probability of the implementation of relevant events C.
Let us make the necessary calculations on the influence diagram and set next expert estimates of unconditional and conditional probabilities of all events: p(C3) = 0.2; p(C4) = 0.1; p(C5) = 0.3; p(C6) = 0.3; p(C7) = 0.25; p(C8) = 0.4; p(C1|C3,C4,C5) = 0.3; p(C2|C6,C7,C8) = 0.4.
Since events represent a converging type of connection, based on (2), the total probabilities of events C1 and C2 been calculated.
So, we have: p(C1) = p(C3) p(C4) p(C5) p(C1| C3, C4, C5) = 0.002; p(C2) = p(C6) p(C7) p(C8) p(C2| C6, C7, C8) = 0.012.
Next, taking into account the numerical values presented in Τable 1 (S1 = 10.0; S2 = 14.4; S3 = 1.5; S6 = 3.0; S7 = 1.9; S11 = 6.6), the value node V value has been calculated: V = (10.0 + 14.4 + 1.5) 0.002 + (3.0 + 1.9 + 6.6) 0.012 = 0.1898 million cu.
Thus, the given probability of cost reduction for the planned costing items make it possible to expect a price reduction of only 0.188 million cu. Therefore, the decision to purchase of a ship is not made. 4
The ID satisfies a number of practical requirements: in a clear and understandable form it reflects all the many factors of the problem and their relationship; provides all relevant information; allows the implementation of the most appropriate decision-making procedure. However, the most effective may be the complex application of ID and decision tree. Any decision-making situation at the initial stage can be modeled by an influence diagram. For a better understanding of the situation, certain ID fragments can be transformed into a decision tree, which gives a more detailed presentation of the problem under consideration.
Acknowledgment. This research was partially supported by the state research project “Development of information and communication decision support technologies for strategic decision-making with multiple criteria and uncertainty for military-civilian use” (research project no. 0117U007144, financed by the Government of Ukraine).