Business Modeling Lecturer: Ing. Martina Hanová, PhD.

- by Nature of the Environment:  Stochastic - means that some elements of the model are random. So called Probabilistic models developing for real-life systems having an element of uncertainty.  Deterministic - model parameters are completely defined and the outcomes are certain. In other words, deterministic models represent completely closed systems and the results of the models assume single values only.

Stochastic Processes Xj(t) j = 1, 2,...n - the realization stochastic process

states E1, E2,.... Em - random phenomena - states Markov property the distribution for the variable depends only on the distribution of the previous state Markov chain – Finite Markov chain

states E1, E2,.... Em - random phenomena - states Markov property the distribution for the variable depends only on the distribution of the previous state Markov chain – Finite Markov chain

 transition matrix of conditional probabilities after k-steps: States: 1. transient 2. recurrent (refundable): - periodic (with regular return) - aperiodic (irregular return) 3. absorbent (non-refundable)

 The probabilities of weather conditions (modeled as either rainy or sunny), given the weather on the previous day, can be represented by a transition matrix:transition matrix  The matrix P represents the weather model in which a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day.

 The weather on day 0 (today) is known to be sunny. This is represented by a vector in which the "sunny" entry is 100%, and the "rainy" entry is 0%: The weather on day 1 (tomorrow) can be predicted by: Thus, there is a 90% chance that tomorrow will also be sunny.

Company placed on the market a new product and explores its success, in terms of sales which can be characterized as follows: - product is considered to be successful if in specified time sells more than 70% of the production - product is deemed to have failed, if in specified time sell less than 70% of production.

 E1 - the product is successful  E2 - the product is unsuccessful Changes to the success of the product examine after months, or step = 1 month. Suppose that it is a finite Markov chain with states E1, E2,... Em.

If the product is successful in the first month, with probability 0.5 and remain successful in the next month. If not, with probability 0.2 will become successful in the next month. Transition matrix: E1 E2

 ergodic Markov chains  absorbing Markov chain Example: At the beginning in the first month, found 75% of the success of the product

vector of the absolute probabilities after 1-month: vector of the absolute probabilities after 2-month: vector of the absolute probabilities after 3-month:

 revenue matrix R  mean values ​​ of the immediate revenue  expected total revenue after k-steps

k v1nv1n 0710,412,7214,71616,61518,48420,345 v2nv2n 0-0,21,042,7124,5146,3548,20610,062 From the results above, we see that the difference between revenues of states E1 and E2 after specified number of steps is close to the value of Could be interpreted as the initial state of a successful product brings in each month at units higher revenue than the initial state of an unsuccessful product. At the same time we can see a constant increase in the values of expected returns, which reached to the level of This feature is related with the limit properties described process.

State i Alternativ hE1E1 E2E2 E1E1 E2E2 E 1 (succesfull) 1no advertising0, advertising0,80,2142 E 2 (unsuccessful) 1no advertising0,20,83 2advertising0,30,72-3 Enterprise started an advertising campaign. Successfulness of successful product in the first month increase to 80%. On the other hand, if the sale was at the beginning unsuccessful, its success is increased just to 30%. The task is to determine the optimal alternatives that lead to the highest expected revenues. Input data are as follows:

Final step: determine the optimal path vector of corresponding alternatives From the results we can see that after 3 th step the process is stabilized, so that optimal decision is uses the second alternative - advertising. For the enterprise is optimal decision to implement an advertising campaign to increase the success of a new product. kk ,620,8428,66435,75642, ,21,966,12411,38617,

A major goal of the decision tree analysis is to determine the best decisions.  a model for a sequential decision problems under uncertainty  describes graphically the decisions to be made, the events that may occur, and the outcomes associated with combinations of decisions and events  probabilities are assigned to the events, and values are determined for each outcome.

Decision trees offer advantages over other methods of analyzing alternatives.  Graphic - represent decision alternatives, possible outcomes, and chance events schematically.  Efficient - quickly express complex alternatives clearly, compare how changing input values affect various decision alternatives.  Revealing - compare competing alternatives—even without complete information—in terms of risk and probable value.  Complementary - use decision trees in conjunction with other project management tools: evaluate project schedules. Expected Value (EV) term combines relative investment costs, anticipated payoffs, and uncertainties into a single numerical value. The EV reveals the overall merits of competing alternatives.

Decision tree Decision tree is a diagram of nodes and connecting branches. Nodes Nodes indicate decision points, chance events, or branch terminals. Branches Branches correspond to each decision alternative or event outcome emerging from a node.

Decision trees have three kinds of nodes and two kinds of branches. 1. Decision node - Root node represents the first set of decision alternatives. Is shown as a square, lead to two or more possible outcomes.  decision branches The set of alternatives must be:  mutually exclusive  collectively exhaustive

2. Chance node - Event node is a point where uncertainty is resolved. Is shown as a circle, lead to two or more possible outcomes.  event branches - representing one of the possible events that may occur at that point. probability. Each event is assigned a subjective probability. The sum of probabilities for the events in a set must equal one.

 decision nodes and branches represent the controllable factors in a decision problem;  chance nodes and branches represent uncontrollable factors in a decision problem.  decision nodes and chance nodes are arranged in order of subjective chronology.

3. Terminal node – Endpoint is the endpoint of a decision tree, shown as a triangle or rhombus.  the payoff value at the endpoint represents the value of the decision-making criteria in three ways: - directly, the exacting value - simple formula - model

 the method known as rollback determines the single best strategy.  The rollback algorithm, sometimes called backward induction or "average out and fold back,"  determining the certain equivalent rollback values for each node.

 At a terminal node, the rollback value equals the terminal value.  At an event node, the rollback value for a risk neutral decision maker is determined using expected value (probability-weighted average); the branch probability is multiplied times the successor rollback value, and the products are summed.  At a decision node, the rollback value is set equal to the highest rollback value on the immediate successor nodes.