Simulation for Optimization in Logistics

According to T.H. Taylor, “Simulation is a numerical technique for conducting experiments on digital computers, which involves certain types of mathematical and logical relationships necessary to describe the behaviour and structure of a complex real-world system over an extended period of time.” When decisions are to be taken under conditions of uncertainty, simulation can be used. Simulation provides trial and error movement towards optimal solution. Simulation as a quantita­tive method requires the setting up of a mathematical model that would represent the interrelation­ships between the variables involved in the actual situation in which a decision is to be taken. Then, a number of trials or experiments are conducted with the model to determine the results that can be expected, when the variables assume various values. Simulation can therefore be defined as a procedure whereby one can draw conclusions about the behaviour of a given system, by examining the behaviour of a corresponding model whose cause and effect relationships are similar to those in the actual system.

Simulation can serve as a “pre-test” to try out new decision rules for a system. It can anticipate problems and bottlenecks that may arise while operating a system. There are a few basic concepts that must be understood before applying the simulating techniqe.

System: It is the segment to be studied or understood to draw conclusions. In the product-market system, the market for the products together with the firm’s production process constitutes the relevant system. The variable can be identified only after defining the system. The variables that interact with one another in the system and establish their relationships mathematically are given below:

Decision Variables: Decision variables are those variables whose value is to be determined through the process of simulation.

Environmental Variables: These are the variables that describe the environment. In marketing, the environmental variables are the competitors’ average price, consumer preferences and demand and so forth.

Endogenous Variables: These variables are generated within the system itself. In the marketing context, the quantity sold, sales revenue, total cost and profit are endogenous variables.

Criterion Function: Any variable can be used as the criterion function for evaluating the perfor­mance of the system. In marketing, profit is used as the criterion function.

For example, let us assume that the competitors’ average price is P_c and the price charged by the firm is P. If the quantity sold is Q, then, as the quantity sold depends upon the firm’s price P and the competitors’ average price P_c, we can then express this relationship mathematically as:

If we assume the total cost of the quantity sold is C, then C = g(Q).

The total cost is a function of the quantity sold. If 1t is the profit earned by the firm, then

Profit = 1t = PQ – C = Pf(P, P_c) – g(Q)

The above equation is a mathematical model of the system. It contains (in equation form) the interrelationships among the endogenous, decision and environmental variables. This math­ematical model is also the criterion function. Mathematical modelling requires the setting up of mathematical relationships that would represent the system. Although some relationships can be expressed as equations, other relationships or constraints on the criterion function may be expressed only as inequalities (as we have seen in linear programming). If the mathematical model set up could always be optimized by the analytical approach, then there would be no need for simu­lation. It is only when the interrelationships are too complex, or there is uncertainty regarding the values that could be assumed by the variables, or both, that we have to resort to simulation.

The Monte Carlo method is a technique that involves using random numbers and probability to solve problems. S. Ulam and Nicholas Metropolis coined the term “Monte Carlo” in reference to the games of chance that are a popular attraction in Monte Carlo, Monaco.

Computer simulation has to do with using computer models to imitate real life or make pre­dictions. When you create a model with a spreadsheet like in Excel, you have a certain number of input parameters and a few equations that use those inputs to give you a set of outputs (or response variables). This type of model is usually deterministic, which means that you get the same results no matter how many times you recalculate.

The Monte Carlo simulation is a method for alliteratively evaluating a deterministic model using sets of random numbers as inputs. This method is often used when the model is complex, nonlinear, or involves more than just a couple of uncertain parameters. A simulation can typically involve over 10,000 evaluations of the model, a task that in the past was only practical using super computers.

By using random inputs, you are essentially turning the deterministic model into a stochastic model.

The Monte Carlo method is just one of many methods for analyzing uncertainty propagation, where the goal is to determine how random variation, lack of knowledge or error affects the sensitivity, performance, or reliability of the system that is being modelled. The Monte Carlo simulation is catego­rized as a sampling method, because the inputs are randomly generated from probability distributions to simulate the process of sampling from an actual population. So, we try to choose a distribution for the inputs that most closely matches data we already have, or best represents our current state of knowledge. The data generated from the simulation can be represented as probability distributions (or histograms) or converted to error bars, reliability predictions, tolerance zones and confidence intervals.

All we need to do is follow the five simple steps listed below:

Step 1: Create a parametric model, y = f(x₁, x₂, . . . , x_q).

Step 2: Generate a set of random inputs, x_t1, x ₂, . . . , x _q.

Step 3: Evaluate the model and store the results as y..

Step 4: Repeat steps 2 and 3 for i = 1 to n.

Step 5: Analyze the results using histograms, summary statistics, confidence intervals, and so on.

A model usually refers to a particular level in the managerial decision-making context and the decisions of the higher levels (if any) involving multiple variables. The process of modelling follows the managerial hierarchy of strategic, tactical and operational concerns. For example, an operation­al model will have certain constraints imposed by a tactical-level decision. The objective associated with the model is likely to be derived again from a higher level decision. Full-scale mathematical modelling is probably most useful for tactical decision making, with a medium time horizon under consideration, and where a certain level of aggregation of data is possible. For higher level decision making, by its very nature, data is uncertain, intangibles are many and hard modelling is unlikely to be very useful in decision making. Other options, such as soft systems modelling, some models of discrete decision analysis and cognitive maps are possibilities for strategic decision making.

For operational decisions, simple automated rule-based systems are likely to be effective, and the models used will need to be simple to validate. Fast computational procedures will be the major consideration. Information technology for decision support and automation of data processing is needed in such situations, rather than complicated models.

Models are useful to describe the interrelationships between different quantities of interest and sometimes to derive certain optimal or good policies. The models based on mathematical pro­gramming, inventory theory, routing and scheduling theory can directly address a sharp decision area, or can be part of the quantitative assessment for more aggregate decisions.

Source: Sople V.V (2013), Logistics Management, Pearson Education India; Third edition.