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Operations Research Models and Methods
Models Section

Variable Definitions

The model is a mathematical description of the steps of the process. Variables are defined for each quantitative factor that affects the production. In the following we generalize the discussion to use the language of serial production systems. Rather than boys we identify the steps of production as stations. We identify a play with the parameter t. In a production process a play might be equivalent to a one day interval and t would measure the number of days. Matches become units of product with raw material amounts, work in process and finished production all measured in this unit. Note, however, that the unit of product changes as it passes through the system, starting as a raw material and ending as a finished good ready for distribution and sale. The following defines the notation for a general production system.


The model relates the variables to one another. The model for the production system is described below.

In equation 1, we see that the production in station 1 is the same as its capacity. Equation 2 determines the WIP available at time t as a function of variables at previous time. Specifically, the WIP available is equal to the WIP available at the previous time plus any new production from station 1 less any withdrawals for production at station 2.

In equation 3, we limit the production for stations 2 through n to the stations capacity or the amount of material available from WIP or new production. Note that this equation has the effect that the WIP will never become negative. Equation 4 is like equation 2, but computes the WIP for all but the last station of production.

Equation 5 computes the production at the last station. We include no WIP equation for the last station, because all units produced leave the system. The production at the last station is the production for the entire system.

For the equations to have meaning for t = 1, values for all variables must be specified for t = 0. There is no upper limit for the value of t, but normally some maximum T is specified. We call this the time horizon.

If the capacities of the stations were known for all times within the time horizon, and we assume initial values of the variables for time 0, the system of equations defined above could be easily solved for any finite time horizon. The values of the WIP variables and production variables are completely determined by the capacities. Since each variable at time t depends only on variables at time t - 1 and selected additional variables at time t, we can solve the equations for all the variables for t =1, then solve for the variables at t = 2, and so on until t = T. For this solution to be easily obtained the variables for any given t must be ordered so the value of one variable depends only on the variables earlier in the order. For this example the order is:

For any given t, the variables are easily determined if considered from left to right.

Random Variables

In the case of the production example, as in most simulation models, some of the variables are not known with certainty. Here, the capacities are not known initially, rather they are random variables drawn from specified distributions. In the case of the boys producing matches, capacities were determined with the throw of a die. Thus the capacity for each boy for each play is a random variable with a discrete-uniform distribution that ranges from 1 to 6.

For the more general case we assume that each capacity is drawn from a specified distribution. In the following we say that the capacity for station i in period t is a realization of a random variable that has a given distribution. The distribution might depend on the station index.

Running the Simulation

To run the simulation, we select some initial condition that defines the variables at time 0. For the match example on the previous page we chose 0 for all the variables.

We the draw the capacities for the first period from uniform distributions (throwing the die).

Now we use the model equations to solve for the variables in period 1.

We simulate the second play by again drawing the capacity values from the uniform distribution and solving the equations for t = 2.

We repeat the process for play 3.

The process continues in this sequential fashion until the time horizon is reached. Again we show the results for the match example for a 10 plays,


The Excel Model

Simulations of the type described in this section are easily implemented in Excel. The Simulation add-in produces multiline simulations as illustrated below. The particular example shows a simulation of the match problem. The boxes show the equations in row 1 of the simulation. The cells holding the RV_sim function are simulating random variables taken from a uniform distribution. These are cells I16, L16 and O16. The RV_sim function is a user-defined function provided by the Random Variables add-in. Cell J16 holds equation 1 of the model, the production for the first boy. Cell K16 holds equation 2, computing the WIP of the first boy. Cell M16 holds equation 3 giving the production for the second boy. Cell N16 holds equation 4 computing the WIP for the second boy. Cell P16 uses equation 5 to compute the production of the third boy. Finally, Q16 is equal to P16 and holds the system output.

It is easy to run the simulation, by simply copying the equations in row 1 to as many rows as desired below row 1. Excel automatically creates formulas in the other rows using relative addressing. Excel provides the fill down command for this purpose. The example shows 25 plays of the game. The Simulation add-in can easily and quickly run simulations of 1000's of plays.

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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved

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