A generator for test instances of scheduling problems concerning cranes in transshipment terminals

Abstract

We present a test data generator that can be used for simulating processes of cranes handling containers. The concepts originate from container storage areas at seaports, but the generator can also be used for other applications, particularly for train terminals. A key aspect is that one or multiple cranes handle containers, that is, they store containers, receiving the containers in a designated handover area; retrieve containers, handing the containers over in the handover area; or reshuffle containers. We present a generic model and outline what is captured by the test data itself and what is left to be estimated by the user. Furthermore, we detail how data are generated to capture the considerable variety of container characteristics, which can be found in major terminals. Finally, we present examples to illustrate the variety of research projects supported by our test data generator.

Appendix

1.1 Distributions

The test data generator offers the following distribution functions (version 1.08).

• Uniform distribution function:

  Input: $a,b \in \mathbb {Z}, a \le b$

  Let [a, b] be the interval that limits the output. We generate a random value $x'$ using uniform distribution ${\mathcal {U}}[a-1,b]$. To achieve integer numbers, we round this value $x = \lfloor x' \rfloor + 1$. Output: $x \in \{a,a+1,...,b\}$

• Normal distribution function:

  Input: $\mu ,\sigma ^2 \in \mathbb {R}^+$

  We generate a random value $x'$ using normal distribution $\mathcal {N} (\mu ,\sigma ^2)$. To achieve integer numbers, we round this value $x = \lfloor x' + 0,5 \rfloor$. Output: $x \in \mathbb {Z}$

• Truncated normal distribution function:

  Input: $\mu ,\sigma ^2 \in \mathbb {R}^+$ and $a,b \in \mathbb {Z}, a \le b$

  Let [a, b] be the interval that limits the output. We generate a random value $x'$ using normal distribution $\mathcal {N} (\mu ,\sigma ^2)$. To achieve integer numbers, we round this value $x = \lfloor x' + 0,5 \rfloor$. We reject x and restart the function if $x<a$ or $x>b$ applies. Output: $x \in \{a,a+1,...,b\}$

• Arbitrary distribution function:

  Input: $p_1,p_2,...,p_C \in \mathbb {R}^+$ and $v_1,v_2,...,v_C$

  Let C be the number of classes, where each class $c=1,...,C$ has a probability $p_{c}$ and a value $v_{c}$. The value can be a number or an alphanumeric value. First, we calculate normalized probabilities $p_{c}^n = p_{c} / \sum _{i=1}^{C}p_i$$\forall c=1,...,C$. In the next step, the value $v_c$ is assigned to x with the probability of occurrence $P(X=v_{c})=p_{c}^n$$\forall c=1,...,C$. Output: $x \in \{ v_1, v_2,...,v_C\}$

We use Mersenne Twister (MT 19937) to generate pseudo-random 32-bit integer numbers. These numbers are converted into uniformly distributed numbers in the interval [0, 1). Afterward, we modify the numbers to obtain the required probability distributions (e.g., ${\mathcal {U}}[a,b]$ or $\mathcal {N} (\mu ,\sigma ^2)$). Obviously, this procedure is simple for uniform distribution. However, to obtain normal distribution we use the Box–Muller transformation (Box and Muller 1958). For the arbitrary distribution, interval [0, 1) is divided into C intervals. Each interval $c=1,\ldots ,C$ represents a class where the width of the interval is equal to the probability of occurrence $p^n_c$. To determine output value $v_c$, we identify the interval in which the random number is located.