PROBLEM: Your company is interested in bidding a project with a large earthmoving component. An estimated 3 million cubic yards of material is to be loaded with a 992K wheel loader and 777 rigid frame haul trucks. You are responsible for selecting the appropriate equipment spread and estimating the unit cost. The 992K loader has a bucket capacity of 14 cy and the 777 haul trucks have a capacity of 75 cy. The internal rental rates for the 992K and 777 are $300/hr and $191/hr, respectively. You have worked through the Cat Performance Manual and estimated the loader cycle time to be 0.7 mins and the following for the truck cycle: - Haul: 11 mins - Dump: 2.7 mins - Return: 8 mins Overall, the earthmoving fleets work an average of 50 mins per hour. You have prepared the attached analysis based on this information. What is the lowest deterministic unit cost for this operation? Your answer is deterministic in nature and Steve, the senior estimator, believes some variability should be considered in the loader capacity and the equipment cycle times. He is very experienced in earthmoving operations and suggests: 1. The loader capacity is influenced by the material type and is most likely 14 cy with little opportunity to be much more, but with opportunity to be as little as about 13.5 cy. 2. The loader cycle time is likely to be anywhere between 0.65 and 0.75 mins. 3. The average haul and return times will be influenced by haul road conditions and will likely be as estimated. He expects the times to be normally distributed with a standard deviation that is 10 percent of the average value. 4. Dump time is most likely 2.7 mins with little opportunity to be much less, but with opportunity to be as high as about 2.85 mins. He has suggested that you revise your analysis to consider this variability, and to gain insight into the distribution of unit costs and production rates. You have given Steves comments much thought and agree that it will be very beneficial to incorporate variability into the analysis. You also see an opportunity to go a step further and consider correlation between some of the variables. In particular, you expect that: 1. There is a strong positive correlation between loader capacity and load time. That is, conditions that lead to large load quantities will also result in longer loader cycle times, and vice versa. 2. There is a positive correlation between haul and return times. Update your analysis in light of this additional information and report to Steve your insights on the distribution of unit costs and production rates.

Monte Carlo Simulation Your answer from previous calculations is deterministic in nature, and Steve, the senior estimator, believes some variability should be considered in the loader capacity and the equipment cycle times. He is very experienced in earthmoving operations and suggests:

The loader capacity is influenced by the material type and is most likely 14 cy with little opportunity to be much more, but with opportunity to be as little as about 13.5 cy. The loader cycle time is likely to be anywhere between 0.65 and 0.75 mins. The average haul and return times will be influenced by haul road conditions and will likely be as estimated. He expects the times to be normally distributed with a standard deviation that is 10 percent of the average value. Dump time is most likely 2.7 mins with little opportunity to be much less, but with opportunity to be as high as about 2.85 mins. He has suggested that you revise your analysis to consider this variability and to gain insight into the distribution of unit costs and production rates.

Next, we will review the probability distributions to identify the best distributions for each of these variables.

Defining Probability Distributions in Earthmoving Operation

You may wonder which probability distribution best captures the variability of, for example, the loader capacity, given the provided information:

"The loader capacity is influenced by the material type and is most likely 14 cy with little opportunity to be much more, but with opportunity to be as little as about 13.5 cy."

Is the Beta Distribution a Good Fit for Loader Capacity?

The Beta distribution is especially suitable for modeling random variables that lie within a defined range, especially when we know the most likely value and the plausible range of values (minimum and maximum). In this case, we have a loader capacity that has both an upper and lower boundary (13.5 cy and a value slightly above 14 cy), and a most likely value (14 cy). The Beta distribution can capture this type of variability quite well, taking into account the likelihood of the variable nearing its bounds and its most likely value.

What are the best distributions for other variables presented in the problem?

Carefully review the given information and define the best-fit distributions for loader cycle time, haul and return times, and dump time.

How to define a Beta Distribution in @Risk

• Define Distribution: In the @RISK ribbon, go to "Define" panel and click on the "Distribution" and dropdown, then "Define..." This will give you a list of probability distributions you can choose from.
• Select BetaGeneral. This will open up a dialog box to help you define the Beta distribution.
• Define the parameters:
  • (1) and (2): These are shape parameters and should be adjusted based on the desired shape of your distribution (e.g., how peaked or flat it is around the most likely value). Here you can define (1 = 5) and (2= 1.75)
  • Minimum and Maximum values: Enter 13.5 for the minimum value and a value slightly above 14 (say, 14.1 or whatever you deem appropriate) for the maximum value.
• Insert the distribution function: Click OK to insert the function into the selected cell. Now, this cell will produce a random value based on the Beta distribution you have defined each time a new Monte Carlo simulation is run.

How to Define Correlations in @Risk

(The following material is adapted from the Palisade Help Resources)Links to an external site.

Defining correlations (mutual relationships or connections) between input distributions in @RISK can be an important step in building a model. Input distributions are correlated when the values sampled for one (or more) distributions rely to some extent on the values returned by another distribution; they move, to some degree, in tandem - either in the same direction (positive correlation) or in opposite directions (negative correlation). When two or more input distributions are related, defining a correlation can help to ensure that a simulation does not return unrealistic results for the combination of distributions.

Adding a correlation to input distributions adds the RiskCorrmatLinks to an external site. property function to the @RISK distribution function. For example, the following input:

=RiskNormal(0.055,0.085)

Is updated as follows when added to a correlation matrix named "MyCorrelation":

=RiskNormal(0.055,0.085,RiskCorrmat(MyCorrelation,1))

Specifying Inputs

When creating a new correlation matrix, the first window that opens is the Specify Inputs window (Figure below). The Specify Inputs window includes two options:

• Specify Inputs - Create a correlation matrix for the currently selected cells, as listed in the text box. Click the Select Cells button to the right of the text box to change which cells will be included in the correlation.
• Define New Correlation Matrix and Specify Inputs Later - Create an empty correlation matrix with two elements. Each element can be configured individually, and additional elements can be added. See Edit CorrelationsLinks to an external site. for more information.

Once inputs have been specified, or if the option 'Define New Correlation Matrix and Specify Inputs Later' is chosen, the Define Correlation window will open. See Define Correlation WindowLinks to an external site. for more information.

Please read through the PROBLEM, follow the instructions, and your work in an Excel file with three sheets: (1) deterministic calculations, (2) Probabilistic Monte Carlo Simulation, (3) Probabilistic Monte Carlo Simulation with Correlations.

PLEASE SHOW ALL WORK, ALL EQUATIONS AND RESULTS.

Duantity Moved: Loader Rate: \begin{tabular}{|c|c|c|c|} \hline 5 & Pass Loa & & CYITRI \\ \hline Loader Caycl & le Time & 0.7 & MN \\ \hline & & 3.5 & \\ \hline Truck Cycle & & & \\ \hline Load & & 3.5 & MN \\ \hline Haul & & 11 & MN \\ \hline Dump & & 2.7 & MN \\ \hline Return & & 8 & MN \\ \hline Total Truck [ & Eycle & 25.2 & MN \\ \hline Optimum Tru & Jek C & 7.2 & \\ \hline Worked MINI & & 50 & MINHA \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|} \hline 6 & Pass Loading & & CYITRK \\ \hline Loader & Caycle Time & 0.7 & MN \\ \hline Load Tir & & 4.2 & MN \\ \hline Truck & Cucle & & \\ \hline Load & & 4.2 & MN \\ \hline Haul & & 11 & MN \\ \hline Dump & & 2.7 & MN \\ \hline Return & & 8 & MN \\ \hline Total T & Truck Cycle & 25.9 & \\ \hline Optimu & Jm Truck Count & 6.2 & \\ \hline Worke & d MINUH & 50 & MINUHA \\ \hline \end{tabular} The Earthmoving roject can achieve its lowest unit cost at \$1.66 per cubic yard with a fleet of 6 trucks. However, the maximum productivity of 1,000 cubic yards per hour can be attained with 8 trucks. The selection of the appropriate number of trucks hinges on the specific requirements of the project. Discussion: The analytical findings indicate that the most economical approach is realized with a fleet of 6 trucks. Conversely, the highest productivity levels are attained with a fleet of 8 trucks. The determination of the optimal number of trucks to deploy will be contingent upon the unique demands of the project. If cost minimization is paramount, a fleet of 6 trucks is advised. Conversely, for projects prioritizing maximum productivity, an 8-truck fleet is recommended. B.1 \begin{tabular}{|c|c|c|c|} \hline 7 & Trucks-Trucks Co & atrol & \\ \hline Dp Prod & & 972.2 & CYIHA \\ \hline Duration & & 3,085.71 & HRS \\ \hline Loader & & 3.085 .71 & HRS \\ \hline Trucks & & 21,600.00 & HRS \\ \hline Cost & & & \\ \hline Loader & & 925,714 & \\ \hline Trucks & & $4,125,600 & \\ \hline Total Cost & & $5,051,314 & \\ \hline Unit Cost & & 1.68 & NCY \\ \hline \end{tabular} B.1 Opting for this scenario entails A.1 This option offers a cost-effective solution with the second-highest production rate, enabling timely completion while optimizing expenses. Among the four scenarios, this stands out as the preferred choice when adhering to the established schedule. A. 2 \begin{tabular}{|c|c|c|c|} \hline 8 & Trucks-Loader Co & trol & \\ \hline Dp Prod & & 1000.0 & CYIHA \\ \hline Duration & & 3,000.00 & HRS \\ \hline Loader & & 3,000.00 & HRS \\ \hline Trucks & & 24,000.00 & HRS \\ \hline \multicolumn{4}{|l|}{ Cost } \\ \hline Loader & & $900,000 & \\ \hline Trucks & & $4,584,000 & \\ \hline Total Cost & & $5,484,000 & \\ \hline Unit Cost & & 1.83 & 1CY \\ \hline \end{tabular} B. 2 A.2 In cases where the production schedule is behind and there is a need to expedite operations, despite its higher cost, this scenario facilitates the highest quantity output compared to the alternatives. Therefore, while not as cost-effective, it is a time-sensitive option. prioritizing cost-effectiveness, with a longer timeframe for completion. Thi choice is particularly advantageous when time constraints are not a primary concern. B.2 Among the four available options, this is the least favorable choice. This is attributed to its lower production rate and higher production costs, resulting in a compromised outcome in both cost and productivity when