Statistics and Operational Research for Logistics
Tagged: Business & Management
Coursework
Question 1
The statland mining company digs for gold. The amount purified and ready for transport at the end of each day has the following probability distribution:
| Gold (kg) | 5 | 6 | 7 | 8 |
| Percentage of days | 15 | 24 | 47 | 14 |
It is proposed to load the purified gold onto two armoured motorcycles for transfer to a ship in the nearby harbour. For security reasons, each motorcycle can only carry up to 7kg of gold.
Although the mine itself is operational for seven days a week, the ships leave the harbour on Monday, Wednesday and Friday evenings only. Gold bars are stockpiled until the next ship is due to leave and any which cannot be transported by motorcycle are put into storage for the next sailing.
Calculate the expected gold production per day and hence the comment on the feasibility of the shipping arrangements.
Answer:
The expected gold production per day (Department of Statistics Online Programs, 2016) = where i=1 to 4
Xi denotes the random variable and P(Xi) denotes the probability of the random variable Xi.
Table 1: Probability distribution for gold production
| Gold (X) | % of days | P(X) | X.P(X) |
| 5 | 15 | 0.15 | 0.75 |
| 6 | 24 | 0.24 | 1.44 |
| 7 | 47 | 0.47 | 3.29 |
| 8 | 14 | 0.14 | 1.12 |
Therefore, expected gold production per day =
=X1P(X1) + X2P(X2) + X3P(X3) + X4P(X4)
= 0.75 + 1.44 + 3.29 + 1.12
= 6.6
Hence, the expected gold production per day is 6.6 kg and the present arrangement is feasible as the maximum capacity of motorcycles to carry is up to 7 kg of gold and the given data shows that the distribution is concentrated between 6 and 7.
Question 1(b)
Simulate a seven day period of the mine’s operation and transport arrangements starting on a Saturday. Show clearly how you use random numbers, the amounts carried by the motorcycles and the amount of gold in the queue.
Answer:
Table 2: Cumulative probability distribution for gold production
| Gold (X) | % of days | Cumulative
probability |
Random no. interval |
| 5 | 15 | 15 | 0-15 |
| 6 | 24 | 39 | 16-39 |
| 7 | 47 | 86 | 40-86 |
| 8 | 14 | 100 | 87-100 |
As per the given the data, the cumulative probability is calculated and the random number interval is generated based on the cumulative probability. With reference to the random number table, the daily demand is predicted which is presented in the table no.4 (Knill, 2009).
Table 3: Random Number Generator Table
| 93 | 22 | 53 | 64 | 39 | 7 | 10 | 63 | 76 | 35 |
| 78 | 76 | 58 | 54 | 74 | 92 | 38 | 70 | 96 | 92 |
| 61 | 81 | 31 | 96 | 82 | 0 | 57 | 25 | 60 | 56 |
| 42 | 88 | 7 | 10 | 5 | 24 | 98 | 65 | 8 | 21 |
| 77 | 94 | 30 | 5 | 33 | 28 | 10 | 99 | 0 | 27 |
From the random number table, select first seven numbers; i.e., the numbers are 93, 22, 53, 64, 39, 7 and 10.
Table 4: Daily demand for the gold based on the simulation data
| Day | Random No | Daily production |
| Saturday | 93 | 5 |
| Sunday | 22 | 8 |
| Monday | 53 | 9 |
| Tuesday | 64 | 5 |
| Wednesday | 39 | 6 |
| Thursday | 7 | 4 |
| Friday | 10 | 4 |
| Total demand for the week | 41 | |
The random number 93 falls under the random number interval 87-100 (refer table 2), which involves to the gold production of 7kg. The same manner, 22 falls under the random number interval 16-39 which pertain to the gold production of 5kg. Also the random numbers 53 and 64 fall under the random number interval 40-86 which involve to the gold production of 6kg. This information is presented in the table 4. From the table, it is noticed that the total gold production for the week is 37kg.
Question 1(c)
Explain how you would use your simulation to estimate the average amount of gold which would be left behind each Friday evening after the motorcycles have gone.
Answer:
Estimated average amount of gold which would be left behind each Friday evening after the motor cycles have gone = (Refer table no.4)
= 5.857 kg
Question 2
Demand for gold bars over the last four weeks in a bank was as follows:
| Gold bars demanded | Week | |||
| Day | 1 | 2 | 3 | 4 |
| Monday | 46 | 40 | 38 | 42 |
| Tuesday | 50 | 51 | 47 | 46 |
| Wednesday | 52 | 49 | 45 | 55 |
| Thursday | 42 | 43 | 44 | 50 |
| Friday | 70 | 68 | 79 | 67 |
| Saturday | 49 | 52 | 47 | 43 |
| Sunday | 35 | 33 | 38 | 42 |
Question 2(a)
Compute appropriate moving average to eliminate any seasonality in the data
Answer:
The 7 order moving average is the most suitable method for the given data. Hence the forecast is done and interpreted in the table no.5 (Grandell, 2016):
Table 5: Moving average method
| Demand | 7 day moving average | ||
| Week 1 | Monday | 46 | |
| Tuesday | 50 | ||
| Wednesday | 52 | ||
| Thursday | 42 | 49.14 | |
| Friday | 70 | 48.29 | |
| Saturday | 49 | 48.43 | |
| Sunday | 35 | 48.00 | |
| Week 2 | Monday | 40 | 48.14 |
| Tuesday | 51 | 47.86 | |
| Wednesday | 49 | 48.29 | |
| Thursday | 43 | 48.00 | |
| Friday | 68 | 47.71 | |
| Saturday | 52 | 47.14 | |
| Sunday | 33 | 46.57 | |
| Week 3 | Monday | 38 | 46.71 |
| Tuesday | 47 | 48.29 | |
| Wednesday | 45 | 47.57 | |
| Thursday | 44 | 48.29 | |
| Friday | 79 | 48.86 | |
| Saturday | 47 | 48.71 | |
| Sunday | 38 | 50.14 | |
| Week 4 | Monday | 42 | 51.00 |
| Tuesday | 46 | 49.29 | |
| Wednesday | 55 | 48.71 | |
| Thursday | 50 | 49.29 | |
| Friday | 67 | ||
| Saturday | 43 | ||
| Sunday | 42 |
Question 2(b)
Use simple exponential smoothing on the moving averages to estimate the value of the average daily demand
Answer:
The formula for simple exponential smoothing on the moving averages = αY1 + (1-α)M1 (Falk, 2012) where α is the smoothing constant, Y1 is the actual gold bar demand at current period and M1 is the forecasted value of the previous period.
By assuming a = 0.1, the exponential smoothing technique is applied and the forecast is done for the entire 4 weeks:
Table 6: Simple exponential smoothing on the 7day moving averages
| Demand | 7 day moving
Average (Y) |
Simple exponential smoothing
on the moving averages (M) |
||
| Week 1 | Monday | 46 | ||
| Tuesday | 50 | |||
| Wednesday | 52 | |||
| Thursday | 42 | 49.14 | 49.14 | |
| Friday | 70 | 48.29 | 49.06 | |
| Saturday | 49 | 48.43 | 48.99 | |
| Sunday | 35 | 48.00 | 48.89 | |
| Week 2 | Monday | 40 | 48.14 | 48.82 |
| Tuesday | 51 | 47.86 | 48.72 | |
| Wednesday | 49 | 48.29 | 48.68 | |
| Thursday | 43 | 48.00 | 48.61 | |
| Friday | 68 | 47.71 | 48.52 | |
| Saturday | 52 | 47.14 | 48.38 | |
| Sunday | 33 | 46.57 | 48.20 | |
| Week 3 | Monday | 38 | 46.71 | 48.05 |
| Tuesday | 47 | 48.29 | 48.08 | |
| Wednesday | 45 | 47.57 | 48.03 | |
| Thursday | 44 | 48.29 | 48.05 | |
| Friday | 79 | 48.86 | 48.13 | |
| Saturday | 47 | 48.71 | 48.19 | |
| Sunday | 38 | 50.14 | 48.39 | |
| Week 4 | Monday | 42 | 51.00 | 48.65 |
| Tuesday | 46 | 49.29 | 48.71 | |
| Wednesday | 55 | 48.71 | 48.71 | |
| Thursday | 50 | 49.29 | 48.77 | |
| Friday | 67 | |||
| Saturday | 43 | |||
| Sunday | 42 |
On the basis of the table 6, the sum of forecasted values (M) for four weeks is 1067.79kg gold. Hence, the average daily gold demand is 48.54kg (1067.79/22).
Figure 1: Forecasting the gold demand using simple exponential smoothing model
Calculate percentage deviations to describe the daily seasonality.
Table 7: Percentage deviations for observed and forecasted gold demand
The percentage deviations can be gauged through the following formula:
Percentage deviations (Student Watershed Research Project, n.d.) = [(Observed – Forecasted) / (Forecasted)] * 100
| Demand | Simple exponential smoothing model | Percentage deviation | ||
| Week 1 | Monday | 46 | ||
| Tuesday | 50 | |||
| Wednesday | 52 | |||
| Thursday | 42 | 49.14 | -14.53 | |
| Friday | 70 | 49.06 | 42.69 | |
| Saturday | 49 | 48.99 | 0.01 | |
| Sunday | 35 | 48.89 | -28.42 | |
| Week 2 | Monday | 40 | 48.82 | -18.07 |
| Tuesday | 51 | 48.72 | 4.67 | |
| Wednesday | 49 | 48.68 | 0.66 | |
| Thursday | 43 | 48.61 | -11.54 | |
| Friday | 68 | 48.52 | 40.14 | |
| Saturday | 52 | 48.38 | 7.47 | |
| Sunday | 33 | 48.20 | -31.54 | |
| Week 3 | Monday | 38 | 48.05 | -20.92 |
| Tuesday | 47 | 48.08 | -2.24 | |
| Wednesday | 45 | 48.03 | -6.30 | |
| Thursday | 44 | 48.05 | -8.43 | |
| Friday | 79 | 48.13 | 64.13 | |
| Saturday | 47 | 48.19 | -2.47 | |
| Sunday | 38 | 48.39 | -21.47 | |
| Week 4 | Monday | 42 | 48.65 | -13.66 |
| Tuesday | 46 | 48.71 | -5.57 | |
| Wednesday | 55 | 48.71 | 12.91 | |
| Thursday | 50 | 48.77 | 2.52 | |
| Friday | 67 | |||
| Saturday | 43 | |||
| Sunday | 42 |
Utilize your findings to project Wednesday demand. Comment briefly on the time series’ nature assumptions that you used in your study as well as how trustworthy you think your forecasting approach.
Answer:
The forecasted gold demand for a Wednesday of the second week is 48.68, the third week is 48.03 and fourth week is 48.71 (refer table no.6).
The assumptions of the exponential smoothing model (Montgomery, Jennings and Kulahci, 2008):
The data is normally distributed.
The variance is homogenous.
No outliers in the data.
Question 3
Using the responses you came up with for the previous question, determine the average weekly demand for the gold bars. Any quantity of gold bars can be ordered for £20, with a lead time of one and a half weeks and a weekly maintenance cost of 3p for each bar.
Question 3(a)
Determine the overall stock handling cost per bar and suggest to the manager a least expensive re-order level stock control approach, assuming that demand is consistent every week. To demonstrate how your plan will operate, draw a graph of the stock level over time.
Answer:
Assuming that demand is constant every week, the re-order level when demand is constant will be calculated as:
Estimated weekly demand = 45.67*7 = 319.69
= 320 Kg Gold bars
The Economic Order Quantity (EOQ) minimizes the total stock handling cost hence
Demand is constant so no need for safety stock
EOQ =
Where
p=Cost of placing order for any number of gold bars = £ 20
d= Average demand = 320
s=Cost of Storage = 3p
EOQ =
EOQ = 653.20 = 653 gold bars
Re order level = 0.5 (Half a week) * 320
= 160 Kg gold bars
Our advice is to order 653 Kg gold bars whenever stock level falls to 160 Kg of gold bars.
Question 3(b)
Assume now that the demand for the bars is random in time. The manager believes that the lost sales penalty for a bar is 50p. Advise on a re-order stock control strategy that gives an 80% level of service.
Answer:
Demand is random so we need to calculate safety stock.
Safety stock = Z *
Where,
Z = 80% Service level.
= 0.84 *
= 0.84 * 12.65
= 10.63
= 11 Kg gold bars
Re order level = Average demand during lead time + Safety Stock
= 160 + 11
= 171 Kg gold
This means ordering 653 gold bars whenever the stock level falls to 171 Kg gold bars should give us 80% level of service.
Question 3(c)
The senior administrator in head office has decreed that orders for the bars can be placed only on Monday for delivery on the Thursday of the same week. Advise on a re-order cycle strategy to give an 80% level of service.
Answer:
Re order level for 4 days (Monday to Thursday) is = 48.50362 * 3.5 + 0.9978 * 0.84 = 160.68
Re order level for 4 days (Monday to Thursday) is calculated as 160.68.
Question 3(d)
Comment on the various stock control strategies you have devised in view of the time series analysis you have performed on the demand pattern for the gold bars.
Answer:
The reorder level is calculated based on the assumption for (a) as demand is constant and for (b) & (c) as random.
The forecasting for the given data is done using time series based on the assumptions of normality, homogeneity.
However, the inventory model also undergoes the assumptions that the demand is deterministic, lead time is known, replenishment is instantaneous, unit of carrying the inventory and ordering are known and each item is independent.
Overall clarity and presentation
- The clear picture of the formulae
- Application of the formulae
- Structure of the solution
- Neat presentation
- Clarity of thought should be visible in the solution part
- Perfection in the calculation part of the solution
Through the assignment, it is concluded that the expected gold production in Statland mining company is 5.6k.g. The total gold production for a week is the 37kg and 5.28kg amount of gold which would be left behind each Friday evening after the motor cycles have gone. However, the estimated gold demand for a day is 48.54kg. The forecasted gold demand for a Wednesday of the second week is 48.68, the third week is 48.03 and the fourth week is 48.71. As per the stock control strategy, re order level for Monday to Thursday is 160.68.
References
- Department of Statistics Online Programs (2016) 5.3 – Expected Value of a Discrete Random Variable2016
- Falk, M. (2012) A First Course on Time Series Analysis: Examples with SAS Falk, M., Marohn, F., Michel, R., Hofmann, D., Macke, M., Spachmann, C., Englert, S. and Dinges, P. (eds.)
- Grandell, J. (2016) Time Series Analysis2016 1–121.
- Knill, O. (2009) Probability Theory and Stochastic Processes with Applications NewDelhi, Overseas Press India Private Limited.
- Montgomery, D.C., Jennings, C.L. and Kulahci, M. (2008) Introduction to Time Series Analysis and Forecasting New Jersey, John Wiley & Sons. Inc.
- Student Watershed Research Project (no date) Percent Deviation Halleyhosting.
