# OR-Notes

## J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

#### Forecasting tutorial question

The owner of a local computer store rents printers to some of her preferred customers. She is interested in arriving at a forecast of rentals so that she can order the correct quantities of ink cartridges that go with the printers. Data for the last 10 weeks is shown below:

```Week   Rentals      Week   Rentals
1      25           6      22
2      28           7      26
3      30           8      24
4      26           9      20
5      27           10     23```

Generate a forecast for week 11 using a five-week moving average. Generate a forecast for week 11 using single exponential smoothing with a smoothing constant of 0.8. Which of the two forecasts for week 11 do you prefer and why?

What is the best forecast you can get using single exponential smoothing?

The manager of a public library must schedule employees to reshelve books and periodicals checked out of the library. The number of items checked out will determine the labour requirements. The following data reflects the number of items checked out of the library for the past three years:

```Month        Year 1        Year 2       Year 3
January      1847          2045         1986
February     2669          2321         2564
March        2467          2419         2635
April        2432          2088         2150
May          2464          2667         2201
June         2378          2122         2663
July         2217          2206         2055
August       2445          1869         1678
September    1894          2441         1845
October      1922          2291         2065
November     2431          2364         2147
December     2274          2189         2451```

The manager needs a time series forecasting method for forecasting the number of items to be checked out during the next month (January of year 4). Find the best exponential (single or double/with or without trend) smoothing forecast you can.

By best here we mean the forecast corresponding to the minimum value of MSD.

Compare this forecast with that obtained using Holt-Winters additive model and a seasonal cycle of 12 (months).

The Central Building Supply Company serves the building industry in the Midwest region. Many items are stocked, and close inventory control is necessary to assure customers of efficient service. Recently, business has been increasing and management is concerned about stockouts. A forecasting method that will estimate demand next month is required so that adequate replenishment quantities can be purchased. An example of the sales growth experienced over the last 50 months is the demand for item J785, a common door hinge. This growth in demand is shown below:

```Month        J785         Month         J785         Month         J785
sales                      sales                      sales
1            80            2            132          3             143
4            180           5            200          6             168
7            212           8            254          9             397
10           385           11           472          12            397
13           476           14           699          15            545
16           837           17           743          18            722
19           735           20           838          21            1057
22           930           23           1085         24            1090
25           1218          26           1296         27            1199
28           1267          29           1300         30            1370
31           1489          32           1499         33            1669
34           1716          35           1603         36            1812
37           1817          38           1798         39            1873
40           1923          41           2028         42            2049
43           2084          44           2083         45            2121
46           2072          47           2262         48            2371
49           2309          50           2422```
• Plot the data - does this plot tell you anything useful? Hint the data can be plotted by choosing any forecasting method and then plotting the results after solution.
• Investigate different forecasting methods for forecasting demand for J785.
• Which of these methods do you consider to be the "best" method and why.
• Hence forecast demand for month 51.

Again by best here we mean the forecast/method that corresponds to the minimum value of MSD.