![[UW shield]](UWlogo.gif)
Megan Loppe
3A Operations Research / Teaching Option
ID# 95126468
Bill Pinchak
3B Combinatorics & Optimization
ID# 95206238
Currently on hand at the Record is Kevin Vaters, Single Copy Manager. His job is to plan out exactly the number of papers to deliver to each sales location. He needs to give enough papers so that locations will not sell out (loss of revenue), but will also return the least number of papers possible. Currently, a program is being used that accounts for the sales history of a single location - number of copies given, number returned - and suggests the number of newspapers to give on that day. Mr. Vaters takes these suggestions, and then adjusts the numbers according to other factors not accounted for in the program.
We were approached by the Record to help with initial research into improving the existing program. A new program would eventually account for the added factors considered by Mr. Vaters. Meetings with Mr. Vaters outlined the needed improvements for the new program. Asked of us was to provide information on how computer suggestions would change, given newly introduced factors.
| # PAPERS TO GIVE: |
GROCERIES | CONVENIENCE | 24 HOURS | BOXES |
|---|---|---|---|---|
| SLOW DAY, OFF SEASON |
|
|
|
|
| SLOW DAY, UP SEASON |
|
|
|
|
| SLOW DAY, DOWN SEASON |
|
|
|
|
| REGULAR DAY, OFF SEASON |
|
|
|
|
| REGULAR DAY, UP SEASON |
|
|
|
|
| REGULAR DAY, DOWN SEASON |
|
|
|
|
| GOOD DAY, OFF SEASON |
|
|
|
|
| GOOD DAY, UP SEASON |
|
|
|
|
| GOOD DAY, DOWN SEASON |
|
|
|
|
| VERY GOOD DAY, OFF SEASON |
|
|
|
|
| VERY GOOD DAY, UP SEASON |
|
|
|
|
| VERY GOOD DAY, DOWN SEASON |
|
|
|
|
| INSANE DAY, OFF SEASON |
|
|
|
|
| INSANE DAY, UP SEASON |
|
|
|
|
| INSANE DAY, DOWN SEASON |
|
|
|
|
1. The current software takes into account sales history of each sales location.
2. There are four main classes of newspaper sales locations: grocery stores, convenience stores, 24-hour stores, and newspaper boxes.
3. The main concern for Mr. Vaters is that any new programs will account for two new factors when deciding on the number of papers to give to each location. These concerns are to go on top of all concerns already being met by the current program.
5. There is no upper limit for the number of newspapers that can be given to any location. However, a lower limit of five newspapers is mandatory.
6. Factors such as weather and the day of the week do not cause alarming disturbances in sales or the number of returns.
7. The demographics of location need not be considered for the problem.
The data used is pure hypothetical. Since the current program uses a history of sales for each location, we have assumed that it is available for use in the model. The largest assumption made is that each single location within a class has the same average history and the same last day history. There will be no restocking of newspapers at their selling locations. The history of past sales at each location is a good indication of the number of sales at that location for the present time. The following ranking scales and outcomes are used for the new factors:
| Rank | Meaning | Outcome |
|---|---|---|
| 1 | |
|
| 2 | |
|
| 3 | |
|
| 4 | |
|
| 5 | |
|
| Rank | Meaning | Outcome |
|---|---|---|
| 0 | |
|
| 1 | |
|
| 2 | |
|
Index Sets:
Variables:
Data:
Program:
An attempt was made to receive "real" history and last day data. Unfortunately, this data was not made available in time for this report. Any further inquest into this problem needs to use solid, accurate historical information. This information can be provided by Mr. Vaters, Single Copy Manager. Restocking of stores only occurs sporadically and in special cases, and thus the assumption made that it does not occur still makes this problem fairly accurate. Past sales are a good indicator to future sales and so this assumption is completely valid and very appropriate. The ranking scales used (and their outcomes) were only estimated from discussions with Mr. Vaters. Any further research should require another meeting with Mr. Vaters so he may set up the scale outcomes for his fancy. The greatest assumption made was that all single locations within a class are the same. For an initial, base model and core problem, this assumption is sensible. All locations grouped into classes are done so because their sales history and number of papers given out to each are very similar. However, for a more complete problem and solution, this assumption should be discarded.
Assessment of Model Choice:
The model used to solve the problem includes all information provided by Mr. Vaters and also takes into account all additional factors requested by Mr. Vaters. The only criticism of the model choice would be through the assumption that all single locations within a class are exactly the same. Although this assumption is acceptable for the core problem, a more accurate model of the entire problem should be used. A more appropriate model that ignores this assumption may use new index sets for numbered single locations. For example, g1, g2, could represent grocery store 1 and 2, etc. The same logic could be used for other classes.
The GAMS output in the Appendices provides ranges for changes in the right hand sides of the constraints, plus ranges for changes in the objective coefficient. The output from GAMS is summarized below:
Change in Cost of a Non-Basic Variable:
Since one requirement of the problem was that every location gets at least five newspapers, all decision variables are basic. Thus, there are no values for this section.
Change in Cost of a Basic Variable:
The ranges for objective coefficients change only with regards to the type of location. The story rank and season rank does not affect these ranges. These ranges imply that the average fraction of newspapers returned from each location can range accordingly, and the optimal values recorded will remain the same.
-0.070 <= C grocery <= infinity
-0.100 <= C convenience <= infinity
-0.125 <= C 24hour <= infinity
-0.150 <= C box <= infinity
Changing the Right-Hand-Side Values:
The ranges for the right-hand-side constraints depend on all three indexes: location, story rank, and season rank. These ranges imply that the right-hand-side values of a constraint can vary a particular amount, and keep the optimal solution.
More Appropriate Model:
Based on suggestions given in the Solution Validity section, an extended model was created. The new model takes into account single locations of the four classes. Thus, single location sales history can be used instead of general class history. This is how the existing program used by the Record conducts its assessments. Please note that the new model is still a simplified model, using only a small number of single locations, (the number of single sales locations for the Record is over eight hundred).
Changes in the existing model include an edited index set and parameters. The index set:
locations := grocery, convenience, 24hour, box was changed to:
locations := gro1, gro2, gro3, gro4, gro5, con1, con2, con3, con4, con5, tf1, tf2, tf3, box1,
box2, box3, box4, box5, box6, box7
where gro# represents a specific numbered grocery store with its own sales history. Similarily,
con#, tf#, and box# represent specific convenience stores, 24 hour stores, and boxes,
respectively. To accommodate for the lengthened location set, the l(location) parameter was
changed to include the number of papers given yesterday to new locations i. The c(location)
parameter also uses new average fractions of returns in a day for new locations i.
The GAMS output for the extended programmed model gives the optimal solution of 1292 returned papers, the number of papers to be delivered to achieve this solution, and sensitivity analysis on the constraints, shadow prices, and objective coefficients. Given the enormity of the solution, a summary will not be provided. Please refer to the GAMS link to view the output of this extended problem.
Last Updated November 19, 1997
Questions?
Comments? 