1. MATH143 Lec 10 Quantitative Methods
Charles W Jackson
Andrew Nunekpeku
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

2. Agenda for today Solve A3 and Quiz 2 Issues with identifying D.V.
Transposition – Make Story Easy to tell
Mixture Problems – Volume Fractions
Transportation Problems – Matrix D.V.
Sensitivity Analysis
Integer Programs
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

3. A3: Oxbridge Computer Scheduling
Beryl needs to cover 8am-10pm each day at minimum cost while guaranteeing 8 or 7 hrs/wk to workers, depending on grade.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

4. Oxbridge Computer Scheduling
Beryl’s “program” assigns each worker to various shifts to cover the need, so there are 30 numbers required to spec the program, not five or six.
Layout D.V. and supplied availability constraints in same directions to ease explanations and solver setup.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

5. Oxbridge Computer Scheduling
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

6. Oxbridge Computer Scheduling
Tell everyone their hours to work each day of the week
Minimize total payroll bill
Cover 14 hrs each day
Without exceeding available hours per worker
Or working negative hours per shift
While keeping everyone sharp by working a minimum number of hours, based on rank
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

7. Test 2: Nunekpeku § - Transportation
A company ships frozen orange juice concentrate from processing plants A and B to distributors C, D, and E.
Each plant can produce 20 tons of concentrate each week. The company has just received orders of 10 tons from C for the coming week, 15 tons from D, and 10 tons from E. The cost per ton for supplying each of the distributors from each of the processing plants is shown in the following table.
C D E
A $260 $220 $290
B $230 $240 $310
The company wants to determine the least costly plan for filling their orders for the coming week.
(a) Formulate a Linear Programming model for this problem. [6 marks]
(b) Implement the model in a spreadsheet and solve it. [12 marks]
(c) What is the optimal solution? [2 marks]
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

8. Test 2: Nunekpeku § - Transportation
Algebraic statement:
Minimize 260 AC AD AE BC BD BE
Subject to:
AC+AD+AE ≤ 20
BC+BD+BE ≤ 20
AC+BC ≥ 10
AD+BD ≥ 15
AE+BE ≥ 10
AC, AD, AE, BC, BD, BE ≥ 0
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

9. Test 2: Nunekpeku § - Transportation
Spreadsheet statement and solver parameters:
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

10. Quiz 2 Jackson § : “Trouble”
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

11. Quiz 2 Jackson § 4-way Mixture
The Slick Spread Paint Co. has a custom order for 5,000 gallons of red paint from a customer planning to paint the town. The paint must meet the following specifications.
For fire safety during application, the combustion point must be no lower than 225 degrees Fahrenheit. For proper application, the specific weight may be no more than 1.0. The customer is particularly concerned that the paint not fade, but retain its vivid hue. Thus, he has restricted the proportions of the common, but fugitive, pigments Red-1 and Red-3. The Red-1 content may be no more than 10% by volume and the Red-3 content no more than 1% by volume.
The paints approaching these specifications that are available to Slick Spread are riot red, alarm crimson, racy rouge and cherry. The table below gives their costs and relevant characteristics. When the paints are mixed, these characteristics are assumed to combine linearly, that is you compute the mixture’s value of a characteristic by multiplying the volume fraction of each component by its characteristic and summing the products.
If the required paint can be mixed using these stock paints, determine their proportions to produce the minimum material cost.
MATH143 – Fall 2011 –Tuesday Week 4
5/29/2018

12. Quiz 2 Jackson § 4-way Mixture
Riot Red Alarm Crimson Racy Rouge Cherry
Combustion point 200◦ ◦ ◦ ◦
Specific Weight
Red-1 Content % % % %
Red-3 Content % % % %
Cost per gallon $ $ $ $0.75
Commentary
“Mixture” problems. Blend a combination of ingredients to obtain characteristics not found in any of them, but some sort of weighted average value. Often found in oil refineries, continuous processing plants. Decision variables are the recipe ratios, either weight or volume fractions for each of the components.
There is always a hidden, or implicit, constraint in these problems, where the sum of the proportions of all the ingredients must be 100%.
MATH143 – Fall 2011 –Tuesday Week 4
5/29/2018

13. Quiz 2 Jackson § Solution
MATH143 – Fall 2011 –Tuesday Week 4
5/29/2018

14. Sensitivity Reports - Transportation
Red-circled entries
indicate we do not
have a unique solution,
more can be found
by moving about the rectangle ABDE
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

15. Finding Alternate Solution - Transportation
Push on one of the corners in direction allowed
While not letting the optimum cost shift from what we already had
Fix objective, then shove traffic onto other lanes, making checkerboard adds and subtracts to find other solution.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

16. Found Alternate Solution - Transportation
Notice we moved 5 tons into AD and BE from AE and BD without changing total cost. This is because = =530.
MATH143 – Fall 2011 – Tuesday Week 3
5/29/2018

17. Sensitivity Reports - Transportation
Shadow price column shows impact to Objective at the margin for one more of each resource.
The -20 tells us that we save 20 by increasing output from A by one ton, and shifting off the B paths, which are more costly.
The 230/240/310 for deliveries to C, D, and E show that at the margin, we send from B, which has slack.
Shadow price of zero on the output from B means we wouldn’t use any more B if we got some, it’s already got slack.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

18. Sensitivity Reports - Transportation
Reduced Cost column shows impact to Objective at the margin for one more of each activity.
This reduced cost is zero because of the slack available from B.
Reduced Cost on the lane A to C means we would pay 50 more if we pulled one more ton that way, with adjustments to the cells in its rectangle.
These reduced costs are zero because of the alternate solutions.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

19. Sensitivity Reports - Transportation
Reduced Cost column shows impact to Objective at the margin for one more of each activity.
Reduced Cost on the lane A to C means we would pay 50 more if we pulled one more ton that way, with adjustments to the cells in its rectangle.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

20. Sensitivity Reports - Transportation
Reduced Cost column shows impact to Objective at the margin for one more of each activity.
Constraining AC to 1 and re-running, the total cost rises from $8600 to $8650, a change of $50, the reduced cost on lane AC. This is the net of paying =500 and avoiding paying =450.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

21. Araba’s Bakeshop
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

22. Araba’s Bakeshop – Sensitivity Report
We won’t make any custards until they sell for $16.65, $6.65 more than currently.
We would make $1.24 more from each egg up to 23.2 more. Likewise, $0.21 per minute cook time up to 443 minutes more, and $0.084 per cup of flour up to cups more.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

23. Araba’s Bakeshop We don’t make any custard.
We are left with some sugar.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

24. Valu-Comm Layout after Solving
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

25. Valu-Comm Solver Specification
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

26. Valu-Comm Solution Report
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

27. Valu-Comm – Sensitivity Report
Use Reduced Costs to consider entry/exit of activities
Watch % variation of objectives still at same corner to identify “tipping point” sensitivity.
Use Shadow Prices to consider “bite” of various constraints
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018

28. Integer Programs
Situations where the decision variables are naturally unitized.
We had seen this in Araba’s Bakeshop, where you do not waste fractional hours on partial recipes, just as you would not suggest a family plan on having 3.2 children!
Solutions could be guessed by rounding, or exhaustively searched for via integer constraints, although cannot do sensitivity with those.
Special case – binary variables as in assignment problems and the like where the decision is yes/no or go/no go.
MATH143 – Fall 2011 – Tuesday Week 4
5/29/2018