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

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

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

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

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

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

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

Test 2: Nunekpeku § - Transportation Algebraic statement: Minimize 260 AC + 220 AD + 290 AE + 230 BC + 240 BD + 310 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

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

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

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

Quiz 2 Jackson § 4-way Mixture Riot Red Alarm Crimson Racy Rouge Cherry Combustion point 200◦ 225◦ 350◦ 200◦ Specific Weight 0.92 1.08 1.10 0.92 Red-1 Content 9.8% 9.6% 9.6% 10.4% Red-3 Content 0.9% 0.8% 1.2% 0.9% Cost per gallon $0.90 $0.80 $0.60 $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

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

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

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

Found Alternate Solution - Transportation Notice we moved 5 tons into AD and BE from AE and BD without changing total cost. This is because 220+310=240+290=530. MATH143 – Fall 2011 – Tuesday Week 3 5/29/2018

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

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

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

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 260+240=500 and avoiding paying 220+230=450. MATH143 – Fall 2011 – Tuesday Week 4 5/29/2018

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

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 20.37 cups more. MATH143 – Fall 2011 – Tuesday Week 4 5/29/2018

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

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

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

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

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

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