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LP Formulation Set 2

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2 Ardavan Asef-Vaziri June-2013LP-Formulation Agricultural planning : narrative Three farming communities are developing a joint agricultural production plan for the coming year. Production capacity of each community is limited by their land and water. CommunityLand (Acres)Water (Acres Feet) 1400600 2600800 3300375 The crops suited for this region include sugar beets, cotton, and sorghum. These are the three being considered for the next year. Information regarding the maximum desired production of each product, water consumption, and net profit are given below

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3 Ardavan Asef-Vaziri June-2013LP-Formulation Agricultural planning : narrative Crop Max desired Water consumptionNet return (Acres) (Acre feet / Acre) ($/Acre) 1 600 3 1000 2 500 2 750 3 325 1 250 Because of the limited available water, it has been agreed that every community will plant the same proportion of its available irritable land. For example, if community 1 plants 200 of its available 400 acres, then communities 2 and 3 should plant 300 out of 600, and 150 out of 300 acres respectively. However, any combination of crops may be grown at any community. Goal : find the optimal combination of crops in each community, in order to maximize total return of all communities

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4 Ardavan Asef-Vaziri June-2013LP-Formulation Agricultural planning : decision variables x 11 = Acres allocated to Crop 1 in Community 1 x 21 = Acres allocated to Crop 2 in Community 1 x 31 = Acres allocated to Crop 3 in Community 1 x 12 = Acres allocated to Crop 1 in Community 2 x 22 = Acres allocated to Crop 2 in Community 2 x 32 = Acres allocated to Crop 3 in Community 2 …………….. x ij = Acres allocated to Crop i in Community j i for crop j for community, we could have switched them Note that x is volume not portion, we could have had it as portion

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5 Ardavan Asef-Vaziri June-2013LP-Formulation Agricultural planning : Formulation Land x 11 +x 21 +x 31 400 x 12 +x 22 +x 32 600 x 13 +x 23 +x 33 300 Water 3x 11 +2x 21 +1x 31 600 3x 12 +2x 22 +1x 32 800 3x 13 +2x 23 +1x 33 375

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6 Ardavan Asef-Vaziri June-2013LP-Formulation Agricultural planning : Formulation Crops x 11 + x 12 + x 13 600 x 21 +x 22 +x 23 500 x 31 +x 32 +x 33 320 Proportionality of land use x 11 +x 21 +x 31 x 12 +x 22 +x 32 400 600 x 11 +x 21 +x 31 x 13 +x 23 +x 33 400 300

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7 Ardavan Asef-Vaziri June-2013LP-Formulation Agricultural planning : Formulation Crops x 11 + x 12 + x 13 600 x 21 +x 22 +x 23 500 x 31 +x 32 +x 33 320 Proportionality of land use x 11 +x 21 +x 31 x 12 +x 22 +x 32 400 600 x 11 +x 21 +x 31 x 13 +x 23 +x 33 400 300

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8 Ardavan Asef-Vaziri June-2013LP-Formulation Agricultural planning : all variables on LHS Proportionality of land use 600(x 11 +x 21 +x 31 ) - 400(x 12 +x 22 +x 32 ) = 0 300(x 11 +x 21 +x 31 ) - 400(x 13 +x 23 +x 33 ) = 0 600x 11 + 600 x 21 + 600 x 31 - 400x 12 - 400 x 22 - 400 x 32 = 0 300x 11 + 300 x 21 + 300 x 31 - 400x 13 - 400 x 23 - 400 x 33 = 0 x 11, x 21,x 31, x 12, x 22, x 32, x 13, x 23, x 33 0

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9 Ardavan Asef-Vaziri June-2013LP-Formulation SAVE-IT Company : Narrative A reclamation center collects 4 types of solid waste material, treat them, then amalgamate them to produce 3 grades of product. Techno-economical specifications are given below Grade Specifications ProcessingSales price cost / pound/ pound M1 : 30% of total A M2 : 40% of total 38.5 M3 : 50% of total M4 : exactly 20% M1 : 50% of total B M2 : 10% of total2.57 M4 : exactly 10% C M1 : 70% of total25.5

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10 Ardavan Asef-Vaziri June-2013LP-Formulation SAVE-IT Company : Narrative Availability and cost of the solid waste materials M1, M2, M3, and M4 per week are given below MaterialPounds available / weekTreatment cost / pound M130003 M220006 M340004 M410005 Due to environmental considerations, a budget of $30000 / week should be used to treat these material. Furthermore, for each material, at least half of the pounds per week available should be collected and treated.

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11 Ardavan Asef-Vaziri June-2013LP-Formulation SAVE-IT Company : Mixture Specification A1: weight of solid waste 1 in grade A A1, A2, A1, A4, B1, B2, B3, B4, C1, C2, C3, C4 Mixture Specifications: Grade A: A1 0.3 (A1+A2+A3+A4) A2 0.4 (A1+A2+A3+A4) A3 0.5 (A1+A2+A3+A4) A3 = 0.2 (A1+A2+A3+A4) Grade B: B1 0.5(B1+B2+B3+B4) B2 0.1(B1+B2+B3+B4) B4 = 0.1(B1+B2+B3+B4) Grade C: C1 0.3 (C1+C2+C3+C4)

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12 Ardavan Asef-Vaziri June-2013LP-Formulation SAVE-IT Company : Material Availability and ussage Availability of material A1+B1+C1 3000 A2+B2+C2 2000 A3+B3+C3 4000 A4+B4+C4 1000 At least half of the material treated A1+B1+C1 1500 A2+B2+C2 1000 A3+B3+C3 2000 A4+B4+C4 500

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13 Ardavan Asef-Vaziri June-2013LP-Formulation SAVE-IT Company : Treatment and Processing Costs, and Profit Spend all the treatment budget 3(A1+B1+C1)+6(A2+B2+C2)+4(A3+B3+C3)+5(A4+B4+C4) = 30000 Maximize profit Z (8.5-3)(A1+A2+A3+A4)+(7-2.5) (B1+B2+B3+B4)+(5.5-2) (C1+C2+C3+C4) – 3(A1+B1+C1)-6(A2+B2+C2)-4(A3+B3+C3)-5(A4+B4+C4)) A1, A2, A1, A4, B1, B2, B3, B4, C1, C2, C3, C4 0

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14 Ardavan Asef-Vaziri June-2013LP-Formulation Problem (From Hillier and Hillier) Strawberry shake production Several ingredients can be used in this product. Ingredient calories from fat Total calories Vitamin Thickener Cost ( per tbsp) (per tbsp) (mg/tbsp) (mg/tbsp) ( c/tbsp) Strawberry flavoring 1 50 20 3 10 Cream 75 100 0 8 8 Vitamin supplement 0 0 50 1 25 Artificial sweetener 0 120 0 2 15 Thickening agent 30 80 2 25 6 This beverage has the following requirements Total calories between 380 and 420. No more than 20% of total calories from fat. At least 50 mg vitamin. At least 2 tbsp of strawberry flavoring for each 1 tbsp of artificial sweetener. Exactly 15 mg thickeners. Formulate the problem to minimize costs.

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15 Ardavan Asef-Vaziri June-2013LP-Formulation Decision variables Decision Variables X 1 : tbsp of strawberry X 2 : tbsp of cream X 3 : tbsp of vitamin X 4 : tbsp of Artificial sweetener X 5 : tbsp of thickening

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16 Ardavan Asef-Vaziri June-2013LP-Formulation Constraints Objective Function Min Z = 10X 1 + 8X 2 + 25 X 3 + 15 X 4 + 6 X 5 Calories 50X1 + 100 X2 + 120 X4 + 80 X5 380 50X1 + 100 X2 + 120 X4 + 80 X5 420 Calories from fat X1 + 75 X2 + 30 X5 0.2(50X1 + 100 X2 + 120 X4 + 80 X5) Vitamin 20X1 + 50 X3 + 2 X5 50 Strawberry and sweetener X 1 2 X 4 Thickeners 3X 1 + 8X 2 + X 3 + 2 X 4 + 2.5 X 5 = 15 Non-negativity X 1, X 2, X 3, X 4, X 5 0

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17 Ardavan Asef-Vaziri June-2013LP-Formulation Capital budgeting : Narrative representation We are an investor, and there are 3 investment projects offered to the public. We may invest in any portion of one or more projects. Investment requirements of each project in each year ( in millions of dollars) is given below. The Net Present Value (NPV) of total cash flow is also given. YearProject 1Project 2Project 3 0408090 1608060 2908020 3107060 NPV457050

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18 Ardavan Asef-Vaziri June-2013LP-Formulation Capital budgeting : Narrative representation If we invest in 5% of project 1, then we need to invest 2, 3, 4.5, and 0.5 million dollars in years 0, 1, 2, 3 respectively. The NPV of our investment would be also equal to 5% of the NPV of this project, i.e. 2.25 million dollars. YearProject 15% of Project 1 0402 1603 2904.5 310.5 NPV452.25

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19 Ardavan Asef-Vaziri June-2013LP-Formulation Capital budgeting : Narrative representation Based on our budget forecasts, Our total available money to invest in year 0 is 25M. Our total available money to invest in years 0 and 1 is 45M Our total available money to invest in years 0, 1, 2 is 65M Our total available money to invest in years 0, 1, 2, 3 is 80M To clarify, in year 0 we can not invest more than 25M. In year 1 we can invest 45M minus what we have invested in year 0. The same is true for years 2 and 3. The objective is to maximize the NPV of our investments

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20 Ardavan Asef-Vaziri June-2013LP-Formulation proportion x 1 = proportion of project 1 invested by us. proportion x 2 = proportion of project 2 invested by us. proportion x 3 = proportion of project 3 invested by us. Maximize NPV Z = 45x 1 + 70 x 2 + 50 x 3 subject to Year 0 : 40 x 1 + 80 x 2 + 90 x 3 25 Year 1 : Investment in year 0 + Investment in year 1 45 Capital budgeting : Formulation

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21 Ardavan Asef-Vaziri June-2013LP-Formulation Investment in year 0 = 40 x 1 + 80 x 2 + 90 x 3 Investment in year 1 = 60 x 1 + 80 x 2 + 60 x 3 Year 1 : 60 x 1 + 80 x 2 + 60 x 3 + 40 x 1 + 80 x 2 + 90 x 3 45 Year 1 : 100 x 1 + 160 x 2 + 150 x 3 45 Year 2 : 90x 1 + 80x 2 + 20 x 3 + 100x 1 + 160 x 2 + 150 x 3 65 Year 2 : 190x 1 + 240x 2 + 170 x 3 65 Year 3 : 10x 1 + 70x 2 + 60 x 3 + 190x 1 + 240x 2 + 170 x 3 80 Year 3 : 200x 1 + 310x 2 + 230 x 3 80 x 1, x 2, x 3 0. Capital budgeting : Formulation

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22 Ardavan Asef-Vaziri June-2013LP-Formulation An airline reservations office is open to take reservations by telephone 24 hours per day, Monday through Friday. The number of reservation officers needed for each time period is: The union requires all employees to work 8 consecutive hours. Therefore, we have shifts of 12am-8am, 4am-12pm, 8am-4pm, 12pm-8pm, 4pm-12am, 8pm-4am. Hire the minimum number of reservation agents needed to cover all requirements. Personnel scheduling problem : Narrative representation PeriodRequirement 12am-4am11 4am-8am15 8am-12pm31 12pm-4pm17 4pm-8pm25 8pm-12am19

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23 Ardavan Asef-Vaziri June-2013LP-Formulation The union contract requires all employees to work 8 consecutive hours. We have shifts of 12am-8am, 4am-12pm, 8am-4pm, 12pm-8pm, 4pm-12am, 8pm- 4am. Hire the minimum number of reservation agents needed to cover all requirements. If there were not restrictions of 8 hrs sifts, then we could hire as required, for example 11 workers for 4 hors and 15 workers for 4 hours. Personnel scheduling problem : Narrative representation

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24 Ardavan Asef-Vaziri June-2013LP-Formulation Personnel scheduling problem : Pictorial representation 12 am to 4 am 4 am to 8 am 8 am to 12 pm 12 pm to 4 pm 4 pm to 8 pm 8 pm to 12 am PeriodShift 123456 11 15 31 17 25 19

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25 Ardavan Asef-Vaziri June-2013LP-Formulation x 1 = Number of officers in 12 am to 8 am shift x 2 = Number of officers in 4 am to 12 pm shift x 3 = Number of officers in 8 am to 4 pm shift x 4 = Number of officers in 12 pm to 8 pm shift x 5 = Number of officers in 4 pm to 12 am shift x 6 = Number of officers in 8 pm to 4 am shift Personnel scheduling problem : Decision variables

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26 Ardavan Asef-Vaziri June-2013LP-Formulation Min Z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 12 am - 4 am : x 1 +x 6 11 4 am - 8 am : x 1 +x 2 15 8 am - 12 pm : +x 2 + x 3 31 12 pm - 4 pm : +x 3 + x 4 17 4 pm - 8 pm : +x 4 + x 5 25 8 pm - 12 am : +x 5 + x 6 19 x 1, x 2, x 3, x 4, x 5, x 6 0. Personnel problem : constraints and objective function

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27 Ardavan Asef-Vaziri June-2013LP-Formulation Personnel scheduling problem : excel solution

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28 Ardavan Asef-Vaziri June-2013LP-Formulation Aggregate Production Planning : Narrative PM Computer Services assembles its own brand of computers. Production capacity in regular time is 160 computer / week Production capacity in over time is 50 computer / week Assembly and inspection cost / computer is $190 in regular time and $260 in over time. Customer orders are as follows Week123456 Orders105170230180150250 It costs $10 / computer / week to produce a computer in one week and hold it in inventory for another week. The Goal is to satisfy customer orders at minimum cost.

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29 Ardavan Asef-Vaziri June-2013LP-Formulation Refresh We need to lease warehouse space. The estimated required space ( in 1000 sq ft) is given below. Month 1 2 3 4 5 Space required 3020401050 If the leasing cost was fixed the best strategy was to lease as needed. But this is not the case Leasing period (months) 12345 Cost per sq-feet leased 65 100 135 160 190 Now it may be more economical to lease for more than one month and take advantage of the lower rates for longer periods. Find the optimal leasing strategy to minimize leasing costs.

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30 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables X ij spaced leased in month i and kept until month j months. i = 1, 2, 3, 4, 5. j = i, i+1, …, 5 Min z = 65 X 11 +100 X 12 +135 X 13 +160 X 14 +190 X 15 + 65 X 22 +100 X 23 +135 X 24 +160 X 25 + 65 X 33 +100 X 34 +135 X 35 + 65 X 44 +100 X 45 + 65 X 55

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31 Ardavan Asef-Vaziri June-2013LP-Formulation Constraints X 11 + X 12 + X 13 + X 14 + X 15 30,000 X 12 + X 13 + X 14 + X 15 + X 22 + X 23 + X 24 + X 25 20,000 X 13 + X 14 + X 15 + X 23 + X 24 + X 25 + X 33 + X 34 + X 35 40,000 X 14 + X 15 + X 24 + X 25 + X 34 + X 35 + X 44 + X 45 10,000 X 15 + X 25 + X 35 + X 45 + X 55 50,000 X 11, X 12, X 13, X 14, X 15, X 22, X 23, X 24, X 25, X 33, X 34, X 35 X 44, X 45, X 55 0

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32 Ardavan Asef-Vaziri June-2013LP-Formulation excel; Format 1

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33 Ardavan Asef-Vaziri June-2013LP-Formulation excel; Format 2

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34 Ardavan Asef-Vaziri June-2013LP-Formulation excel; Best Format (Ctrl)

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35 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : narrative This is a good example to show that the statement of a problem could be complicated. But as soon as we define the correct decision variables, things become very clear Two sources of pollution: Open furnace and Blast furnace Three types of pollutants : Particulate matter, Sulfur oxides, and hydrocarbons. ( Pollutant1, Pollutant2, Pollutant3). Required reduction in these 3 pollutants are 60, 150, 125 million pounds per year. ( These are RHS) Three pollution reduction techniques : taller smokestacks, Filters, Better fuels. ( these are indeed our activities). We may implement a portion of full capacity of each technique. If we implement full capacity of each technique on each source, their impact on reduction of each type of pollutant is as follows

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36 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : narrative Pollutant Taller Filter Better fuel smokestacks B.F.O.F B.F.O.F. B.F.O.F. Particulate 12 9 2520 1713 Sulfur 35 42 1831 5649 Hydrocarb. 37 53 2824 2920 The cost of implementing full capacity of each pollutant reduction technique on each source of pollution is as follows Pollutant Taller Filter Better fuel smokestacks B.F.O.F B.F.O.F. B.F.O.F. Cost 12 9 25 20 17 13

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37 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : Decision Variables How many techniques?? How many sources of pollution?? How many constraints do we have in this problem??? How many variables do we have Technique i source j

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38 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : Decision Variables x 11 = Proportion of technique 1 implemented of source 1 x 12 = Proportion of technique 1 implemented of source 2 x 21 = Proportion of technique 2 implemented of source 1. x 22 = Proportion of technique 2 implemented of source 2 x 31 = Proportion of technique 3 implemented of source 1 x 32 = Proportion of technique 3 implemented of source 2.

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39 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : Formulation Min Z= 12x 11 +9x 12 + 25x 21 +20x 22 + 17x 31 +13x 32 Particulate; 12x 11 +9x 12 + 25x 21 +20x 22 + 17x 31 +13x 32 60 Sulfur; 35x 11 +42x 12 + 18x 21 +31x 22 + 56x 31 +49x 32 150 Hydrocarbon; 37x 11 +53x 12 + 28x 21 +24x 22 + 29x 31 +20x 32 125 x11, x12, x21, x22, x31, x32 ???? Pollutant Taller Filter Better fuel smokestacks B.F.O.F B.F.O.F. B.F.O.F. Particulate 12 9 2520 1713 Sulfur 35 42 1831 5649 Hydrocarb. 37 53 2824 2920

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