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MT 2351 Chapter 6 Integer Linear Programming

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MT 2352 Integer Linear Programming All-Integer Linear Program AAll variables must be integers Mixed-Integer Linear Program SSome, but not all variables must be integers 0-1 Integer Linear Program IInteger variables must be 0 or 1, also known as binary variables

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MT 2353 Integer Programming – All Integers Northern Airlines is a small regional airline. Management is now considering expanding the company by buying additional aircraft. One of the main decisions is whether to buy large or small aircraft to use in the expansion. The table below gives data on the large and small aircraft that may be purchased. As noted in the table, management does not want to buy more than 2 small aircraft, while the number of large aircraft to be purchased is not limited. How many aircraft of each type should be purchased in order to maximize annual profit? SmallLargeCapital Available Annual profit$1 million$5 million Purchase cost$5 million$50 million$100 million Maximum purchase quantity2No maximum

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MT 2354 Define Variables - Northern Airlines Let: S = # of Small Aircraft L = # of Large Aircraft

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MT 2355 General Form - Northern Airlines Max 1S + 5L s.t. 5S + 50L <= 100 S <= 2 S, L >= 0 & Integer

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MT 2356 Northern Airlines – Graph Solution Small AC LP Relaxation (2, 1.8) Budget

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MT 2357 Northern Airlines – Graph Solution Small AC Rounded Solution (2, 1) Budget

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MT 2358 Northern Airlines – Graph Solution Budget Small AC Optimal Solution (0, 2)

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MT Integer Linear Programming All-Integer Linear Program AAll variables must be integers Mixed-Integer Linear Program SSome, but not all variables must be integers 0-1 Integer Linear Program IInteger variables must be 0 or 1, also known as binary variables

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MT Integer Programming – Mixed Integer Hart Manufacturing, a mixed integer production problem: Hart Manufacturing makes three products. Each product goes through three manufacturing departments, A, B, and C. The required production data are given in the table below. (All data are for a monthly production schedule.) Production DepartmentProduct 1Product 2Product 3Hours available A (hours/unit) B (hours/unit) C (hours/unit) Profit Contributions per Unit$25$28$30 Setup Costs per production run$400$550$600 Max Production per production run (Units)

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MT General Form – Hart Manu. Let: X 1 = units of product 1 X 2 = units of product 2 X 3 = units of product 3 Y 1 = 1 if production run, else = 0 Y 2 = 1 if production run, else = 0 Y 3 = 1 if production run, else = 0

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MT General Form – Hart Manu. Max 25X X X 3 – 400Y 1 – 550Y 2 – 600Y 3 s.t. 1.5X 1 + 3X 2 + 2X 3 <= 450 Dept. A 2X 1 + X X 3 <= 350 Dept. B.25X +.25X +.25X <= 50 Dept. C X 1 <= 175Y 1 X 2 <= 150Y 2 X 3 <= 140Y 3 X i >= 0 Y i = integer 0,1

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MT Integer Linear Programming All-Integer Linear Program AAll variables must be integers Mixed-Integer Linear Program SSome, but not all variables must be integers 0-1 Integer Linear Program IInteger variables must be 0 or 1, also known as binary variables

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MT Integer Linear Program (Binary Integer Programming) Assists in selection process 11 corresponding to undertaking 00 corresponding to not undertaking

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MT Integer Linear Program (Binary Integer Programming) Allows for modeling flexibility through: MMultiple choice constraints k out of n alternatives constraint MMutually exclusive constraints CConditional & co-requisite constraint

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MT Integer Programming - Binary Integer Programming CAPEX Inc. is a high technology company that faces some important capital budgeting decisions over the next four years. The company must decide among four opportunities: 1. Funding of a major R&D project. 2. Acquisition of an existing company, R&D Inc. 3. Building a new plant, and 4. Launching a new product. CAPEX does not have enough capital to fund all of these projects. The table below gives the net present value of each item together with the schedule of outlays for each over the next four years. All values are in millions of dollars. R&D Project Acquisition of R&D Inc.New Plant Launch New Product Capital Available Net Present Value (NPV) Year Year Year Year

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MT General Form – CAPEX Inc. Let: X 1 = 1 if R&D Project funded, else = 0 X 2 = 1 if acquire company, else = 0 X 3 = 1 if build new plant, else = 0 X 4 = 1 if launch new project, else = 0

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MT General Form – CAPEX Inc Max 100X X X X 4 s.t. 10X X 2 + 5X X 4 <= 40 Yr 1 15X 1 + 0X 2 + 5X X 4 <= 60 Yr 2 15X 1 + 0X 2 + 5X X 4 <= 80 Yr 3 20X 1 + 0X 2 + 5X X 4 <= 70 Yr 4 X i = 0,1

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MT Review Problems Electrical Utility Distribution Co. Alpha Airlines

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MT Integer Programming - Review Electrical Utility, a mixed integer set-up problem: A problem faced by an electrical utility each day is that of deciding which generators to start up in order to minimize total cost. The utility in question has three generators with the characteristics shown in the table below. There are two periods in a day, and the number of megawatts needed in the first period is The second period requires 3900 megawatts. A generator started in the first period may be used in the second period without incurring an additional startup cost. All major generators (e.g. A, B, and C) are turned off at the end of the day. (Assume all startups occur in time period 1.) GeneratorFixed Startup Cost Cost Per Period Per Megawatt Used Maximum Capacity In Each Period (MW) A$3,000$52,100 B$2,000$41,800 C$1,000$73,000

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MT General Form – Electrical Utility Let: X A1 = Power from Gen A in Period 1 X B1 = Power from Gen B in Period 1 X C1 = Power from Gen C in Period 1 X A2 = Power from Gen A in Period 2 X B2 = Power from Gen B in Period 2 X C2 = Power from Gen C in Period 2 Y A = 1 if Generator A started; else = 0 Y B = 1 if Generator A started; else = 0 Y C = 1 if Generator A started; else = 0

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MT General Form – Electrical Utility Min 5(X A1 +X A2 ) + 4(X B1 +X B2 ) + 7(X C1 +X C2 ) Y A Y B Y C s.t. X A1 + X B1 + X C1 >= 2900 X A2 + X B2 + X C2 >= 3900 X A1 <= 2100Y A X A2 <= 2100Y A X B1 <= 1800Y B X B2 <= 1800Y B X C1 <= 3000Y C X C2 <= 3000Y C X ij >= 0 Y i = 0, 1

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MT Integer Programming - Review Distribution Company, a integer transportation problem: A distribution company wants to minimize the cost of transporting goods from its warehouses A, B, and C to the retail outlets 1, 2, and 3. The costs (in $’s) for transporting one unit from warehouse to retailer are given in the following table. The fixed cost of operating a warehouse is $500 for A, $750 for B, and $600 for C, and at least two of them have to be open. The warehouses can be assumed to have adequate storage capacity to store all units demanded, ie., assume each warehouse can store 525 units. Retailer Warehouse123 A$15$32$21 B$9$7$6 C$11$18$5 Demand

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MT General Form – Distribution Co. Let: X ij = units shipped from i to j Y A = 1 if warehouse A opens, else = 0 Y B = 1 if warehouse B opens, else = 0 Y C = 1 if warehouse C opens, else = 0

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MT General Form – Distribution Co. Min 500Y A + 750Y B + 600Y C + 15X A1 + 32X A2 + 21X A3 + 9X B1 + 7X B2 + 6X B3 + 11X C1 + 18X C2 + 5X C3 s.t. X A1 + X B1 + X C1 = 200 X A2 + X B2 + X C2 = 150 X A3 + X B3 + X C3 = 175 X A1 + X B1 + X C1 <= 525Y A X A2 + X B2 + X C2 <= 525Y B X A3 + X B3 + X C3 <= 525Y C Y A + Y B + Y C >= 2 X ij >= 0 Y i = 0, 1

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MT Integer Programming - Review Alpha Airlines, a integer scheduling problem: Alpha Airlines wishes to schedule no more than one flight out of Chicago to each of the following cities: Columbus, Denver, Los Angeles, and New York. The available departure slots are 8 A.M., 10 A.M., and 12 noon. Alpha leases the airplanes at the cost of $5000 before and including 10 A.M. and $3000 after 10 A.M., and is able to lease at most two per departure slot. Also, if a flight leaves for New York in a time slot, there must be a flight leaving for Los Angeles in the same time slot. The expected profit contribution before rental costs per flight is shown below (in K$) Time Slot Cities 8:00 AM10:00 AM12:00 Noon Columbus1066 Denver9109 Los Angeles New York181510

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MT General Form – Alpha Airlines Let: X ij = 1 if flight to i occurs in time slot j, else = 0 Y j = number of planes leased for time slot j

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MT General Form – Alpha Airlines Max 10X C1 + 6X C2 + 6X C3 + 9X D1 + 10X D2 + 9X D3 + 14X L1 + 11X L2 + 10X L3 + 18X N1 + 15X N2 + 10X N3 – 5Y 1 – 5Y 2 – 3Y 3 s.t. X C1 + X C2 + X C3 <= 1 X D1 + X D2 + X D3 <= 1 X L1 + X L2 + X L3 <= 1 X N1 + X N2 + X N3 <= 1 X C1 + X D1 + X L1 + X N1 = Y 1 X C2 + X D2 + X L2 + X N2 = Y 2 X C3 + X D3 + X L3 + X N3 = Y 3 Y 1 <= 2 Y 2 <= 2 Y 3 <= 2 X N1 <= X L1 X N2 <= X L2 X N3 <= X L3 X ij = 0,1 Y j = INTEGER

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