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**Integer Linear Programming**

Chapter 6 Integer Linear Programming MT 235

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**Integer Linear Programming**

All-Integer Linear Program All variables must be integers Mixed-Integer Linear Program Some, but not all variables must be integers 0-1 Integer Linear Program Integer variables must be 0 or 1, also known as binary variables MT 235

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**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? Small Large Capital Available Annual profit $1 million $5 million Purchase cost $5 million $50 million $100 million Maximum purchase quantity 2 No maximum MT 235

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**Define Variables - Northern Airlines**

Let: S = # of Small Aircraft L = # of Large Aircraft MT 235

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**General Form - Northern Airlines**

Max 1S + 5L s.t. 5S + 50L <= 100 S <= 2 S, L >= 0 & Integer MT 235

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**Northern Airlines – Graph Solution**

LP Relaxation (2, 1.8) Budget Small AC MT 235

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**Northern Airlines – Graph Solution**

Budget Small AC Rounded Solution (2, 1) MT 235

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**Northern Airlines – Graph Solution**

Optimal Solution (0, 2) Budget Small AC MT 235

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**Integer Linear Programming**

All-Integer Linear Program All variables must be integers Mixed-Integer Linear Program Some, but not all variables must be integers 0-1 Integer Linear Program Integer variables must be 0 or 1, also known as binary variables MT 235

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**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 Department Product 1 Product 2 Product 3 Hours available A (hours/unit) 1.5 3 2 450 B (hours/unit) 1 2.5 350 C (hours/unit) 0.25 50 Profit Contributions per Unit $25 $28 $30 Setup Costs per production run $400 $550 $600 Max Production per production run (Units) 175 150 140 MT 235

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**General Form – Hart Manu.**

Let: X1= units of product 1 X2= units of product 2 X3= units of product 3 Y1= 1 if production run, else = 0 Y2= 1 if production run, else = 0 Y3= 1 if production run, else = 0 MT 235

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**General Form – Hart Manu.**

Max 25X1 + 28X2 + 30X3 – 400Y1 – 550Y2 – 600Y3 s.t. 1.5X1 + 3X2 + 2X3 <= Dept. A 2X1 + X X3 <= Dept. B .25X + .25X + .25X <= Dept. C X1 <= 175Y1 X2 <= 150Y2 X3 <= 140Y3 Xi >= 0 Yi = integer 0,1 MT 235

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**Integer Linear Programming**

All-Integer Linear Program All variables must be integers Mixed-Integer Linear Program Some, but not all variables must be integers 0-1 Integer Linear Program Integer variables must be 0 or 1, also known as binary variables MT 235

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**0-1 Integer Linear Program (Binary Integer Programming)**

Assists in selection process 1 corresponding to undertaking 0 corresponding to not undertaking MT 235

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**0-1 Integer Linear Program (Binary Integer Programming)**

Allows for modeling flexibility through: Multiple choice constraints k out of n alternatives constraint Mutually exclusive constraints Conditional & co-requisite constraint MT 235

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**Integer Programming - Binary**

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) 100 50 30 Year 1 10 5 40 Year 2 15 60 Year 3 80 Year 4 20 70 MT 235

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**General Form – CAPEX Inc.**

Let: X1= 1 if R&D Project funded, else = 0 X2= 1 if acquire company, else = 0 X3= 1 if build new plant, else = 0 X4= 1 if launch new project, else = 0 MT 235

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**General Form – CAPEX Inc**

Max 100X1 + 50X2 + 30X3 + 50X4 s.t. 10X1 + 30X2 + 5X3 + 10X4 <= Yr 1 15X1 + 0X2 + 5X3 + 10X4 <= Yr 2 15X1 + 0X2 + 5X3 + 10X4 <= Yr 3 20X1 + 0X2 + 5X3 + 10X4 <= Yr 4 Xi = 0,1 MT 235

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**Review Problems Electrical Utility Distribution Co. Alpha Airlines**

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**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.) Generator Fixed Startup Cost Cost Per Period Per Megawatt Used Maximum Capacity In Each Period (MW) A $3,000 $5 2,100 B $2,000 $4 1,800 C $1,000 $7 3,000 MT 235

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**General Form – Electrical Utility**

Let: XA1 = Power from Gen A in Period 1 XB1 = Power from Gen B in Period 1 XC1 = Power from Gen C in Period 1 XA2 = Power from Gen A in Period 2 XB2 = Power from Gen B in Period 2 XC2 = Power from Gen C in Period 2 YA = 1 if Generator A started; else = 0 YB = 1 if Generator A started; else = 0 YC = 1 if Generator A started; else = 0 MT 235

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**General Form – Electrical Utility**

Min 5(XA1+XA2) + 4(XB1+XB2) + 7(XC1+XC2) YA YB YC s.t. XA1 + XB1 + XC1 >= 2900 XA2 + XB2 + XC2 >= 3900 XA1 <= 2100YA XA2 <= 2100YA XB1 <= 1800YB XB2 <= 1800YB XC1 <= 3000YC XC2 <= 3000YC Xij >= 0 Yi = 0, 1 MT 235

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**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 Warehouse 1 2 3 A $15 $32 $21 B $9 $7 $6 C $11 $18 $5 Demand 200 150 175 MT 235

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**General Form – Distribution Co.**

Let: Xij = units shipped from i to j YA = 1 if warehouse A opens, else = 0 YB = 1 if warehouse B opens, else = 0 YC = 1 if warehouse C opens, else = 0 MT 235

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**General Form – Distribution Co.**

Min 500YA + 750YB + 600YC + 15XA1 + 32XA2 + 21XA3 + 9XB1 + 7XB2 + 6XB3 + 11XC1 + 18XC2 + 5XC3 s.t. XA1 + XB1 + XC1 = 200 XA2 + XB2 + XC2 = 150 XA3 + XB3 + XC3 = 175 XA1 + XB1 + XC1 <= 525YA XA2 + XB2 + XC2 <= 525YB XA3 + XB3 + XC3 <= 525YC YA + YB + YC >= 2 Xij >= 0 Yi = 0, 1 MT 235

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**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 AM 10:00 AM 12:00 Noon Columbus 10 6 Denver 9 Los Angeles 14 11 New York 18 15 MT 235

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**General Form – Alpha Airlines**

Let: Xij= 1 if flight to i occurs in time slot j, else = 0 Yj = number of planes leased for time slot j MT 235

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**General Form – Alpha Airlines**

Max 10XC1 + 6XC2 + 6XC3 + 9XD1 + 10XD2 + 9XD3 + 14XL1+ 11XL2 + 10XL3 + 18XN1 + 15XN2 + 10XN3 – 5Y1 – 5Y2 – 3Y3 s.t. XC1 + XC2 + XC3 <= 1 XD1 + XD2 + XD3 <= 1 XL1 + XL2 + XL3 <= 1 XN1 + XN2 + XN3 <= 1 XC1 + XD1 + XL1 + XN1 = Y1 XC2 + XD2 + XL2 + XN2 = Y2 XC3 + XD3 + XL3 + XN3 = Y3 Y1 <= 2 Y2 <= 2 Y3 <= 2 XN1 <= XL1 XN2 <= XL2 XN3 <= XL3 Xij= 0,1 Yj = INTEGER MT 235

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