Integer Linear Programming

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Chapter 6

• Integer Linear Programming

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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

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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  
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Define Variables - Northern Airlines

• Let
• S of Small Aircraft
• L of Large Aircraft

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General Form - Northern Airlines

• Max
• 1S 5L
• s.t.
• 5S 50L lt 100
• S lt 2
• S, L gt 0 Integer

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Northern Airlines Graph Solution
LP Relaxation (2, 1.8)
Budget
Small AC
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Northern Airlines Graph Solution
Budget
Small AC
Rounded Solution (2, 1)
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Northern Airlines Graph Solution
Optimal Solution (0, 2)
Budget
Small AC
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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

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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) 2 1 2.5 350
C (hours/unit) 0.25 0.25 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  
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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

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General Form Hart Manu.

• Max
• 25X1 28X2 30X3 400Y1 550Y2 600Y3
• s.t.
• 1.5X1 3X2 2X3 lt 450
  Dept. A
• 2X1 X2 2.5X3 lt 350
  Dept. B
• .25X .25X .25X lt 50
  Dept. C
• X1 lt 175Y1
• X2 lt 150Y2
• X3 lt 140Y3
• Xi gt 0
• Yi integer 0,1

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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

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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

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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

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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 RD project.
• 2. Acquisition of an existing company, RD 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.

  RD Project Acquisition of RD Inc. New Plant Launch New Product Capital Available
Net Present Value (NPV) 100 50 30 50  
Year 1 10 30 5 10 40
Year 2 15 0 5 10 60
Year 3 15 0 5 10 80
Year 4 20 0 5 10 70
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General Form CAPEX Inc.

• Let
• X1 1 if RD 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

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General Form CAPEX Inc

• Max
• 100X1 50X2 30X3 50X4
• s.t.
• 10X1 30X2 5X3 10X4 lt 40
  Yr 1
• 15X1 0X2 5X3 10X4 lt 60
  Yr 2
• 15X1 0X2 5X3 10X4 lt 80
  Yr 3
• 20X1 0X2 5X3 10X4 lt 70
  Yr 4
• Xi 0,1

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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 2900.
  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
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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

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General Form Electrical Utility

• Min
• 5(XA1XA2) 4(XB1XB2) 7(XC1XC2) 3000YA
  2000YB 1000YC
• s.t.
• XA1 XB1 XC1 gt 2900
• XA2 XB2 XC2 gt 3900
• XA1 lt 2100YA
• XA2 lt 2100YA
• XB1 lt 1800YB
• XB2 lt 1800YB
• XC1 lt 3000YC
• XC2 lt 3000YC
• Xij gt 0
• Yi 0, 1

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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
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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

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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 lt 525YA
• XA2 XB2 XC2 lt 525YB
• XA3 XB3 XC3 lt 525YC
• YA YB YC gt 2
• Xij gt 0
• Yi 0, 1

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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 800 AM 1000 AM 1200 Noon
Columbus 10 6 6
Denver 9 10 9
Los Angeles 14 11 10
New York 18 15 10
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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

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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 lt 1
• XD1 XD2 XD3 lt 1
• XL1 XL2 XL3 lt 1
• XN1 XN2 XN3 lt 1
• XC1 XD1 XL1 XN1 Y1
• XC2 XD2 XL2 XN2 Y2
• XC3 XD3 XL3 XN3 Y3
• Y1 lt 2
• Y2 lt 2
• Y3 lt 2
• XN1 lt XL1
• XN2 lt XL2
• XN3 lt XL3
• Xij 0,1
• Yj INTEGER

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