Integer Programming

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Integer Programming

Introduction to Integer Programming (IP)

Difficulties of LP relaxation

IP Formulations

Branch and Bound Algorithms

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Integer Programming Model

An Integer Programming model is a linear programming problem where some or all of the variables are required to be non-negative integers.

These models are in general substantially harder than solving linear programming models.

Network models are special cases of integer programming models and are very efficiently solvable.

We will discuss several applications of integer programming models.

We will study the branch and bound technique, one of the most popular algorithm to solve integer programming models.

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Classifications of IP Models

Pure IP Model: Where all variables must take integer values.

Maximize z = 3x1 + 2x2

subject to x1 + x2 6 x1, x2 0, x1 and x2 integer

Mixed IP Model: Where some variables must be integer while others can take real values.


subject to x1 + x2 6x1, x2 0, x1 integer

0-1 IP Model: Where all variables must take values 0 or 1 .

Maximize z = x1 - x2

subject to x1 + 2x2 2 2x1 - x2 1, x1, x2 = 0 or 1

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Classifications of IP Models (contd.)

LP Relaxation: The LP obtained by omitting all integer or 0-1 constraints on variables is called the LP relaxation of IP.

IP:Maximize z = 21x1 + 11x2

subject to 7x1 + 4x2 13 x1, x2 0, x1 and x2 integer

LP Relaxation:Maximize z = 21x1 + 11x2

subject to 7x1 + 4x2 13 x1, x2 0

Result:Optimal objective function value of IP

Optimal objective function value of LP relaxation

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IP and LP Relaxation

x x xx

xx

xx

x1

x2

1 2 3

1

3

2

7x1 + 4x2= 13

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Simple Approaches for Solving IP

Approach 1:

Enumerate all possible solutionsDetermine their objective function valuesSelect the solution with the maximum (or, minimum) value.

Any potential difficulty with this approach?-- may be time-consuming

Approach 2:

Solve the LP relaxationRound-off the solution to the nearest feasible integer

solution

Any potential difficulty with this approach?-- may not be optimal solution to the original IP

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X 10 If a new health care plan is adopted If it is not

X 1 If a new police station is built downtown0 If it is not

X 1 If a particular constraint must hold0 If it is not

Any decision situation that can be modeled by “yes”/“no”, “good”/“bad” etc., falls into the binary category.

To illustrate

The use of binary variables in constraints

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Example A decision is to be made whether each of three plants should be built (Yi = 1)

or not built (Yi = 0)

Requirement Binary Representation

At least 2 plants must be built Y1 + Y2 +Y3 2

If plant 1 is built, plant 2 must not be built Y1 + Y2 1

If plant 1 is built, plant 2 must be built Y1 – Y2

One, but not both plants must be built Y1+ Y2 = 1

Both or neither plants must be built Y1 – Y2 =0

Plant construction cannot exceed $17 milliongiven the costs to build plants are $5, $8, $10 million 5Y1+8Y2+10Y3 17

The use of binary variables in constraints

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Capital Budgeting Problem

Stockco Co. is considering four investments

It has $14,000 available for investment

Formulate an IP model to maximize the NPV obtained from the investments

IP:

Maximize z = 16x1 + 22x2 + 12x3 + 8x4

subject to

5x1 + 7x2 + 4x3 + 3x4 14x1, x2,,x3, x4 0, 1

Investmentchoice

1 2 3 4

Cashoutflow

$5000 $7000 $4000 $3000

NPV $16000 $22000 $12000 $8000

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Fixed Charge Problem

Gandhi cloth company manufactures three types of clothing: shirts, shorts, and pants

Machinery must be rented on a weekly basis to make each type of clothing. Rental Cost:

$200 per week for shirt machinery $150 per week for shorts machinery $100 per week for pants machinery

There are 150 hours of labor available per week and 160 square yards of cloth

Find a solution to maximize the weekly profit

Labor hr Cloth yd Price Shirts 3 4 $6 Shorts 2 3 $4 Pants 6 4 $8

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Fixed Charge Problem (contd.)

Decision Variables:

x1 = number of shirts produced each weekx2 = number of shorts produced each week x3 = number of pants produced each week

y1 = 1 if shirts are produced and 0 otherwisey2 = 1 if shorts are produced and 0 otherwisey3 = 1 if pants are produced and 0 otherwise

Formulation:

Max. z = 6x1 + 4x2 + 8x3 - 200y1 - 150 y2 - 100y3

subject to3x1 + 2x2 + 6x3 150

4x1 + 3x2 + 4x3 160

x1 M y1, x2 M y2, x3 M y3

x1, x2,,x3 0, and integer; y1, y2,,y3 0 or 1

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Either-Or Constraints

Dorian Auto is considering manufacturing three types of auto: compact, midsize, large.

Resources required and profits obtained from these cars are given below.

We have 6,000 tons of steel and 60,000 hours of labor available.

If any car is produced, we must produce at least 1,000 units of that car.

Find a production plan to maximize the profit.

Compact Midsize Large

Steel Req. 1.5 tons 3 tons 5 tons

Labor Req. 30 hours 25 hours 40 hours

Profit $2000 $3000 $4000

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Either-Or Constraints (contd.)

Decision Variables:

x1, x2, x3 = number of compact, midsize and large cars producedy1, y2, y3 = 1 if compact , midsize and large cars are produced or

not

Formulation:

Maximize z = 2x1 + 3x2 + 4x3

subject tox1 My1; x2 My2; x3 My3

1000 - x1 M(1-y1)1000 - x2 M(1-y2)

1000 - x3 M(1-y3)1.5 x1 + 3x2 + 5x3 600030 x1 + 25x2 + 40 x3 60000

x1, x2, x3 0 and integer; y1, y2, y3 = 0 or 1

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Set Covering Problems

Western Airlines has decided to have hubs in USA.

Western runs flights between the following cities: Atlanta, Boston, Chicago, Denver, Houston, Los Angeles, New Orleans, New York, Pittsburgh, Salt Lake City, San Francisco, and Seattle.

Western needs to have a hub within 1000 miles of each of these cities.

Determine the minimum number of hubs

Cities within 1000 milesAtlanta (AT) AT, CH, HO, NO, NY, PIBoston (BO) BO, NY, PIChicago (CH) AT, CH, NY, NO, PIDenver (DE) DE, SLHouston (HO) AT, HO, NOLos Angeles (LA) LA, SL, SFNew Orleans (NO) AT, CH, HO, NONew York (NY) AT, BO, CH, NY, PIPittsburgh (PI) AT, BO, CH, NY, PISalt Lake City (SL) DE, LA, SL, SF, SESan Francisco (SF) LA, SL, SF, SESeattle (SE) SL, SF, SE

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Formulation of Set Covering Problems

Decision Variables:

xi = 1 if a hub is located in city ixi = 0 if a hub is not located in city i

Minimize xAT + xBO + xCH + xDE + xHO + xLA + xNO + xNY + xPI + xSL + xSF + xSE

subject to

AT BO CH DE HO LA NO NY PI SL SF SE Required

AT 1 0 1 0 1 0 1 1 1 0 0 0 xAT >= 1BO 0 1 0 0 0 0 0 1 1 0 0 0 xBO >= 1CH 1 0 1 0 0 0 1 1 1 0 0 0 xCH >= 1DE 0 0 0 1 0 0 0 0 0 1 0 0 xDE >= 1HO 1 0 0 0 1 0 1 0 0 0 0 0 xHO >= 1LA 0 0 0 0 0 1 0 0 0 1 1 0 xLA >= 1NO 1 0 1 0 1 0 1 0 0 0 0 0 xNO >= 1NY 1 1 1 0 0 0 0 1 1 0 0 0 xNY >= 1PI 1 1 1 0 0 0 0 1 1 0 0 0 xPI >= 1SL 0 0 0 1 0 1 0 0 0 1 1 1 xSL >= 1SF 0 0 0 0 0 1 0 0 0 1 1 1 xSF >= 1SE 0 0 0 0 0 0 0 0 0 1 1 1 xSE >= 1

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Additional Applications

Location of fire stations needed to cover all cities

Location of fire stations to cover all regions

Truck dispatching problem

Political redistricting

Capital investments

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Branch and Bound Algorithm

Branch and bound algorithms are the most popular methods for solving integer programming problems

They enumerate the entire solution space but only implicitly; hence they are called implicit enumeration algorithms.

A general-purpose solution technique which must be specialized for individual IP's.

Running time grows exponentially with the problem size, but small to moderate size problems can be solved in reasonable time.

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Example:

s.t.

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Solved as LP by Simplex method

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An Example

Telfa Corporation makes tables and chairs

A table requires one hour of labor and 9 square board feet of wood

A chair requires one hour of labor and 5 square board feet of wood

Each table contributes $8 to profit, and each chair contributes $5 to profit.

6 hours of labor and 45 square board feet is available

Find a product mix to maximize the profit


subject to x1 + x2 6; 9x1 + 5x2 45; x1, x2 0; x1, x2 integer

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Feasible Region for Telfa’s Problem

Subproblem 1 : The LP relaxation of original

Optimal LP Solution: x1 = 3.75 and x2 = 2.25 and z = 41.25

Subproblem 2: Subproblem 1 + Constraint x1 4


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Feasible Region for Subproblems

Branching : The process of decomposing a subproblem into two or more subproblems is called branching.

Optimal solution of Subproblem 2:

z = 41, x1 = 4, x2 = 9/5 = 1.8



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Feasible Region for Subproblems 4 & 5

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The Branch and Bound Tree

Subproblem 1z = 41.25x1 = 3.75x2 = 2.25


z = 40.05, x1 = 4.44, x2 = 1



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Subproblem 2z = 41x1 = 4

x2 = 1.8

Subproblem 3

Subproblem 4Infeasible

Subproblem 5

x1 4 x1 3

x2 2 x2 1

1

2

4

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Feasible Region for Subproblems 6 & 7


z = 37, x1 = 4, x2 = 1


z = 40, x1 = 5, x2 = 0

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Subproblem 1z = 41.25x1 = 3.75x2 = 2.25


x2 = 1.8


x2 = 3,

Subproblem 4Infeasible

Subproblem 5z = 40.55x1 = 4.44

x2 = 1

x1 4 x1 3

x2 2 x2 1


x2 = 0,

Subproblem 7z = 37x1 = 4x2 = 1

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2

3 4

7

6 5

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Solving Knapsack Problems

Max z = 16x1+ 22x2 + 12x3 + 8x4

subject to

5x1+ 7x2 + 4x3 + 3x4 14xi = 0 or 1 for all i = 1, 2, 3, 4

LP Relaxation:

Max z = 16x1+ 22x2 + 12x3 + 8x4

subject to

5x1+ 7x2 + 4x3 + 3x4 14 0 xi 1 for all i = 1, 2, 3, 4

Solving the LP Relaxation: Order xi’s in the decreasing order of ci/ai where ci are the cost

coefficients and ai’s are the coefficients in the constraint ( Here: x1→x2 → x3 → x4) Select items in this order until the constraint is satisfied with

equality

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x3 = 0 x3 = 1

x2 = 0 x4 = 1

Subproblem 1z = 44

x1 = x2 = 1x3 =.5

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7 2

8 9

Subproblem 5z = 43.6

x1 =.6, x2=x3=1 x4 = 0, LB = 36

Subproblem 4z = 36

x1 = x3=1x2 = 0, x4 =1

Subproblem 3z = 43.7

x1 =x3= 1, x2 = .7, x4=0

Subproblem 6z = 42

x1 =0, x2=x3=1 x4 = 1, LB = 42

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Subproblem 7LB = 42

Infeasible 6

x2 = 1

x1 = 0 x1 = 1

Subproblem 2z = 43.3, LB=42

x1 = x2=1x3 = 0, x4 =.67

Subproblem 8z = 38, LB=42

x1 = x2=1x3 = x4 = 0

Subproblem 9 z= 42.85, LB=42

x1 = x4 =1x3 = 0, x2 = .85

x4 = 0

3 4

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Strategies of Branch and Bound

The branch and bound algorithm is a divide and conquer algorithm, where a problem is divided into smaller and smaller subproblems. Each subproblem is solved separately, and the best solution is taken.

Lower Bound (LB): Objective function value of the best solution found so far.

Branching Strategy : The process of decomposing a subproblem into two or more subproblems is called branching.

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Strategies of Branch and Bound (contd.)

Upper Bounding Strategy: The process of obtaining an upper bound (UB) for each subproblem is called an upper bounding strategy.

Pruning Strategy: If for a subproblem, UB LB, then the subproblem need not be explored further.(Illustrate how to fathom nodes in a search tree )

Searching Strategy: The order in which subproblems are examined. Popular search strategies: LIFO and FIFO.

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11.6 分枝界限法及其在二位元整數規劃的應用

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分枝界限法的步驟

分枝界限洞悉

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分枝

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界限

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洞悉

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洞悉測試摘要

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BIP 分枝界限演算法摘要

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完成例題解題過程

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應用分枝界限法的其他選擇

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11.7 混合整數規劃之分枝界限演算法

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MIP分枝界限演算法摘要

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Integer Programming