Chapter 3 Introduction to Linear Programming

College of Management, NCTU

Operation Research I

Fall, 2008

Chapter 3 Introduction to Linear Programming Usually (at least for now) deal with the problems of allocating limited resources among competing activities in the best possible (optimal) way. Requires all functions to be linear functions.

Example: The WYNDOR GLASS CO. 9 Two new products ¾ Product 1: aluminum-frame window ¾ Product 2: wood-frame window 9 Three plants ¾ Plant 1 produces aluminum frames ¾ Plant 2 produces wood frames ¾ Plant 3 produces the glass and assembles the products 9 Interview and gathering data yield the following information: Plant Product 1 Product 2 Capacity Available 1 0 4 1 0 2 12 2 3 2 18 3 3 5 Profit (thousands) 9 Defining variables and formulating the model

Jin Y. Wang

Chap3-1



Fall, 2008

How to solve a LP problem – The Graphic Solution 9 Recall the Wyndor Problem Max Z = 3x1 + 5x2 ≤ 4 S.T. x1 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1, x2 ≥ 0 9 The first step is to identify the values of (x1, x2) that are permitted by the restrictions. 9 The resulting region of permissible values of (x1, x2), called the feasible region.

9 Then, we need to pick out the point in this feasible region that maximizes the value of Z = 3x1 + 5x2.

Take a look at the OR Tutor in the CD-ROM 9 Install the program and try it out. Jin Y. Wang

Chap3-2



Fall, 2008

The Linear Programming Model 9 The key terms are resources and activities. ¾ m denotes the number of resources; n denotes the number of activities. 9 Once again, the most common type of application of linear programming involves allocating resources to activities. 9 Back to our Wyndor example Wyndor Example General Problem 3 Plants m resources 2 products n activities Production rate of product j, xj Level of activity j, xj Profit Z Overall measure of performance Z Our Standard Form of the Model 9 Some commonly used symbols: ¾ Z = value of overall measure of performance ¾ xj = level of activity j ¾ cj = increase in Z that would result from each unit increase in level of activity j ¾ bi = amount of resource i that is available for allocation to activities ¾ aij = amount of resource i consumed by each unit of activity j Max Z = c1x2 + c2x2 +…+cnxn S.T. a11x1 + a12x2 +…+ a1nxn ≤ b1 a21x1 + a22x2 +…+ a2nxn ≤ b2 ……………

am1x1 + am2x2 +…+ amnxn ≤ bm x1, x2,…, xn ≥ 0 OR n

Max

∑c j =1

j

xj

n

S.T.

∑a j =1

ij

x j ≤ bi , i = 1, 2, …,m

xj ≥ 0,

j = 1, 2, …, n

9 Objective function, constraints, functional constraints, and nonnegativity constraints.

Jin Y. Wang

Chap3-3



Fall, 2008

Other Forms 9 Minimizing rather than maximizing the objective function:

9 Some functional constraints with a greater-than-or-equal-to inequality:

9 Some functional constraints in equation form:

9 Deleting the nonnegativity constraints for some decision variables:

Some Terminology 9 Solution—any specification of values for the decision variables is called a solution (this is a bit different from what you are used to).

9 Feasible Solution—a solution for which all the constraints are satisfied. 9 Infeasible Solution—is a solution for which at least one constraint is violated.

9 Feasible Region—is the collection of all feasible solutions.

9 Note that it is possible to have no feasible solution.

Jin Y. Wang

Chap3-4



Fall, 2008

9 Optimal Solution—is a feasible solution that that has the most favorable value of the objective function.

9 Most problems will have just one optimal solution. However, it is possible to have more than one. 9 Multiple optimal solutions—have an infinite number of them, each with the same optimal value of the objective function.

9 No optimal solutions—occurs only if (1) it has no feasible solutions or (2) the constraints do not prevent improving objective value (unbounded Z).

9 Corner-Point Feasible (CPF) Solution—is a solution that lies at a corner of the feasible region.

Relationship between optimal solution and CPF solution 9 Consider any LP problem with feasible solutions and a bounded feasible region. optimal solution. 9 This problem must possess CPF solutions and . That is, if a problem 9 The best CPF solution must be has exactly one optimal solution, it must be . must be CPF 9 If the problem has multiple optimal solutions, at lease solutions.

Jin Y. Wang

Chap3-5



Fall, 2008

Assumptions of Linear Programming 9 Proportionality ¾ The contribution of each activity to the value of the objective function Z is proportional to the level of the activity xj. ¾ Similarly, the contribution of each activity to the left-hand side of each functional constraint is proportional to the level of the activity xj.

9 Additivity ¾ Every function in a linear programming model is the sum of the individual contributions of the respective activities.

9 Divisibility ¾ Decision variables in a LP model are allowed to have any values, including noninteger values, that satisfy the functional and nonnegativity constraints.

9 Certainty ¾ The value assigned to each parameter of a LP model is assumed to be a known constant.

Jin Y. Wang

Chap3-6



Fall, 2008

Example—Radiation Therapy Fraction of Entry Dose Absorbed by Area (Average) Area

Restriction on total Average Dosage, Kilorads

Beam 1

Beam 2

Health Anatomy

0.4

0.5

Critical tissues

0.3

0.1

≤ 2.7

Tumor region

0.5

0.5

=6

Minimize

Center of tumor 0.6 0.4 ≥ 6 9 Problem: Determine the number of doses required for beam 1 and beam2.

Jin Y. Wang

Chap3-7


Example —Region Planning Farms Acres 1 400 2 600 3 300 Corp

Max quota (acres) Sugar beets 600 Cotton 500 Sorghum 325


Fall, 2008

Water allocated in acre-foot 600 800 375 Water consumption (acre-foot/acre) 3 2 1

Net return ($/acre) 1,000 750 250

¾

Political Concern: each farm will plant the same proportion of land. ¾ Problem: Determine how many acres of each crop should be planted on each farm.

Jin Y. Wang

Chap3-8



Fall, 2008

Example—Air Pollution Control Pollutants Required Reduction Rate Particulate (1) 60 Sulfur oxides (2) 150 Hydrocarbons (3) 125 9 There are two major sources of pollution; each has three possible methods to reduce its emissions. Pollutants Method 1 Method 2 Method 3 Blast Open Blast Open Blast Open furnaces furnaces furnaces furnaces furnaces furnaces 1 12 9 25 20 17 13 2 35 42 18 31 56 49 3 37 53 28 24 29 20 9 Annual Cost Method Blast furnaces Open furnaces 1 8 10 2 7 6 3 11 9 9 Problem: Determine what fraction of each method to be applied to each type of furnace.

Jin Y. Wang

Chap3-9



Fall, 2008

Example—Reclaiming Solid Wastes Grade Specification A

Amalgamation Selling Price Cost per Pound per Pound

Material 1: Not more than 30% of total Material 2: Not less than 40% of total

3.00

8.50

2.50

7.00

2.00

5.50

Material 3: Not more than 50% of total Material 4: Exactly 20% of total B

Material 1: Not more than 50% of total Material 2: Not less than 10% of total Material 4: Exactly 10% of total

C

Material 1: Not more than 70% of total

Material

Pounds per Week Available

Treatment Cost per Pound

Additional Restrictions

1

3,000

3.00

2

2,000

6.00

1. For each material, at least half of the pound per week available should be collected and treated.

3

4,000

4.00

4

1,000

5.00

Jin Y. Wang Chap3-10

2. $30,000 per week should be used to treat these materials.



Fall, 2008

Example—Personnel Scheduling 9 Minimize number of customer service agents that need to be on duty at different times of the day to provide a satisfactory level of service. Time Periods Covered Minimum Number Shift Time Periods of Agents Needed 1 2 3 4 5 6:00am~8:00am V 48 8:00am~10:00am V V 79 10:00am~12:00am V V 65 12:00am~2:00pm V V V 87 2:00pm~4:00pm V V 64 4:00pm~6:00pm V V 73 6:00pm~8:00pm V V 82 8:00pm~10:00pm V 43 10:00pm~12:00pm V V 52 12:00pm~6:00am V 15 Daily cost per agent $170 $160 $175 $180 $195




Example—Distributing Goods Through a Distribution Network


Fall, 2008



Fall, 2008

Bonus Example—A relief agency is sending agricultural experts to a certain country to increase food production. Each full project undertaken in country 1 will increase food production to feed 2000 additional people; In country 2, 3000 people. Amount used per project Resources Availability Country 1 Country 2 0 5 20 Equipment 1 2 10 Expert 60 20 300 Money 9 Assumed that we can carry out the fractional project. 9 Problem: maximize the min increase in food production in both countries.
