Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Operations Research Introduction to Linear Programming Ling-Chieh Kung Department of Information Management National Taiwan University Introduction to Linear Programming 1 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation A promise is a promise ◮ If you produce foods, what are important in getting an order from restaurants and retailers? ◮ Customers ask “When may I get them?” and “How much may I get?” ◮ You need to give accurate answers immediately. ◮ You need to promise and keep your promise. ◮ Why difficult? ◮ You have more than 8000 customers sharing your capacity and inventory. ◮ Once you promise one customer, you need to immediately update the availability information that are needed elsewhere. ◮ And updating requires a lot of planning and calculations. ◮ Read the application vignette in Section 3.1 and the article on CEIBA. Introduction to Linear Programming 2 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Introduction ◮ We need a very powerful way of planning. ◮ In the next five weeks, we will study Linear Programming (LP). ◮ It is used a lot in practice. ◮ It also provides important theoretical properties. ◮ It is good starting point for all OR subjects. ◮ We will study: ◮ What kind of practical problems can be solved by LP. ◮ How to formulate a problem as an LP. ◮ How to solve an LP. ◮ Any many more. ◮ Read Chapter 3 for this lecture! ◮ Read Sections 3.1 to 3.3 thoroughly. ◮ Read Section 3.4 for many examples (some may be discussed later). ◮ Skip Sections 3.5 and 3.6. Introduction to Linear Programming 3 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Road map ◮ Terminology . ◮ The graphical approach. ◮ Three types of LPs. ◮ Simple LP formulations. ◮ Compact LP formulations. Introduction to Linear Programming 4 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Introduction ◮ Linear Programming is the process of formulating and solving linear programs (also abbreviated as LP). ◮ An LP is a mathematical program with some special properties. ◮ Let’s first introduce some concepts of mathematical programs. Introduction to Linear Programming 5 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Basic elements of a program ◮ In general, any mathematical program can be expressed as min f ( x ) ( objective function ) s.t. g i ( x ) ≤ b i ∀ i = 1 , ..., m ( constraints ) x j ∈ R ∀ j = 1 , ..., n. ( decision variable ) ◮ There are m constraints and n variables. x 1 . ∈ R n is a vector. ◮ x = . . x n ◮ f : R n → R and g i : R n → R are all real-valued functions. ◮ Mostly we will omit x j ∈ R . Introduction to Linear Programming 6 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Transformation ◮ How about a maximization objective function? ◮ max f ( x ) ⇔ min − f ( x ). ◮ How about “=” or “ ≥ ” constraints? ◮ g i ( x ) ≥ b i ⇔ − g i ( x ) ≤ − b i . ◮ g i ( x ) = b i ⇔ g i ( x ) ≤ b i and g i ( x ) ≥ b i , i.e., − g i ( x ) ≤ − b i . max − min − x 1 + x 1 x 2 x 2 s.t. − 2 x 1 + x 2 ≥ − 3 s.t. 2 x 1 − x 2 ≤ 3 ⇔ x 1 + 4 x 2 = 5 . x 1 + 4 x 2 ≤ 5 − x 1 − 4 x 2 ≤ − 5 . Introduction to Linear Programming 7 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Sign constraints ◮ For some reasons that will be clear in the next week, we distinguish between two kinds of constraints: ◮ Sign constraints : x i ≥ 0 or x i ≤ 0. ◮ Functional constraints : all others. ◮ For a variable x i : ◮ It is nonnegative if x i ≥ 0 . ◮ It is nonpositive if x i ≤ 0. ◮ It is unrestricted in sign (urs.) or free if it has no sign constraint. Introduction to Linear Programming 8 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Feasible solutions ◮ For a mathematical program: ◮ A feasible solution satisfies all the constraints. ◮ An infeasible solution violates at least one constraint. ◮ Feasible? min 2 x 1 + x 2 ◮ x 1 = (2 , 3). s.t. ≤ 10 x 1 ◮ x 2 = (6 , 0). + 2 x 2 ≤ 12 x 1 ◮ x 3 = (6 , 6). − 2 x 2 ≥ − 8 x 1 ≥ 0 x 1 ≥ 0 . x 2 Introduction to Linear Programming 9 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Feasible region and optimal solutions ◮ The feasible region (or feasible set ) is the set of feasible solutions. ◮ The feasible region may be empty. ◮ An optimal solution is a feasible solution that: ◮ Attains the largest objective value for a maximization problem. ◮ Attains the smallest objective value for a minimization problem. ◮ In short, no feasible solution is better than it. ◮ An optimal solution may not be unique. ◮ There may be multiple optimal solutions. ◮ There may be no optimal solution. Introduction to Linear Programming 10 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Binding constraints ◮ At a solution, a constraint may be binding : 1 Definition 1 Let g ( · ) ≤ b be an inequality constraint and ¯ x be a solution. g ( · ) is binding at ¯ x if g (¯ x ) = b . ◮ An inequality is nonbinding at a point if it is strict at that point. ◮ An equality constraint is always binding at any feasible solution. ◮ Some examples: ◮ x 1 + x 2 ≤ 10 is binding at ( x 1 , x 2 ) = (2 , 8). ◮ 2 x 1 + x 2 ≥ 6 is nonbinding at ( x 1 , x 2 ) = (2 , 8). ◮ x 1 + 3 x 2 = 9 is binding at ( x 1 , x 2 ) = (6 , 1). 1 Binding/nonbinding constraints are also called active /inactive constraints. Introduction to Linear Programming 11 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Strict constraints? ◮ An inequality may be strict or weak : ◮ It is strict if the two sides cannot be equal. E.g., x 1 + x 2 > 5. ◮ It is weak if the two sides may be equal. E.g., x 1 + x 2 ≥ 5. ◮ A “practical” mathematical program’s inequalities are all weak . ◮ With strict inequalities, an optimal solution may not be attainable! ◮ What is the optimal solution of min x s.t. x > 0? ◮ Think about budget constraints. ◮ You want to spend ✩ 500 to buy several things. ◮ Typically, you cannot spend more than ✩ 500. ◮ But you can spend exactly ✩ 500. Introduction to Linear Programming 12 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Linear Programs ◮ For a mathematical program min f ( x ) s.t. g i ( x ) ≤ b i ∀ i = 1 , ..., m, if f and g i s are all linear functions, it is an LP. ◮ In general, an LP can be expressed as ◮ Or expressed by matrices: n c T x min � min c j x j s.t. Ax ≤ b. j =1 n � s.t. A ij x j ≤ b i ∀ i = 1 , ..., m. ◮ A ∈ R m × n . ◮ b ∈ R m . j =1 ◮ c ∈ R n . ◮ A ij s: the constraint coefficients . ◮ x ∈ R n . ◮ b i s: the right-hand-side values ( RHS ). ◮ c j s: the objective coefficients . Introduction to Linear Programming 13 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Summary ◮ The decision variables, objective function, and constraints. ◮ Functional and sign constraints. ◮ Feasible solutions and optimal solutions. ◮ Binding constraints. Introduction to Linear Programming 14 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Road map ◮ Terminology. ◮ The graphical approach . ◮ Three types of LPs. ◮ Simple LP formulations. ◮ Compact LP formulations. Introduction to Linear Programming 15 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Graphical approach ◮ For LPs with only two decision variables, we may solve them with the graphical approach . ◮ Consider the following example: max 2 x 1 + x 2 s.t. x 1 ≤ 10 x 1 + 2 x 2 ≤ 12 x 1 − 2 x 2 ≥ − 8 x 1 ≥ 0 x 2 ≥ 0 . Introduction to Linear Programming 16 / 55 Ling-Chieh Kung (NTU IM)
Terminology Graphical approach Three types of LPs Simple formulation Compact formulation Graphical approach ◮ Step 1: Draw the feasible region. ◮ Draw each constraint one by one, and then find the intersection. max 2 x 1 + x 2 s.t. ≤ 10 x 1 + 2 x 2 ≤ 12 x 1 − 2 x 2 ≥ − 8 x 1 ≥ 0 x 1 x 2 ≥ 0 . Introduction to Linear Programming 17 / 55 Ling-Chieh Kung (NTU IM)
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