Linear Programming Topics General optimization model LP model and assumptions Manufacturing example Characteristics of solutions Sensitivity analysis Excel add-ins

Most of the deterministic OR models can be formulated as mathematical programs. "Program" in this context, has to do with a “plan” and not a computer program. Mathematical Program Maximize / Minimize z = f ( x 1, x 2,…, x n ) Subject to {    } b i, i =1,…, m x j ≥ 0, j = 1,…, n g i ( x 1, x 2,…, x n ) Deterministic OR Models

x j are called decision variables. These are things that you control {    } bi bi are called structural (or functional or technological) constraints x j ≥ 0 are nonnegativity constraints f ( x 1, x 2,…, x n ) is the objective function g i ( x 1, x 2,…, x n ) Model Components

( x 1.. n x A feasible solution x = satisfies all the constraints (both structural and nonnegativity) The objective function ranks the feasible solutions; call them x 1, x 2,..., x k. The optimal solution is the best among these. For a minimization objective, we have z * = min{ f ( x 1 ), f ( x 2 ),..., f ( x k ) }.. ) Feasibility and Optimality

A linear program is a special case of a mathematical program where f ( x ) and g 1 ( x ),…, g m ( x ) are linear functions Linear Program: Maximize/Minimize z = c 1 x 1 + c 2 x c n x n Subject to a i 1 x 1 + a i 2 x a in x n {    } bi bi, i = 1,…, m x j  u j, j = 1,…, n x j  0, j = 1,…, n Linear Programming

x j  u j are called simple bound constraints x = decision vector = "activity levels" a ij, c j, b i, u j are all known data  goal is to find x = ( x 1, x 2,…, x n ) T (the symbol “ T ” means) LP Model Components

(i) proportionality (ii) additivity linearity (iii) divisibility (iv) certainty Linear Programming Assumptions

(i) activity j ’s contribution to objective function is c j x j and usage in constraint i is a ij x j both are proportional to the level of activity j (volume discounts, set-up charges, and nonlinear efficiencies are potential sources of violation) (ii) 1 2 no “cross terms” such as x 1 x 5 may not appear in the objective or constraints. Explanation of LP Assumptions

(iii)Fractional values for decision variables are permitted (iv)Data elements a ij, c j, b i, u j are known with certainty Nonlinear or integer programming models should be used when some subset of assumptions (i), (ii) and (iii) are not satisfied. Stochastic models should be used when a problem has significant uncertainties in the data that must be explicitly taken into account [a relaxation of assumption (iv)]. Explanation of LP Assumptions (cont’d)

Product Structure for Manufacturing Example

Machine data Product data Data for Manufacturing Example

Data Summary PQ Selling price/unit Raw Material cost/unit 4540 Maximum sales Minutes/unit on A 2010 B 1228 C 15 6 D 1015 Machine Availability: 2400 min/wk Operating Expenses = $6,000/wk (fixed cost) Decision Variables x P = # of units of product P to produce per week x Q = # of units of product Q to produce per week x R = # of units of product R to produce per week R Structural coefficients

max z = 45 x P + 60 x Q + 50 x R – 6000 Objective Function s.t. 20 x P +  x P + 28 x Q + 16 x R  x P + 6 x Q + 16 x R  x P + 15 x Q + 0 x R  2400 demand Are we done? nonnegativity Structural constraints x P  0, x Q  0, x R  0 x P  100, x Q  40, x R  x Q + 10 x R Are the LP assumptions valid for this problem? Optimal solution x * P = 81.82, x * Q = 16.36, x * R = 60 LP Formulation

Optimal objective value is $7,664 but when we subtract the weekly operating expenses of $6,000 we obtain a weekly profit of $1,664. Machines A & B are being used at maximum level and are bottlenecks. There is slack production capacity in Machines C & D. How would we solve model using Excel Add-ins ? Discussion of Results for Manufacturing Example

Solution to Manufacturing Example

A Graphical Solution Procedure (LPs with 2 decision variables can be solved/viewed this way.) 1. Plot each constraint as an equation and then decide which side of the line is feasible (if it’s an inequality). 2. Find the feasible region. 3. Plot two iso-profit (or iso-cost) lines. 4. Imagine sliding the iso-profit line in the improving direction. The “last point touched” as the iso-profit line leaves the feasible region region is optimal. Characteristics of Solutions to LPs

Two-Dimensional Machine Scheduling Problem -- let x R = 60 max z = 45 x P + 60 x Q Objective Function s.t. 20 x P +  x P + 28 x Q  x P + 6 x Q  x P + 15 x Q  2400 demand nonnegativity Structural constraints x P  0, x Q  0 x P  100, x Q  x Q

Feasible Region for Manufacturing Example

Iso-Profit Lines and Optimal Solution for Example

3. Infeasible : feasible region is empty; e.g., if the constraints include x 1 + x 2  6 and x 1 + x 2  7 4. Unbounded :Max 15 x x 2 (no finite optimal solution) s.t. 2. Multiple optimal solutions : Max 3 x x 2 s.t. x 1 + x 2  1 x 1, x 2  0 1. Unique Optimal Solution Note: multiple optimal solutions occur in many practical (real-world) LPs. x 1 + x 2  1 x 1, x 2  0 Possible Outcomes of an LP

Example with Multiple Optimal Solutions

Bounded Objective Function with Unbound Feasible Region

Inconsistent constraint system Constraint system allowing only nonpositive values for x 1 and x 2

Shadow Price (dual variable) on Constraint i Amount object function changes with unit increase in RHS, all other coefficients held constant Objective Function Coefficient Ranging Allowable increase & decrease for which current optimal solution is valid RHS Ranging Allowable increase & decrease for which shadow prices remain valid Sensitivity Analysis

Solution to Manufacturing Example

Sensitivity Analysis with Add-ins

What You Should Know About Linear Programming What the components of a problem are. How to formulate a problem. What the assumptions are underlying an LP. How to find a solution to a 2-dimensional problem graphically. Possible solutions. How to solve an LP with the Excel add-in.