Operations Research Models

Deterministic Models •Linear Programming •Network Optimization •Integer Programming •Nonlinear Programming Stochastic Models •Discrete-Time Markov Chains •Continuous-Time Markov Chains •Queueing •Decision Analysis

Deterministic models assume all data are known with

certainty.

Stochastic models explicitly represent uncertain data via random variables or stochastic processes. Deterministic models involve optimization.

Stochastic models characterize / estimate system performance.

Most of the deterministic OR models can be formulated as mathematical programs.

"Program", in this context, has to do with a plan (not a computer program).

Mathematical Program Maximize / Minimize Subject to g i z = f (x 1 , x 2 ,…, x n ) { (x 1 , x 2 ,…, x n )   = } b i i =1,…,m x j ≥ 0 j = 1,…,n

• x j are called decision variables . These are things that you control.

• g i (x 1 ,x 2 ,…, x n ) {   = } b i are called structural (or functional or technological) constraints.

• x j • f (x 1 , …, x n ) is the objective function.

• A feasible solution x = ( x .

.

.

x 1 n ) satisfies the constraints (both structural and nonnegativity) • The objective function ranks the feasible solutions.

Linear Programming A linear program is a special case of a mathematical program, i.e., f and g 1 ,…, g m are

linear

functions.

Linear Program: Maximize/Minimize z = c 1 x 1 + c 2 x 2 + • • • + c n x n Subject to a i1 x 1 + a i2 x 2 + … + a in x n {   = } b i, i = 1,…,m x j  u j , j = 1,…,n x j ≥ 0 , j = 1,…,n

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 Our goal is to find

X

.

(the one that is feasible and maximizes/minimizes z)

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

(i) activity j’s contribution to obj fcn is usage in constraint i c j x j 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) no "cross terms", e.g., 1 2 x x 5 the objective or constraints.

, may not appear in

(iii) fractional values for decision variables are permitted (iv) the data in a ij , c j , b i , and u j certainty are

known

with Nonlinear programming and integer programming are needed when we cannot satisfy some combination of (i), (ii), and (iii).

Stochastic models must be used when a problem has significant uncertainties in the data that must be explicitly incorporated in the model ( a relaxation of assumption (iv) )

Example Machines: A, B, C, D Available times differ Oper expenses not including raw materials: $3000/week Purchase Part $5/U C 9 min/unit P $90/unit 100 unit/wk D 10 min/unit C 6 min/unit Q $100/unit 40 unit/wk Products D 15 min/unit B 16 min/unit A 20 min/unit RM1 $20/U B 12 min/unit RM2 $20/U A 10 min/unit RM3 $20/U

Data Summary P Q Selling price/unit Raw Material cost/unit Demand (maximum) mins/unit on A B C D 90 45 100 20 12 15 10 100 40 40 10 28 6 15 Machine Availability: A  C  1800 min/wk; B  1440 min/wk, 2040 min/wk, and D  2400 min/wk Operating Expenses = $3000/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

profit Formulation max 45 s.k. 20 12 15 10 X p x p x p x p x p x p + 60 x Q + 10 x Q  + 28 x Q  + 6 x Q  + 15 x Q   100, x Q 1800 1440 2040 2400  Are we done?

X p  0 x Q  0 Objective Function Structural constraints demand nonnegativity Are the LP assumptions valid for this problem?

Optimal Solution X * p = 81.82 x * Q = 16.36

Characteristics of Solutions to LPs A Graphical Solution Procedure (LP’s 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 direction of improving objective. The “last point touched” as the sliding iso-profit line leaves the feasible region is optimal.

P

240 200 160 120 80 40 0 0 Max Q D Max P A 40 B C 80 120 160 200 240 280 320 360

Q P

120 100 80 60 40 20 0 0 Optimal solution = (16.36, 81.82) 10 20 30 Max Q A

Z

= $4664 40 B 50

Z

= $3600 60

Q

max 45 s.k. 20 12 15 10 X p X p x p x p x p x p x p + 60 x Q + 10 x Q  + 28 x Q  + 6 + 15 x Q   x Q   100, x Q 0 x Q 1800 1440 2040 2400   0 (A) (B) (C) (D)

Optimal objective value is $4664 but when subtract $3000 weekly operating expenses we obtain a weekly profit of $1664.

Machines A & B are being used at maximum level and are

bottlenecks .

There is slack production capacity in Machines C & D.

How can we increase the profit? (selling price,…..) How would we solve model using Excel Add-ins ?

Excel Worksheet with Optimal Solution

LINDO Input File Example

Possible Outcomes of an LP 1. Infeasible – 2. Unbounded feasible region is empty; e.g., if the constraints include x 1 s.t.

+ x 2  Max 15x x x 1 1 6 and x 1 +15x + x 2 , x 2 2   1 + x 2  7 (no finite optimal solution) 1 0 3. Multiple optimal solutions max 3x s.t.

x x 1 1 1 + 3x + x , x 2 2 2  1  0 4. Unique Optimal Solution.

Note: multiple optimal solutions occur in many practical (real-world) LPs .

x

2

z

1

z

2

z

3 4 3 2 1 0 0 1 2 3 4

x

1 Maximize

z

= 3

x

1 –

x

2 subject to 15

x

1 – 5

x

2  30 10

x

1 + 30

x

2  120

x

1  0,

x

2  0 Figure 6. Example with alternate optimal solutions

x

2 4 3 2 1 0 0 1

z

3

z

2

z

1 2 3 4 Maximize

z

= –

x

1 +

x

2

x

1 subject to –

x

1 + 4

x

2  10 –3

x

1 + 2

x

2  2

x

1  0,

x

2  0 Figure 7. Bounded objective function with an unbounded feasible region

x

2 4 3 2 1 0 0 1 2 3 4

x

1 Maximize

z

=

x

1 +

x

2 subject to 3

x

1 +

x

2  6 3

x

1 +

x

2  3

x

1  0,

x

2  0 Figure 10. Inconsistent constraint system

x

1 –4 –3 –2 –1 0 0 –1 –2 –3 –4

x

2 Maximize

z

=

x

1 +

x

2 subject to

x

1 – 2

x

2  0 –

x

1 +

x

2  1

x

1  0,

x

2  0 Figure 11. Constraint system allowing only nonpositive values for

x

1 and

x

2

Excel Worksheet with Optimal Solution

Shadow Price on Constraint i Amount object function changes with

unit

increase in RHS, all other coefficients held constant RHS Ranges Allowable increase & decrease for which shadow prices remain valid Objective Function Coefficient Ranges Allowable increase & decrease for which current optimal solution is valid

Sensitivity Analysis

How much you should pay for extra 1 hour available time on mach. A?

How about mach. C?