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

2. Deterministic Models
Most of the deterministic OR models can be formulated as mathematical programs.
"Program" in this context, refers to a plan and not a computer program.
Mathematical Program
Maximize / minimize z = f(x1, x2, , xn)
subject to gi(x1, x2, , xn) = bi, i = 1,…,m
xj  0, j = 1,…,n

3. Notation xj = decision variables (under control of decision maker)
f(x1, x2, , xn) = objective function
gi(x1, x2, , xn) = bi, structural constraints or technological constraints (may be written as inequalities:  or 
xj  0, nonnegativity constraints
Feasible solution: vector x = (x1, x2, , xn) that satisfies all the constraints.
Objective function ranks all feasible solutions

4. Linear Programming
A linear program is a special case of a mathematical program in which all the functions are linear:
Maximize z = c1x1 + c2x cnxn
subject to a11x1 + a12x a1nxn = b1
a21x1 + a22x a2nxn = b2
: :
am1x1 + am2x amnxn = bm
xj  uj, j = 1,…,n
xj  0, j = 1,…,n

5. Linear Programming Notation
xj  uj are called simple bound constraints
x = decision vector (activity levels)
cj, aij, bi, uj are all known data
Goal  find x

6. Linear Programming Assumptions(
i) Divisibility
(ii) Proportionality
Linearity
(iii) Certainty

7. Explanation of Assumptions
(i) Divisibility: fractional values for decision variables are permitted.
(ii) Proportionality: contribution of activity j to
(a) objective function = cjxj
(b) constraint i = aijxj
Both are proportional to the level of activity j. No cross-terms can appear in the model; e.g., 3x1x2.
Volume discounts, setup charges, and nonlinear efficiencies are similarly not permitted.

8. (iv) Certainty: the data cj, aij, bi, uj are known and deterministic.
Note:
Integer or nonlinear programming must be used when either assumption (i) or (ii) cannot be justified.
Stochastic models must be used when a problem has significant uncertainties in the data that must be taken into account in the analysis.

9. Example P Q P ur c has e P a rt $5 / U $ 9 / unit $ 1 0/ unit
Machines: A, B, C, D $ 9 /
unit $ 1 0/
unit
Available times differ
100 unit/wk
40 unit/wk
Operating expenses not including raw materials: $3000/week D
10 min/unit D
15 min/unit P ur c
has e P a rt $5 / U C
9 min/unit C
6 min/unit B
16 min/unit A
20 min/unit B
12 min/unit A
10 min/unit R M 1 R M 2 R M 3 $ 2 / U $ 2 / U $ 2 / U

10. Data Summary Decision Variables P Q Selling price/unit90
100
Raw Material cost/unit 45 40
Demand (maximum)
100 40
mins/unit on A 20 10 B 12 28 C 15 6 D 10 15
Machine Availability: A  1800 min/wk; B  1440 min/wk,
C  2040 min/wk, and D  2400 min/wk
Operating Expenses = $3000/wk (fixed cost)
Decision Variables
xP = # of units of product P to produce per week
xQ = # of units of product Q to produce per week

11. Model Formulation Maximize subject to 45 + 60 = z Objective functionp x Q 20 x + 10 x
1800
Structural p Q 12 x + 28 x
1440
constraints p Q 15 x + 6 x
2040 p Q 10 x + 15 x Q
2400 p
xP £ 100, xQ £ 40
demand
Are we done?
xP ³ 0, xQ ³ 0
nonnegativity
Optimal solution
Are the LP assumptions valid for this problem?
xP* = 81.82, xQ* = 16.36
z* = 4664

12. Graphical Solutions to LPs
Linear programs with 2 decision variables can be solved with a graphical procedure.
Plot each constraint as an equation and then decide which side of the line is feasible (if it’s an inequality).
Find the feasible region.
Plot two iso-profit (or iso-cost) lines.
Imagine sliding the iso-profit line in the improving direction. The “last point touched” in the feasible region when sliding iso-profit line is optimal.

14. Solution to Production Planning Problem
Optimal objective value is $4664 but when $3000 in weekly operating expenses is subtracted, we obtain a weekly profit of $1664.
Machines A & B are being used at their maximum levels and are bottlenecks.
There is slack production capacity in Machines C & D.
How would we solve model using Excel Add-ins ?

16. Possible Outcomes of an LP1.
Infeasible –
feasible region is empty; e.g., if the
constraints include
x1 + x2  6 and x1 + x2  7
2. Unbounded -
Max
15x1 + 15x2
(no finite optimal solution)
s.t. x1 + x2  1
x1  0, x2  0 3.
Multiple optimal solutions -
Max 3x1 + 3x2
s.t. x1 + x2  1
x1  0, x2  0
4. Unique optimal solution.
Note: multiple optimal solutions occur in many practical (real-world) LPs.

17. 1 x 2 3 4 z

18. Figure 10. Inconsistent constraint system
Maximize z = x + x 1 2
subject to 3 x + x
 6 1 2 3 x + x
 3 1 2 x
 0, x
 0 1 2
Figure 10. Inconsistent constraint system

19. Sensitivity Analysis and Ranging
Shadow Price (dual variable) 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
current optimal solution is valid

21. Interpreting Sensitivity Analysis Results