Deterministic Models Stochastic Models •Linear Programming •Discrete-Time Markov Chains •Network Optimization •Continu

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 + c2x2 + . . . + cnxn subject to a11x1 + a12x2 + . . . + a1nxn = b1 a21x1 + a22x2 + . . . + a2nxn = b2 : : am1x1 + am2x2 + . . . + amnxn = bm xj  uj, j = 1,…,n xj  0, j = 1,…,n

5. Linear Programming Notation xj  ujare called simple bound constraints x = decision vector (activity levels) cj,aij, bi, ujare 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 Q P Machines: A, B, C, D $ 9 0 / unit $ 1 0 0/ unit Available times differ 40 unit/wk 100 unit/wk Operating expenses not including raw materials: $3000/week D 15 min/unit D 10 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 3 R M 2 $ 2 0 / U $ 2 0 / U $ 2 0 / U

10. Data Summary P Q Selling price/unit 90 100 45 40 Raw Material cost/unit 100 40 Demand (maximum) 20 10 mins/unit on A 12 28 B 15 6 C 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. x Q Model Formulation = z Objective function 45 + 60 Maximize subject to x x p Q £ 20 + 10 1800 x x Structural p Q £ constraints 12 + 28 1440 x x p Q £ 15 + 6 2040 x x p Q £ 10 + 15 2400 x p demand xP £ 100, xQ£ 40 Are we done? xP³ 0, xQ³ 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 LP 1. Infeasible – feasible region is empty; e.g., if the constraints include x1 + x2 6 and x1 + x2 7 (no finite optimal solution) 2. Unbounded - Max 15x1 + 15x2 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. x 2 z z z 2 3 1 4 3 2 1 0 x 0 1 2 3 4 1

18. z x x Maximize = + 1 2 6 x x subject to 3 + 1 2 x x 3 +  3 1 2 x x  0,  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 Allowable increase & decrease for which current optimal solution is valid

21. Interpreting Sensitivity Analysis Results