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

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

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

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

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

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

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.

(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.

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

Data Summary Decision Variables P Q Selling price/unit 90 100 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

Model Formulation Maximize subject to 45 + 60 = z Objective function p 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

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.

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 ?

Possible Outcomes of an LP 1. 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.

1 x 2 3 4 z

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

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

Interpreting Sensitivity Analysis Results