Addressing uncertainty in a transshipment problem Model coefficients are generally not known with certainty and are estimated The mean value is often used How useful is traditional sensitivity analysis if multiple coefficients are not known and are estimated? Sensitivity analysis must be conducted one coefficient at a time – no insight into simultaneous changes Not applicable to LHS constraint coefficients

A basic transshipment LP Maximize Z = c11x11 + c12x12 + … + cnmxnm   Subject to constraints a11x11 + a12x12 + … + a1nx1n ≤ b1 a21x21 + a22x22 + … + a2nx2n ≤ b2 . am1xm1 + am2xm2 + … + amnxmn ≤ bm and xij ≥ 0

Transformed pulp used as input into final products Waste paper input Stage 1 Stage 2 Recycling process to generate paper pulp Transformed pulp used as input into final products Waste paper input

Multi-stage production model The mean yield each associated with each input for each of the recycling processes is given by the % at the right of each flow arrow The mean yield associated with processing each ton of newspaper in recycling process #1 is 90% (we lose 10% of the initial input material due to quality issues, waste, etc.) The mean yield associated with processing each ton of cardboard in recycling process #2 is 85% (we lose 15%)

Multi-stage production model The waste paper inputs end up being processed by a particular recycling process (#1 or #2) and the recycled paper pulp generated by the recycling process is then further treated, and is used in one of three different final paper products

Multi-stage production model There is a demand or production requirement associated with each of the final paper products Newsprint pulp (60 tons) node 7 Packing paper pulp (40 tons) node 8 Printing stock pulp (50 tons) node 9

Multi-stage production model Objective: minimize the cost of the recycling operations given inputs while meeting demand

Multi-stage production model MIN: 13X15 + 12X16 + 11X25 + 13X26 + 9X35 + 10X36 + 13X45 + 14X46 + 5X57 + 6X58 + 8X59 + 6X67 + 8X68 + 7X69   Subject To: X15 + X16 ≤ 70 X25 + X26 ≤ 50 X35 + X36 ≤ 30 X45 + X46 ≤ 40 0.9X15 + 0.8X25 + 0.95X35 + 0.75X45 - X57 - X58 - X59 ≥ 0 0.85X16 + 0.85X26 + 0.9X36 + 0.85X46 - X67 - X68 - X69 ≥ 0 0.95X57 + 0.9X67 ≥ 60 0.9X58 + 0.95X68 ≥ 40 0.9X59 + 0.95X69 ≥ 50 Xi j ≥ 0 for all i and j

Multi-stage production model X15 + X16 ≤ 70 SUPPLY CONSTRAINT: (the quantity of waste newspaper (node 1) flowing into each recycling process (nodes 5 and 6) must be less than or equal to the amount of waste newspaper collected / supplied 0.9X15 + 0.8X25 + 0.95X35 + 0.75X45 - X57 - X58 - X59 ≥ 0 BALANCE CONSTRAINT: (the quantity of raw input material flowing into Recycling Process #1 (node 5) from all four input sources (nodes 1 – 4) must be greater than or equal to the quantity of pulp flowing out of Recycling Process #1 to all three final paper products

Multi-stage production model 0.95X57 + 0.9X67 ≥ 60 DEMAND CONSTRAINT: (the quantity of recycled material flowing from each recycling process to the newsprint final paper product (node 7) must be adjusted for yield, and that value must be greater than or equal to demand for the final product

Deterministic Solution

Stochastic approach Treat all outlined parameter values as random variables which follow some type of probability distribution based on observed data Here, I assume a truncated normal distribution

Stochastic approach All 14 cost parameters (OF coefficients), all 14 yield parameters (LHS constraint coefficients), and all three demand parameters (RHS constraint coefficients) (31 parameters in all) are all assumed to vary simultaneously Estimates for each parameter are generated using a random number generator

Stochastic approach In this example, we run 25 scenarios, where each scenario consists of 31 randomly generated parameter values

Stochastic approach Solver is run separately for each scenario The solution is either optimal (for that scenario) or infeasible If the solution is infeasible, a new set of random parameter values is automatically generated and the scenario is rerun until there is an optimal solution I end up with 25 different optimal solutions

Stochastic approach The optimal solutions (all optimal allocations for all 14 decision variables x15, x16, … x69) and all 31 randomly generated parameter values (each set of 31 parameters is unique for each scenario) are stored in an m x n matrix (here m = 45 unique values and n = 25 scenarios) 25 slightly different values for each of the 14 decision variables x15, x16, … x69

Single scenario example 14 optimal values for THIS SPECIFIC scenario 31 randomly generated parameter values for THIS SPECIFIC scenario

Scenario output

Scenario output

Reevaluation Once all 25 scenarios are solved, each scenario is reevaluated using the other 24 sets of randomly generated parameter values (the numbers outlined in RED)

Base Scenario 25 14 optimal values for scenario #25 31 parameter values for scenario #25

Scenario 25 reevaluated with parameters from Scenario 1 14 optimal values for scenario #25 Output or flow values change Total cost changes Surplus of inputs and final products changes 31 parameter values for scenario #1

Scenario 25 reevaluated with parameters from Scenario 2 14 optimal values for scenario #25 Output or flow values change Total cost changes Surplus of inputs and final products changes 31 parameter values for scenario #2

Reevaluation This allows the decision maker to determine how robust a particular optimal solution is in situations where multiple parameters are uncertain In other words, how well does a particular solution set work for different sets of parameter values

Reevaluation I can see how changes to yield and cost parameters impact each of the 25 optimal solution sets: The flow of raw input material into each recycling process The pulp outputs from each recycling process for each of the final products The total cost Surplus or slack

Reevaluation Level of Service (LOS) = mean output / mean demand Strictly speaking, LOS cannot exceed 100%, but here LOS > 100% means output exceeds demand – I have more than I need and a surplus of the final product is produced

Summary statistics

LOS

Deterministic solution vs scenario #6