1. 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

2. 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

3. 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

4. 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%)

6. 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

7. 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

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

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

11. 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
0.9X X X X45 - 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

12. Multi-stage production model
0.95X X67 ≥ 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

13. Deterministic Solution

14. 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

15. 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

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

17. 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

18. 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

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

20. Scenario output

21. Scenario output

22. 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)

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

24. 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

25. 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

26. 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

27. 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

28. 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

29. Summary statistics

30. LOS

31. Deterministic solution vs scenario #6