6.4 Problems
6.4.2 Single-Commodity Triangular Distributed Cost Example
We consider again the network of the single-commodity uncertain cost exam- ple, where the cost values of arcs (2, 4), (2, 5) and (3, 6) shall be subject to uncertainty with a possible extra cost of 10 per arc. This time, we consider a triangular distribution of the cost values on these arcs. The following table lists the minimum, maximum and most probable cost values on these arcs.
arc minimum cost value most probable cost value maximum cost value (2, 4) 3 5 13 (2, 5) 4 12 14 (3, 6) 2 4 12
Table 6.7: Single-commodity triangular distributed cost example: Triangular distri-
bution of the cost values
The algorithm of Bertsimas and Sim only considers cost data given in the form of an interval [cij, cij+ dij]. Hence, given distributions of the cost values within
the interval bounds are not used by the algorithm. However, even though the algorithm of Bertsimas and Sim a priori is not perfectly suitable for solving problems with given distributions, we want to test the algorithm of Bertsimas and Sim on the triangular distribution and compare the results to the results of the ant algorithm.
In our example, arcs (2, 4), (2, 5) and (3, 6) all have a possible extra cost of 10. For arcs (2, 4) and (3, 6), however, the most probable cost value is much lower than for arc (2, 5). The following figure illustrates the flow between node layers 2 and 3 (subgraph given by nodes 2 to 6) of the example network for the optimal solution of the algorithm of Bertsimas and Sim for different values of Γ.
CHAPTER 6. ROBUST OPTIMIZATION IN NETWORK FLOWS: UNCERTAIN COSTS 2 4 5 3 6 80 70 50 (a) Γ = 0 2 4 5 3 6 36.7 36.7 89.9 36.7 (b) Γ = 1 2 4 5 3 6 25 25 35 90 25 (c) Γ = 1.5 2 4 5 3 6 25 25 35 90 25 (d) Γ = 2 2 4 5 3 6 50 60 90 (e) Γ = 3
Figure 6.11: Single-commodity triangular distributed cost example: Flows between
node layers 2 and 3 for Bertsimas and Sim
As we would expect, the algorithm of Bertsimas and Sim does not respect the fact that the most probable cost value for arc (2, 5) is much higher than the one for arcs (2, 4) and (3, 6). Therefore, even in the most robust case (Γ = 3), 50 flow units are sent on arc (2, 5) and none on arcs (2, 4) and (3, 6).
Now we modify the ant algorithm such that it generates cost data that corre- spond to the triangular distribution for arcs (2, 4), (2, 5) and (3, 6). This results in the following flow between node layers 2 and 3 of the example network:
2 4 5 3 6 100 10 90
Figure 6.12: Single-commodity triangular distributed cost example: Flows between
node layers 2 and 3 for the ant algorithm
6.4. PROBLEMS
The following tables list the best case cost, most probable case cost, expected case cost and worst case cost for the algorithms of Bertsimas and Sim and the ant algorithm. Note that the expected cost for arc (2, 4) is 7, for arc (2, 5) is 10 and for arc (3, 6) is 6.
Γ best case most probable case
expected case worst case
0 21680 22500 22620 23680
1 21990 22430 22679 23090
1.5 22165 22290 22515 22915
2 22165 22290 22515 22915
3 22240 22640 22540 22740
(a) Bertsimas and Sim
best case most probable case
expected case worst case
22000 22200 22400 23000
(b) Ant algorithm
Table 6.8: Single-commodity triangular distributed cost example: Comparison of the
results of Bertsimas and Sim and the ant algorithm
For all values of Γ, the algorithm of Bertsimas and Sim produces a solution that has a most probable cost which is higher than for the ant algorithm. If we compare the solution of Bertsimas and Sim for Γ = 1.5, which has the lowest most probable cost, to the solution of the ant algorithm, we can see that both best case and most probable case cost are higher for Bertsimas and Sim, while the worst case cost is slightly lower. For Γ = 3, the most robust case of Bertsimas and Sim, the most probable case cost is significantly higher than for the ant algorithm.
6.4.3
Summary
It can be clearly seen that the ant algorithm is more flexible and adaptable than the algorithm of Bertsimas and Sim, if more information about the distribution of the uncertain cost values is known. This is due to the fact that the algorithm of Bertsimas and Sim only considers the uncertainty interval data, but ignores information about distributions, if given.
An important advantage of the ant algorithm in comparison to the algorithm of Bertsimas and Sim is the computational time for network flow problems with a
CHAPTER 6. ROBUST OPTIMIZATION IN NETWORK FLOWS: UNCERTAIN COSTS
large number of arcs having uncertain input data. Considering a supply chain network with a fixed number of arcs, we first examine the problem for a small number of arcs actually having uncertain input data and then the same network with a large number of arcs with uncertain input data. While the running time of the algorithm of Bertsimas and Sim increases immensely, it remains almost constant for the ant algorithm: The number of arcs having uncertain input data influences the ant algorithm only in the way that at the beginning of each iteration, a specific cost value has to be generated for every uncertain arc, which is not computationally intensive.
However, the fact that the ant algorithm does not guarantee to find a global optimal solution has to be kept in mind.
6.5
Summary and Conclusions
In this chapter, network flow problems with uncertain cost values on the arcs were examined. For every arc subject to uncertainty, a nominal cost value and a maximum extra cost value were considered. Both an exact and a heuristic solution to this kind of problem were developed:
In Section 6.2, the Robust Optimization based method for network flow prob- lems with uncertain costs, which was proposed by D. Bertsimas and M. Sim in [14], was extended from single-commodity to multicommodity problems. In Section 6.3, an ACO based heuristic method for network flow problems with uncertain costs is presented. The introduction of uncertain cost data is in line with the random-based stochastic components of the ant algorithm in the so- lution construction process. The presented method is suitable for both single- commodity and multicommodity problems.
For both approaches, the level of robustness can be controlled via a robustness parameter Γ.
The presented approaches were applied to example problems in 6.4. A main difference of the two approaches is the fact that the algorithm of Bertsimas and Sim does not consider the distribution of the uncertainty values, while the ant algorithm can easily be adapted to diverse probability distributions. Concerning the computational time, the ant algorithm was superior to the algorithm of Bertsimas and Sim in our test runs. However, it has to be kept in mind that the ant algorithm does not guarantee to find a global optimal solution.
Chapter 7
Transformation of
Uncertain Demands to
Uncertain Costs
In the previous chapter, we presented algorithms for supply chain problems with uncertain cost values. However, there is another important type of uncertainty in supply chains: Uncertainty in demand.
In the context of supply chain management, the demand is situated at the end of the supply chain in the form of a warehouse or a customer, for example. We assume wlog that the demand arises at a warehouse and will therefore use the term “warehouse” as a synonym for a demand node in the following.
If the demand in a supply chain is subject to uncertainty, we want to find a production and transportation strategy which can cope best with the uncertain demand values - that means that we have to find a compromise between a full safety stock production and the risk of deficiency at the warehouses. As deficiency can be “worse” - i.e. more expensive - at some warehouses, we have to decide which warehouses should be delivered with extra product units. In this chapter, we describe the transformation of uncertain demands to uncer- tain costs in a given network G = (N , A). Our goal is to apply the algorithms for supply chain problems with uncertain costs, which we presented in Chapter 6, to supply chain problems with uncertain demands.
We assume in the following that G is a layered network with positive costs only, where all demand nodes (warehouses) are situated in the last layer. For the
7.1. TRANSFORMATION USING ADDITIONAL INFORMATION: PENALTY AND STORAGE COSTS
definition of the term layered network, see Section 4.2.
7.1
Transformation using Additional Informa-
tion: Penalty and Storage Costs
This section describes the transformation of uncertain demands to uncertain costs using additional information. The additional information is given in the form of penalty costs in case of deficiency and storage costs for surplus delivery. Let G = (N , A) be a given single-source supply chain network with source node s. Let i be a demand node with nominal demand bi. Now we consider uncertain
demand in node i. Let zi be the possible extra demand in node i.
s . . . . . . . . . i ⇒ ր−→ ց ց −→ ր ⇒ [|b i|, |bi| + |zi|]
Figure 7.1: Network with uncertain demand at node i
In a fixed time-period, a warehouse is supplied with a specific amount of goods. If not all units of this delivery can be sold, they have to be stored at the warehouse. The warehouse charges a storage cost of cstorage per unit to be
stored. However, if the current demand at the warehouse is higher than the amount of goods which were supplied, the warehouse charges a penalty cost of cpenalty per unavailable unit.