Extended wait-and-wee solutions for ISPO

v^maxandvl˜^max. This is done by traversing the while-loop in Step15. After these steps we have a feasible solution for Problem12. To obtain the optimum we greedily choose a branch and size with highest positive additional revenue. We increase the correspond-ing supplyIb⁰,s⁰by one. This is done in the while-loop in Step22. If there is no branch and size with positive additional revenue, we are already optimal. Otherwise the last step is repeated as long as the maximum additional profitac(b, s)is positive or until we meet the upper boundI¯for the overall supply.

In the algorithm the concavity of the expected single supply revenues is exploited.

Because we know that the additional revenue for one additional item does not increase for growingn, greedily choosing the size and branch which highest positive additional revenue yields the optimal solution of Problem12.

Remark 6(runtime of Algorithm 8). By storing the additional revenuesac(b, s)in heaps the complexity of Algorithm8is given by

O(|B||S|+|B|(vl^min−v^min)log(|S|) + (I− |B|vl^min)log(|B||S|)). (8.38)

8.2.2 Extended wait-and-wee solutions for ISPO

In the enumeration tree for ISPO, see Section7.4, we fix a price trajectoryt^eto every scenarioe ∈ E according to the set of mapsW^E. To obtain the optimal first stage decision for a particular scenarioewe solve an SLDP(W^E0), Problem7, withE⁰={e}

and objective function Notation 5(extended wait-and-see solution for ISPO). We denote the extended wait-and-see solution for ISPO with fixed price trajectories according to the setW^Eof maps forE⁰ ⊆EbyWS˜ ^E

0

(W^E).

Observation 5(extended wait-and-see solution for ISPO). We consider a set of maps W^E where each scenario e ∈ E is mapped to a price trajectory and a subset of scenariosE⁰ ⊆ E. The optimal objective values for theSLDP({e→t^e})withe ∈ E\E⁰are denoted byz_SLDP({e→t^∗ e})and for theSLDP(W^E0)byz_SLDP(W^∗ _E0).

Then the extended wait-and-see solutionWS˜ ^E

0

Because – as we will see in Chapter9– the SLDP(W^E0)s for real-world instances are hard to solve, we only use relaxed (extended) wait-and-see solutions in our cus-tomized Branch&Bound solver. On the one side we use LP relaxations, i.e. we solve

the related SLDP-LP(W^E0), on the other side single supply relaxations obtained by solving the SLDP-CB(W^E0) via Algorithm8.

Notation 6(LP-relaxed extended wait-and-see solution – LPB, ELPB). We denote the LP-relaxed extended wait-and-see solution or extended LP bound ELPB for ISPO with fixed price trajectories for each scenario according to the setW^E0 of maps for the set E⁰ of scenarios withE⁰ 6=∅byWS^E_lp⁰(W^E0). ForE⁰ =∅the LP-relaxed extended wait-and-see solution equals the LP-relaxed wait-and-see solution and is denoted by LPB.

Observation 6(LP-relaxed extended wait-and-see solution – LPB, ELPB). We con-sider a set of mapsW^E where each scenarioe ∈ E is mapped to a price trajectory and a subset of scenariosE⁰⊆E.

Then the LP-relaxed extended wait-and-see solutionWS˜ ^E

0 Notation 7(single-supply-relaxed extended wait-and-see solution, (extended) combi-natorial bound – CB, ECB). We denote the single-supply-relaxed extended wait-and-see solution or extended combinatorial bound ECB for ISPO with fixed price trajecto-ries for the set of mapsW^EforE⁰6=∅byWS^E_cb⁰(W^E). ForE⁰=∅the single-supply-relaxed extended and-see solution equals the single-supply-single-supply-relaxed classical wait-and-see solution and is denoted by CB.

Observation 7(single-supply-relaxed extended wait-and-see solution, (extended) com-binatorial bound – CB, ECB). We consider a set of mapsW^E where each scenario e∈Eis mapped to a price trajectory and a subset of scenariosE⁰⊆E.

Then the single-supply-relaxed extended wait-and-see solutionWS˜ ^E

0

We can mix up combinatorial and LP-relaxations for the computation of our relaxed (extended) wait-and-see solution. Both, the optimal objective value of the LP relaxation SLDP-LP(W^E0) and the optimal objective value of the SLDP-CB(W^E0), yield dual bounds for the SLDP(W^E0). For all subsets of scenariosE1andE2withE1⊆E\E⁰, are dual bounds for the SLDP(W^E).

In our exact Branch&Bound solver, see Chapter9, we will mix up combinatorial and LP relaxations. Because – as we will see in the next section – the computation of the (extended) combinatorial bounds is much faster than solving the LP relaxations of the SLDP(W^E0), we will solve LP relaxations only on demand if combinatorial bounds are not tight enough to prune the tree.

0 1 2 3 4 5 6

mean gap(%)

instance

CB ECB LPB ELPB ILPB

Figure 8.1: Goodness of dual bounds

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

mean time(s)

instance

CB ECB LPB ELPB ILPB

Figure 8.2: Computation time of dual bounds

8.2.3 Computational results

We compared the proposed bounds on the first 79 instances of our test set I6, see AppendixE.

We computed the bounds for all leaves of the enumeration tree of ISPO. In Fig-ure8.1and Figure8.2we depicted results in terms of optimality gaps and computation time.

We denoted the bound resulting from the standard wait-and-see solution of the related SLDP(W^E) by “ILPB”. For the other names see the previous notations.

For the extended bounds ECB and ELPB we always pooled all three scenarios.

That means ELPB related toW^Ein our case is the same as the optimal objective value of the SLDP-LP(W^E).

Averaged over all instances and rounded to two decimal figures we get optimality gaps of1.74%for CB,1.26%for ECB,1.53%for LPB,0.16%for ELPB and1.47%

for ILPB. The mean runtime is0.00seconds for CB and ECB,0.80seconds for LPB, 0.49seconds for ELPB and2.53seconds for ILPB. Please note that for an LPB the numbers of the SLDP-LP(W^E0)s that have to be solved amounts to|E| = 3, for an ELPB just one, namely SLDP-LP(W^E).

The wait-and-see solutions based on single-supply-relaxed bounds, as well the combinatorial bounds CB as the extended combinatorial bounds ECB, are as a rule weaker than the wait-and-see solutions LPB and ELPB based on LP relaxations, but can be solved much faster. ECB and ELPB in many cases also beat the standard

wait-and-see solution ILPB. For ELPB this is even always the case. As already mentioned ELPB forE⁰=Eis the same as the LP relaxation SLDP-LP(W^E) of the SLDP(W^E).

This means that the SLDP(W^E) has a small integrality gap, i.e. the optimal objective value of SLDP-LP(W^E) is not far from the optimal objective value of the SLDP(W^E).

8.3 Conclusion of the chapter

We introduced new bounds for two-stage stochastic programs based on the wait-and-see solution from stochastic programming. The extended wait-and-wait-and-see solutions are tighter than the classical wait-and-see solution. We applied these bounds on the leaves of the enumeration tree of ISPO – here each leaf corresponds to an SLDP(W^E). Be-cause the underlying binary programs are hard to solve we relax them – either by allowing single supply instead of lot-types or by the LP relaxation. The results for a set of small test instances show that the bounds based on the single supply relaxation can be obtained very fast. The relaxed extended wait-and-see solutions are always faster to compute than the classical wait-and-see solution. In some cases they even beat the clas-sical wait-and-see solution in terms of the optimality gap; for the LP-relaxed extended wait-and-see solution this is even always the case.

Chapter 9

Solving the Integrated Size and Price Optimization Problem

Now we present two of the main results of this thesis: our solvers for the Integrated Size and Price Optimization problem ISPO. Since the MIP formulation of ISPO pre-sented in Chapter6for real instances cannot be solved directly by state-of-the-art MIP solvers, we developed two approaches: An exact algorithm for benchmarking and a fast heuristic for practical use. In the exact algorithm we exploit the fact that for fixed price trajectories ISPO reduces to an SLDP, Chapter7. Dual bounds on the base of the wait-and-see solution, see Chapter8, allow us to prune the enumeration tree from Section7.4. We outline the resulting Branch&Bound algorithm named ISPO-BAB in Section9.1. The heuristic solver, ISPO-PingPong, is presented in Section9.2. It ex-ploits the fact that for every valid second stage decision, i.e. for every set of maps

“scenario¸to price trajectory” for all scenarios there exists a valid first stage decision, a valid supply policy. We call this propertyreversible recourse. We present compu-tational results for both solvers on real-world instances in Section9.3and give some remarks about the goodness of our proposed heuristic in Section9.4, before we con-clude this chapter in Section9.5.

9.1 An exact Branch&Bound approach

Now we present our exact solver for ISPO – a customized Branch&Bound algorithm.

In Chapter7, Section7.4, we already showed how to solve ISPO by enumerating all possible combinations of assignments of price trajectories to scenarios. We adopt this idea and develop it further by applying the dual bounds presented in Chapter8. The re-sult is our exact Branch&Bound algorithm ISPO-BAB. A node at Depthjcorresponds to all maps “scenario to price trajectory” with the images of the firstjscenarios fixed.

The leaves are the maps with fixed images for all scenarios. In the branching step we extend a partially defined set of maps at a node by maps to all valid price trajectories for the next scenario.

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