The former DISPO-team enhanced the LDP to the Stochastic Lot-type Design Problem, the SLDP. The SLDP can be understood as an intermediate model between the deter-ministic LDP and the final stochastic model ISPO integrating price optimization. In the SLDP different scenarios with scenario probabilities and scenario dependent demands estimated from historical data are treated. While in the LDP abstract costs in form of theL1-norm are considered and over- and undersupply are treated the same, now monetary asymmetric costs are imposed for the deviation between supply and demand.
With these costs the SLDP can be seen as a first step in integrating the sales process in the size optimization. Monetary measurement now allows also to take other costs into account. On the one hand pick costs which arise from arranging the lot-types to lots and on the other hand lot-opening costs which arise from the fact that each additional supplied lot-type leads to higher logistic effort.
2.3.1 Problem specification
We consider an article with a given setSofsizes. We want to deliver each branch from a setBofbrancheswith one lot-type from a setLoflot-typesin a multiplicity from the setM ={1, . . . , mmax}ofmultiplicities. At the maximumκdifferent lot-types are allowed to use for supply. For theith supplied new lot-type from the setLlot-opening costδiarise . For every handgrip needed for putting together the lot-types to lotspick costpcostarise. The overall supply must lie in between alower boundIand anupper bound I. Now we consider a set¯ E of differentscenarioswithscenario probabilites Prob(e),∀e∈E. With given demandsdeb,s for each sizes, branch b and scenario e an oversupply is penalized byacquisiton price apminussalvage valueπpmax, an undersupply bystarting priceπ0 minusap. This means that it is assumed, that each undersupply would lead to a loss of the full starting price while each oversupplied item can just be sold for the salvage value. The aim is to minimize the expected overall costs, i.e. the sum of the handling costs – lot-opening and pick cost – together with the expected costs for oversupply and undersupply. In terms of demand estimation and the estimation of the probabilitiesProb(e), e∈Esee Chapter3.
2.3.2 Modelling the SLDP
Before we introduce the entire model we first focus on the coefficients in the ob-jective: The expected dependent demand db,s for Branch b and Size sis given by db,s=P
e∈EProb(e)·deb,s, wheredb,sequals the dependent demand in the LDP. Now asymmetric costs for over- and undersupply are introduced. An oversupply is penal-ized by acquisition price minus salvage valueap−πpmax, an undersupply by starting
price minus acquisition priceπ0−ap. Then the cost arising by suppling Branchbwith Lot-type`in Multiplicitymare given bydistSLDPb,`,m which is defined as
distSLDPb,`,m :=X
s∈S
(max{m·ls−db,s,0}·(ap−πpmax)+max{db,s−m·ls,0}·(π0−ap)).
(2.11) The Stochastic Lot-type Design Problem SLDP is modeled as follows:
Problem 2(SLDP). Most constraints are similar to the ones of the LDP. At this point we explain only the differences and refer the reader to Problem1.
To take handling costs into account we introduce the binary variableszi, i= 1. . . , κ which indicate if at leastidifferent lot-types are opened. Constraint (2.15) links the variablesziwithyl. By Constraint (2.16) it is ensured thatzican take value one only ifzi−1also does. The additional costs for opening new lot-types are added in the ob-jective function and for every delivered lot pick costsm·pcostarise in addition to the costs for over- and undersupply (2.12).
Corollary 1(Complexity of the SLDP). The SLDP is NP-hard.
Proof. If we setδi to zero for i = 1, . . . , κin the SLDP, we obtain an LDP with changed objective coefficients because the constraints (2.15) and (2.16) in this case are equivalent to Constraint (2.3). That means, we can reduce the LDP in polynomial time to the SLDP. Because the LDP is NP-hard – as stated in Remark2– the SLDP is, too.
2.3.3 Solving the SLDP by the LDP
The SLDP simplifies to an LDP if we set all lot-opening costs to zero. We now show that in similar way we are able to determine the optimal solution of the SLDP as the best solution resulting from solvingκLDPs.
Fori= 1, . . . , κwe consider the following formulation of the LDP. The LDP(i) is an LDP with the restriction that at mostiinstead ofκdifferent lot-types are allowed for supply, Constraint (2.25). The coefficients of the variablesxb,`,m
in the objective function are these from the SLDP. Opening-costs for new lot-types are not regarded.
Having solved the LPD(i) with the optimal solution(x∗(i), y∗(i))we can compute the corresponding overall opening costs by adding
P
`∈Ly`∗(i)
X
j=1
δj
to the optimal objective value z∗LDP(i). Thus, by solving the LDP(i) for each1 ≤ i ≤ κseparately we obtain the optimal supply for each possible allowed number of different lot-types. Adding the opening costs to the related objective value yields the optimal objective value of the SLDP. The LDP(i) for which the objective value plus the corresponding opening costs is minimal among1≤i≤κthen yields the optimal solution of the SLDP.
Theorem 1(Deducing the optimal solution of the SLDP from the LDP). WithzLDP(i)∗ we denote the optimal objective value of theLDP(i). The corresponding optimal so-lutions are denoted byx∗(i)andy∗(i). WithzSLDP∗ we denote the optimal objective value of theSLDPand withx∗,y∗andz∗the related values of the variables. We define
Then the optimal objective function value of theSLDPis given by
Proof. It isi+ the number of used lot-type according to the optimal solution of the SLDP. If we would setκ=i+in the SLDP this would yield the same optimal solution.
We call the SLDP restricted to maximalκ=i+different lot-types SLDP(i+). The cor-responding optimal objective value is denoted byzSLDP(i∗ +). The LDP(i+) yields a sup-plyx∗(i+)that minimizesP
for maximali+different lot-types not regarding lot-opening costs. Because theziand the lot-opening costs are independent from the selected lot-types and depend only on the number of them the LDP(i+) yields the same optimal solutions in terms of the supply than the SLDP(i+). To obtain the same solution we could set the lot-opening costsδiin the SLDP(i+) to zero, compute the optimal supply and later on add the costs δifor thei+ used lot-types. This is the same as solving the LDP(i+) and adding the corresponding lot-opening costs. Overall that means
z∗SLDP=zSLDP(i∗ +)=zLDP(i∗ +)+ SLDP would yield an optimal solution with less or more thani+lot-types.
By settingi∗=i+the claim follows.
Remark 3. In order to compute the optimal solution of theSLDPwe propose to solve theLDP(i)s in orderingi=κ, . . . ,1. If theLDP(i)yielded a supply policy with just i− < ilot-types we would not have to solve theLDP(j)fori−≤j < i. TheLDP(j)s would yield the same optimal solution as theLDP(i). So traversing theLDP(i)s in orderi=κ, . . . ,1may reduce the computational effort.
By reducing the SLDP to the LDP now it is possible to apply solving methods for the LDP – as the described SFA heuristic – to the SLDP. Later on, in Chapter9, we will mention how this property can be exploited when solving the Integrated Size and Price Optimization Problem ISPO which is discussed in Chapter6.
2.3.4 A column generation approach
In [KKR11a] an exact column generation approach for the LDP is presented by the DISPO-team. The approach is guided by two main ideas1.
• Considering the restricted master problem (RMP) with only a subsetL0 ⊂Lof lot-types
1For further information about column generation we refer the reader to [LD05],[LD11] or [L¨ub10]
• Solving the LDP for themost promisingsubset of lot-typesL¯⊆L0exactly We will sketch the main parts of the approach. For further details we refer the reader to [KKR11a].
A restricted master problem RMP of the LP relaxation of the LDP is considered.
The only difference to the LP relaxation is that only a subset of the lot-types are con-sidered. Thus, because the optimal solution may not be contained, the restricted master problem yields an upper bound for the LP relaxation of the original problem.
1. At first, a starting solution(x∗, y∗)of the LDP is determined. This can be done via an adapted version of the SFA heuristic. The used lot-types, i.e. lot-types withy`∗= 1then are added to the initial subsetL0of lot-types. Additionally the three best fitting lot-types for each branch – as they result from the score-step of the SFA-heuristic, see2.2.3– are added toL0. The lot-types from the setL0 are the only lot-types that are considered in the RMP at the beginning.
2. With(xRMP, yRMP)we denote the optimal solution of the RMP. The set of most promising lot-typesL¯is the subset of all lot-types from the setL0withylRMP≥ε whereεis a small constant, for exampleε= 0.15. If the optimal objective value of the RMP is smaller than the objective value of the current best integer solution (x∗, y∗)the LDP restricted toL¯is solved exactly and possibly the currently best integer solution(x∗, y∗)is updated. (If the RMP yields an optimal value higher than the to(x∗, y∗)corresponding objective value the setL0cannot contain the optimal subset of lot-types. Because the RMP is a relaxation of the LDP that contains only the lot-typesL0 the optimal objective value is a lower bound for the LDP restricted to the setL0 of lot-types. Such, we are not able to obtain a better integer solution than(x∗, y∗)by only regarding the lot-types from the setL0.) Cover cuts are added to the RMP to forbid that the optimal solution of the RMP yieldsL¯ as the set of most promising lot-types again. This implies a branching on the setL¯ and the rest of lot-typesL0 currently considered in the RMP.
3. Whenever the optimal function value of the RMP is higher than or equals the objective value of the current best solution(x∗, y∗)the pricing step is performed in which – if possible – new lot-types are added to the RMP – i.e.L0is updated and the RMP is solved again and so on. Whenever the optimal objective function value of the RMP is smaller than the to(x∗, y∗)related objective function value of the LDP, then we update the subset of most promising lot-typesL0and branch on this subset, i.e. perform Step2. If the optimal objective function value of the RMP exceeds or equals the objective value of the LDP corresponding to(x∗, y∗) and no more lot-types are/can be added to the RMP than we end up at this point and return(x∗, y∗)as optimal solution.
The results in [GKR09] show that for real-world instances the maximum amount of time for solving can be reduced from 36 minutes to 4 seconds. Even very large instances – for which state-of-the-art MIP solvers fail – can be solved in less than 16 minutes.