ROBUST OPTIMIZATION OF MULTI-PERIOD PRODUCTION PLANNING UNDER DEMAND UNCERTAINTY. A. Ben-Tal, B. Golany and M. Rozenblit

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1 ROBUST OPTIMIZATION OF MULTI-PERIOD PRODUCTION PLANNING UNDER DEMAND UNCERTAINTY A. Ben-Tal, B. Golany and M. Rozenblit Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel ABSTRACT Real-world production planning problems are modeled as inventory control problems over a finite horizon. Here we address a multi-period production planning under demand uncertainty which is only known to drift around a nominal demand trajectory. We employ the robust optimization (RO) methodology, and compare our solutions to those obtained employing alternative methods. The results show that the dynamic version of the RO method yields on one hand a guaranteed lower bound on the optimal profit, but on the other hand provides better average profit than the non-robust methods. 1 INTRODUCTION AND PROBLEM FORMULATION In this paper we apply the Robust Optimization (RO) methodology to a single-product multiperiod production planning problem in which the inventories are managed periodically over a finite horizon of T periods. The demand for the product is uncertain and is only known to drift around a nominal trajectory ( d 1, d 2,..., d T ), which follows a typical life-cycle pattern. At the beginning of each period t, the decision maker produces a quantity q t at a unit cost c t starting with inventory level I t 1 at the end of period t 1. The actual demand d t is then realized, and the supplied goods are sold at a unit selling price m t. Unsatisfied demand can be delivered in later periods (at a unit selling price m t ), except for the last period T. A holding unit cost h t and shortage unit cost p t incur for each unit of surplus or shortage, respectively. Surplus left at the end of period T can be salvaged at a unit price s. The aim is to determine the production quantities so as to maximize total profit over the entire horizon. The inventory level I t at the end of each period t is given by I t = k + t (q i d i ) where k i=1

2 is the initial inventory. If the demands d 1, d 2,..., d T were known the problem could be cast as a deterministic piecewise linear problem (where [δ] + max[0, δ]): T max {F (q, d) = m t d t m T [ I T ] + + s[i T ] + q 0 t=1 T (c t q t + h t [I t ] + + p t [ I t ] + )} (1) t=1 Previous studies where based mainly on dynamic programming (DP), or stochastic programming (SPR). Both methods suffer from severe computational difficulties, in particular when T is more than 3-5. This raised a need for a tractable method designed for distribution-free large scale multi-period problems, one such recent method is Robust Optimization (RO). RO was developed by Ben-Tal and Nemirovski [1]. It assumes that the demand can only reside within a known uncertainty set U, and find the solution to the following problem: max min q 0 d U F (q, d) (2) This approach guarantees that the solution is feasible for any realization of d within U, and provides a guaranteed value for the objective function. 2 KEY METHODOLOGY We assume the actual demand d t is uncertain and drifting around a nominal trajectory ( d 1, d 2,..., d T ). The simplest user-defined uncertainty set has a box shape: U box = {d R T d t = d t + θ t ζ t t = 1,..., T where ζ 1}. (3) Where ρ = θ t d t is the uncertainty level. RO was originally designed to deal with static problems where all decision variables should be determined at the beginning of the horizon, before the uncertain data is revealed. This static version of RO is named robust counterpart (RC). The RC model of our problem is similar to model (2) with U = U box (3). Nevertheless, in most real-world problems some decisions can and should depend on past realizations of the uncertain data. Recognizing the need to address such dynamic environments, RO was extended into the affinely adjustable robust counterpart (AARC) by Ben-Tal et al. [2]. In AARC the dependence of the adjustable decision variables on past realized data is restricted to be linear. This restriction is imposed to achieve tractability. In our AARC model, it was assumed that decision on the production quantity q t in period t can be delayed to the beginning of period t, in

3 which case q t can be adjusted according to past demand realizations d 1, d 2,..., d t 1. In our model, the dependence of the adjustable decision variables q t on the past demand is a linear decision rule (LDR): t 1 q t = αt 0 + αt r d r r=1 (4) In order to explicitly write the AARC model we need to substitute q t in model (1) with its LDR (4), and solve problem (2). Thus in the AARC formulation, the coefficients αt r become the actual decisions variables. Therefore, the implementation of the AARC solution is dynamic: at the beginning of each period t we calculate q t according to LDR (4). Both methods were previously applied to the field of operation management by Ben-Tal et al. [3]. This research also used a folding horizon robust counterpart approach (RCF) where, instead of solving the problem once at the beginning of the entire planning horizon, the problem is resolved afresh at the start of each period after adjusting the revealed parameters of the previous periods. A common critic of RO is that it is conservative. Its guaranteed objective value might be inferior as compared to a nominal solution, i.e., the solution produced by replacing the uncertain parameters by their expected (nominal) values. The difference between the nominal and the robust objective values is the so called price of robustness (POR). For example, the left hand side part of Table 1 shows the objective value of the optimal nominal solution q NOM and the guaranteed objective value regarding the AARC method q AARC as a function of the uncertainty level ρ. The positive values of the POR seems to imply that the nominal solution objective value is substantially better than the AARC guaranteed objective value. However, the POR is in fact inappropriate measure, since in practice it is rare that the demand indeed takes the worst possible scenario. A better measure is to base the comparison on average performance. Let the average profit over L simulated demand trajectories corresponding to the L production policy A be AP (q A ) = 1 F (q L A, d l ), and define the actual price of robustness (APOR) for method A as the difference between AP (q NOM ) and AP (q A ). l=1 3 SIMULATIONS RESULTS We generated three sets of L=100 demand vectors, all belonging to U box (3), for several uncertainty levels ρ. Each vector consisting of T =12 entries to represent a twelve months planning horizon. Each of the three sets of demand vectors was generated according to a Beta distribution

4 with specific shape parameters α, and β. The first set was generated according to a skewed left distribution (α = 2, and β = 5), the second according to a uniform distribution (α = β = 1), and the third according to a skewed right distribution (α = 5, and β = 2). Figure 1 describes the simulation results for the demand vectors that were generated from the left skewed Beta distribution. 100 Average profit 50 0 RC Monte Carlo NOM AARC RCF PH Uncertainty level ρ (in %) Figure 1: Left skewed Beta distribution simulation results. The figure shows the average profit as a function of the uncertainty level ρ for the RO methods and the nominal solution. We also included the results of a sampled based (Monte Carlo) method that maximizes an approximation of the expected profit by simulating uniformly demand vectors within the uncertainty set U box (3). In addition, the figure includes the perfect hindsight (PH) profit, i.e., the objective value that would have been obtained to model (1) if it was possible to know exactly the realizations of demand at the beginning of the horizon. PH is in fact the ultimate upper bound on the profit. It is easy to see that the dynamic RO methods, namely AARC and RCF, has very similar results, and outperform all other practical methods studied. In contrast, the static RC achieved the worst results. The reason for that is that RC plans for the worst case scenario in which the demand vectors takes the highest values within U box (3), but the beta distribution in study is skewed to the left. The results of the nominal solution and the Monte Carlo method resembles, but are better compared with the RC results. Based on the simulation study results we calculated the actual price of robustness (APOR). The right hand side part of Table 1 shows the APOR according to the AARC production policy, for the skewed left Beta distribution of the demand, as a function of the uncertainty level ρ. The negative values of the more realistic APOR indicates that on average AARC outperform considerably the nominal solution.

5 Table 1: AARC POR and APOR according to the left skewed Beta distribution. Optimal Uncertainty level ρ (in %) Simulation Uncertainty level ρ (in %) solution skewed left F (q NOM, d) AP (q NOM ) F (q AARC ) AP (q AARC ) P OR AP OR CONCLUSIONS A production planning problem is in fact composed of two stages: planning and operation. The latter relate to the time period of the entire horizon, while the first relate to the point in time right before the first period. Accordingly, in the planning stage we can use the offline results of the RC and AARC methods as well as the nominal solution. The RC and AARC guaranteed objective values can be used to compare them to revenues that can be obtained by using the resources for alternative investment opportunities. They can also be used in negotiations, e.g., over the selling price, or the shortage penalty (which is sometimes considered as customer compensation). The AARC objective value can be employed together with the nominal solution to estimate a horizon length after which it becomes unprofitable to operate the system. In the operation stage we can use the dynamic RO methods, namely AARC or RCF. As shown by our simulation study these methods, although based on suboptimal decision rules (LDR), achieved on average, solutions that are surprisingly close (82% %) to the ultimate PH profit. 5 REFERENCES 1. A. Ben-Tal and A. Nemirovski, Robust Optimization - Methodology and Applications, Mathematical Programming (Series B) 92, pp , A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski, Adjustable Robust Solutions of Uncertain Linear Programs, Mathematical Programming 99, pp , A. Ben-Tal, B. Golany, A. Nemirovski, and J.P. Vial, Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach, M &SOM 7(3), pp , 2005.