Hnich, B.

Prestwich, S.

Rossi, R.

Tarim, S.A.

Wageningen Yield 

A Decision Support System for Computing
Optimal (R,S) Policy Parameters?
S. Armagan Tarim,1 Brahim Hnich,2 Roberto Rossi,3 and Steven Prestwich3
Department of Management, Hacettepe University, Ankara, Turkey1
armagan.tarim@hacettepe.edu.tr
Faculty of Computer Science, Izmir University of Economics, Turkey2
brahim.hnich@ieu.edu.tr
Cork Constraint Computation Centre - CTVR, University College, Cork, Ireland3
{r.rossi,s.prestwich}@4c.ucc.ie
Retail replenishment is a high-value activity. According to the US Commerce
Department, $1.1 trillion in inventory supports $3.2 trillion in annual US retail
sales [...]. Improving distribution centre efficiency of just a few percentage points
through advanced automation and real-time replenishment may deliver significant
savings and require less capital to be tied up in inventory.1
An interesting class of production/inventory control problems is the one that
considers the single-location, single-product case under non-stationary stochas-
tic demand, fixed production/ordering cost and per-unit holding cost. Exact and
efficient approaches for computing optimal production/replenishment decisions
are a key factor for achieving profitability in retail business. One of the possible
policies that can be adopted to manage stocks is the replenishment cycle policy
[6]. In this policy the inventory review times are set under a here-and-now strat-
egy at the beginning of the planning horizon. These decisions are not affected by
the actual demand realized in each period. On the other hand, for each inven-
tory review we observe the actual demand realized in former periods to compute
the actual order quantity. This provides an effective means of damping plan-
ning instability (deviations in planned orders, also known as nervousness) and
coping with demand uncertainty. We developed Constraint Programming (CP)
[1] models augmented with dedicated cost-based filtering [2] strategies in order
to improve efficiency. Our approach is now the state-of-the-art technique for
computing optimal replenishment cycle policy parameters [7, 4]. An extension
considering a stochastic delivery lag is under development [5]. These techniques
have been embedded in a Decision Support System (DSS) (Fig. 1) developed
in Java on the top of Choco [3], an open source CP solver. Our DSS can run
both as a web or desktop application. Our system receives the following inputs:
a planning horizon of N periods and a demand dt for each period t ∈ {1, . . . , N},
which is a normally distributed random variable with probability density func-
tion gt(dt). We assume that demands in different periods are independent. A
fixed delivery cost a is considered for each order and also a linear holding cost
?
Roberto Rossi is supported by Science Foundation Ireland under Grant No. 03/CE3/I405 as
part of the Centre for Telecommunications Value-Chain-Driven Research (CTVR) and Grant No.
05/IN/I886. S. Armagan Tarim and Brahim Hnich are supported by the Scientific and Techno-
logical Research Council of Turkey (TUBITAK) under Grant No. SOBAG-108K027.
1
“The Future of Retail Replenishment”, Manhattan Associates c©, 2006.
h is considered for each unit of product carried in stock from one period to the
next. Demands occurring when the system is out of stock are assumed to be
back-ordered and satisfied as soon as the next replenishment order arrives. We
assume that it is not possible to sell back excess items to the vendor at the end
of a period. Our DSS finds a replenishment plan that minimizes the expected
total cost, which is composed of ordering costs, holding costs and penalty cost,
over the N -period planning horizon. A second implementation considers service
level constraints instead of penalties for stock-outs. As a service level constraint
we require that, with a probability of at least a given value α, at the end of each
period the net inventory will be non-negative.
Fig. 1. Our Decision Support System. The main window is divided into 4 parts. The top-left win-
dow receives the inputs from the user and provides buttons for controlling the DSS. The bottom-left
window plots the expected demand as a bar graph. The top-right window plots the optimal replen-
ishment cycle policy: order periods (red dots) and the respective order-up-to-levels (red figures). In
the bottom-right window the system is simulated and simulation statistics — number of runs and
realized average cost per run — are reported. Finally, for each period the realized average service
level per run is also reported.
References
1. K. Apt. Principles of Constraint Programming. Cambridge University Press, Cambridge, UK,
2003.
2. F. Focacci, A. Lodi, and M. Milano. Optimization-oriented global constraints. Constraints,
7:351–365, 2002.
3. F. Laburthe and the OCRE project team. Choco: Implementing a cp kernel. Technical report,
Bouygues e-Lab, France, 1994.
4. R. Rossi, S. A. Tarim, B. Hnich, and S. Prestwich. Replenishment planning for stochastic inven-
tory systems with shortage cost. In Proceedings of the International Conference on Integration
of AI and OR techniques in Constraint Programming for Combinatorial Optimization Prob-
lems CP-AI-OR 2007, pages 229–243. Springer Verlag, 2007. Lecture Notes in Computer Science
No. 4510.
5. R. Rossi, S. A. Tarim, B. Hnich, and S. Prestwich. Computing replenishment cycle policy under
non-stationary stochastic lead time. Technical report, Cork Constraint Computation Centre,
2008.
6. E. A. Silver, D. F. Pyke, and R. Peterson. Inventory Management and Production Planning
and Scheduling. John-Wiley and Sons, New York, 1998.
7. S. A. Tarim, B. Hnich, R. Rossi, and S. Prestwich. Cost-based filtering techniques for stochastic
inventory control under service level constraints. Constraints, 2008 (forthcoming).


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