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© 2010 Elsevier B.V. All rights reserved.

Inventory Management MILP Modeling for Tank

Farm Systems

Susana Relvas,

a

Ana Paula F.D. Barbosa-Póvoa,

a

Henrique A. Matos

b

a

CEG-IST, UTL, Av. Rovisco Pais, 1049-001, Lisbon, Portugal, susanaicr@ist.utl.pt, apovoa@ist.utl.pt

b

CPQ, IST, UTL, Av. Rovisco Pais, 1049-001, Lisbon, Portugal,henrimatos@ist.utl.pt

Abstract

Process industries face complex problems when dealing with the optimization of production, control and operational tasks. While production and control are widely covered, the side operations related with logistics, supply and demand are usually underestimated, since they are not core activities. However, their contribution to business efficiency and margins maximization is crucial, because they guarantee process inputs and outputs at the desired quantities, timings and quality. One of these activities relies on inventory management. This paper presents MILP model that can be integrated with other process models, which represents a flexible storage tank farm. Under this general tank view, different logistic structures can be modeled: process feeding tanks, intermediate tanks or a final product tanks. The proposed MILP model is tested for two example problems.

Keywords: Tank, Inventory Management, MILP, flexible model

1.

Introduction

Supply chain inventory management is a critical issue to address in order to enhance service level, competitive advantage and cost reduction, being a common source of mismanagement (Lee and Billington, 1992 and You and Grossmann, 2008). Given the benefits related with inventory management, process industries should give more focus to logistic issues when designing and planning their supply chain.

Inventory management policies are strongly developed for discrete products (e.g. components, parts) either accounting for the ordering or for inventory management. In the first case there are policies from traditional approaches as the Economic Order Quantity to tailor-made methodologies. Inventory management is based on warehouse optimization approaches that take into account flows and cost minimization. On the other hand, for continuous quantities stored in tank farms the problem is rather complex to represent, model and optimize. This is due to the high dependency that exists between inventory, process, product specificities and requirements, which may lead to complex models to be coupled to the complete process optimization models.

The published literature on inventory management for process industries appears on two directions: i) storage policies applied to operations scheduling models, such as zero wait storage, unlimited intermediate storage, non-intermediated storage, finite intermediate storage (as revised in Mendez et al., 2006) or using simple capacity limitations (taking as example supply chain planning models) and ii) operation specific models, such as the ones proposed for the petroleum industry (Relvas et al. 2006). The present work develops a tank farm scheduling model, where inventory is managed under common constraints related with process industries.

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2.

Problem Statement

The main system dealt in this work is a common inventory tank farm that can be located either in a chemical plant or other supply chain node, such as distribution centers or ports. For the purpose of the paper, we will consider that the tank farm is connected to a supply system that provides inputs and to a demand system where outputs are supplied, as presented in Figure 1. A tank farm requires some operations’ management related to tank service, storage capacity, settling periods, quality tasks and fulfilling demands. The inputs required are product quantities and timings of arrival while outputs are the product demand and timing.

Supply System

Demand System

Figure 1 – Multiproduct tank farm system

The tank farm is connected through internal lines to the supply and demand systems. It will be considered that only one product is received at a given instant. At any point this assumption can be levered. On the other hand, the demand system has an independent delivery schedule per product. It is considered that each tank as a fixed service, i.e., the allocation of products to tanks is a problem data, which is a common procedure for chemical products. The tanks can store either liquid products or gases. For this end, the operating volume of each tank is assumed as its operating capacity.

For this system, it is required to determine the scheduling of storage activities at the tank farm that minimizes product mixing with origin in different receiving batches. Other objectives can be implemented for specific problem systems.

3.

Inventory Tank Farm MILP Modeling

At the tank level, some usual procedures imply that each tank, at any instant, can only have a maximum of one active connection, i.e., inputs from the supply system and outputs for the demand system. For this reason, it is defined as rotation scheme the procedure that permits knowing, during the current time horizon and at any time point, the schedule of tanks’ states, for each tank of each product. The states are defined from the tank operational cycle, which is represented in Figure 2. Each tank must accomplish this cycle in normal operation at the tank farm. It starts to be filled up until the total storage capacity is reached. At this point the settling period is carried out, to meet quality standards. Whenever the settling period is completed, the product stored is now available for clients’ satisfaction, until the tank is completely empty.

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The modeling challenges consist on the representation of the dynamic states (being filled up or being discharged), the boundary states detection (completely full or completely empty) and the model flexibility.

The proposed model formulation is described as follows.

Sets: ∈ set of arriving batches of products, tT set of tanks,

p

P

set of products

and sSset of tank states (ss = supplying state, rs = receiving state, fs = full state and es = empty state).

Note: time points are referred to the arrival of batch i to the tank farm, thus set I can either represent batches or time points. Both descriptions will be used throughout the model details.

Parameters: hmaxmaximum time horizon extent, δ small value,

C

itime scale referred to the arrival of batch i, Di,pvolume of arriving batch i of product p, MDp,imarket demand of product p at time point i, 0

,t p

Tset initial settling time for product p in storage tank t, TCapt,ptank capacity of tank t of product p,

0 ,t p

ID initial inventory at tank t of product p, 0

, ,ts p

ts initial state s of tank t of product p, Trepp,tminimum settling period of product p at tank t.

Positive variables: Intp,t,ivolume of product p assigned to tank t at time interval i,

i t p

Outt ,,volume that is supplied to fulfill market demand of product p from tank t at time interval i, IDp,t,iinventory level of product p at tank t at time interval i,

i t p

Tset ,, settling time of product p stored at tank t at time interval i.

Binary variables: tsp,t,s,i=1 if tank t of product p is at state s at time interval i,

1

, ,ti= p

ba if batch i of product p is assigned to tank t, mdfp,t,i =1if tank t supplies the market of product p at time i.

Model Constraints: rs s ss s i t p ts tsp,t,s,i+ p,t,s,'i=1, ∀ , , , = , '= (1) fs s es s i t p ts tsp,t,s,i+ p,t,s,'i≤1, ∀ , , , = , '= (2) fs s ss s rs s i t p ts ts tsp,t,s,i1+ p,t,s,'i ≤1+ p,t,s','i1, ∀ , , >1, = , '= , ''= (3) fs s ss s rs s i t p ts ts tspts+ ptsi ≤1+ 0pts, ∀ , , =1, = , '= , ''= '' , , ,' , , 0 , , (4) es s rs s ss s i t p ts ts tsp,t,s,i1+ p,t,s,'i ≤1+ p,t,s','i1, ∀ , , >1, = , '= , ''= (5) es s rs s ss s i t p ts ts tspts+ ptsi≤1+ 0pts, ∀ , , =1, = , '= , ''= '' , , ,' , , 0 , , (6)

Equations (1) and (2) establish the occurrence of simultaneous tank states whereas equations (3) to (6) ensure how tanks can transit between states.

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t p Int D t i t p i p, =

,,, ∀ , (7) i t p Tcap ba Intp,t,ip,t,i× p,t, ∀ , , (8) i p MD Outt pi t i t p,, = ,, ∀ ,

(9) i t p Tcap mdf Outtp,t,ip,t,i× p,t, ∀ , , (10) Equations (7) and (8) determine batch assignments to tanks, regarding the product. Equation (9) ensures outputs to fulfill market demands whereas equation (10) captures whether tank t supplies the market with product p at time i.

rs s i t p Tcap ts Intp,t,ip,t,s,i× p,t, ∀ , , , = (11) ss s i t p Tcap ts Outtp,t,ip,t,s,i× p,t, ∀ , , , = (12) Equations (11) and (12) ensure the relation between tank state and input reception and output supply. 1 , , , , , , , 1 , , , , =ID − +IntOuttp t i> IDpti pti pti pti (13) 1 , , , , , , , 0 , , , =ID +IntOuttp t i= IDpti pt pti pti (14) i t p Tcap IDp,t,ip,t, ∀ , , (15)

Equations (13) and (14) calculate the current inventory level at each tank whereas equation (15) limits inventory to maximum tank capacity.

(

Tcap

)

p t i s fs ts IDp,t,ip,t,s,i× p,t

δ

, ∀ , , , = (16)

(

ts

)

p t i s fs Tcap IDp,t,ip,t

δ

×1− p,t,s,i, ∀ , , , = (17)

(

ts

)

p t i s es Tcap IDp,t,i

δ

+ p,t×1− p,t,s,i, ∀ , , , = (18) es s i t p ts Tcap IDp,t,i

δ

p,t× p,t,s,i, ∀ , , , = (19) Equations (16) to (19) identify inventory level values that allow changing to full and empty tank states. For these equations a small difference from the inventory boundaries is used. fs s i t p ts Tsetp,t,ip,t,s,i×hmax, ∀ , , , = (20)

(

ts

)

p t i s fs C C Tset Tsetp,t,ip,t,i−1+ ii−1−1− p,t,s,i ×hmax, ∀ , , >1, = (21)

(

ts

)

p t i s fs C Tset Tsetptipt+ i−1− p,t,s,i ×hmax, ∀ , , =1, = 0 , , , (22) 1 , , , 1 1 , , , , ≤Tset − +CC− ∀p t i> Tsetpti pti i i (23) 1 , , , 0 , , , ≤Tset +Cpt i= Tsetpti pt i (24)

(

ts ts

)

p t i s fs s ss Trep Tsetp,t,ip,t× p,t,s,i−1+ p,t,s,'i−1, ∀ , , >1, = , '= (25)

(

ts ts

)

pt i s fs s ss Trep Tsetptpt× pts+ p,t,s,'i−1, ∀ , , =1, = , '= 0 , , , 0 , (26)

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ss s i t p ts Trep Tsetp,t,i1p,t+ p,t,s,i×hmax, ∀ , , >1, = (27) ss s i t p ts Trep Tsetp0tsp,t+ p,t,s,i×hmax, ∀ , , =1, = , , (28)

Equation (20) ensures that the settling time is accounted when the tank is full. Equations (21) to (24) update the settling period for each tank at each time interval. Equations (25) to (28) ensure that whenever the minimum settling period is achieved, the tank state can change from full to supply.

        × −       +         −      

∑∑∑

∑∑∑

ba I mdf P I p t i i t p p t i i t p,, ,, min (29)

Equation (29) represents the objective function which intends to minimize the difference between total number of batch assignments to tanks and the cardinality of set I, so as to avoid that one batch is assigned to several tanks and, therefore, that each tanks receives several batches before settling. The second term minimizes the difference between number of supplies from different tanks and total number of market demands. This objective function is relevant when product quality traceability is important.

4.

Implementation and Results

Two scenarios (S1 and S2) are proposed to evaluate model performance and model size. S1 represents a motivation problem with 3 products, 3 tanks per product and 10 arriving batches. S2 represents a larger problem and accounts for 5 products, 25 tanks distributed by all the products and 25 arriving batches. S1 addresses a time horizon of 10 days whereas S2 has 25 days. The model was implemented in GAMS 22.8, CPLEX 11.1 on an Intel Pentium Core 2 Duo P9400, 4 GB Ram. The objective is to obtain either the optimal solution or a solution in 600 s (plus 120 s of solution polishing). Table 1 summarizes the model statistics and performance.

Table 1. Model Performance S1 S2 Time (s) 2.598 720.854 Objective function (OF) 3 12

OF Term 1 2 2 OF Term 2 1 10 Relative gap (%) 0.0 97.52 Iterations 14879 1991790 Nodes 495 41633 Continuous Variables 361 2501 Integer variables 540 3750 Constraints 2311 15876

From table 1 it can de seen how model size increases with the size of the scenario. Additionally, in S1 only two batch splits occurred whereas for a larger problem as in S2, two batch splits were necessary to accommodate all the incoming products. Regarding tank supply, in S1 there was only one market demand fulfillment that required two tanks whereas in S2 this situation occurred ten times. However, as it can be seen, there is space for improvement in the model performance. Despite locating solutions in an early stage of the tree search, the lower bound improvement is slow. The relaxed solution was 0.0 and after a high number of iterations and node exploration, the lower bound improved to 0.2976.

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Figure 3 represents the inventory profile of tank 1 of product P1 for example S2. Table 2 summarizes the monitoring of this tank cycle, which has a capacity of 20000 m3 and a minimum settling period of 24 h. The dynamic and boundary states are matched against the inventory profile evolution, so as to verify the effectiveness of the model. The boundary states only occur when the inventory is as its limits and the settling period is referred to the full state.

0 4000 8000 12000 16000 20000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 In v e n to ry ( m 3) Days

Figure 3 – Inventory profile for tank 1 of product P1, S2 Table 2. Tank cycle monitoring: S2, P1, tank 1

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Boundary ES ES ES ES ES FS FS Dynamic RS RS RS RS RS RS RS RS RS RS RS SS SS SS SS SS SS SS SS SS SS SS SS SS SS Settling 0 24 Input 10000 10000 Inventory 0 0 0 0 0 10000 10000 10000 100002000020000 16350 16350 1270090509050540054005400 1750 1750 1750 1750 1750 1750 Output 3650 36503650 3650 3650

5.

Conclusions and Future Work

In this paper it was presented a flexible MILP model that can be coupled to scheduling and/or planning models for process industries, where liquid or gas products must be stored and managed under the requirements of a precedent supply system and a demand system ahead. The model accounts for tank farm constraints such as settling periods but can be further extended to account for other tank farm requirements that may describe other type of systems.

As future work, the authors propose to integrate the model with real world scheduling models for further testing. The development of the model should account for the option of having different time scales for the supply and demand systems. The formulation can also be revised so as to become tighter. Research on alternative system types requiring other constraints is also an objective of the authors.

6.

Acknowledgements

The authors acknowledge financial support provided by Companhia Logística de Combustíveis.

References

H.L. Lee; C. Billington, 1992, MIT Sloan Management Review, v.33, 3 (Spring), 65-73;

C.A. Mendez; J. Cerda; I.E. Grossmann; I. Harjunkoski; M. Fahl, 2006, Computers and Chemical

Engineering, v.30, issue 6-7, 913-946;

S. Relvas; H.A. Matos; A.P.F.D. Barbosa-Póvoa; J. Fialho; A.S. Pinheiro, 2006, Ind. Eng. Chem. Res., v.45 (23), 7841-7855.

Figure

Figure 2 - Individual tank’s operational cycle
Table 1. Model Performance
Figure 3 represents the inventory profile of tank 1 of product P1 for example S2. Table  2 summarizes the monitoring of this tank cycle, which has a capacity of 20000 m 3  and a  minimum settling period of 24 h

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