Optimization and Engineering

, Volume 19, Issue 1, pp 97–123

# Simultaneous optimization of production planning and inventory management of polyurethane foam plant

• Maria Analia Rodriguez
• Gabriela Corsano
• Aldo Vecchietti
• Jorge Marcelo Montagna
Article

## Abstract

In this work, the management of a polyurethane foam plant is tackled through a mixed integer linear programming model that simultaneously solves production and inventory planning problems. The production process considers the foaming stage where large polyurethane blocks are produced as well as the curing step where the blocks are dried. The proposed formulation takes into account several tradeoffs involved in the overall production process. The daily production planning is tightly related to production requirements, available space for the curing and stored elements. Moreover, the required time to dry blocks introduces a delay that must be appropriately considered in order to allow an adequate operation of downstream operations. Thus, an integrated approach where all these problems are jointly addressed is proposed using a mathematical programming model. Several study cases provided by a local company are tested to demonstrate the model performance.

### Keywords

Production planning Inventory management Polyurethane foam MILP

### Indices

i

Block widths

j

Block densities

k

Block lengths

h

Rows in the curing area

g

Groups of blocks

### Sets

Blocks

Set of possible foam blocks of width i, density j, and length k

$$Cart$$

Set of long blocks that can be cured on special carts since they have a low density j

$$Long$$

Set of long blocks

$$Orders$$

Set of special orders to produce blocks of width i, density j, and length k that are made to order

$$Groups$$

Set of blocks of width i, density j, and length k belonging to groups g

### Positive variables

$$dif_{ijk}$$

The difference between the final stock and the minimal stock for the block of width i, density j and length k

$$sf_{ijk}$$

Final stock for the block of width i, density j and length k

$$sm_{ijk}$$

Intermediate stock for the block of width i, density j and length k

### Binary variables

$$u_{ijk}$$

Indicates if there is no unsatisfied demand for the block of width i, density j and length k

$$w_{jh}$$

Indicates if density j is produced and assigned to be cured in row h

$$x_{ijk}$$

Indicates if external order blocks of width i, density j and length k are produced

$$y_{i}$$

Indicates if the width i is selected to be produced

$$z_{jkh}$$

Indicates if any block of density j and length k is placed on row h

### Integer variables

$$n_{ijkh}$$

Number of blocks produced of width i, density j, and length k, placed in row h

$$n1_{ijkh}$$

Number of long blocks of width i and density j cured on the special carts (note that the purpose of keeping index h in this variable is given by Eq. (7) but it has no physical meaning)

$$n2_{ijkh}$$

Number of long blocks of width i and density j placed on row h of the floor

### Parameters

$$BM_{o}$$

“Big M” parameter, where o = 0, 1, 2,…,7

$$cw_{i}$$

Width of the curing area when width i is selected

$$cl_{i}$$

Length of the curing area when width i is selected

$$d_{ijk}$$

Demand for blocks of width i, density j, and length k

$$fl_{ijk}$$

Length of block of width i, density j, and length k occupied on the floor in the curing stage

$$bw_{i}$$

Width of blocks from set i

$$fs$$

Minimal space that must be left between blocks to allow air flow in the curing area

$$l_{ijk}$$

Length of block of width i, density j, and length k

$$l\_min$$

Minimal length to produce for each density

$$ml_{i}$$

Minimal length to be produced for width i

$$np$$

Number of places available in the carts for the curing stage of long blocks

$$Rows_{i}$$

Number of available rows in the curing area when width i is selected

$$s_{ijk}$$

Number of blocks of width i, density j, and length k in stock at the beginning of the day

$$sc_{g}$$

Stock capacity of group g

$$smax_{ijk}$$

Maximal stock level (stock capacity) for block of width i, density j, and length k

$$smin_{ijk}$$

Minimal stock level for block of width i, density j, and length k

$$soq_{ijk}$$

Number of special blocks ordered of width i, density j, and length k

## Notes

### Acknowledgement

The authors would like to acknowledge Limansky S.A. for their financial support as well as the information provided to test the mathematical models. We also appreciated financial support from CONICET, Universidad Tecnologica Nacional and ANPCyT to develop the research activities through their projects PIP 0682, PID 25/O152 and PICT-2012-2484, respectively.

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## Authors and Affiliations

• Maria Analia Rodriguez
• 1
• Gabriela Corsano
• 2
• Aldo Vecchietti
• 2
• Jorge Marcelo Montagna
• 2
1. 1.IPQA (CONICET-UNC)CórdobaArgentina
2. 2.INGAR (CONICET-UTN)Santa FeArgentina