A combined bi-level approach for the spatial design of rack storage area

Abstract

This paper presents the problem of arrangement of storage racks for non-uniform load units in a storage area space when considering limitations created by the carrying pillars of a warehouse. Three variants of rack section arrangement within the planning grid of carrying pillars are presented. A mathematical formalism of the arrangement within the X, Y, and Z axes is introduced. An exact model for the warehouse building cost minimization and space consumption minimization has been defined. The model has been enriched with numerical examples. Finally, the bi-level approach containing both the mathematical model and the coordinating procedure for the problem solution is presented.

Appendix

One of the most important issues occurring while arranging storage racks consisting of identical rack cells is determining the number of work aisles. In particular, one must decide which configuration of racking system is better according to the criterion of space utilization: the one with even (Figure A1(b)) or odd (Figure A1(a)) number of rack rows. For this purpose it is necessary to present schemes of these two options (Figure A1) and to make assumptions reflecting some technological requirements.

• Assumption A: the dimensions of rack cells are x and y. Width of the working aisle is Ast.

• Assumption B: let n be the number of rack cells to be arranged.

• Assumption C: for each n, x, y, Ast, the following expressions are true:

  • °x, y, Ast> 0; x < y; n> 1;

  • °Ast> 2x this assumption reflects the relation between the width of rack rows and the width of work aisles for most types of forklifts used in conventional warehouses;

  • °x, y, Ast—real numbers;

  • °n—integer.

• Assumption D: width of an aisle between rack rows in a double rack row (equivalent to Ast _2,3 on Figure 5) is 0.

• Assumption E: storage is in one layer.

Figure A1

Storage area designed with: (a) an odd number of rack rows; (b) an even number of rack rows.

Theorem

• With the assumptions A, B, C, D, and E for the fixed number of rack cells, and when using a uniform work aisle width, designing a storage area with an even number of rack rows (Figure A1(b)) consumes less area than designing with an odd number (Figure A1(a)).

Proof

• To prove the Theorem, it is sufficient to show that for any n, x, y, Ast, the following inequality is satisfied:

  

  The rearranged inequality takes the form:

  

  where

  

  which is true for any Ast> 0.

  Assuming that n is even, inequality (23) will be:

  

  hence,

  

  which is true for any Ast> 0.

  Assuming that n is odd, inequality (23) will be:

  

  hence,

  

  Because assumption C restricts n> 1 and Ast> 2x, inequality (28) is always true. □