Maximizing the number of mixed packages subject to variety constraints

• Published in: Computers & operations research Vol. 26; no. 13; pp. 1323 - 1333
• Main Authors: Robb, David J., Trietsch, Dan
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.11.1999; Elsevier Science; Pergamon Press Inc
• Subjects: Algorithms; Applied sciences; Bin packing; Christmas; Exact sciences and technology; Greeting cards; Mathematical programming; Operational research and scientific management; Operational research. Management science; Operations research; Packaging; Polynomial algorithm; Pseudo-polynomial; Seasonal products; Studies; Style goods; Seasonal products; Bin packing; Style goods; Polynomial algorithm; Pseudo-polynomial; Integer programming; Maximization; Constraint; Optimal solution; Bin packing problem; Variety; Polynomial time
• ISSN: 0305-0548; 1873-765X; 0305-0548
• DOI: 10.1016/S0305-0548(98)00105-1

We develop a polynomial-time algorithm to optimise a variant of the one-dimensional bin-packing problem with side constraints. We also develop a pseudo-polynomial procedure to actually implement that optimal solution. The specific application is the allocation of excess of a population of various types of cards (e.g., left over from a previous selling season) to fixed-sized “variety packs” which guarantee a given level of variety (i.e., no more than k of any type of card). Some card types with large numbers (perhaps the most unpopular from the previous season) may have to be discarded to preserve the variety constraint. The method developed employs a test for feasibility of a given number of packs and includes a simple allocation procedure. A numerical example is provided along with (worst-case) complexity calculations. In addition, we solve a practical problem in which an organisation marketing Christmas cards sought to determine the impact of pack size and variety level on the level of unallocated cards. Scope and purpose Organisations involved in the stocking and sale of seasonal “style” items such as greeting cards, may periodically face excess stock of various items (e.g., from previous seasons), and may decide to market the items in packs guaranteed to contain a certain degree of variety. Our objective in this paper is to show how to maximise the number of card packs that can be formed from an assortment of excess stock, where there is a marketing-based variety constraint restricting the number of each type of card in each pack. The solution procedure developed is intuitively appealing and can be easily implemented on a spreadsheet. In addition to presenting a numerical example, we provide results from the application and implementation of the method in the card sales operations of a charitable organisation. The method could also have broader application in other settings where variety is sought – such as in groups or teams.