Solving knapsack sharing problems with general tradeoff functions
- J. Randall Brown
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A knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more “knapsack” constraints (an inequality constraint with all non-negative coefficients). All knapsack sharing algorithms to date have assumed that the objective function is composed of single variable functions called tradeoff functions which are strictly increasing continuous functions. This paper develops optimality conditions and algorithms for knapsack sharing problems with any type of continuous tradeoff function including multiple-valued and staircase functions which can be increasing, decreasing, unimodal, bimodal, or even multi-modal. To do this, optimality conditions are developed for a special type of multiple-valued, nondecreasing tradeoff function called an ascending function. The optimal solution to any knapsack sharing problem can then be found by solving an equivalent problem where all the tradeoff functions have been transformed to ascending functions. Polynomial algorithms are developed for piecewise linear tradeoff functions with a fixed number of breakpoints. The techniques needed to construct efficient algorithms for any type of tradeoff function are discussed.
- S. Agnihothri, U. S. Karmarkar and P. Kubat, “Stochastic allocation rules,”Operations Research 30 (1982) 545–555.
- J.R. Brown, “The knapsack sharing problem,”Operations Research 27 (1979) 341–355.
- J.R. Brown, “The sharing problem,”Operations Research 27 (1979) 324–340.
- J.R. Brown, “The flow circulation sharing problem,”Mathematical Programming 25 (1983) 199–227.
- J.R. Brown, “The linear sharing problem,”Operations Research 32 (1984) 1087–1106.
- J.R. Brown, “Decision utility, Chapter 31,” Kent State University Working Paper (Kent, OH, 1988).
- J.R. Brown, “Sharing (maximin and minimax) constrained optimization,” Kent State University Working Paper (Kent, OH, 1989).
- W. Czuchra, “A graphical method to solve a maximin allocation problem,”European Journal of Operational Research 26 (1986) 259–261.
- H.A. Eiselt, “Continuous maximin knapsack problems with GLB constraints,”Mathematical Programming 36 (1986) 114–121.
- T. Ichimori, “On min-max integer allocation problems,”Operations Research 32 (1984) 449–450.
- S. Jacobsen, “On marginal allocation in single constraint min-max problems,”Management Science 17 (1971) 780–783.
- U.S. Karmarkar, “Equilization of runout times,”Operations Research 29 (1981) 757–762.
- H. Luss, “An algorithm for separable non-linear minimax problems,”Operations Research Letters 6 (1987) 159–162.
- H. Luss and D.R. Smith, “Resource allocation among competing activities: a lexicographic approach,”Operations Research Letters 5 (1986) 227–231.
- H. Mendelson, S. Pliskin and U. Yechiali, “Optimal storage allocation for serial files,”Communications of the ACM 22 (1979) 124–130.
- H. Mendelson, S. Pliskin and U. Yechiali, “A stochastic allocation problem,”Operations Research 28 (1980) 687–693.
- K.M. Mjelde, “Max-min resource allocation,”BIT 23 (1983) 537.
- J.T. Mohat,Elementary Functions: An Introduction (Addison-Wesley, Reading, MA, 1970).
- E.L. Porteus and J.S. Yormark, “More on min-max allocation,”Management Science 17 (1972) 502–507.
- C.S. Tang, “A max-min allocation problem: its solutions and applications,”Operations Research 36 (1988) 359–367.
- Z. Zeitlin, “Integer allocation problems of min-max type with quasiconvex separable functions,”Operations Research 29 (1981) 207–211.
- Solving knapsack sharing problems with general tradeoff functions
Volume 51, Issue 1-3 , pp 55-73
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- Print ISSN
- Online ISSN
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- Maximin programming
- knapsack sharing problems
- multiple-valued objective functions
- staircase objective functions
- Industry Sectors
- J. Randall Brown (1)
- Author Affiliations
- 1. Graduate School of Management, Kent State University, 44242, Kent, OH, USA