Mathematical Programming

, Volume 51, Issue 1–3, pp 55–73

Solving knapsack sharing problems with general tradeoff functions

  • J. Randall Brown


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.

Key words

Maximin programming knapsack sharing problems multiple-valued objective functions staircase objective functions 


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  1. [1]
    S. Agnihothri, U. S. Karmarkar and P. Kubat, “Stochastic allocation rules,”Operations Research 30 (1982) 545–555.Google Scholar
  2. [2]
    J.R. Brown, “The knapsack sharing problem,”Operations Research 27 (1979) 341–355.Google Scholar
  3. [3]
    J.R. Brown, “The sharing problem,”Operations Research 27 (1979) 324–340.Google Scholar
  4. [4]
    J.R. Brown, “The flow circulation sharing problem,”Mathematical Programming 25 (1983) 199–227.Google Scholar
  5. [5]
    J.R. Brown, “The linear sharing problem,”Operations Research 32 (1984) 1087–1106.Google Scholar
  6. [6]
    J.R. Brown, “Decision utility, Chapter 31,” Kent State University Working Paper (Kent, OH, 1988).Google Scholar
  7. [7]
    J.R. Brown, “Sharing (maximin and minimax) constrained optimization,” Kent State University Working Paper (Kent, OH, 1989).Google Scholar
  8. [8]
    W. Czuchra, “A graphical method to solve a maximin allocation problem,”European Journal of Operational Research 26 (1986) 259–261.Google Scholar
  9. [9]
    H.A. Eiselt, “Continuous maximin knapsack problems with GLB constraints,”Mathematical Programming 36 (1986) 114–121.Google Scholar
  10. [10]
    T. Ichimori, “On min-max integer allocation problems,”Operations Research 32 (1984) 449–450.Google Scholar
  11. [11]
    S. Jacobsen, “On marginal allocation in single constraint min-max problems,”Management Science 17 (1971) 780–783.Google Scholar
  12. [12]
    U.S. Karmarkar, “Equilization of runout times,”Operations Research 29 (1981) 757–762.Google Scholar
  13. [13]
    H. Luss, “An algorithm for separable non-linear minimax problems,”Operations Research Letters 6 (1987) 159–162.Google Scholar
  14. [14]
    H. Luss and D.R. Smith, “Resource allocation among competing activities: a lexicographic approach,”Operations Research Letters 5 (1986) 227–231.Google Scholar
  15. [15]
    H. Mendelson, S. Pliskin and U. Yechiali, “Optimal storage allocation for serial files,”Communications of the ACM 22 (1979) 124–130.Google Scholar
  16. [16]
    H. Mendelson, S. Pliskin and U. Yechiali, “A stochastic allocation problem,”Operations Research 28 (1980) 687–693.Google Scholar
  17. [17]
    K.M. Mjelde, “Max-min resource allocation,”BIT 23 (1983) 537.Google Scholar
  18. [18]
    J.T. Mohat,Elementary Functions: An Introduction (Addison-Wesley, Reading, MA, 1970).Google Scholar
  19. [19]
    E.L. Porteus and J.S. Yormark, “More on min-max allocation,”Management Science 17 (1972) 502–507.Google Scholar
  20. [20]
    C.S. Tang, “A max-min allocation problem: its solutions and applications,”Operations Research 36 (1988) 359–367.Google Scholar
  21. [21]
    Z. Zeitlin, “Integer allocation problems of min-max type with quasiconvex separable functions,”Operations Research 29 (1981) 207–211.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1991

Authors and Affiliations

  • J. Randall Brown
    • 1
  1. 1.Graduate School of ManagementKent State UniversityKentUSA

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