Estimation of the Largest and the Smallest Function Values of a Feasible Solution for the Total Product Rate Variation Problem

The problem of minimizing the total deviations between the actual and the ideal cumulative production of a variety of models of a common base product arises as a sequencing problem in mixed-model just-in-time production systems. This is called the total product rate variation problem. Several pseudo-polynomial exact algorithms and heuristics have been derived for this problem. In this paper, we estimate the largest and the smallest function values of a feasible solution for the problem when the m-th power of the total deviations between the actual and the ideal cumulative productions has to be minimized.


INTRODUCTION
Many companies have changed the assembly lines from paced single-model lines for mass production to mixedmodel assembly lines for mass customization of a variety of models of a common base product.Just-intime production system which requires producing only the necessary products in the necessary quantities at the necessary times usually uses mixed-model assembly lines.Mixed-model just-in-time production systems with negligible change-over costs between the models have been used in order to respond to the customer demands for a variety of models of a common base product without holding large inventories or incurring large shortages.One of the most important problems for the effective utilization of the systems consists in sequencing different models with keeping the rate of usage of all parts used by the assembly lines as constant as possible.This problem is known as the mixed-model just-in-time sequencing problem (abbreviated as MMJITSP).The problem of minimizing the variation in the rate at which different models are produced on the line is called the product rate variation problem (abbreviated as PRVP).The latter problem is the singlelevel case of MMJITSP.The problem of minimizing the total deviations between the actual cumulative productions from the ideal one is called the total PRVP (abbreviated as TPRVP), see Kubiak (1993).This problem has been widely investigated in the literature since it has a model with a strong mathematical base and wide real-world applications, see Dhamala and Khadka (2009), a recent survey and therein.In Kubiak (1993), Kubiak solved the TPRVP with a general objective in pseudo-polynomial time     .The problem is transformed into an equivalent assignment problem.Moreover, several heuristics also exist in the literature for near to optimal solutions, see Dhamala and Khadka (2009).In this paper, we propose a lower and an upper bound for TPRVP.We also establish an explicit lower bound of the problem.The remainder of the paper is as follows.In the second section, we present a non-linear integer programming formulation.In the third section, we estimate the largest and the smallest function values of a feasible solution of the problem which is the major contribution of this paper.First, the level curves are investigated, then the largest function value and finally the smallest function value.The last section concludes the paper.

NON-LINEAR INTEGER PROGRAMMING FORMULATION
Let  be the total demand of n different models with . The time horizon is partitioned into  equal time units under the assumption that each copy of a model  be the demand rate.Let   and    be the actual and the ideal cumulative productions, respectively, of model  produced during the time units  through  .An inventory holds if          , and a shortage incurs if          .We assign the same cost for both inventory and shortage.Miltenburg (1989) and Kubiak and Sethi (1991) gave an integer programming formulation for TPRVP as follows with  being a positive integer:

ESTIMATIONS Level curve
There exist  deviations between the actual and the ideal cumulative productions of  copies of  models.The value of the actual cumulative production A sequencing time unit           , means that a copy of a model is produced during the time units from    to .One seeks smaller value of  so that the total deviations between actual and the ideal cumulative productions can be reduced with the sequencing time units not exceeding the planning horizon.
It is important to establish the largest and the smallest function values of a feasible solution of the problem so that one can minimize the total deviations in a reasonable time.A sequence corresponding to the minimum value, denoted as   , which satisfies the inequality is optimal for TPRVP.A necessary and sufficient condition for the existence of a feasible sequence for the product rate variation problem with the objective of minimizing intersects with the time interval within which copy   is sequenced, see Brauner and Crama (2004).Theorem 1: Let   be the largest function value of a feasible solution for TPRVP.Then Given   be the largest function value of a feasible solution for the problem.Then the value   satisfies the inequality Let   be the largest function value of a feasible solution for the problem with the objective function The value   satisfies the inequality Then, we can write and The two inequalities (1) and ( 2) show that  is one of the largest function values of a feasible solution for TPRVP with the objective function

Smallest functions value
If an instance has a feasible sequence at the smallest function value of a feasible solution, the sequence is optimal.However, not all instances are even feasible at this value.respectively.These bounds can be used to develop an     exact solution procedure recently given by Khadka and Werner (2014) which improves the known exact algorithm by Kubiak from (1993) with a complexity of     .
Fig. Level curves   for the instance            Largest function valueWe set a horizontal line with a suitable value   intersecting the level curve for each copy                      , of the objective function of TPRVP on the planning horizon  .The horizontal line with the value  is called a bound for TPRVP.The intersecting points of the level curve of the objective function for each copy and the value B are important to determine the sequencing time units for all copies of all models.A sequencing time unit           , means that a copy of a model is produced during the time units from    to .One seeks smaller value of  so that the total deviations between actual and the ideal cumulative productions can be reduced with the

Theorem 2 :
If   be the smallest function value of a feasible solution for TPRVP, then                    must be sequenced at the time unit   .So, one can replace   by  at   .Which can be written as                        The inequality consists of  terms of        .Thus,          Hence, the smallest function value of a feasible solution for TPRVP is           