Donald C. Aucamp

Nonlinear freight costs in the EOQ problem

Abstract This note gives a solution to a qualification of the standard EOQ problem in which freight costs are at least partially determined by the integer number of carloads required to fill the order. The model also applies to a broad class of problems in which there are multiple set-up costs.

A table for the calculation of safety stock

Abstract Safety stock is often calculated to accommodate a given service level (SL), where service is defined in some manner by the user. In this regard, the two favorite definitions of service are the percent of good cycles and the fraction of demand satisfied by off-the-shelf inventory. Judging from its increased appearance in the literature, especially in textbooks, the latter definition apparently is gaining in popularity. Unfortunately, this definition requires a little more effort to use. A procedure originally worked out by Brown (Browns Method) shows how the calculations are to proceed. The partial expectation, E(z), is first calculated as a function of the expected lead time demand (μ), the standard deviation (σ) of lead time demand, and the required service level. Once E(z) is found, a table is consulted (Browns Table) to determine z. Finally, the safety stock is calculated from the formula, zσ. There are several problems with using Browns Table. First, z is given as the dependent variable rather than the independent variable, so the user has to search a bit to find the nearest pair of entries that straddle the required value of E(z). Then the user must perform an awkward interpolation, something many business students find very confusing. An additional problem with the table is that it is both sparse and inaccurate for high values of z. In fact, there are no entries at all for z 4.50, and there is only one significant digit of accuracy for z 3.93. Finally, Brown does not supply a computer program that calculates z directly from E(z). While this lack of a program is not important to the casual user of the tables, it can be important when one wishes to set safety stock levels automatically and internally by the computer. This article presents a new table, in which E(z) is the dependent variable rather than the independent variable. The computer program used to evaluate the entries yields solutions that are accurate to about fourteen decimal places. With this kind of precision the authors were able to extend Browns Table to high values of z, while at the same time providing for virtually any number of significant digits. Incidentally, the entry density in this table is nonuniform (somewhat logarithmic) to accommodate usage over various orders of magnitude of z and E(z). The article also includes a simpler version of our computer program (single precision), which yields about four decimal place accuracy. The double precision program actually used is available on request, but it is somewhat complicated and depends on the computer used (the authors used a TRS-80).

A solution to the multiple set-up problem

This note finds the optimal order quantity for the multiple set-up problem. The solution is obtained by a direct method.

Implied Backorder Costs

Abstract Management often has a policy concerning the maximum allowed backorder delay. In effect, this policy tacitly implies a certain backorder cost, which is developed in this paper. This implied cost can be used to ascertain whether the current policy is in the right ballpark. By calculating the implied backorder cost when the allowed delay is set to the smallest value of practical interest (say, one day), management is able to determine the critical backorder cost. An actual value higher than this means that backorders should not be planned.

A test for the difference of means

This paper offers A new test for the difference of means when the ratio of variances is unknown (the Behrens Fisher problem). The proposed test is akin to the familiar z-test, which is sometimes used when the sample sizes are large, with the exception (hat there is a variance correction factor. Evidence is presented which indicates the proposed test significantly outperforms the test.

Graphical analysis of duality and the Kuhn-Tucker conditions in linear programming

Abstract Careful inspection of the geometry of the primal linear programming problem reveals the Kuhn-Tucker conditions as well as the dual. Many of the well-known special cases in duality are also seen from the geometry, as well as the complementary slackness conditions and shadow prices. The latter at demonstrated to differ from the dual variables in situations involving primal degeneracy. Virtually all the special relationships between linear programming and duality theory can be seen from the geometry of the primal and an elementary application of vector analysis.

The Transportation Problem with Penalty Costs

Abstract The transportation problem with excess total demand and varying penalty costs for not meeting the individual demands is converted to a standard transportation problem.

Aggregate Backorder Exchange Curves

In an effort to avoid estimating backorder costs in developing an inventory strategy, this paper derives an exchange curve which can be used by upper management to decide on backorder policy. Specifically, the exchange curve shows the trade-off between average dollar value of aggregate inventory versus the maximum allowed backorder delay.

Doubling-up in craps and other games of chance

This note formalizes the popular doubling-up (DU) betting strategy. We develop expressions for success probability and the corresponding number of wins necessary to achieve a specific wealth goal, as well as the expected number of plays given that the goal is achieved. Three popular games of chance provide numerical illustration of the process. The doubling-up (DU) betting strategy has been a favorite tactic employed by gamblers throughout the ages. Assume the player is involved in a game which possesses two outcomes (win or lose) and pays even money. He begins by making an initial betB 1 If he wins, the playing sequence terminates, and he starts the next sequence with a second initial wagerB 2. If he loses on the first bet, he bets 2B 1 on the next game (i.e. he“doubles up”). If he loses again, the next wager is 4B 1 and so on until the sequence terminates with a success or bankruptcy. It is clear that the player enters each playing sequence with the hope of winning an amount equal to his initial wager. ...

A tutorial proof of the integrality of transportation solutions

Abstract A simple, verbal proof is offered that shows transportation algorithms automatically yield integral solutions when supplies and demands are integral. Thus, there is no need to avoid this aspect of the theory in introductory texts or in classroom presentations to non-mathematically sophisticated audiences. A fallout of the proof is the fact that tree solutions are basic and vice-versa.