Multiple Setup Costs Research Articles

Dynamic Multi-Product Procurement with Joint and Individual Setup Costs: Theory and Insights

In practice, it is common, especially for online retailers, to bundle different products together during procurement to save transportation and handling costs. It is important to understand and theorize how to manage dynamic procurement by taking advantage of joint ordering in the presence of joint and individual setup costs. We characterize the structure of optimal policy for a periodic review multi-product inventory system with multiple setup costs, including a joint setup cost and individual setup costs for each product. By proposing the notion of ($\mathcal K,\bm \eta$)-quasi-convexity, we show that an optimal procurement policy for such a system follows the so-called ($\sigma,\omega,\bm S$) policy when demands are stochastically increasing over time: order up to $\bm S$ for states in the region $\sigma$, do not order for states in the region $\omega$, and order certain quantities for states neither in $\sigma$ nor in $\omega$. We also obtain ordering monotonicity properties that characterize the structure of the optimal policy, and we provide some bounds for the parameters of the optimal policy. Our results can also accommodate inter-product demand correlation, such as customer choice for different products. Leveraging these operational insights, we propose three simple heuristic policies: the independent ($\bm s,\bm S$) policy, vector ($\bm s,\bm S$) policy and linear interpolation ($\bm s,\bm S$) policy. Extensive numerical experiments indicate that the linear interpolation ($\bm s, \bm S$) policy outperforms the other two heuristics, with the performance gap being less than 3.5% on average and only 0.5% in some cases. Finally, we show that how our results can be extended to systems with more complex setup cost functions, such as time-varying, set-based, and quantity-dependent setup costs.

Heuristic algorithms for inventory replenishment with perishable products and multiple transportation modes

ABSTRACTThis study extends classic economic lot-sizing problems to permit the replenishment of age-dependent perishable inventories via multiple transportation modes. Inventory replenishment costs include a multiple-setup cost function that considers order setup, purchase, and cargo container costs. The objective is to identify the timing of orders, order quantities, and a choice from among I transportation modes that minimizes the cost of replenishing perishable inventories during a planning horizon of length T. We present a mixed-integer programming formulation of this problem and characterize properties of optimal solutions. We propose a primal-dual heuristic algorithm that runs in O(IT2). In addition, we provide heuristic algorithms for two special cases of the problem involving one or two replenishment modes. For the single replenishment mode problem, we propose (i) a dynamic programming algorithm that explores solutions that satisfy the Zero Inventory Ordering Policy and runs in O(T2) and (ii) a dynamic programming algorithm that explores solutions that satisfy the Less-than-Truckload first positioning property and runs in O(T3). For the two replenishment mode problem, we present a knapsack-based algorithm that identifies the minimum number of cargo containers required to meet demand. The running time of this algorithm is O(T2). We evaluate the quality of the solutions generated by these different approaches via extensive numerical analyses.

Optimality of (s, S) Inventory Policies under Renewal Demand and General Cost Structures

We study a single‐stage, continuous‐time inventory model where unit‐sized demands arrive according to a renewal process and show that an ( s, S) policy is optimal under minimal assumptions on the ordering/procurement and holding/backorder cost functions. To our knowledge, the derivation of almost all existing ( s, S)‐optimality results for stochastic inventory models assume that the ordering cost is composed of a fixed setup cost and a proportional variable cost; in contrast, our formulation allows virtually any reasonable ordering‐cost structure. Thus, our paper demonstrates that ( s, S)‐optimality actually holds in an important, primitive stochastic setting for all other practically interesting ordering cost structures such as well‐known quantity discount schemes (e.g., all‐units, incremental and truckload), multiple setup costs, supplier‐imposed size constraints (e.g., batch‐ordering and minimum‐order‐quantity), arbitrary increasing and concave cost, as well as any variants of these. It is noteworthy that our proof only relies on elementary arguments.

Stochastic Dynamic Inventory Problem Under Explicit Inbound Transportation Cost and Capacity

We study a practical generalization of the classical stochastic dynamic inventory problem where privately owned trucks with limited cargo capacity are used to transport the replenishment quantity. The resulting replenishment cost function also includes the traditional fixed setup cost, and it is known as a multiple setup cost structure, which leads to complicated cost-to-go functions in the problem of interest. We introduce the concepts of non-(Δ, C)-decreasing and [Formula: see text], and we develop a sophisticated replenishment policy called the [Formula: see text] policy. We examine the sufficient conditions under which the new policy is optimal. Our results offer a detailed characterization of optimal policies and generalize the existing theory on the concepts of K-convexity and non-K-decreasing. In doing so, we are able to investigate a traditional stochastic inventory problem which has remained open in the literature for more than four decades. The online appendix is available at https://doi.org/10.1287/opre.2017.1625 .

Optimal Policies for Brownian Inventory Systems With a Piecewise Linear Ordering Cost

We consider a stochastic inventory system with general piecewise linear ordering cost. The cumulative demand is modeled as a Brownian motion process. The ordering cost function is neither convex nor concave; it may not be monotone; and it is not even necessarily continuous, and it includes most ordering cost functions studied in the literature, e.g., economies-of-scale or dis-economies of scale, all-unit discount or incremental discount, and multiple setup costs, as special cases. In addition to ordering cost, the system incurs the usual holding/shortage cost, and the objective is to minimize the average system cost per unit of time. Despite the complexity in the ordering cost function, we show that an optimal control policy is very simple: it is either an $(s,S)$ policy or a one-sided singular control policy.

Inventory Control with Multiple Setup Costs

We consider an infinite-horizon, periodic-review, single-item production/inventory system with random demand and backordering, where multiple setups are allowed in any period and a separate fixed cost is associated for each setup. Contrary to the majority of the literature on this topic, we do not restrict the order quantities to be integer multiples of the exogenously given batch size and instead allow the possibility of partial batches, in which case the fixed cost for ordering the batch is still fully charged. We build a model that particularly takes the batch-ordering cost structure into account. We introduce an alternative cost-accounting scheme to analyze the problem, which we use to develop a computationally efficient optimal solution method and several properties of the optimal solution. In addition, we propose two heuristic policies, both of which perform extremely well computationally.

Uncertainty estimation for multiposition form error metrology

We analyze a general multiposition comparator measurement procedure that leads to partial removal of artifact error for a class of problems including roundness metrology, measurement of radial error motions of precision spindles, and figure error metrology of high-accuracy optical components. Using spindle radial error motion as an explicit example, we present a detailed analysis of a complete test with N orientations of a test ball with respect to the spindle. In particular, we show that (1) all components of the ball roundness error average to zero except those with frequencies of kN cycles/revolution, where k is a positive integer; and (2) the combined standard uncertainty of the measurement is proportional to 1/√N. We then show how a complete set of measurements for an N-position test can be synthesized from only two measurements, and we derive a general expression for the combined standard uncertainty as a function of the number of positions n (2 ≤ n ≤ N) actually measured in an N-position test. This uncertainty can serve as a useful guide to measurement design, involving trade-offs between multiple setup cost and complexity and required levels of angular harmonic resolution and combined standard measurement uncertainty.

Uncertainty estimation for multiposition form error metrology

We analyze a general multiposition comparator measurement procedure that leads to partial removal of artifact error for a class of problems including roundness metrology, measurement of radial error motions of precision spindles, and figure error metrology of high-accuracy optical components. Using spindle radial error motion as an explicit example, we present a detailed analysis of a complete test with N orientations of a test ball with respect to the spindle. In particular, we show that (1) all components of the ball roundness error average to zero except those with frequencies of kN cycles/revolution, where k is a positive integer; and (2) the combined standard uncertainty of the measurement is proportional to 1/√ N. We then show how a complete set of measurements for an N-position test can be synthesized from only two measurements, and we derive a general expression for the combined standard uncertainty as a function of the number of positions n (2 ≤ n ≤ N) actually measured in an N-position test. This uncertainty can serve as a useful guide to measurement design, involving trade-offs between multiple setup cost and complexity and required levels of angular harmonic resolution and combined standard measurement uncertainty.

The only correct method to evaluate roundness

We analyze a general multiposition comparator measurement procedure that leads to partial removal of artifact error for a class of problems including roundness metrology, measurement of radial error motions of precision spindles, and figure error metrology of high-accuracy optical components. Using spindle radial error motion as an explicit example, we present a detailed analysis of a complete test with N orientations of a test ball with respect to the spindle. In particular, we show that (1) all components of the ball roundness error average to zero except those with frequencies of kN cycles/revolution, where k is a positive integer; and (2) the combined standard uncertainty of the measurement is proportional to 1/√N. We then show how a complete set of measurements for an N-position test can be synthesized from only two measurements, and we derive a general expression for the combined standard uncertainty as a function of the number of positions n (2 ≤ n ≤ N) actually measured in an N-position test. This uncertainty can serve as a useful guide to measurement design, involving trade-offs between multiple setup cost and complexity and required levels of angular harmonic resolution and combined standard measurement uncertainty.

A Solution to the Multiple Set-Up Problem with Dynamic.Demand

Abstract In this note, we study an important practical problem, the classic dynamic lot size model with set-up cost including a fixed cost and freight cost, where the freight cost is proportional to the number of containers used. This freight cost is the additional multiple set-up cost. We solve the T-period problem in O(T44) effort using dynamic programming, Handled by the Editor-in-Chief.

Nonlinear freight costs in the EOQ problem

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.

OPTIMAL BATCH (s , S) POLICIES FOR THE MULTIPLE SET-UP ORDERING COSTS IN INVENTORY PROBLEM

Consider the dynamic inventory problem when the ordering cost functioYl is linear with multiple set-up costS. In general, an optimal inventory policy is sensitive to the form of the ordering cost, so that the purpose of this note is to define a new policy, ie, a batch (s,S) policy and to show the sufficient conditions under which this policy is optimal.

Economic Order Quantities and Multiple Set-Up Costs

We consider the problem of scheduling the production of a single product at each instant during a time horizon of length T (≦ ∞) so as to minimize the average cost per unit time; backlogging of demand and disposal of stock are not allowed. Two types of costs are incurred; the holding cost per unit time is assumed to be proportional to the inventory level while the ordering cost is that associated with the multiple set-up cost function (see (3) and Figure 1). By studying a special case (K = 0 in (3)) in §2, the analysis of the problem practically becomes that of the standard economic lot size model so that we can rigorously derive some not too widely known results about the lot size model while exploring our model. In particular, we obtain planning horizon theorems and characterize the form of an optimal production schedule for T < ∞ and T = ∞. In §3 the special assumption is removed and similar, but slightly weaker, theorems are established. For example, an optimal production schedule may have several orders of two different sizes in contrast to the special case where every order is the same size. In addition, simple algorithms for finding optimal policies for both T < ∞ and T = ∞ are given.

Optimal Inventory Policy with Multiple Set-Up Costs

We consider a deterministic, single product, discrete review, finite time horizon inventory problem, called the multiple set-up cost problem. The holding cost in each period is a nondecreasing (and sometimes concave) function. The distinguishing feature of our model is the ordering cost function which is neither concave nor convex. The ordering cost is such that a natural interpretation of it consists in assuming that the order in period i is delivered in trucks with capacity Mi and that the cost of delivery for each truck is a nondecreasing concave function of the amount delivered by that truck. We establish the existence of an optimal production schedule such that for each period (1) there are no partially filled trucks in period i if the inventory entering period i is positive and (2) the inventory at the end of period i is less than Mi. Exploiting this information, an efficient algorithm is developed. In part II, we study the stationary, infinite horizon version of the multiple set-up cost problem. We single out a countable set S of schedules, each of which possess a periodic property in addition to properties (1) and (2) above, and we show that S contains a schedule with minimal cost per unit time. Moreover, if the ratio of demand per period to Mi is rational, then S contains a schedule with minimal discounted cost.

Optimal Inventory Policy with Subadditive Ordering Costs and Stochastic Demands

We consider a discrete review, single product, dynamic inventory model. With the exception of the ordering cost function, we make the usual assumptions (see [8]). The distinguishing feature of our model is that we merely assume that the ordering cost function for each period is non-decreasing and subadditive. Denoting the smallest minimizer of the expected costs for period i by $\bar y_i $, we establish that when x is the inventory level entering period i, the optimal stock level $y_i (x)$ after ordering in period i is x if $x\geqq \bar y_i $. Moreover, if the ordering cost (apart from a linear term) is eventually constant, then we can show that $y_i ( \cdot )$ is lower semicontinuous and satisfies $y_i (x) = S_i $ for $x < s_i $ and $y_i (x)\leqq S_i $ for $s_i \leqq x < \bar y_i $ where $s_i \leqq \bar y_i \leqq S_i $. Also, simple conditions are given which imply (the planning horizon theorem) $S_i = \bar y_i $. Upon assuming that the ordering cost is the multiple set-up cost, a partial characterizatio...