Minimizing tardiness

The occurrence of the set-up procedure in period i is denoted by the binary variable Wi (0 = no, 1 = yes). The production costs per batch are denoted by p = 1.0 and the cost for a set-up is y = 3.0. Demands di that are satisfied in the same period as requested result in a regular sale Mi with a full revenue of a = 2.0 per unit of product. Demands that are satisfied with a tardiness of one period result in a late sale Mf with a reduced revenue of aL = 1.5 per unit. Demands which are not satisfied in the same or in the next period result in a deficit Bf with a penalty of a = 0.5 per unit. The surplus production of each period is stored and can be sold later. The amount of batches stored at the end of a period is denoted by Mf and the storage costs are a+ =0.1 per unit. The objective is to maximize the profit over a horizon of H periods. The cost function P contains terms for sales revenues, penalties, production costs, and storage costs. For technical reasons, the model is reformulated as a minimization problem ... 

As an example, consider a problem of sequencing three jobs on a single machine to minimize the sum of weighted tardiness for all jobs, where tardiness is defined as the difference between the completion time of a job and its due date if this difference is positive, and zero otherwise. Job processing times, due dates, and delay penalties for an instance of this problem are shown in Table 10.1. 

Table 1. Tardiness minimization (T(,pu) eomputation time in CPU sec., (Tgjgt) computation time required to reach the first feasible solution, (Obj.) objeetive value, (Opt.sol.nodes) total number of nodes explored to reach the optimal solution, (dise.) number of discrete variables, (eont.) number...

Pm rj,SjJi2wjTj denotes m identical machines in parallel, n jobs with different release dates, different due dates, and different weights. The jobs are subject to sequence-dependent setup times and the objective is to find a sequence that minimizes the sum of the weighted tardinesses. 

Emmons, H. (1969), One-Machine Sequencing to Minimize Certain Functions of Job Tardiness, Operations Research, Vol. 17, pp. 701-715. 

Jackson, J. R. (1955), Scheduling a Production Line to Minimize Maximum Tardiness, Research Report 43, Management Science Research Project, University of California, Los Angeles. 

Consider for example a production unit supplied by one or more raw material [5]. An order cannot be released before aU the required has arrived. That is an order j has an earliest possible starting time (release date rj), a committed shipping date dj, and a priority factor or weight Wj. Every time a machine switches over from one type of item to another, a setup may be required and a setup cost may be incurred. The supply chain model here is composed of a model within each production unit depending on its structure, job shop, flowshop, single machine, etc., and an objective function that links the two models. The objective to be minimized may include the minimization of the total setup times and the total tardiness 7 denoted as,... 

Consider a supply chain of three levels in series shown in Fig. 3.10. The first and most upstream level (Level 1) has two factories in parallel producing two major products, FI and F2, in full production capacity of 168 h (24 x 7) a week. They both feed a distribution center (DC) in Level 2 and deliver to a common customer in Level 3. Products can also be delivered to the customer by the distribution center. Both factories have no room for finished goods storage and the customer does not want to receive any early deliveries. The medium term planning production timing and quantities that minimizes the total cost of production cost, storage cost, transportation cost, tardiness cost for the whole supply chain over a 4 week time horizon, (the unit of time being one week). The transportation time from any one of the two factories to the DC, firom any one of the two factories to the customer, and from the DC to the customer all transportation times are assumed to be identical and equal to one week. 

Minimize the total of the production costs, storage costs, transportation costs, tardiness costs, and penalty costs for non-delivery over a horizon of 4 weeks. The problem is formulated as a Mixed Integer Program as follows ... 

Baker and Scudder, 1990). In a JIT enviromnent, both earliness and tardiness must be discouraged, since jobs finished early increase inventory cost while late jobs lead to customers dissatisfaction and loss of business goodwill. Thus an ideal schedule is one in which all jobs finish within the assigned due dates. The objectives of early/tardy (E/T) scheduhng could be interpreted in different ways, for example minimizing total absolute deviation from due dates, job-dependent earhness and tardiness penalties, non-linear penalties, and so forth (see Baker and Scudder, 1990 for a comprehensive survey). 

Bagchi, U., Sullivan, R.S. and Chang, Y.L., 1987. Minimizing mean square deviation of completion times about a common due date. Management Sciences, 33, 894-906. Baker, K.R. and Scudder, G.D., 1990. Sequencing with earliness and tardiness penalties a review. Operations Research, 38(1), 22-36. 

De, P, Ghosh, J.B. and Wells, C.E., 1991. Scheduling to minimize weighted earliness and tardiness about a common due-date. Computers Operations Research, 18(5), 465 75. De, R, Ghosh, J.B. and Wells, C.E., 1993. On the general solution for a class of early/tardy problems. Computers Operations Research, 20(2), 141-149. 

Considering the problem description given in Sect. 2, the purpose of the proposed model is the determination of repairing equipment assignment to candidate repair facilities and material flows between 1) collection centers and repair facilities, 2) repair facilities and production plants, and 3) repair facilities and disposal centers in order to minimize total fixed costs and transportation costs as well as total tardiness of shipping returned products back to collection centers after the... 

Objective function (26.2) minimizes the total weighted tardiness of returning repairable products and new products to the collection centers. 

A related problem, where customers have a common preferred due date A, is studied in [87]. A lead time penalty is charged for lead time (or due date) delay, Aj = max 0, dj — A, which is the amount of time the assigned due date of a job exceeds the preferred due date, A. The objective is to minimize the weighted sum of earliness, tardiness and lead time penalty. The authors propose a simple policy for setting the due dates and show that this policy is optimal when used in conjunction with the SPT rule. 

Cheng [25] studies the SLK due date rule under the objective of minimizing the (weighted) flow allowance plus the maximum tardiness. He shows that the optimal sequence is EDD and derives a simple function to compute the optimal SLK due dates. Gordon [45] studies a generalized version of this problem with... 

Chamsirisakskul et al. [16] take a profit-maximization rather than a cost-minimization perspective and consider order acceptance in addition to scheduling and due-date setting decisions. In their model, an order is specified by a unit revenue, arrival time, processing time, tardiness penalty, and preferred and latest acceptable due dates. While they allow preemption, they assume that the entire order has to be sent to a customer in one shipment, i.e., pieces of an order... 

One of the earliest papers that studies DDM is by Conway [27]. He considers four due date policies, CON, NOP, TWK and RND, and tests the performance of nine priority dispatching rules for each of these policies in a job shop. For the objective of minimizing average tardiness, the author finds that under FCFS sequencing, the four due date policies exhibit similar, mediocre... 

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