Hindawi Publishing Corporation ξ e Scientiο¬c World Journal Volume 2015, Article ID 804968, 7 pages http://dx.doi.org/10.1155/2015/804968
Research Article Integrated Production and Distribution Scheduling Problems Related to Fixed Delivery Departure Dates and Weights of Late Orders Shanlin Li and Maoqin Li Department of Mathematics, Taizhou University, Taizhou, Zhejiang 317000, China Correspondence should be addressed to Shanlin Li;
[email protected] Received 29 July 2014; Accepted 2 February 2015 Academic Editor: Dar-Li Yang Copyright Β© 2015 S. Li and M. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider an integrated production and distribution scheduling problem faced by a typical make-to-order manufacturer which relies on a third-party logistics (3PL) provider for finished product delivery to customers. In the beginning of a planning horizon, the manufacturer has received a set of orders to be processed on a single production line. Completed orders are delivered to customers by a finite number of vehicles provided by the 3PL company which follows a fixed daily or weekly shipping schedule such that the vehicles have fixed departure dates which are not part of the decisions. The problem is to find a feasible schedule that minimizes one of the following objective functions when processing times and weights are oppositely ordered: (1) the total weight of late orders and (2) the number of vehicles used subject to the condition that the total weight of late orders is minimum. We show that both problems are solvable in polynomial time.
1. Introduction An increasing number of companies now adopt make-toorder business models in which products are custom-made and delivered to customers within a very short lead time directly from the factory. Consequently, there is little or no finished product inventory in the supply chain such that production and outbound distribution are intimately linked and must be scheduled jointly. A majority of the companies worldwide rely on the 3PL providers for their daily distribution and other logistics needs [1]. 3PL providers often follow a fixed daily or weekly schedule for serving their customers. For example, many package delivery service providers such as UPS and FedEx have daily fixed package pickup times; and most 3PL rail, ocean, and air freight service providers have a fixed weekly schedule for a specific origin-destination pair. In this paper, we study integrated production and outbound distribution scheduling decisions commonly faced by many manufacturers that operate in a make-to-order mode and rely on a 3PL provider for finished product delivery to customers where the 3PL provider follows a fixed delivery schedule. Examples of such manufacturers include most
high-end custom-made consumer electronics product manufacturers based in Asia that rely on air flights (which have fixed departure times) to deliver finished products to the US and European markets. The production and distribution scheduling problem faced by such a manufacturer can be described as follows. At the beginning of a planning horizon, the manufacturer has received a set π½ = {π½1 , π½2 , . . . , π½π } of π independent orders from its customers to be processed on a single assembly line. Order π½π has a processing time ππ , a weight π€π and a desired due date ππ which is negotiated and agreed on by the manufacturer and the customer who placed the order. Finished orders are delivered by vehicles which have fixed departure times. In the planning horizon, there are π§ possible vehicle departure time instants π1 , π2 , . . . , ππ§ , whereby at time ππ , 1 β€ π β€ π§, there are Vπ vehicles available for delivery. In the air flight case, each vehicle represents an air freight container. Based on a contractual agreement between the manufacturer and the 3PL provider, the manufacturer can use a certain number (e.g., Vπ ) of containers available on a given flight with departure time ππ . Usually the 3PL provider charges the manufacturer a fixed transportation cost for each air freight container used. Thus, the total transportation cost
2 is represented by the total number of vehicles (i.e., total number of containers) used. Each order is packaged into a standard-size pallet for delivery convenience regardless of the order size. Each vehicle can deliver at most πΆ orders (e.g., in the air flight case, each container can hold up to πΆ pallets). The Vπ vehicles can only deliver orders that are completed by time ππ . A feasible schedule is one in which each order has completed processing and delivered by one of the available vehicles. Without loss of generality, we may assume that ππ§ β₯ βππ=1 ππ ; otherwise, there is at least one order that cannot be delivered and hence there is no feasible schedule. In a given feasible schedule, if order π½π is delivered at time ππ and ππ > ππ , we define ππ to be 1; if order π½π is delivered at time ππ and ππ β€ ππ , we define ππ to be 0. We say in a given feasible schedule an order π½π is early if ππ = 0 and late if ππ = 1. The minimum total weight of late orders βππ=1 π€π ππ measures the delivery timeliness relative to the customers desired due dates and is one of the most commonly used measurements in practice. The problem of minimizing βππ=1 π€π ππ is NP-hard as it contains the NP-hard classical single-machine total weight of late orders scheduling problem [2] as a special case when the delivery part is not considered. In this paper, we consider the special case of the problem in which processing times and weights are oppositely ordered; that is, if π€π < π€π , then ππ β₯ ππ for all π½π , π½π β π½. That processing times and weights are oppositely ordered is the case in many practical applications. For example, when production resources are comparative shortage, one would prefer the order with shorter processing time. Our problem is to find a feasible schedule that minimizes one of the following objective functions: (1) βππ=1 π€π ππ and (2) the number of vehicles used subject to the condition that βππ=1 π€π ππ is minimum. We show that all two problems are solvable in polynomial time. Research on integrated production and outbound distribution scheduling problems is relatively recent but has attracted a rapidly growing interest in the last several years [3]. In most of the problems considered in the literature, vehicle departure times are not fixed and need to be determined along with other decisions. Only a handful of problems considered in the literature involve fixed vehicle departure times. Such problems can be classified into two types based on vehicle availability. One type assumes that there are infinite number of vehicles available at each departure time, whereas the other type assumes that there are a limited number of vehicles available at each departure time. Stecke and Zhao [4], Melo and Wolsey [5], and Zhong et al. [6] all consider similar problems with an infinite number of vehicles where each order has a deadline which has to be satisfied and the objective is to minimize the total transportation cost. Their problems differ slightly in the structure of the transportation cost. Since the focus of this paper is on problems with a finite number of vehicles, we do not review these papers in detail. Li et al. [7β9] and Zandieh and Molla-AlizadehZavardehi [10] study several similar problems with a finite number of vehicles at each departure time which are all motivated by applications involving synchronizing assembly operations of consumer electronics products such as PCs and air transportation schedules. Orders may have different sizes and the capacity of a vehicle is measured by the total size
The Scientific World Journal (weight or volume) of orders that it can carry. There is an earliness or tardiness penalty if an order is delivered earlier or later than the due date. The objective is to minimize the total transportation cost and total weighted earliness and tardiness penalty. Li et al. [9] consider the case where all the orders are processed on a number of parallel production lines, whereas the other papers consider the case with a single production line. The problems are strongly NP-hard as they contain the strongly NP-hard classical single-machine total weighted tardiness scheduling problem [11] as a special case when the delivery part is not considered. These papers propose various heuristics for solving their problems. Wang et al. [12] study a problem with a finite number of vehicles which involves coordinating mail processing and distribution schedules at a mail processing and distribution center. The objective is to minimize the total unused vehicle capacity. The authors show that this problem is strongly NP-hard and propose dispatching rules and heuristics. Fu et al. [13] consider a problem where there is a limit on the total delivery capacity at each departure time. Each order has a delivery departure deadline, a production window, a size, and a profit. The problem is to select a subset of orders to accept so as to maximize the total profit of the accepted orders under the constraint that each accepted order is processed within its production window, the delivery of this order is departed by its delivery departure deadline, and the total size of the orders delivered at each departure time does not exceed the available vehicle capacity limit. The problem is strongly NP-hard as it contains the bin packing problem as its special case when only the delivery part is considered. The authors propose a polynomial time approximation scheme for the problem. Leung and Chen [14] discuss an integrated production and distribution scheduling problem. In the beginning of a planning horizon, the manufacturer has received a set of orders to be processed on a single production line. Completed orders are delivered to customers by a finite number of vehicles provided by the 3PL company which follows a fixed daily or weekly shipping schedule such that the vehicles have fixed departure dates. The problem is to find a feasible schedule that minimizes one of the following objective functions: (a) the maximum lateness of orders, (b) the number of vehicles used subject to the condition that the maximum lateness is minimum, and (c) the weighted sum of the maximum lateness and the number of vehicles used. They show that all three problems are solvable in polynomial time. The remainder of this paper is organized as follows. In Section 2, we give a simple algorithm to check the feasibility of a given instance of the problem. In Sections 3 and 4, we give polynomial time algorithms to solve problems (1) and (2), respectively. We conclude the paper in Section 5.
2. Feasibility Given an instance of the problem, we first need to determine whether there is any feasible schedule. The following Algorithm FD come from Leung and Chen [14] and are stated. The idea is to schedule the orders in Smallest-Processing-Time
The Scientific World Journal first (SPT) order. Let π be a SPT schedule. Let π(ππ , ππ ) denote the set of orders completed in the interval (ππ , ππ ] in π. Let ππ§+1 be any integer greater than ππ§ . We then assign the orders to the vehicles by the following algorithm. Algorithm FD Input. An SPT schedule π. For each departure time ππ , for 1 β€ π β€ π§, there are Vπ vehicles available for delivery at ππ . Output. βYesβ if it is possible to deliver all the orders in the schedule π, βNoβ otherwise. Method (1) π(ππ§ , ππ§+1 ) := 0; π0 := 0. (2) For π = 1 to π§ do the following:
3 Method (1) π = βππ=1 ππ ; π‘ := 0. π0 := 0. π0,0 := 0. (2) For π = π§ down to 1 do the following. (2.1) Let π
π‘,π be all the orders in ππ‘,π , arranged in decreasing order of their weights. (2.2) If Vπ πΆ < |π
π‘,π |, then update ππ‘,πβ1 := ππ‘,πβ1 βͺ πΉπ‘,π and π
π‘,π := π
π‘,π \ πΉπ‘,π , where πΉπ‘,π are the first |π
π‘,π | β Vπ πΆ orders from π
π‘,π . (2.3) π := βπ½π βπ
π‘,π ππ . (2.4) Schedule the orders in π
π‘,π from time π β π to π. These orders will be delivered by the vehicles available at time ππ . (2.5) π := π β π. (2.6) If π > ππβ1 , find π½ππ‘ β ππ‘,ππ such that π€ππ‘ = πβ1
(a) assign the orders in π(ππβ1 , ππ ) to one of the Vπ vehicles available at time ππ ; after an order is assigned to a vehicle, it is removed from π(ππβ1 , ππ ); (b) if all of the Vπ vehicles are full and there is still at least one unassigned order in π(ππβ1 , ππ ), then put all the unassigned orders into π(ππ , ππ+1 ); (3) if π(ππ§ , ππ§+1 ) = 0 then return βYes,β else return βNo.β If the algorithm returns βYes,β then there is a feasible schedule; otherwise, there is no feasible schedule. The overall running time of the algorithm is π(π log π) time. Let (π1 , π2 ) be a SPT schedule of the instance of the problem. Clearly, there is no feasible schedule for the instance of the problem if |π2 | = βπ§π=π Vπ πΆ and βπ½π βπ1 ππ > ππβ1 , where 1 β€ π β€ π§. In the remainder of this paper, we will assume that there is a feasible schedule for the given instance.
3. Total Weight of Late Orders In this section we give a polynomial-time algorithm to solve the total weight of late orders problem. For each 1 β€ π β€ π, let ππ = max{ππ | ππ β€ ππ , 1 β€ π β€ π§} and for each 1 β€ π β€ π§, let π0,π = {π½π | ππ = ππ , 1 β€ π β€ π}. Clearly, an order π½π β π0,π is early if and only if it is completed and delivered by time ππ for a given schedule. The following algorithm decides whether there is a feasible schedule that minimizes the total weight of late orders. To break ties when sequencing the orders in decreasing order of their weights, we employ the last-in first rule; that is, we arrange order π½π before order π½π if π½π is merged into a set of orders π and π€π = π€π , where π½π β π. Algorithm WF Input. π½ and π0,1 , . . . , π0,π§ . Output. A feasible schedule that minimizes the total weight of late orders.
min{π€π | π½π β βπ=0 ππ‘,π }, where 0 β€ ππ β€ π β 1. Update ππ‘,0 := ππ‘,0 , ππ‘,1 := ππ‘,1 , . . . , ππ‘,ππ β1 := ππ‘,ππ β1 , ππ‘,ππ := ππ‘,ππ \ {π½ππ‘ }, ππ‘,ππ +1 := := ππ‘,πβ1 , ππ‘,π := ππ‘,ππ +1 , . . . , ππ‘,πβ1 π
π‘,π , . . . , ππ‘,π§β1 := π
π‘,π§β1 , ππ‘,π§ := π
π‘,π§ βͺ {π½ππ‘ }, and π‘ := π‘ + 1; return to (2).
(3) Stop. The schedule (π
π‘,1 , . . . , π
π‘,π§ ) is an optimal feasible schedule, where the orders in π
π‘,π will be delivered by the vehicles available at time ππ and βπ‘β1 π=0 π€ππ = min β π€π ππ . Algorithm WF consists of π‘ + 1 iterations. For ease of presentation, we present the detailed output data obtained by the π‘ + 1 iterations as follows: (π
0,π0 , . . . , π
0,π§ ) , π½π0 , π€π0 ; . . . ; (π
π‘β1,ππ‘β1 , . . . , π
π‘β1,π§ ) , π½ππ‘β1 , π€ππ‘β1 ; (π
π‘,1 , . . . , π
π‘,π§ ) , where for π = 0, 1, . . . , π‘ β 1, 1 β€ ππ β€ π§, π β βπ½π ββπ§π=π βπ§π=ππ
π
π
π,π
(1) ππ >
πππ β1 , π½ππ β π½ \ π
π,π , and π€ππ = min{π€π | π½π β π½ \ βπ§π=ππ π
π,π }. The following lemma describes some properties of the data obtained by Algorithm WF. Lemma 1. Let the data in (1) be obtained by Algorithm WF. All of the following hold: (i) π0 β₯ π1 β₯ β
β
β
β₯ ππ‘ = 1, (ii) π€π0 β€ π€π1 β€ β
β
β
β€ π€ππ‘β1 , (iii) for each 0 β€ π β€ π‘ β 1 and π + 1 β€ π β€ π‘, π½ππ β βπ§π=ππ π
π,π and is a late order if the orders in π
π,π are delivered at time ππ . Proof. Let (π
0,π0 , . . . , π
0,π§ ), π½π0 , and π€π0 be the output data obtained by the first iteration of Algorithm WF. Then, we have that π β βπ½π ββπ§π=π π
0,π ππ β€ ππβ1 , |π
0,π | β€ Vπ πΆ for π = π§, π§ β 1, . . . , π0 + 1, and π β βπ½π ββπ§π=π π
0,π ππ > ππ0 β1 , |π
0,π0 | β€ Vπ0 πΆ. 0 We now run the second iteration on the data
4
The Scientific World Journal
(π1,0 , π1,1 , . . . , π1,π§ ). It follows from Steps (2.1) and (2.2) that π
1,π§ = π
0,π§ βͺ {π½π0 } if |π
0,π§ | < Vπ§ πΆ and π
1,π§ = π
0,π§ βͺ {π½π0 } \ {π½π } otherwise, where π€π = max{π€π | π½π β π
0,π§ βͺ {π½π0 }}. Since order processing times and order weights are oppositely ordered, we have ππ = min{ππ | π½π β π
0,π§ βͺ {π½π0 }}. In either case, βπ½π βπ
1,π§ ππ β₯ βπ½π βπ
0,π§ ππ . This implies that π β βπ½π βπ
1,π§ ππ β€ π β βπ½π βπ
0,π§ ππ β€ ππ§β1 . Further, we have that for π = π§ β 1, . . . , π0 , π§
π§
π=π
π=π
π§
π§
π=π
π=π
β π
1,π = β π
0,π βͺ {π½π0 }
π§ π§ σ΅¨ σ΅¨ if β σ΅¨σ΅¨σ΅¨σ΅¨π
0,π σ΅¨σ΅¨σ΅¨σ΅¨ < β Vπ πΆ, π=π
β π
1,π = β π
0,π βͺ {π½π0 } \ {π½π }
π=π
(2)
π§ π§ σ΅¨ σ΅¨ if β σ΅¨σ΅¨σ΅¨σ΅¨π
0,π σ΅¨σ΅¨σ΅¨σ΅¨ = β Vπ πΆ, π=π
π=π
(3) where ππ = min{ππ | π½π β βπ§π=π π
0,π βͺ {π½π0 }}. This implies that for π = π§ β 1, . . . , π0 + 1, π β βπ½π ββπ§π=π π
1,π ππ β€ π β βπ½π ββπ§π=π π
0,π ππ β€ ππβ1 . Thus, π0 β₯ π1 holds. The proof of the following inequalities in (i) is similar to that of the first inequality. We conclude that (i) holds. By Algorithm WF, we have that π½π0 β π½ \ βπ§π=π0 π
0,π and π½π1 β π½\βπ§π=π1 π
1,π , where π€π0 = min{π€π | π½π β π½\βπ§π=π0 π
0,π } and π€π1 = min{π€π | π½π β π½ \ βπ§π=π1 π
1,π }. Due to (i), π0 β₯ π1 . By (2), βπ§π=π0 π
1,π = βπ§π=π0 π
0,π βͺ {π½π0 } if βπ§π=π0 |π
0,π | < βπ§π=π0 Vπ πΆ. In the case, since π½\βπ§π=π0 π
0,π β π½\βπ§π=π1 π
1,π , we have that π€π0 β€ π€π1 . By (3), βπ§π=π0 π
1,π = βπ§π=π0 π
0,π βͺ {π½π0 } \ {π½π } if βπ§π=π0 |π
0,π | = βπ§π=π0 Vπ πΆ, where π€π = max{π€π | π½π β βπ§π=π0 π
0,π βͺ {π½π0 }}. In the case, since π½ \ βπ§π=π0 π
0,π β {π½π } βͺ π½ \ βπ§π=π1 π
1,π , along with π€π β₯ π€π0 , we have that π€π0 β€ π€π1 . Thus, the first inequality π€π0 β€ π€π1 in (ii) holds. The proof of the following inequalities in (ii) is similar to that of the first inequality. We conclude that (ii) holds. For any π β {0, 1, . . . , π‘ β 1}, suppose π½ππ β π0,π0 . Clearly, π½ππ is a late order if it is delivered at time ππ , where π0 < π β€ π§. To show (iii), there are two cases to consider: (a) π0 < ππ and (b) π0 β₯ ππ . (a) Consider π0 < ππ . Assume that π½ππ β βπ§π=ππ π
π+1,π does not hold. It follows from the argument of (i) that βπ§π=ππ |π
π,π | = βπ§π=ππ Vπ πΆ, βπ§π=ππ π
π+1,π = βπ§π=ππ π
π,π , and π€ππ = max{π€π | π½π β βπ§π=ππ π
π,π βͺ{π½ππ }}; that is, πππ = min{ππ | π½π β βπ§π=ππ π
π,π βͺ{π½ππ }}. This, along with π€ππ = min{π€π | π½π β π½ \ βπ§π=ππ π
π,π }, that is, πππ = max{ππ | π½π β π½ \ βπ§π=ππ π
π,π }, implies that, for any π½π1 β π½ \ βπ§π=ππ π
π,π and π½π2 β βπ§π=ππ π
π,π , ππ1 β€ ππ2 . Thus, we have that there is a SPT schedule (π1 , π2 ) such that |π2 | = βπ§π=ππ Vπ πΆ and βπ½π βπ1 ππ > πππ β1 , where π1 = π½ \ βπ§π=ππ π
π,π and π2 = βπ§π=ππ π
π,π . This implies that there is no feasible schedule for the instance of the problem, which contradicts the assumption that there is a feasible schedule for the given instance. Thus, π½ππ β βπ§π=ππ π
π+1,π and is a late order if the orders in π
π+1,π are delivered at time ππ . Further, it follows from the argument of (i) that βπ§π=ππ π
π+2,π = βπ§π=ππ π
π+1,π βͺ {π½ππ+1 } if βπ§π=ππ |π
π+1,π | < βπ§π=ππ Vπ πΆ and βπ§π=ππ π
π+2,π = βπ§π=ππ π
π+1,π βͺ {π½ππ+1 } \ {π½π } if βπ§π=ππ |π
π+1,π | = βπ§π=ππ Vπ πΆ, where
π€π = max{π€π | π½π β βπ§π=ππ π
π+1,π βͺ {π½ππ+1 }}. This, along with π€ππ β€ π€ππ+1 and π½ππ merged into βπ§π=ππ π
π+1,π before π½ππ+1 , implies that π½ππ β βπ§π=ππ π
π+2,π and is a late order if the orders in π
π+2,π are delivered at time ππ . Similarly, we can show π½ππ β βπ§π=ππ π
π,π and is a late order if the orders in π
π,π are delivered at time ππ for π = π + 3, . . . , π‘. (b) Consider π0 β₯ ππ . Note the fact that order π½ππ is pushed by the iterations from π0,π0 into π½\βπ§π=ππ π
π,π . By Steps (2.1) and (2.2), we have that |π
π,π | = Vπ πΆ for π = ππ , . . . , π0 π
0 and π€ππ = max{π€π | π½π β βπ=π π
π,π }. This implies π½ππ β π π§ π§ βπ=π0 +1 π
π+1,π (β βπ=ππ π
π+1,π ). Otherwise, it follows from the argument of (i) that βπ§π=π0 +1 |π
π,π | = βπ§π=π0 +1 Vπ πΆ, βπ§π=π0 +1 π
π+1,π = βπ§π=π0 +1 π
π,π , and π€ππ = max{π€π | π½π β βπ§π=π0 +1 π
π,π βͺ {π½ππ }}, and then we have that βπ§π=ππ |π
π,π | = βπ§π=ππ Vπ πΆ, βπ§π=ππ π
π+1,π = βπ§π=ππ π
π,π , and π€ππ = max{π€π | π½π β βπ§π=ππ π
π,π βͺ {π½ππ }}. Similar to the argument of case (a), we can derive a contradiction. Similarly, we can show π½ππ β βπ§π=ππ π
π,π and is a late order if the orders in π
π,π are delivered at time ππ for π = π + 2, . . . , π‘. This ends the proof for (iii).
Theorem 2. Algorithm WF correctly finds a feasible schedule that minimizes the total weight of late orders in π(π§π2 log π) time. Proof. We first prove that βπ§π=ππ π
π,π has not only exactly π late orders {π½π0 , . . . , π½ππβ1 } but also the maximum total processing time and the minimum total weight βπβ1 π=0 π€ππ of late orders among all schedules for π = 0, 1, . . . , π‘ β 1, where {π½π0 , . . . , π½ππβ1 } = 0 and βπβ1 π=0 π€ππ = 0 if π = 0. By the definition of π0,π§ and Steps (2.1) and (2.2) of Algorithm WF, we have that π
0,π§ is a set of early orders delivered at time ππ§ with the maximum total processing time among all schedules. Similarly, we have that βπ§π=π π
0,π is a set of early orders with the maximum total processing time among all schedules for π = π§ β 1, . . . , π0 . By (2) and (3), we have that βπ§π=π0 π
1,π = βπ§π=π0 π
0,π βͺ{π½π0 } if π§ βπ=π0 |π
0,π | < βπ§π=π0 Vπ πΆ and βπ§π=π0 π
1,π = βπ§π=π0 π
0,π βͺ {π½π0 } \ {π½π } if βπ§π=π0 |π
0,π | = βπ§π=π0 Vπ πΆ where π€π = max{π€π | π½π β βπ§π=π0 π
0,π βͺ {π½π0 }}. This, along with (iii) of Lemma 1, π€π0 = min{π€π | π½π β π½ \ βπ§π=π0 π
0,π } and βπ§π=π0 π
0,π being a set of early orders with the maximum total processing time among all schedules, implies that βπ§π=π0 π
1,π has not only exactly a late order π½π0 but also the maximum total processing time and the minimum weight π€π0 among all schedules. Further, by Steps (2.1) and (2.2) of Algorithm WF, we have that βπ§π=π π
1,π has not only exactly a late order π½π0 but also the maximum total processing time and the minimum weight π€π0 among all schedules for π = π0 β 1, . . . , π1 . Similarly, we can show the result for π = 2, . . . , π‘ β 1. We below prove that (π
π‘,1 , . . . , π
π‘,π§ ) is a feasible schedule minimizing the total weight of late orders. By Algorithm WF, the feasibility is obvious. For any feasible schedule π = (π
1 , . . . , π
π§ ), where the orders in π
π are delivered at time ππ for π = 1, . . . , π§, we have that βπ§π=π0 π
π contains at least
The Scientific World Journal
5
a late order. Otherwise, by βπ½π ββπ§π=π π
π ππ β€ βπ½π ββπ§π=π π
0,π ππ , 0 0 we have that βπ½π βπ\βπ§π=π π
π ππ β₯ π β βπ½π ββπ§π=π π
0,π ππ > ππ0 β1 . 0
0
This contradicts the feasibility of π. Given that βπ§π=π0 π
π contains at least a late order, βπ§π=π1 π
π contains at least two late orders. Otherwise, βπ§π=π1 π
π contains a late order. By βπ½π ββπ§π=π π
π ππ β€ βπ½π ββπ§π=π π
1,π ππ , we have that βπ½π βπ\βπ§π=π π
π ππ β₯ 1 1 1 π β βπ½π ββπ§π=π π
1,π ππ > ππ1 β1 . This contradicts the feasibility of π. 1
Similarly, we can show that βπ§π=ππ π
π contains π + 1 late orders at least for π = 2, . . . , π‘ β 1. Thus, (π
π‘,1 , . . . , π
π‘,π§ ) is a feasible schedule minimizing the number of late orders. This, along with βπ‘β1 π=0 π€ππ being the minimum total weight, implies that Algorithm WF correctly finds a feasible schedule that minimizes the total weight of late orders. We now show that the algorithm can be implemented to run in π(π§π2 log π) time. The algorithm consists of π+1 iterations at most since there are π late orders at most. To calculate the complexity of the algorithm we first calculate in a preprocessing step the sets (π1 , . . . , ππ ) and (π0,1 , . . . , π0,π§ ). The calculation of the set (π1 , . . . , ππ ) takes π(π log π§). The calculation of the set (π0,1 , . . . , π0,π§ ) takes π(π log π§) also. Step (1) takes π(π) time. Step (2) is iterated π§ times. Within each iteration, the most time-consuming step is Step (2.1), sorting the orders in decreasing order of their weights, which takes π(π log π) time. Hence Step (2) takes π(π§π log π) time. Thus, the overall running time of the algorithm is π(π§π2 log π) time.
Method (1) π0 := 0. V0 := 0. (2) For π = π§ down to 1 do the following. (a) Let ππσΈ be all the orders in π
π‘,π , arranged in nondecreasing order of their processing times, and |ππσΈ | = ππΆ+π, where π and π are nonnegative integers, and 0 β€ π < πΆ and πΉ is the set of the first π orders from ππσΈ . (b) If βπ½ ββͺπβ1 π
βͺπΉ ππ > ππβ1 , then update π΄ := 0 π
π=1 π‘,π
and ππ := ππσΈ , and the orders in ππ will be delivered by the vehicles at time ππ and proceed to the next π. (c) Let π΄σΈ be the first π0 πΆ + π orders from ππσΈ , where π0 is the maximal nonnegative integer such that βπ½ ββͺπβ1 π
βͺπ΄σΈ ππ β€ ππβ1 . π
π=1 π‘,π
(d) If β|π
π‘,πβ1 βͺ π΄σΈ |/πΆβ β€ Vπβ1 πΆ, then update π΄ := π΄σΈ , π
π‘,πβ1 := π
π‘,πβ1 βͺ π΄, and ππ := ππσΈ \ π΄ and the orders in ππ will be delivered by the vehicles available at time ππ and proceed to the next π. (e) Call Subalgorithm CA to computer π΄. Then update π
π‘,πβ1 := π
π‘,πβ1 βͺ π΄ and ππ := ππσΈ \ π΄, and the orders in ππ will be delivered by the vehicles at time ππ and proceed to the next π. Subalgorithm CA
4. Number of Vehicles Used In this section we show that the problem of minimizing the number of vehicles used subject to the constraint that the total weight of late orders is minimum can be solved in polynomial time. We assume that we have found the schedule that minimizes the total weight of late orders using the algorithm given in the previous section. The schedule (π
π‘,1 , . . . , π
π‘,π§ ) was obtained by Algorithm WF. By the fact that πβ1 if |π
π‘,π | < Vπ πΆ, then each order in βπ=1 π
π‘,π is an early order and will be a late order if we push the order to be delivered by vehicles at ππ . Thus, either there is no feasible schedule or the total weight of late orders will be increased if we push some order in π
π‘,π to be delivered by vehicles at ππ+1 , . . . , ππ§ . By the optimality of βπ‘β1 π=0 π€ππ , we have that the total weight of late orders will not be decreased if we push some order in π
π‘,π to be delivered by vehicles at π1 , . . . , ππβ1 . The following algorithm finds a solution with a minimum number of vehicles used under the constraint that the total weight of late orders is minimum by pushing some orders to an earlier departure time for delivery. Algorithm MV Input. π§ sets of orders π
π‘,1 , . . . , π
π‘,π§ given by Algorithm WF. Output. A vehicle assignment: π1 , . . . , ππ§ , where the orders in ππ will be delivered by the vehicles available at time ππ , so that the number of vehicles used is minimum.
(1) π΄σΈ := the first π0 πΆ + π orders from ππ , arranged in nondecreasing order of their processing times. (2) Sort the orders in π
π‘,πβ1 βͺ π΄σΈ in ascending order of their processing times. (3) If |π
π‘,πβ1 βͺ π΄σΈ | β€ Vπβ1 πΆ, then stop and output π΄ = π΄σΈ . σΈ := π
π‘,β . (4) For β = 1 to π β 2, let π
π‘,β
(5) π΅ := the first |π
π‘,πβ1 βͺπ΄σΈ |βVπβ1 πΆ orders from π
π‘,πβ1 βͺπ΄σΈ . (6) For β = π β 2 down to 1 do the following. σΈ βͺ π΅ in ascending order of (a) Sort the orders in π
π‘,β their processing times. (b) If βπ½π ββͺβπ=1 π
π‘,π σΈ βͺπ΅ ππ > πβ ,
(b1) if |π΄σΈ | < πΆ, stop and output π΄ = 0; (b2) update π΄σΈ := the first |π΄σΈ | β πΆ orders from π΄σΈ and return (2). σΈ (c) If |π
π‘,β βͺ π΅| β€ Vβ πΆ, then stop and output π΄ = π΄σΈ .
(d) If β = 1, stop and output π΄ = 0 if |π΄σΈ | < πΆ and update π΄σΈ := the first |π΄σΈ | β πΆ orders from π΄σΈ and return (2) else. σΈ (e) Update π΅ := the first |π
π‘,β βͺ π΅| β Vβ πΆ orders from σΈ π
π‘,β βͺ π΅ and proceed to the next β. Theorem 3. Algorithm MV finds an optimal solution with the minimum number of vehicles used under the constraint that the total weight of late orders is minimum in π(π§2 π2 log π) time.
6 Proof. We first point out that the solution by Algorithm MV does not change the optimality of the total weight of late orders, since the solution is found by pushing some orders in π
π‘,2 , . . . , π
π‘,π§ to an earlier departure time for delivery. Let ππ§ and π΄ be the output data obtained by the first iteration of Algorithm MV, where ππ§ = ππ§σΈ \ π΄ and π΄ is the set of the first |π΄| orders from ππ§σΈ (the orders of ππ§σΈ = π
π‘,π§ have been arranged in nondecreasing order of their processing times). Step (2)(b) corresponds to the case where βπ½π ββͺπ§β1 ππ > ππ§β1 , where πΉ the set of the first |πΉ| orders π=1 π
π‘,π βͺπΉ
from ππ§σΈ and 0 < |πΉ| < πΆ. In the case, It is impossible to deliver all orders in πΉ by the vehicles at π1 , . . . , ππ§β1 . This, along with πΉ consisting of the first |πΉ| orders from ππ§σΈ , means that the number of vehicles used at ππ§ cannot decrease by one. Thus, β|ππ§σΈ |/πΆβ is the minimal number of vehicles used at ππ§ . However, we cannot push part of orders in πΉ to be delivered by vehicles at ππ§β1 , . . . , π1 , since, if we do that, it not only does not decrease the number of vehicles used at ππ§ but also increases the amount of orders delivered by vehicles at ππ§β1 , . . . , π1 . In the case, the data ππ§ = ππ§σΈ and π΄ = 0 are ππ β€ ππ§β1 , Step (2)(c) optimal. Under the case of βπ½π ββͺπ§β1 π=1 π
π‘,π βͺπΉ σΈ computes the set of orders π΄ which made β|ππ§σΈ |/πΆβββ|π΄σΈ |/πΆβ the minimal number possibly of vehicles used at ππ§ , where π΄σΈ are the first π0 πΆ + π orders from ππ§σΈ . This is because π0 is the maximal nonnegative integer such that βπ½π ββͺπ§β1 σΈ ππ β€ π=1 π
π‘,π βͺπ΄ ππ§β1 and π΄σΈ consists of some small orders in ππ§σΈ . Step (2)(d) corresponds to the case where β|π
π‘,π§β1 βͺ π΄σΈ |/πΆβ β€ Vπ§β1 πΆ. This means that we can push all of orders in π΄σΈ to be delivered by vehicles at ππ§β1 . Thus, in the case, the output data π΄ = π΄σΈ and ππ§ = ππ§σΈ \ π΄ make β|ππ§ |/πΆβ the minimal number of vehicles used at ππ§ . Under the case of β|π
π‘,π§β1 βͺ π΄σΈ |/πΆβ > Vπ§β1 πΆ, Step (2)(e) calls Subalgorithm CA to decide a set of orders π΄ such that β|ππ§ |/πΆβ is the minimal number of vehicles used at ππ§ , where ππ§ = ππ§σΈ \ π΄. We show below that Subalgorithm CA can really do that. Subalgorithm CA consists of π0 +1 main iterations for the first π0 πΆ + π orders, the first (π0 β 1)πΆ + π orders, . . ., and the first π orders from ππ§σΈ , respectively. We now run the first main iteration for the first π0 πΆ + π orders π΄σΈ from ππ§σΈ . Due to the corresponding case by Step (2)(e), we have that π΄σΈ =ΜΈ 0 and |π
π‘,π§β1 βͺ π΄σΈ | > Vπ§β1 πΆ. We need to proceed through Step (4). σΈ for Step (4) assigns the orders in π
π‘,β to a temporary set π
π‘,β each β from 1 to πβ2. This is necessary since Subalgorithm CA σΈ without changing the content of π
π‘,β . Step (5) operates on π
π‘,β assigns the first |π
π‘,π§β1 βͺ π΄σΈ | β Vπ§β1 πΆ orders from π
π‘,π§β1 βͺ π΄σΈ to a set π΅. If we want to push the orders in π΄σΈ to be delivered by vehicles at ππ§β1 , . . . , π1 , all orders of π΅ are the minimal increment undertaken by vehicles at ππ§β2 , . . . , π1 to deliver. Now we proceed through Step (6). Step (6) consists of π§ β 2 secondary iterations. We now ππ > ππ§β2 , run the first secondary iteration. If βπ½π ββͺπ§β2 σΈ π=1 π
π‘,π βͺπ΅ it is impossible to deliver all orders in π΅ by the vehicles at π1 , . . . , ππ§β2 . This means that the number of vehicles used at ππ§ should to be at least β|ππ§σΈ |/πΆβββ|π΄σΈ |/πΆβ+1. Step (6)(b1) corresponds to the case where |π΄σΈ | < πΆ and outputs π΄ = 0. This, along with π΄σΈ =ΜΈ 0 and ππ§ = ππ§σΈ , implies that β|ππ§ |/πΆβ is the
The Scientific World Journal minimal number of vehicles used at ππ§ . Step (6)(b2) corresponds to the case where |π΄σΈ | β₯ πΆ and starts the second main iteration for the first (π0 β 1)πΆ + π orders π΄σΈ from ππ§σΈ to check whether it is possible to push the orders in π΄σΈ to be delivered by vehicles at ππ§β1 , . . . , π1 . On the other hand, βπ½π ββͺπ§β2 ππ β€ ππ§β2 . Step (6)(c) corresponds to the case σΈ π=1 π
π‘,π βͺπ΅
σΈ βͺ π΅| β€ Vπ§β2 πΆ. This means that we can push all where |π
π‘,π§β2 of orders in π΄σΈ to be delivered by vehicles at ππ§β1 and ππ§β2 . Thus, the output data π΄ = π΄σΈ by Step (6)(c) and ππ§ = ππ§σΈ \ π΄ make β|ππ§ |/πΆβ the minimal number of vehicles used at ππ§ . Otherwise, Step (6)(e) starts the second secondary iteration to check whether it is possible to push the orders in a set σΈ σΈ of the first |π
π‘,π§β2 βͺ π΅| β Vπ§β2 πΆ orders from π
π‘,π§β2 βͺ π΅ to be delivered by vehicles at ππ§β3 , . . . , π1 . Similarly, we can show the result for the following secondary iterations. If needed, we proceed through the last secondary iteration. Suppose that π΅ is a set of orders output by the (π§ β 3)th secondary iteration. In the case of βπ½π βπ
π‘,1 σΈ βͺπ΅ ππ > π1 , Step (6)(b1) outputs π΄ = 0 if
|π΄σΈ | < πΆ and Step (6)(b2) starts the second main iteration for the first (π0 β 1)πΆ + π orders π΄σΈ from ππ§σΈ else. In the σΈ case of βπ½π βπ
π‘,1 σΈ βͺπ΅ ππ β€ π1 , Step (6)(c) outputs data π΄ = π΄
σΈ if |π
π‘,1 βͺ π΅| β€ V1 πΆ. Otherwise, Step (6)(d) outputs π΄ = 0 if |π΄σΈ | < πΆ and starts the second main iteration for the first (π0 β 1)πΆ + π orders π΄σΈ from ππ§σΈ else, since it is impossible σΈ to deliver all orders in the set of the first |π
π‘,1 βͺ π΅| β V1 πΆ σΈ orders from π
π‘,1 βͺ π΅ by the vehicles at π0 = 0. Similarly, we can show the result for the following main iterations in Subalgorithm CA, which can really decide a set orders π΄ such that β|ππ§ |/πΆβ is the minimal number of vehicles used at ππ§ , where ππ§ = ππ§σΈ \ π΄. For the following iterations in Algorithm MV, we can show the results similarly. At last, the algorithm must be able to find an optimal solution with the minimum number of vehicles used. We now look at the time complexity of Algorithm MV. In the algorithm, Step (1) takes constant time. Step (2) is iterated π§ times. Inside the iteration loop, the most time-consuming steps are (2)(a) and (2)(e). Step (2)(a) calls for sorting the orders which takes π(π log π) time. Step (2)(e) calls Subalgorithm CA which takes π(π§π2 log π) time. Thus, the overall time complexity of Algorithm MV is π(π§2 π2 log π).
5. Conclusion In this paper, we have given polynomial-time algorithms for minimizing: (1) the total weight of late orders and (2) the number of vehicles used subject to the condition that the total weight of late orders is minimum. An interesting open question is whether the problem related to release dates is NPhard or not.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
The Scientific World Journal
Acknowledgment This work was supported in part by the Zhejiang Natural Science Foundation of China Grant Y6110054.
References [1] C. J. Langley, E. van Dort, and S. R. Sykes, 2005 Third-Party Logistics: Results and Findings of the 10th Annual Study, Georgia Institute of Technology, 2006. [2] R. M. Karp, βReducibility among combinatorial problems,β in Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, Eds., pp. 85β104, Plenum Press, New York, NY, USA, 1972. [3] Z.-L. Chen, βIntegrated production and outbound distribution scheduling: review and extensions,β Operations Research, vol. 58, no. 1, pp. 130β148, 2010. [4] K. E. Stecke and X. Zhao, βProduction and transportation integration for a make-to-order manufacturing company with a commit-to-delivery business mode,β Manufacturing and Service Operations Management, vol. 9, no. 2, pp. 206β224, 2007. [5] R. A. Melo and L. A. Wolsey, βOptimizing production and transportation in a commit-to-delivery business mode,β European Journal of Operational Research, vol. 203, no. 3, pp. 614β618, 2010. [6] W. Zhong, Z.-L. Chen, and M. Chen, βIntegrated production and distribution scheduling with committed delivery dates,β Operations Research Letters, vol. 38, no. 2, pp. 133β138, 2010. [7] K. P. Li, V. K. Ganesan, and A. I. Sivakumar, βSynchronized scheduling of assembly and multi-destination air-transportation in a consumer electronics supply chain,β International Journal of Production Research, vol. 43, no. 13, pp. 2671β2685, 2005. [8] K. P. Li, V. K. Ganesan, and A. I. Sivakumar, βScheduling of single stage assembly with air transportation in a consumer electronic supply chain,β Computers & Industrial Engineering, vol. 51, no. 2, pp. 264β278, 2006. [9] K. O. Li, A. I. Sivakumar, and V. K. Ganesan, βComplexities and algorithms for synchronized scheduling of parallel machine assembly and air transportation in consumer electronics supply chain,β European Journal of Operational Research, vol. 187, no. 2, pp. 442β455, 2008. [10] M. Zandieh and S. Molla-Alizadeh-Zavardehi, βSynchronizing production and air transportation scheduling using mathematical programming models,β Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 546β558, 2009. [11] J. K. Lenstra, A. H. Rinnooy Kan, and P. Brucker, βComplexity of machine scheduling problems,β in Annals of Discrete Mathematics, vol. 1, pp. 343β362, North-Holland, Amsterdam, The Netherlands, 1977. [12] Q. Wang, R. Batta, and R. J. Szczerba, βSequencing the processing of incoming mail to match an outbound truck delivery schedule,β Computers & Operations Research, vol. 32, no. 7, pp. 1777β1791, 2005. [13] B. Fu, Y. Huo, and H. Zhao, βCoordinated scheduling of production and delivery with production window and delivery capacity constraints,β Theoretical Computer Science, vol. 422, pp. 39β51, 2012. [14] J. Y. Leung and Z.-L. Chen, βIntegrated production and distribution with fixed delivery departure dates,β Operations Research Letters, vol. 41, no. 3, pp. 290β293, 2013.
7