Production and vehicle scheduling for ready-mix operations

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Abstract

This paper deals with the problem of selecting and scheduling the orders to be processed by a ready-mix manufacturing plant for immediate delivery to the customer site. Orders have a fixed due date and must be prepared in a single plant with limited capacity. The objective is to maximize the total value of orders served. We first describe the problem as it occurs in practice in some industrial environments, and then present two scenarios to be considered: arbitrary and uniform profit margins for orders. The first scenario corresponds to the fixed job scheduling problem, which is solved by a minimum cost flow problem. A new scheme that reduces the number of nodes in the graph is proposed in this paper. The second scenario on the other hand requires more computationally expensive approaches as a second objective of minimizing the number of vehicles is to be considered. An exact graph-based method and a heuristic branch-and-bound based scheme are described. Computational experiences show that the optimal approach is not feasible for large problems and that the proposed heuristic approach gives high-quality solutions in short running times.

Introduction

In certain industrial environments the supply of products must be matched with the demand because of the absence of end product inventory. These environments usually involve products with perishable character. A characteristic example is ready-mix concrete manufacturing. Ready-mix manufacturing is a relatively simple operation. Depending on the concrete type, raw material (cement, sand, gravel, water and additives) in different proportions are loaded into a revolving drum mounted on the vehicle which immediately delivers the product to the customer site. Since the product is perishable, there is a limit on the area that can be serviced by a production plant. That is the reason why, it is frequent for the same company to have several plants appropriately distributed along the periphery of a certain metropolitan area.

The production planning horizon of the problem is one day. We assume that all requests are known in advance and correspond to a number of customer orders, each of which specifies the type and amount of concrete to be served and the location and time at which to be delivered. For the manufacturing of orders, we have a plant with limited production capacity. We consider production capacity as the number of orders that can be prepared simultaneously, i.e. each production order is considered as a continuous process that requires one unit of capacity during its processing time. In the distribution stage of an order, three consecutive phases are considered: delivery, unload and return trip. Each vehicle may deliver any order, but no more than one order at a time. We assume that the order size is smaller than the vehicle capacity. Hence, the distribution stage of an order can be considered as a single process, without interruption, that commences immediately after the end of the production stage. A particular feature of ready mix operations is that vehicles are not only required for the distribution of the product but also for the production phase.

Assuming that plant processing time and travel times are deterministic and known, each due date can be translated to a corresponding plant processing start time. Since the plant has limited capacity it may happen that not all orders can be served at the specified times. There are several options available then. One is to allow for some flexibility (i.e. time windows) in the due dates. The other one is to serve only some of the orders, serving the rest from a different plant or simply leaving them for a next scheduling. In this paper, we deal with the second alternative and the problem addressed is the selection of the orders to be served. Hence, we will consider as objective function to maximize the value of orders served. Time window approaches will be presented elsewhere. The complete problem formulation is presented in Section 2.

Two different scenarios have been considered. In Section 3, it is assumed that each order has associated a known profit margin and that an unlimited number of vehicles is available. In Section 4, it is assumed that profit margins are identical. In this case maximizing the total value is equivalent to maximizing the number of orders served. This second scenario becomes interesting because it is very likely that alternative optimal solutions exist. All of these solutions would serve the same (maximum) number of orders but each solution would include different orders. Therefore, among the maximal solutions, the one requiring the minimum number of vehicles should be chosen. In order to solve this problem, an effective method to obtain the exact solution using a graph-based approach and a heuristic based on branch-and-bound are described. The performance of these methods on randomly generated test problems is analysed in Section 5 through computational experiments. Finally, in Section 6, conclusions are drawn.

Section snippets

Problem formulation

The production and vehicle scheduling for ready-mix operations problem can be stated as follows:

Let J={1,2,…n} denote the set of n orders. With each order iJ there is an associated due date di, a weight or profit margin wi, a processing time tpi, travel times tii (one way trip) and tri (return trip) and an unloading time at the customer site tui. Let tdi=tii+tri+tui the total time taken by the vehicle in the distribution phase. Fig. 1 summarises those data.

As plant processing time and travel

Maximize the value of orders served

Assume, without loss of generality, that:

  • a.

    All numerical data are positive integers

  • b.

    s1s2≤⋯≤sn

Let S(i,t)={k∈J:sk≤t&lk>t;i∈Jt∈[si,li−1]} be the set of orders whose production phase overlaps with order i in instant t, with t belonging to the production phase of order i.

Letpi=1iforder Jiisserved0otherwiseThe first scenario can be modelled asMaxi=1nwipis.tk∈S(i,t)pk≤Ci=1…n;t=si…li−1pi∈{0,1}

The objective function is the total value of orders served. Constraints (1) impose the capacity limits at the

Maximize the number of orders served using minimum number of vehicles

This second scenario considers the situation in which either the profit margins are unknown or they are the same for all orders. That case corresponds to assuming wi=1∀i. Let p be the maximum number of orders served, which is calculated using the approach described in Section 3. It will happen very often that there are alternative optimal solutions, all of them serving p orders but requiring a variable number of vehicles. That is why it would be interesting to select the optimal solution

Computational experiences

We have considered the followings parameters in the generation of random test problems:

  • Four problem sizes are considered: 25, 50, 75 and 100 orders. Two different problem sets were generated depending on time horizon. These problem sets will be denoted by PS1 and PS2. The resulting average order overlaps per unit time in the plant (OOP) and in the distribution stage (OOD) obtained from the random generation of the orders are shown in Table 2.

As shown in Table 2, the average order overlaps are

Conclusions

In this paper the production scheduling and vehicle assignment problem in a ready-mix manufacturing plant has been addressed. It has been assumed that customer orders to be served must be delivered at specific, known due times and that there are enough vehicles available. The plant has limited capacity. Two different scenarios have been considered. The first scenario corresponds to the case of known profit margins for each customer order. In this case, the situation can be modelled as linear

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This research has been financed by The Spanish Ministry of Science and Technology under contract no. DPI2000-0567.

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