# 456 modelleme by gegeshandong

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```									  A mathematical programming model and Solution for Scheduling Production
Orders in Shanghai Baoshan Iron and Steel Complex

1. Problem description

In our project the quantities of the production orders are centralized as Q1: U[100, 500], Q2:
U[500,1000], Q3: U[1000, 1500].The steel sheet manufacturing process studied in this paper mainly
consists of 34 operations. Those operations and 708 technology routings constructed by them are
depicted. The unit of time used in the model is one day. The decision horizon is 90 days. Each
production order only involves one product. Only one technology routing is assigned for each
production order. Production orders should be allowed to be split, which is helpful to smooth
operation workloads over time. Each production order must be completed before its fixed delivery
date. Available production capacity of each operation must not be exceeded. Technology routings
must be observed. The average duality gaps and running times tables are below.
The main task is when each production order on each operation should start processing so that the
promised delivery dates for all orders are met and the sum of weighted completion times of all orders
is minimized?

Average duality gaps (%)

Q1:U[100,500]   Q2:U[500,1000]      Q3:U[1000,1500]       Qtotal:U[100,1500]
LRp1            1,352407         1,281581            1,194258                1,239393
LRp2            1,352407          1,39092            1,204024                1,276996
LRp3            2,439522         2,119476            1,392493               1,375323
LRp4            2,544115         2,029682            0,916286               0,965775

Average running times (seconds)

Q1:U[100,500]     Q2:U[500,1000]      Q3:U[1000,1500]     Qtotal:U[100,1500]
LRp1             180,4557           184,9544             261,553              202,1989
LRp2             180,6507           184,6623             239,506              198,3487
LRp3             166,4254           171,4977            237,3244              182,9604
LRp4             166,5109           137,8469            192,3358              133,9383

2. Parameters :
3. Decision variables:

   dij: the production cycle for production order i on operation j, i.e., the time for one unit slab (or
coil) of order i to go through operation j; a graphic representation of di4, the production cycle
for production order i on hot rolling, is given in Fig. 5 as an example. Note that ddije stands
for the minimal integer larger than or equal to dij;
   Rjt: available production capacity of operation j during period t;
   rij: the total amount of the production capacity of operation j needed to complete production
order i;
   lij:the earliest possible starting period of operation j for production order i; it is the earliest
possible period that the materials of operation j for production order i is available;
   uij:the absolute due date of operation j for production order i; the iterative formulae used in
this paper to define uij are described as follows. (Suppose j = Q i(k))

Ui,Qi(Ki) = G

Ui,Qi(k+1) = Ui,Qi(k+1) – di,Qi(k+1) ,   k = 1, . . . , Ki – 1.

It is obvious that each operation of production order i must be completed before its absolute due
date.Otherwise the entire production order will be late.

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