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# A dynamic lot sizing model with production-or-outsourcing decision under minimum production quantities

• In the real-world production process, the firms need to determine the optimal production planning under minimum production quantity constraint in order to achieve economies of scale. However, the inventory cost will hugely increase when there is a very large amount of production in a period and also a large amount of total demands for the next few periods. This paper considers a single-item dynamic lot sizing problem with production-or-outsourcing decisions. In each period, the production level cannot be lower than a given quantity in order to make full use of resources, but the outsourcing is unrestricted. The demands in a period can be backlogged. The production and outsourcing costs are fixed-plus-linear, and the inventory and backlogging costs are linear. We establish a mathematical programming model according to the real problem in the firm. We explore some structural properties of the optimal solution and use them to develop a dynamic programming algorithm to solve the proposed problem. We further present a special case with stationary production and outsourcing costs which can be solved with reduced computational complexities. In the end, we use three numerical instances to show how to obtain the optimal solutions by using the dynamic programming algorithm. Furthermore, we show that the policy of backlogging or outsourcing can reduce the total cost.

Mathematics Subject Classification: Primary: 90C39, 90C90; Secondary: 90B05.

 Citation: • • Figure 1.  The sketch for production and outsourcing decisions

Figure 2.  The sketch for $F(i,j,t)$

Figure 3.  The sketch for $F(i,j,j_1 ,j_1 ',g_2 ,j_2 ,...,g_n ,j_n ,t)$

Figure 4.  The sketch for $F(i,j,g_1 ,j_1 ,...,g_n ,j_n ,t)$

Figure 5.  The sketch for $F(e,i,t)$

Figure 6.  The sketch for $F(e,i,j_1 ,j_1 ',g_2 ,j_2 ,...,g_n ,j_n ,t)$

Figure 7.  The sketch for $F(e,i,g_1 ,j_1 ,...,g_n ,j_n ,t)$

Figure 8.  The sketch for $F(g_n ,j_n ,t)$

Table 1.  The basic notation

 $K_t :$ fixed (setup) costs for production in period $t$, $t=1,2,..,T$; $c_t :$ per unit production cost in period $t$, $t=1,2,..,T$; $k_t :$ fixed (setup) costs for outsourcing in period $t$, $t=1,2,..,T$; $o_t :$ per unit outsourcing cost in period $t$, $t=1,2,..,T$; $h_t :$ per unit holding cost in period $t$, $t=1,2,..,T$; $b_t :$ per unit backlogging cost in period $t$, $t=1,2,..,T$

Table 2.  Summary of Computations of Example 1

 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast$ 30 $X_t^\ast$ 30 0 $X_t^\ast$ 30 0 0 $X_t^\ast$ 30 0 0 30 $X_t^\ast$ 30 0 0 30 0 $X_t^\ast$ 30 0 0 33 0 0 $X_t^\ast$ 30 0 0 38 0 0 0 $X_t^\ast$ 30 0 0 53 0 0 0 0 $X_t^\ast$ 30 0 0 58 0 0 0 0 0 $X_t^\ast$ 30 0 0 38 0 0 0 30 0 0 $F(t)$ 254 286 300 526 532 556 606 786 856 874

Table 3.  Summary of Computations of Example 2

 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast$ 30 $X_t^\ast$ 0 30 $X_t^\ast$ 0 0 30 $X_t^\ast$ 0 0 0 52 $X_t^\ast$ 0 0 0 57 0 $X_t^\ast$ 0 0 0 63 0 0 $X_t^\ast$ 0 0 0 68 0 0 0 $X_t^\ast$ 0 0 0 57 0 0 0 30 $X_t^\ast$ 0 0 0 57 0 0 0 31 0 $X_t^\ast$ 0 0 0 57 0 0 0 37 0 0 $F(t)$ 254 258 268 388 418 466 546 682 688 748

Table 4.  Summary of Computations of Example 3

 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast ,O_t^\ast$ 0, 8 $X_t^\ast ,O_t^\ast$ 0, 14 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 30, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 34, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 40, 0 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 15 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 20 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 26 0, 0 0, 0 $F(t)$ 98 146 236 448 472 532 582 722 762 822
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Tables(4)

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