#### Question

Gandhi Clothing Company produces shirts and pants. Each shirt requires 2 sq yd of cloth, each pair of pants, 3. During the next two months, the following demands for shirts and pants must be met (on time): month 1—10 shirts, 15 pairs of pants; month 2—12 shirts, 14 pairs of pants. During each month, the following resources are available: month 1—90 sq yd of cloth; month 2—60 sq yd. (Cloth that is available during month 1 may, if unused during month 1, be used during month 2.) During each month, it costs $4 to make an article of clothing with regular-time labor and$8 with overtime labor. During each month, a total of at most 25 articles of clothing may be produced with regular-time labor, and an unlimited number of articles of clothing may be produced with overtime labor. At the end of each month, a holding cost of $3 per article of clothing is assessed. Formulate an LP that can be used to meet demands for the next two months (on time) at minimum cost. Assume that at the beginning of month 1, 1 shirt and 2 pairs of pants are available.

#### Solution

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1 of 2Clothing company produces shirts and pants. Each shirt reuqires $2$ square yards for production while pants require $3.$ For first month demands are $10$ shirts and $15$ pants while for the month two demands are $12$ shirts and $14$ pants. Therefore we introduce $ds_1=10,ds_2=12$ and $dp_1=15,dp_2=14.$ Those demands must be met on time. In the first month $90$ square yards of cloth is available while in second month $60$ sq yd is available. However, if unused, cloth from the first month can be used in the second month. During each month production in regular time costs $4\$$ while in overtime it costs$8\$.$ During each month at most $25$ articles of clothing may be produced during reuglar time labor. Unlimited amount of clothing can be produced during the overtime labor. Obviously it doesn't represent the problem for minimization since we want to minimize the cost of the production. Similarly we could argue that we could skip regular time labor and jump immediately to overtime, but this will easily turn out not to be optimal. Holding cost is assesed at the end of the month with price of $3\$$ per article of clothing. Also let us say that at the beggining of month there is$1$shirt and$2$pairs of pants in the inventory.Notice that overtime labour is possible, therefore we introduce$xp_1,xp_2,xs_1$and$xs_2$as a decision variables that represent number of pants or shirts made in overtime labour in their respective month. Let us define the following:$ $s_0=1,s_t=s_{t-1}+S_t+xs_t-ds_{t},$ $p_0=2,p_t=p_{t-1}+P_t+xp_t-dp_t,$ $where$S_t$and$P_t$are number of shirts and pants made in the month$t$(respectively). Also$ds_t$and$dp_t$is demand for shirts and pants in month$t.$Those demands have to be met, ofcourse. \ Since at most$25$articles of clothing can be produced by regular time labour, we simply have:$ $S_1+P_1\leq25, S_2+P_2\leq25.$ $Now let us formulate an LP that minimizes the cost of production that meets demands in time. Let$z$be the objective function. It follows:$ $\min z=4(P_1+P_2+S_1+S_2)+8(xp_1+xp_2+xs_1+xs_2)+3(s_1+s_2+p_1+p_2).$ $We are left to model the constraints which make sure we are using no more than$90$sq yd during month$1$and no more than$60$sq yd during month$2.$2S_1+2xs_1+3P_1+3xp_1\leq90$ $2S_2+3P_2+2xs_2+3xp_2\leq60+90-(2S_1+2xs_1+3P_1-3xp_1).$$ Therefore we are done formulating an LP.