Question #275720

A small business enterprise makes dresses and trousers. To make a dress




requires 1




2




hour of cutting and 20 minutes of stitching. To make a trouser




requires 15 minutes of cutting and 1




2




hours of stitching. The profit on a dress is




P40 and on a pair of trousers is P50. The business operates for a maximum of




8 hours per day. Determine how many dresses and trousers should be made to




maximize profit and what the maximum profit is.

1
Expert's answer
2021-12-06T13:08:15-0500

Let x,y- number of dresses and trousers to done.

Mathematical model is

z=40\cdot x+25\cdot y\rarr max;

50\cdot x+45\cdot y\le 480

(or 10\cdot x+9\cdot y\le 96)

x,y\ge 0

x,y \in Z

Because x is integer x{0,1,2,3,4,5,6,7,8,9}x\in \{0,1,2,3,4,5,6,7,8,9\}

1) x=9,y=[969109]=[0.6]=0;z=409+250=360y=[\frac{96-9\cdot 10}{9}]=[0.6]=0;z=40\cdot 9+25\cdot 0=360

2) x=8,y=968109]=[179]=1;z=408+251=345\frac{96-8\cdot 10}{9}]=[1 \frac{7}{9}]=1;z=40\cdot 8+25\cdot 1=345

3) x=7,y=967109]=[289]=2;z=407+252=330\frac{96-7\cdot 10}{9}]=[2 \frac{8}{9}]=2;z=40\cdot 7+25\cdot 2=330

4) x=6,y=966109]=[4]=4;z=406+254=340\frac{96-6\cdot 10}{9}]=[4]=4;z=40\cdot 6+25\cdot 4=340

5) x=5,y=965109]=[519]=5;z=405+255=325\frac{96-5\cdot 10}{9}]=[5 \frac{1}{9}]=5;z=40\cdot 5+25\cdot 5=325

6) x=4,y=964109]=[629]=6;z=404+256=310\frac{96-4\cdot 10}{9}]=[6 \frac{2}{9}]=6;z=40\cdot 4+25\cdot 6=310

7) x=3,y=963109]=[729]=7;z=403+257=295\frac{96-3\cdot 10}{9}]=[7 \frac{2}{9}]=7;z=40\cdot 3+25\cdot 7=295

cases x=0,1,2 worse than dest case (x=10) because 240+1025=330<360\cdot 40+10\cdot 25=330<360

Now all possibilities are considered and optimal solution (xopt,yopt)=(10,0) with zopt=360.

So xopt=10,yopt=0,zopt=360.


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