Shopping malls

The chain of department stores plans to invest up to 24,000 euros in television advertising. All commercials will be placed on a television station where the broadcast of a 30-second spot costs EUR 1,000 and is watched by 14,000 potential customers, during prime time it costs EUR 2,000 and is watched by 24,000 potential customers, and late at night it costs EUR 1,500 and is watched 18 000 potential customers. The TV station will not accept an order to broadcast more than 15 spots in total at all three times. How many spots should the chain order at each time to maximize the total number of viewers who will watch them? How many potential customers will watch the spots?

Correct result:

n =  296000

Solution:

a 1000+b 2000+c 1500<=24000 a+b+c<15 n=14000 a+24000 b+18000 c  maxn  a1=0;b1=0;c1=7  d1=a1 1000+b1 2000+c1 1500=0 1000+0 2000+7 1500=10500  n1=14000 a1+24000 b1+18000 c1=14000 0+24000 0+18000 7=126000  a2=0;b2=0;c2=8  d2=a2 1000+b2 2000+c2 1500=0 1000+0 2000+8 1500=12000  n2=14000 a2+24000 b2+18000 c2=14000 0+24000 0+18000 8=144000  a3=0;b3=0;c3=9  d3=a3 1000+b3 2000+c3 1500=0 1000+0 2000+9 1500=13500  n3=14000 a3+24000 b3+18000 c3=14000 0+24000 0+18000 9=162000  a4=0;b4=0;c4=10  d4=a4 1000+b4 2000+c4 1500=0 1000+0 2000+10 1500=15000  n4=14000 a4+24000 b4+18000 c4=14000 0+24000 0+18000 10=180000  a5=0;b5=0;c5=11  d5=a5 1000+b5 2000+c5 1500=0 1000+0 2000+11 1500=16500  n5=14000 a5+24000 b5+18000 c5=14000 0+24000 0+18000 11=198000  a6=0;b6=0;c6=12  d6=a6 1000+b6 2000+c6 1500=0 1000+0 2000+12 1500=18000  n6=14000 a6+24000 b6+18000 c6=14000 0+24000 0+18000 12=216000  a7=0;b7=0;c7=13  d7=a7 1000+b7 2000+c7 1500=0 1000+0 2000+13 1500=19500  n7=14000 a7+24000 b7+18000 c7=14000 0+24000 0+18000 13=234000  a8=0;b8=0;c8=14  d8=a8 1000+b8 2000+c8 1500=0 1000+0 2000+14 1500=21000  n8=14000 a8+24000 b8+18000 c8=14000 0+24000 0+18000 14=252000  a9=0;b9=1;c9=13  d9=a9 1000+b9 2000+c9 1500=0 1000+1 2000+13 1500=21500  n9=14000 a9+24000 b9+18000 c9=14000 0+24000 1+18000 13=258000  a10=0;b10=2;c10=12  d10=a10 1000+b10 2000+c10 1500=0 1000+2 2000+12 1500=22000  n10=14000 a10+24000 b10+18000 c10=14000 0+24000 2+18000 12=264000  a11=0;b11=3;c11=11  d11=a11 1000+b11 2000+c11 1500=0 1000+3 2000+11 1500=22500  n11=14000 a11+24000 b11+18000 c11=14000 0+24000 3+18000 11=270000  a12=0;b12=4;c12=10  d12=a12 1000+b12 2000+c12 1500=0 1000+4 2000+10 1500=23000  n12=14000 a12+24000 b12+18000 c12=14000 0+24000 4+18000 10=276000  a13=0;b13=5;c13=9  d13=a13 1000+b13 2000+c13 1500=0 1000+5 2000+9 1500=23500  n13=14000 a13+24000 b13+18000 c13=14000 0+24000 5+18000 9=282000  a14=0;b14=6;c14=8  d14=a14 1000+b14 2000+c14 1500=0 1000+6 2000+8 1500=24000  n14=14000 a14+24000 b14+18000 c14=14000 0+24000 6+18000 8=288000  a15=1;b15=7;c15=6  d15=a15 1000+b15 2000+c15 1500=1 1000+7 2000+6 1500=24000  n15=14000 a15+24000 b15+18000 c15=14000 1+24000 7+18000 6=290000  a16=2;b16=8;c16=4  d16=a16 1000+b16 2000+c16 1500=2 1000+8 2000+4 1500=24000  n16=14000 a16+24000 b16+18000 c16=14000 2+24000 8+18000 4=292000  a17=3;b17=9;c17=2  d17=a17 1000+b17 2000+c17 1500=3 1000+9 2000+2 1500=24000  n17=14000 a17+24000 b17+18000 c17=14000 3+24000 9+18000 2=294000  a18=4;b18=10;c18=0  d18=a18 1000+b18 2000+c18 1500=4 1000+10 2000+0 1500=24000  n18=14000 a18+24000 b18+18000 c18=14000 4+24000 10+18000 0=296000  n=296000a \cdot \ 1000+b \cdot \ 2000+c \cdot \ 1500 <=24000 \ \\ a+b+c < 15 \ \\ n=14000 \cdot \ a+24000 \cdot \ b+18000 \cdot \ c \ \\ \ \\ max n \ \\ \ \\ a_{1}=0; b_{1}=0; c_{1}=7 \ \\ \ \\ d_{1}=a_{1} \cdot \ 1000+b_{1} \cdot \ 2000+c_{1} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+7 \cdot \ 1500=10500 \ \\ \ \\ n_{1}=14000 \cdot \ a_{1}+24000 \cdot \ b_{1}+18000 \cdot \ c_{1}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 7=126000 \ \\ \ \\ a_{2}=0; b_{2}=0; c_{2}=8 \ \\ \ \\ d_{2}=a_{2} \cdot \ 1000+b_{2} \cdot \ 2000+c_{2} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+8 \cdot \ 1500=12000 \ \\ \ \\ n_{2}=14000 \cdot \ a_{2}+24000 \cdot \ b_{2}+18000 \cdot \ c_{2}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 8=144000 \ \\ \ \\ a_{3}=0; b_{3}=0; c_{3}=9 \ \\ \ \\ d_{3}=a_{3} \cdot \ 1000+b_{3} \cdot \ 2000+c_{3} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+9 \cdot \ 1500=13500 \ \\ \ \\ n_{3}=14000 \cdot \ a_{3}+24000 \cdot \ b_{3}+18000 \cdot \ c_{3}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 9=162000 \ \\ \ \\ a_{4}=0; b_{4}=0; c_{4}=10 \ \\ \ \\ d_{4}=a_{4} \cdot \ 1000+b_{4} \cdot \ 2000+c_{4} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+10 \cdot \ 1500=15000 \ \\ \ \\ n_{4}=14000 \cdot \ a_{4}+24000 \cdot \ b_{4}+18000 \cdot \ c_{4}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 10=180000 \ \\ \ \\ a_{5}=0; b_{5}=0; c_{5}=11 \ \\ \ \\ d_{5}=a_{5} \cdot \ 1000+b_{5} \cdot \ 2000+c_{5} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+11 \cdot \ 1500=16500 \ \\ \ \\ n_{5}=14000 \cdot \ a_{5}+24000 \cdot \ b_{5}+18000 \cdot \ c_{5}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 11=198000 \ \\ \ \\ a_{6}=0; b_{6}=0; c_{6}=12 \ \\ \ \\ d_{6}=a_{6} \cdot \ 1000+b_{6} \cdot \ 2000+c_{6} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+12 \cdot \ 1500=18000 \ \\ \ \\ n_{6}=14000 \cdot \ a_{6}+24000 \cdot \ b_{6}+18000 \cdot \ c_{6}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 12=216000 \ \\ \ \\ a_{7}=0; b_{7}=0; c_{7}=13 \ \\ \ \\ d_{7}=a_{7} \cdot \ 1000+b_{7} \cdot \ 2000+c_{7} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+13 \cdot \ 1500=19500 \ \\ \ \\ n_{7}=14000 \cdot \ a_{7}+24000 \cdot \ b_{7}+18000 \cdot \ c_{7}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 13=234000 \ \\ \ \\ a_{8}=0; b_{8}=0; c_{8}=14 \ \\ \ \\ d_{8}=a_{8} \cdot \ 1000+b_{8} \cdot \ 2000+c_{8} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+14 \cdot \ 1500=21000 \ \\ \ \\ n_{8}=14000 \cdot \ a_{8}+24000 \cdot \ b_{8}+18000 \cdot \ c_{8}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 14=252000 \ \\ \ \\ a_{9}=0; b_{9}=1; c_{9}=13 \ \\ \ \\ d_{9}=a_{9} \cdot \ 1000+b_{9} \cdot \ 2000+c_{9} \cdot \ 1500=0 \cdot \ 1000+1 \cdot \ 2000+13 \cdot \ 1500=21500 \ \\ \ \\ n_{9}=14000 \cdot \ a_{9}+24000 \cdot \ b_{9}+18000 \cdot \ c_{9}=14000 \cdot \ 0+24000 \cdot \ 1+18000 \cdot \ 13=258000 \ \\ \ \\ a_{10}=0; b_{10}=2; c_{10}=12 \ \\ \ \\ d_{10}=a_{10} \cdot \ 1000+b_{10} \cdot \ 2000+c_{10} \cdot \ 1500=0 \cdot \ 1000+2 \cdot \ 2000+12 \cdot \ 1500=22000 \ \\ \ \\ n_{10}=14000 \cdot \ a_{10}+24000 \cdot \ b_{10}+18000 \cdot \ c_{10}=14000 \cdot \ 0+24000 \cdot \ 2+18000 \cdot \ 12=264000 \ \\ \ \\ a_{11}=0; b_{11}=3; c_{11}=11 \ \\ \ \\ d_{11}=a_{11} \cdot \ 1000+b_{11} \cdot \ 2000+c_{11} \cdot \ 1500=0 \cdot \ 1000+3 \cdot \ 2000+11 \cdot \ 1500=22500 \ \\ \ \\ n_{11}=14000 \cdot \ a_{11}+24000 \cdot \ b_{11}+18000 \cdot \ c_{11}=14000 \cdot \ 0+24000 \cdot \ 3+18000 \cdot \ 11=270000 \ \\ \ \\ a_{12}=0; b_{12}=4; c_{12}=10 \ \\ \ \\ d_{12}=a_{12} \cdot \ 1000+b_{12} \cdot \ 2000+c_{12} \cdot \ 1500=0 \cdot \ 1000+4 \cdot \ 2000+10 \cdot \ 1500=23000 \ \\ \ \\ n_{12}=14000 \cdot \ a_{12}+24000 \cdot \ b_{12}+18000 \cdot \ c_{12}=14000 \cdot \ 0+24000 \cdot \ 4+18000 \cdot \ 10=276000 \ \\ \ \\ a_{13}=0; b_{13}=5; c_{13}=9 \ \\ \ \\ d_{13}=a_{13} \cdot \ 1000+b_{13} \cdot \ 2000+c_{13} \cdot \ 1500=0 \cdot \ 1000+5 \cdot \ 2000+9 \cdot \ 1500=23500 \ \\ \ \\ n_{13}=14000 \cdot \ a_{13}+24000 \cdot \ b_{13}+18000 \cdot \ c_{13}=14000 \cdot \ 0+24000 \cdot \ 5+18000 \cdot \ 9=282000 \ \\ \ \\ a_{14}=0; b_{14}=6; c_{14}=8 \ \\ \ \\ d_{14}=a_{14} \cdot \ 1000+b_{14} \cdot \ 2000+c_{14} \cdot \ 1500=0 \cdot \ 1000+6 \cdot \ 2000+8 \cdot \ 1500=24000 \ \\ \ \\ n_{14}=14000 \cdot \ a_{14}+24000 \cdot \ b_{14}+18000 \cdot \ c_{14}=14000 \cdot \ 0+24000 \cdot \ 6+18000 \cdot \ 8=288000 \ \\ \ \\ a_{15}=1; b_{15}=7; c_{15}=6 \ \\ \ \\ d_{15}=a_{15} \cdot \ 1000+b_{15} \cdot \ 2000+c_{15} \cdot \ 1500=1 \cdot \ 1000+7 \cdot \ 2000+6 \cdot \ 1500=24000 \ \\ \ \\ n_{15}=14000 \cdot \ a_{15}+24000 \cdot \ b_{15}+18000 \cdot \ c_{15}=14000 \cdot \ 1+24000 \cdot \ 7+18000 \cdot \ 6=290000 \ \\ \ \\ a_{16}=2; b_{16}=8; c_{16}=4 \ \\ \ \\ d_{16}=a_{16} \cdot \ 1000+b_{16} \cdot \ 2000+c_{16} \cdot \ 1500=2 \cdot \ 1000+8 \cdot \ 2000+4 \cdot \ 1500=24000 \ \\ \ \\ n_{16}=14000 \cdot \ a_{16}+24000 \cdot \ b_{16}+18000 \cdot \ c_{16}=14000 \cdot \ 2+24000 \cdot \ 8+18000 \cdot \ 4=292000 \ \\ \ \\ a_{17}=3; b_{17}=9; c_{17}=2 \ \\ \ \\ d_{17}=a_{17} \cdot \ 1000+b_{17} \cdot \ 2000+c_{17} \cdot \ 1500=3 \cdot \ 1000+9 \cdot \ 2000+2 \cdot \ 1500=24000 \ \\ \ \\ n_{17}=14000 \cdot \ a_{17}+24000 \cdot \ b_{17}+18000 \cdot \ c_{17}=14000 \cdot \ 3+24000 \cdot \ 9+18000 \cdot \ 2=294000 \ \\ \ \\ a_{18}=4; b_{18}=10; c_{18}=0 \ \\ \ \\ d_{18}=a_{18} \cdot \ 1000+b_{18} \cdot \ 2000+c_{18} \cdot \ 1500=4 \cdot \ 1000+10 \cdot \ 2000+0 \cdot \ 1500=24000 \ \\ \ \\ n_{18}=14000 \cdot \ a_{18}+24000 \cdot \ b_{18}+18000 \cdot \ c_{18}=14000 \cdot \ 4+24000 \cdot \ 10+18000 \cdot \ 0=296000 \ \\ \ \\ n=296000



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