1. 0 A Toy Production Problem  How many units to produce from each product type in order to maximize the profit? ProductMan-PowerMachineProfit Type A3 h1 h25 SR Type B2 h4 h15 SR Availability70 h110 h

2. 1 A Toy Production Problem x A : number of units of product type A x B : number of units of product type B Total Profit: 25 x A + 15 x B Man-Power availability: 3 x A + 2 x B  70 Machine availability: x A + 4 x B  110 ProductMan-PowerMachineProfit Type A3 h1 h25 SR Type B2 h4 h15 SR Availability70 h110 h

3. 2 A Toy Production Problem The corresponding linear program (LP) is: Max Z = 25 x A + 15 x B Subject to 3 x A + 2 x B  70 x A + 4 x B  110 x A, x B  IN Objective function Constraints Decision Variables

4. 3 A Toy Production Problem The optimal solution of the LP is: x* A = 22, x* B = 2 and Z* = 580 SR The optimal solution of the toy production problem is to produce 22 units of toy A and 2 units of toy B. The optimal profit is 580 SR.

5. 4 A Banking Problem A bank is in the process to allocate 12 million SR to different types of loans. Competition with other financial institutions suggests allocating at least 40% of the funds to farm & commercial loans. To assist in the housing industry, the home loans must be at least 50% of the personal, car and home loans. The bank management requires that the overall bad ratio debt not to exceed 4%. The bank wants to determine the best loan policy that will maximize its profit rate. Type of loanProfit rateBad debt ratio 1Personal0.140.10 2Car0.130.07 3Home0.120.03 4Farm0.1250.05 5Commercial0.100.03

6. 5 A Banking Problem Decision variables: x 1 : amount of personal loans (in million SR) x 2 : amount of car loans (in million SR) x 3 : amount of home loans (in million SR) x 4 : amount of farm loans (in million SR) x 5 : amount of commercial loans (in million SR) Type of loanProfit rateBad debt ratio 1Personal0.140.10 2Car0.130.07 3Home0.120.03 4Farm0.1250.05 5Commercial0.100.03

7. 6 A Banking Problem Objective function: Max Z= 0.14(0.90)x 1 + 0.13(0.93)x 2 + 0.12(0.97)x 3 +0.125(0.95)x 4 +0.1 (0.98)x 5 - 0.1x 1 - 0.07x 2 - 0.03x 3 - 0.05x 4 -0.02x 5 Type of loanProfit rateBad debt ratio 1Personal0.140.10 2Car0.130.07 3Home0.120.03 4Farm0.1250.05 5Commercial0.100.02

8. 7 A Banking Problem Constraints: The total funds shall not exceed 12 million SR: x 1 +x 2 + x 3 + x 4 +x 5  12 Farm and commercial loan constraint: x 4 +x 5  0.4(x 1 +x 2 + x 3 + x 4 +x 5 ) Home loans constraint: x 3  0.5(x 1 +x 2 + x 3 ) Type of loanProfit rateBad debt ratio 1Personal0.140.10 2Car0.130.07 3Home0.120.03 4Farm0.1250.05 5Commercial0.100.03

9. 8 A Banking Problem Constraints: Limit on bad debts rule: (0.1x 1 + 0.07x 2 + 0.03x 3 + 0.05x 4 +0.02x 5 )/( x 1 +x 2 + x 3 + x 4 +x 5 )  0.04 (NONLINEAR) or equivalently, 0.06x 1 + 0.03x 2 - 0.01x 3 + 0.01x 4 - 0.02x 5  0 Type of loanProfit rateBad debt ratio 1Personal0.140.10 2Car0.130.07 3Home0.120.03 4Farm0.1250.05 5Commercial0.100.03

10. 9 A Banking Problem The optimal solution of the LP is: x* 1 = x* 2 = x* 4 = 0, x* 3 = 7.2, x* 5 = 4.8 and Z* = 0.99648 The optimal solution consists in allocating 7.2 M to the home type and 4.8 M to commercial type. The optimal profit is 0.996 M SR.

11. 10 A Telecommunication Problem We have to place transmitters on the sites A,B,…,G. Each transmitter covers the two adjacent zones (e.g. if a transmitter is placed on site D, then it will cover both zones 3 and 4). Each Zone must be covered with at least one transmitter. Zone 4 must be covered by at least 2 transmitters. What is the minimum number of transmitters to be placed? Where should they be placed?

12. 11 A Telecommunication Problem Decision variables x i = 1 if a transmitter is to be placed on site i (i =A,B,…,G) 0 otherwise Objective function Min Z = x A + x B + x C + x D + x E + x F + x G

13. 12 A Telecommunication Problem Constraints x A + x B + x C  1 (Zone 1) x A + x E + x F  1 (Zone 2) x B + x D  1 (Zone 3) x C + x D + x E + x G  2 (Zone 4) x F + x G  1 (Zone 5) x i  {0,1} for all i=A,B,…,G

14. 13 A Telecommunication Problem The optimal solution of the LP is: x * A = x * D = x * G = 1 x * B = x * C = x * E = x * F = 0 Z * = 3 The optimal solution of the telecommunication problem is to place one transmitter at each of the sites A, D, and G.

15. 14 A Transportation Problem All the demands must be satisfied. All the supplies must be delivered. How many units to transport from each source to each destination in order to minimize the total transportation cost? D1D1 D2D2 C B A 7 2 4 6 3 5 Demand 4 5 8 Supply 8 9 SourcesDestinations Unit transportation cost

16. 15 A Transportation Problem x ij : number of units transported from Source i to Destination j i=1,2j=A,B,C Objective function Min Z = 5 x 1A + 3 x 1B + 6 x 1C + 4 x 2A + 7 x 2B + 2 x 2C D1D1 D2D2 C B A 7 2 4 6 3 5 Demand 4 5 8 Supply 8 9 SourcesDestinations Unit transportation cost

17. 16 A Transportation Problem Constraints x 1A + x 1B + x 1C = 8 (Supply of D 1 ) x 2A + x 2B + x 2C = 9 (Supply of D 2 ) x 1A + x 2A = 4 (Demand of A) x 1B + x 2B = 5 (Demand of B) x 1C + x 2C = 8 (Demand of C) x 1A,x 1B, x 1C, x 2A, x 2B, x 2C  IN D1D1 D2D2 C B A 7 2 4 6 3 5 Demand 4 5 8 Supply 8 9 SourcesDestinations

18. 17 A Transportation Problem Optimal solution x* 1A = 3, x* 1B = 5, x* 1C = 0, x* 2A = 1, x* 2B = 0, x* 2C = 8 Z* = 50 D1D1 D2D2 C B A 7 2 4 6 3 5 Demand 4 5 8 Supply 8 9 SourcesDestinations