In the name of Allah 1

Formulation (lecture 5) 2

Crude Oil Refining and Gasoline Blending - Page 52 Shale Oil, located on the island of Aruba, has a capacity of 1,500,000 bbl of crude oil per day. The final products from the refinery include three types of unleaded gasoline with different octane numbers (ON): regular with ON = 87, premium with ON = 89, and super with ON = 92. The refining process encompasses three stages: (1) a distillation tower that produces feedstock (ON = 82) at the rate of .2 bbl per bbl of crude oil, (2) a cracker unit that produces gasoline stock (ON = 98) by using a portion of the feedstock produced from the distillation tower at the rate of .5 bbl per bbl of feedstock, and (3) a blender unit that blends the gasoline stock from the cracker unit and the feedstock from the distillation tower. The company estimates the net profit per barrel of the three types of gasoline to be $6.70, $7.20, and $8.10, respectively. The input capacity of the cracker unit is 200,000 barrels of feedstock a day. The demand limits for regular, premium, and super gasoline are 50,000,30,000, and 40,000 barrels per day. Develop a model for determining the optimum production schedule for the refinery.

Crude Oil Refining and Gasoline Blending - Page 52 Step1(Decision variables) Xij: bbl/day used in stage i to produce oil type j i=1,2 j=1,2,3 Step2 (objective function) amount of produced oil type1(ON 87): x11+x21 amount of produced oil type2(ON 89): x11+x21 amount of produced oil type3(ON 92): x11+x21 Therefore: Max Z=6.70(x11+x21)+7.20(x12+x22)+8.10(x13+x23)

Crude Oil Refining and Gasoline Blending - Page 52 Step3(Constraints) 1. Daily crude oil supply does /lot exceed 1,500,000 bbl/day: 5(x11 + x12 + x13) + 10(x21 + x22 + x23) ≤1,500,000 2. Cracker unit input capacity does not exceed 200,000 bbl/day: 2(x21 + x22 + x23) ≤200,000 3. Daily demand for regular does not exceed 50,000 bbl: x11+ x21 ≤ 50,000 4. Daily demand for premium does not exceed 30,000: x12+ x22≤ 30,000 5. Daily demand for super does not exceed 40,000 bbl: x13 + x23 ≤ 40,000

Crude Oil Refining and Gasoline Blending - Page 52 6. (Average on of regular gasoline) = (Octane stage1×bbl/day + octane stage2 × bbl/day) ÷ (Total bbl/day of regular)= (82x11 + 98x21 )÷ (x11+x21) Therefore: 82x11 + 98x21 ≥87(x11 + x21) 7. 82x12 + 98x22 ≥ 89(x12 + x22) 8. 82x13+98x32 ≥92(x13+x23) xij ≥0 , i=1,2 j=1,2,3

Crude Oil Refining and Gasoline Blending - Page 52 توانينcapacity بةرميلbbl خاوcrude ثالاَوتطاrefinery لة خو طرتنinclude process encompasses قوَناغStage دلَوَثاندنdistillation برج-كةلو tower

Bus Scheduling – page 58 Progress City is studying the feasibility of introducing a mass-transit bus system that will alleviate the smog problem by reducing in-city driving. The study seeks the minimum number of buses that can handle the transportation needs. After gathering necessary information, the city engineer noticed that the minimum number of buses needed fluctuated with the time of the day and that the required number of buses could be approximated by constant values over successive 4 hour intervals. The following figure summarizes the engineer's findings. To carry out the required daily maintenance, each bus can operate 8 successive hours a day only.

Bus Scheduling – page 58 Step 1 (Decision Variables) X1=number of buses starting at 12: 01 A.M. X2=number of buses starting at 4: 01 A.M. X3=number of buses starting at 8: 01 A.M. X4=number of buses starting at 12: 01 A.M. X5=number of buses starting at 4: 01 A.M. X6=number of buses starting at 8: 01 A.M.

Bus Scheduling – page 58 Step2(objective function) Min Z=x1+x2+x3+x4+x5+x6 Step3(constraints) x6+x1 ≥ 4, x1+x2 ≥ 8, x2+x3 ≥ 10, x3+x4 ≥ 7, x4+x5 ≥ 12, x5+x6 ≥ 4, xi ≥0 , i=1,2,3,4,5,6

Bus Scheduling – page 58 لةكردن هاتووfeasibility ئةندازيارengineer ناساندنintroduce نزيكةيapproximate رِيذة طواستنةوة mass-transit بة دواي يةكدا هاتووsuccessive كةم دةكاتةوة alleviate جيَبةجيَدةكاتcarry out دووكةل و مْذ Smog ضاوديَريmaintenance كةم دةكاتةوة reduce كار دةكات Operate ليَدةخورِيَت drive ليَكولينةوةstudying دةطةريَseek ناو شاريin-city بة رِيَوة بردنhandle طواستنةوة و طة ياندنtransportation gathering necessary information کوَكردنةوةي زانيارية ثيَويستية كان