alloys. The composition and prices of these alloys is shown in the table as follows.
Alloy 1 2 3 4 5
%A 10 15 20 30 40
%B 90 85 80 70 60
Available quantity kg 300 400 200 700 450
Price $/kg 6 10 18 24 30
Formulation
Letx1,x2,x3,x4,x5be the number of kg of each of the available alloys melted and mixed to form the needed amount of the required alloy. Then the problem can be formulated as
Minimize 6x1+10x2+18x3+24x4+30x5
Subject to 0.1x1+0.15x2+0.2x3+0.3x4+0.4x5=250 x1+x2+x3+x4+x5=1000
x1≤300, x2≤400, x3≤200, x4≤700, x5≤450 x1≥0, x2≥0, x3≥0, x4≥0, x5≥0
The problem is now ready for solving using the simplex or other methods.
PM3: Refinery Problem
A refinery uses two different crude oils, a light crude costs $40 per barrel, and a heavy crude costs $30 per barrel, to produce gasoline, heating oil, jet fuel, and lube oil. The yield of these per barrel of each type of crude is given in the following table:
Gasoline Heating oil Jet fuel Lube oil
Light crude oil 0.4 0.2 0.3 0.1
Heavy crude oil 0.3 0.45 0.1 0.05
The demand is 8 million barrels of gasoline, 6 million barrels of heating oil, 7 million barrels of jet fuel, and 3 million barrels of lube oil. Determine the amounts of light crude and heavy crude to be purchased for minimum cost.
Formulation
Letx1andx2be the millions of barrels of light and heavy crude purchased, respec- tively. Then the problem can be put in the form
4.4 Problem Modeling 139 Minimize 40x1+30x2
Subject to 0.4x1+0.3x2 ≥8 0.2x1+0.45x2≥6 0.3x1+0.1x2 ≥7 0.1x1+0.05x2≥3 x1 ≥0, x2≥0
This is a simple formulation of petroleum refinery problem. Some of the real prob- lems may have several more variables and more complex structure but they are still LP problems.
PM4: Agriculture
A vegetable farmer has the choice of producing tomatoes, green peppers, or cucum- bers on his 200 acre farm. A total of 500 man-days of labor is available.
Yield $/ acre Labor man-days/acre
Tomatoes 450 6
Green Peppers 360 7
Cucumbers 400 5
Assuming fertilizer costs are same for each of the produce, determine the opti- mum crop combination.
Formulation
Letx1,x2, andx3be the acres of land for tomatoes, green peppers, and cucumbers respectively. The LP problem can be stated as
Maximize 450x1 +360x2+400x3 Subject to x1 + x2+x3 ≤200
6x1 + 7x2+5x3 ≤500 x1 ≥0, x2≥0, x3≥0
PM5: Trim Problem
A paper mill makes jumbo reels of width 1m. They received an order for 200 reels of width 260 mm, 400 reels of width 180 mm, and 300 reels of width 300 mm. These rolls are to be cut from the jumbo roll. The cutter blade combinations are such that
one 300 mm width reel must be included in each cut. Determine the total number of jumbo reels and cutting combinations to minimize trim.
Formulation
First we prepare a table of all combinations with the amount of trim identified in each combination.
Number of Number of Number of
Combination 300 mm 260 mm 180 mm Trim mm
1 3 0 0 100
2 2 1 0 140
3 2 0 2 40
4 1 2 1 0
5 1 1 2 80
6 1 0 3 160
Letxibe the number of jumbo reels subjected to combinationi. The problem is now easy to formulate. We make the first formulation realizing that the best cutting combination leads to least number of total jumbo reels.
Formulation 1 based on minimum total number:
minimize x1 +x2+x3+x4+x5+x6
subject to 3x1+2x2+2x3+x4+x5+x6≥300 x2 +2x4+x5≥200
2x3+x4+2x5+3x6≥400
x1 ≥0, x2≥0, x3≥0, x4≥0, x5≥0, x6 ≥0 Formulation based on minimum trim will require that excess rolls produced in each size be treated as waste. The surplus variable of each constraint must be added to the cost function.
Formulation 2 based on minimum trim:
minimize 100x1+140x2+40x3+80x5+160x6+300x7+260x8+180x9 subject to 3x1 +2x2+2x3+x4+x5+x6−x7=300
x2 +2x4+x5−x8=200 2x3 +x4+2x5+3x6−x9=450
x1 ≥0, x2≥0, x3≥0, x4≥0, x5 ≥0, x6 ≥0, x7≥0, x8≥0, x9≥0
4.4 Problem Modeling 141 The equivalence of the two formulations can be accomplished easily. By elimi- natingx7,x8, andx9using the constraint equations, the objective function becomes 1000(x1+x2+ · · · +x6).
In the aforementioned formulations, the solutions may have fractional values.
We know that each of the variablesximust be an integer. This may be solved later as an integer programming problem. The general strategy in integer programming problems is to start with a solution where this requirement is not imposed on the initial solution. Other constraints are then added, and the solution is obtained iteratively.
PM6: Straightness Evaluation in Precision Manufacture
In the straightness evaluation of a dimensional element, coordinate location along the edgexi in mm and the coordinate measuring machine probe reading yi inμm (micron meter=10−6m) have been recorded as shown.
1 2 3 4 5
ximm 0 2.5 5 7.5 10
yiμm 10 5 19 11 8
Straightness is defined as the smallest distance between two parallel lines between which the measured points are contained. Determine the straightness for the data shown.
Formulation
Letvbe the interpolated value of the measured value given byax+b. Ifviis the calculated value at pointi, then the problem can be posed as a linear programming problem:
Minimize 2z
subject to z≥vi−yi
z≥ −vi+yi i =1to5
The variablesz,a, andbmay be left as unrestricted in sign. Note that the final value ofzis the maximum deviation from the minimum zone best fit straight line.
PM7: Machine Scheduling
A machine shop foreman wishes to schedule two types of parts, each of which has to undergo turning, milling, and grinding operations on three different machines.
The time per lot for the two parts available machine time and profit margins are provided in the table as follows.
Profit
Part Turning Milling Grinding $ per lot
Part 1 12 hrs/lot 8 hrs/lot 15 hrs/lot 120
Part 2 6 hrs/lot 6.5 hrs/lot 10 hrs/lot 90
Machine time available hrs 60 hrs 75 hrs 140 hrs
Schedule the lots for maximizing the profit.
Formulation
Letx1andx2be the number of lots of parts 1 and 2 to be produced, respectively.
Then the LP problem can be formulated as Maximize 120x1 +90x2 Subject to 12x1 +6x2≤60
8x1 +6.5x2≤75 15x1 +10x2≤140
x1 ≥0,x2≥0
Problems of complex nature can be formulated with a careful consideration. Some problems of great importance in transportation, assignment, and networking are of the linear programming category, but have special structures. These problems will be considered in greater detail in a later chapter.
4.5 Geometric Concepts: Hyperplanes, Halfspaces,