5.5Linear Programming Objectives:After completing this section, you should be able to: 1.Solve linear programming problems; 2.Use linear programming to model and solve real-life problems. In Section 4, many of the problems we solved are related to thegeneral type of problems called linearprogrammingproblems.Linearprogrammingisamathematicalprocessthathasbeen developed tohelpmanagementindecisionmaking,andithasbecomeoneofthewidelyusedtoolsinaprocess calledoptimization.Inthissectionwewillstudylinearprogrammingasastrategyforfindingthe optimal value either maximum or minimum of a quantity subject to certain constraints. We will use an intuitivegraphicalapproachbasedon thetechniques wediscussedinSection4toillustratethisprocess for problems involving two variables. GeorgeDantzig(1914-),anAmericanmathematician,wasthefirsttoformulatealinear programming problem in 1947 and introduced a solution technique, called simplex method, that finds the optimalsolutionofcomplexlinearprogrammingproblemsinvolvingthousandsofvariablesand inequalities. This method does not rely on graphing and is well-suited for computer implementation. Linear Programming Model A Linear Programming problemis concerned with finding the optimal value (maximum or minimum value) of a linear objective function of the form z = ax + by where the decision variables x and y are subject to problem constraints stated as a system of linear inequalitiesand to nonnegative constraints x, y 0. The set of points satisfying the system of linear inequalitiesand the nonnegative constraints is called the set of feasible solutions for the problem.The graph of this set is the feasible region. Any point in the feasible region that produces the optimal value of the objective function over the feasible region is called an optimal solution. Unit 5. Systems of Equations and Inequalities Section 5page 2 _____________________________________________________________________________________ Example 5.5.1Linear Programming problem Find the maximum value of z = 2x + 3y subject to >>s +s +0021 2 325 5 2yxy xy x Because every point in the feasible region satisfies each constraint, the question is which of those points will yield a maximum value of z?We cannot check all of the feasible solutions to see which one resultsinthelargestvalueofz.Fortunately,asimplewayhasbeendevelopedtofindthisoptimal solution.Using advanced techniques, it can be shown that If the feasible region is bounded, then one or more of the corner points of the feasible region is an optimal solution to the problem. This means that you can find the maximum value of z by testing z at each of the corner points and choose the corner point that produces the largest value of z. Objective Function Constraints Fundamental Theorem of Linear Programming (Optimal Solution of a Linear programming Problem) LetS be the feasible region for a linear programming problem, and let z = ax + bybe the objective function. If S is bounded, then z has both a maximumand a minimum value on S and each of these occurs at a corner point of S. If S is unbounded, then a maximumor a minimumvalue of z on S may not exist..However if either does exist, then it must occur at a corner point of S. The maximum or minimum value of the objective function z is unique; however, there can be more than one feasible solution that will produce this unique value. Feasible Region 3x + 2y = 21 2x + 5y = 25 Fig. 5.5.1 Unit 5. Systems of Equations and Inequalities Section 5page 3 _____________________________________________________________________________________ Example 5.5.1continued Test the value of z at each corner point. Whydowehavetocheckonlythecornerpointsofthefeasibleregion?Welladdressthis questionbyconsideringafamilyofobjectfunctions.Rewritingtheobjectivefunctiony x z 3 2 + = in slope-intercept form, we obtain.3 32 zx y + =Thisequationrepresentsafamilyofparallellines,eachofslope 32andy-intercept 3z,oneforeach value of z. Of these infinitely many lines, we want the one that intersects the region and gives the largest value of z,i.e., the line that intersects the region and has the largest y-intercept.In Fig. 5.5.2, we set z to thefollowingvalues:0,3,6,9,18,19,21.Lookingatthefigure,weseethattherequiredlinepasses through (5,3), one of the corner points of the region. Corner Point (x, y) Objective Function z = 2x + 3y (0, 0) 0 (7, 0) 14 (5, 3) 19 (0,5) 15 The optimal solution is (5, 3). The maximum value of z on the feasible region is19 which occurs at the corner point (5, 3). Feasible Region (5, 3) z = 2(5) + 3(3) = 19 (7, 0) z = 2(7) + 3(0) = 14 (0, 5)z = 2(0) + 3(5) = 15 (0, 0) z = 2(0) + 3(0) = 0 Maximum value of z Unit 5. Systems of Equations and Inequalities Section 5page 4 _____________________________________________________________________________________ Aszincreasesthroughvalues0,3,6,9,18,and19,thecorrespondinglinepassesthroughthe feasible region.The point at which the family of lines first meets the feasible region gives the minimum value of z, and the point at which the family of lines leaves the feasible region gives the maximum. The idea is to move the line as high as possible but so that it still intersects the feasible region. . Solving a Linear Programming Problem Graphically Step 1.Form a mathematical model for the problem. (a)Introduce decision variables and write a linear objective function. (b)Write problem constraints in the form of linear inequalities. (c)Write nonnegative constraints. Step 2.Graph the feasible region and find the corner points. Step 3.Evaluate the objective function at each corner point to determine the optimal solution. Note:If the optimal value occurs at two corner points, then any point on the line segment joining the two corner points is also an optimal solution. Feasible Region x y z32- 0 = =132- 3 + = = x y z232- 6 + = = x y z332- 9 + = = x y z + = = 732- 21 x y z31932- 19 + = = x y z632- 18 + = = x y zINFEASIBLE (outside the feasible Region) Fig. 5.5.2 Maximum z line Unit 5. Systems of Equations and Inequalities Section 5page 5 _____________________________________________________________________________________ Example 5.5.2 Maximizing Profit Aling Maring sells boxed cassava cakeand espasol.Each box of cassava cake yields a profit of P15perboxandaboxofespasolyieldsaprofitofP20.Consideringtheavailableresourcesandthe weeklymarkettrendforhercakes,AlingMaringobservedthatinaweeksheshouldproducenomore than 120 boxes altogether.She also observed that the production level of cassava cake should be no more than 60 boxes plus three times the production level of a box of espasol.Moreover, the demand for a box of espasol is no more than half of the demand for a box of cassava cake. How many boxes of each type shouldAlingMaringproducetomaximizeherweeklyprofit,assumingallherproductscanbesold?What is her maximum profit? Solution: To formulate this as a linear programming problem: 1. Identify the variables. 2.Write the objective function. 3.Write the constraints. Decision Variables Let x = number of boxes of cassava cake produced per week y = number of boxes of espasol produced per week Objective Function Express the profit as a function of x and y.

y x P 20 15 + = Constraints Express the constraints as linear inequalities. Production level constraint: s + ss + s +60 3 3 60120 120y x y xy x y x Demand constraint:0 221> s y x x yNonnegative constraints:Because neither x nor y can be negative, we also the two nonnegative constraints >>00yx. Function to be maximized Unit 5. Systems of Equations and Inequalities Section 5page 6 _____________________________________________________________________________________ Mathematical Model: We now have a mathematical model for the problem under consideration. Maximizey x P 20 15 + =Subject to >> s s +0 ,0 260 3120y xy xy xy x Examining the values in the table, Aling Maring should produce 80 boxes of cassava cake and 40 boxes of espasol per week for a maximum profit of P2000. (80, 40) (105, 15) (60, 0)(0, 0) x + y = 120 x - 3 y = 60 x - 2 y = 0 Corner Point (x, y) Objective Function z = 2x + 3y (0, 0) 0 (60, 0) 900 (80, 40) 2000 (105,15) 1875 P = 15(0) + 20(0) = 0P = 15(60) + 20(0) = 900 P = 15(105) + 20(15) = 1875 P = 15(80) + 20(40) = 2000 Time to think! A poultry cooperative mixes two brands of chicken feed.Brand X costs P25 per kilo and contains two units of nutritional element A, two units of element B, and two units of element C.Brand Y costs P20 per kilo and contains one unit of nutritional element A, nine units of element B , and three units of element C.The minimum requirements of nutrients A, B, and C are 12 units, 36 units, and 24 units, respectively.Find the number of kilos of each brand that should be mixed to produce a mixture having a minimum cost.What is the minimum cost? Unit 5. Systems of Equations and Inequalities Section 5page 7 _____________________________________________________________________________________ Example 5.5.3 Minimizing and Maximizing an Objective Function Minimize and maximizey x z 15 5 + =Subject to >s > +s +0 ,01060 3y xy xy xy x Solution: This problem is a combination of two liner programming problems- a minimization problem and a maximization problem.Because the feasible region is the same for both problems, we can solve these problems together. Weseeinthetablethattheminimumvalueofzonthefeasibleregionis100which occurs at (5, 5).Hence (5, 5) is the optimal solution to the minimization problem.On the other hand, the maximum value of z on the feasible regionis 300, which occurs at (0, 20) and (15, 15).Thus, the maximization problem has multiple optimal solutions. In general, If two corner points are both optimal solutions of the same type (both producethe same maximum value or both produce the same minimum value) to a linearprogramming problem, then any point on the line segment joining the twocorner points is also an optimal solution of that type. Corner Point (x, y) Objective Function z = 5x + 15y (0, 10) z = 5(0) + 15(10) = 150 (0, 20) (15, 15) z = 5(0) + 15(20)= 300 z = 5(15) + 15(15) = 300 (5,5) z = 5(5) + 15(5) = 100 Minimum value (15, 15) (5, 5) (0, 10) (0, 20) Maximum value (multiple optimal solutions) Unit 5. Systems of Equations and Inequalities Section 5page 8 _____________________________________________________________________________________ Example 5.5.3Unbounded Region The minimum daily requirements from the liquid portion of a diet are 300 calories, 36 units ofvitaminA,and90 unitsof vitamin C.A cupof dietary drink X costsP12 and provides 60 calories, 12 unitsofvitaminA,and10unitsofvitaminC.AcupofdietarydrinkYcostsP15andprovides60 calories,6unitsofvitaminA,and30unitsofvitaminC.Howmanycupsofeachdrinkshouldbe consumed each day to minimize the cost and still meet the daily requirements? Solution: Let x = number of cups of dietary drink Xy = number of cups of dietary drink X Express the cost as a function of x and y to obtain the objective function Minimizey x C 15 12 + =To meet the minimum daily requirements, the following inequalities must be satisfied: >> +> +> +0 ,90 30 1036 6 12300 60 60y xy xy xy x

Time to think! Sketch the region determined by the constraints.Then find the maximum and minimum values of the objective function and where they occur, subject to the indicated constraints.Objective Function:y x z 5 4 + =Constraints:

>s +s > +016 49 36 3 2xy xy xy x For calories For vitamin A For vitamin C Nonnegative constraints Unit 5. Systems of Equations and Inequalities Section 5page 9 _____________________________________________________________________________________ Examining the values in the table, we see that the minimum cost is P66 per day, and this occurs when 3 cups of drink X and two cups of drink Y are consumed each day. Noticethat although the region determinedby theconstraints isunbounded,thereisaminimum value of C.However, for this unbounded region, there is NO maximum value of C. Time to think! Sketch the region determined by the following constraints: >s +s +0 ,15 310 2y xy xy x The corner points for the feasible region determined by these problem constraints areO=(0,0),A=(5,0),B=(3,4),andC=(0,5).Iftheobjectivefunctionisoftheform by ax z + =and a, b > 0,determine conditions on a and b that ensure that the maximum value of z occurs only at B.

(9, 0) (3, 2) (1, 4) (0, 6) Corner Point (x, y) Objective Function C = 12x + 15y (0, 6) C = 12(0) + 15(6) = 90 (1, 4) C = 12(1) + 15(4) = 72 (3, 2) C = 12(3) + 15(2) = 66Minimum value of C (9, 0) C = 12(9) + 15(0) = 108 Unit 5. Systems of Equations and Inequalities Section 5page 10 _____________________________________________________________________________________ Exercise 5.5. A. Sketchtheregiondeterminedbytheconstraints.Thenfindtheminimumandmaximumvaluesof the objective function and where they occur, subject to the indicated constraints. 1.Minimize and maximizez = 4x + 5y Subject to>> +> +0 ,830 5 3y xy xy x 2.Minimize and maximizez = 15x + 30y Subject to>s +s > +0 ,200 20 2100 2y xy xy xy x 3.Minimize and maximizez = 25x + 15y Subject to>sss +> +0 ,4560240 4 3100 5 4y xyxy xy x B.Eachoftheselinearprogrammingproblemshasanunusualcharacteristic.Sketchagraphofthe feasible region for the problem and describe the unusual characteristic.Find the maximum value of the objective function and where it occurs. 1.Maximizez = x + y Subject to >s + s + 0 ,4 21y xy xy x 2.Maximizez = x + y Subject to >> + s + 0 ,3 30y xy xy x 3.Maximizez = x + 2y Subject to >s +s +0 ,4 24 2y xy xy x Unit 5. Systems of Equations and Inequalities Section 5page 11 _____________________________________________________________________________________ C.1.A manufacturer of cell phones makes a profit of P50 on a deluxe model and P75 on standardmodel.Thecompanywishestoproduceatleast80deluxemodelsandatleast100standard modelsperday.Tomaintainhighquality,thedailyproductionshouldnotexceed200cell phones. How many of each type should be produced daily in order to maximize the profit? 2.A stationery company makes two types of notebooks.A deluxe notebook with subject dividers,which sells for P40 and a regular notebook which sells for P30.The production cost is P32 for each deluxe and P26 for each regular notebook.The company has the facilities to manufacture between 2000 and 3000 deluxe and between 3000 and 6000 regular notebooks, but not more than 7000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production costs? 3.Rolys pancake recipe includes corn meal and whole wheat flour.Corn meal has 2.4 grams of linoleic acid and 2.5 milligrams of niacin per cup.Whole wheat flour has 0.8 grams of linoleic acid and 5.2milligrams of niacin per cup.These two dry ingredients do not exceed 3 cups total.They combine for at least3.2 grams of linoleic acid and 10 milligrams of niacin.Minimizethe number of calories possible in the recipe if corn meal has 433 calories per cup and whole wheat flour has 400 calories per cup. 4.Siza requires 1 hour of cutting and 2 hours of sewing to make a Batman costume.She requires 2 hours of cutting and 1 hour of sewing to make a Wonder Woman costume. At most 10 hours per day are available for cutting and at most 8 hours per day are available for sewing.At least one costume must be made each day to stay in business. Find Sizas maximum income from selling one days costume if a Batman costume costs P500 and a Wonder Woman costume costs P700. 5.Because of new government regulations on pollution, a chemical plant introduced a new process to supplement or replace an older process used in the production of a particular chemical.The older process emitted 20 grams of sulfur dioxide and 40 grams ofparticulate matter into the atmosphere for each gallon of chemical produced.The new process emits 5 grams of sulfur dioxide and 20 grams ofparticulate matter for each gallon of chemical produced.The company makes a profit of P60 per gallon and P20 per gallon on the old and new processes, respectively.a)If the regulations allow the plant to emit no more than 16,000 grams of sulfur dioxide and 30,000 grams ofparticulate matter daily, how many gallons of the chemical should be produced by each process to maximize daily profit? What is the maximum daily profit? b)Discuss the effect on the production schedule and the maximum profit of the regulationsrestrict emissions of sulfur dioxide to 11,500 grams daily and all other data remain unchanged. c)Discuss the effect on the production schedule and the maximum profit of the regulationsrestrict emissions of sulfur dioxide to 7,200 grams daily and all other data remain unchanged. Unit 5. Systems of Equations and Inequalities Section 5page 12 _____________________________________________________________________________________ 6.A town council voted to conduct a study on the towns community problems. A nearby university was contacted to provide a maximum of 20 sociologists and research assistants. Allocation of time and cost per week are given in the table. Sociologist (Labor-Hours) Research Assistant (Labor-Hours) Minimum Labor-Hours Needed per Week Fieldwork1030280 Research center 3010360 Cost per weekP2500P2000 a)Howmanysociologistsandresearchassistantsshouldbehiredtomeettheweeklylabor-hour requirements and minimize the weekly cost?What is the weekly cost? b)Discuss the effect on the solution inpart A if the council decides that they should not hire more sociologists than research assistantsand all other data remain unchanged. 7.An owner of a calamansi plantation hires a crew of workers to prune at least 25 of his 50 calamansi trees. Each newer tree requires one hour to prune, while each older tree needs one-and-a-half hours. The crew contracts to work for at least 30 hours and charge P30 for each newer tree and P50 for each oldertree.Tominimizehiscost,howmanyofeachkindoftreewilltheplantationownerhave pruned?What will be the cost?