Mathematics for Management Science
Mathematics for Management Science MT 235
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Mathematics for Management Science Notes 04 prepared by Professor Jenny Baglivo Jenny A Baglivo 2002 All rights reserved Application type 1 blending problems Blending problems arise when a manager must decide how to blend two or more resources to produce one or more products Blending problems occur in the petroleum industry eg crude oil to gasoline chemical industry eg chemical components to fertilizers and the food industry e g ingredients to soups Each product indexed using i1 2 etc is produced by blending resources indexed using j1 2 etc We need to make decisions about how much of resource is in product i for all possible i and j Decision variables have double subscripts and can be visualized using a contingency table The following table shows the table when there are i2 products and j3 components In the table 1 x is the total production of product i 2 x is the total amount of resource j used in making all products 3 x is the total production taking all products into account The sum variables are often called auxiliary or de ned variables They can be used in defining the model and in the spreadsheet implementation page 1 of 24 Exercise 1 The Grand Strand Oil Company produces regular and premium blends for independent service stations in the southeastern United States The Grand Strand refinery manufactures the gasolines by blending three petroleum components The gasolines are sold at different prices and the petroleum components have different costs The first table gives costs for the three components and supplies in the next production cycle The second table gives sales prices for the two blends and blending specifications annnnnnf Costgallon Available 1 050 5000 gallons 2 060 10000 gallons 3 084 10000 gallons Product Revenuegallon sherifirafinn regular 100 at most 30 component at least 40 component at most 20 component premium 108 at least 25 component at most 40 component at least 30 component le WNI n Current commitments to distributors require Grand Strand to produce at least 10000 gallons of regular blend in the next production cycle The firm wants to determine how to mix or blend the three components into the two gasoline products in order to maximize profits 0 Summarize the problem 0 Define the decision variables precisely page 2 of 24 0 Completely Specify the LP made page 3 of 24 0 Clearly state the optimal solution 0 x11 equals zero in the optimal solution but the reduced cost is zero Why 0 If GrandiStrand could obtain 000 gallons of one of the components at current costs which should it buy Why 0 If the commitmentfor regular gas was only 9000 gallons how would the pro t change Be speci c page 4 of 24 Exercise 1 solution and sensitivitV report Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable page 5 of 24 Formulas worksheet page 6 of 24 Application type 2 financial planning problems Many financial planning problems involve planning for multiple periods of time since the amount of money invested or spent at one point directly affects the amount available in subsequent time periods In the problems considered here the manager must determine the levels of initial funding initial investments and reinvestrnents necessary to meet obligations at fixed periods of time over a fixed time horizon For example a construction fund could be set aside now so that scheduled monthly payments for a new building can be made over the next two years The initial funds include the obligation for the first period and the amount to be invested to meet the future obligations By rewriting we get Initial Funds Initial Investments Obligation for period 1 For subsequent time periods Returns on Investments ReInvestments Obligation for the given period In general we write Cash In Cash Out Obligation The manager would like to satisfy all obligations using the minimum level of initial funding Notes 0 There will be ow constraints for each time period 0 There may be other decisions variables and there may be additional constraints page 7 of 24 Exercise 2 Hewlitt Corporation has established an early retirement program as part of its corporate restructuring At the close of the voluntary signup period sixtyeight employees had elected early retirement As a result of these early retirements the company has incurred the following obligations over the next eight years Cash requirements in thousands of dollars are due at the beginning of each year Year 1 2 3 4 5 6 7 8 Reguirement 430 210 222 231 240 195 225 255 The corporate treasurer must determine how much money has to be set aside today to meet the eightyear financial obligations as they come due The financial plan for the retirement program includes investments in government bonds as well as savings The investments in government bonds are limited to three choices PriceUnit Rate Maturity Bond 1 1150 8875 5 years Bond 2 1000 5500 6 years Bond 3 1350 11750 7 years The government bonds have a par value of 1000 that is they pay 1000 at maturity even with different purchase prices The rates shown are based on the par value For purposes of planning the treasurer has assumed that any funds not invested in bonds at the beginning of this period will be placed in savings and eam interest at an annual rate of 4 Hewlitt would like to minimize the total dollars that must be set aside now to meet the eightyear obligation Note Units of bonds are purchased The units do not have to be whole numbers 0 Summarize the problem 0 Define the decision variables precisely page 8 of 24 0 Completely Specify the LP model 0 Clearly State the optimal solution page 9 of 24 Exercise 2 solution and sensitivitV renorts Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable page 10 of 24 Exercise 2 formulas sheet continued on next page page 11 of 24 continuation page 12 of 24 Application type 3 makeorbuy problems In make or buy problems the manager must determine how much of several components the company should manufacture or make and how much it should purchase from an outside supplier or buy There may be other decisions to make as well The components are indexed using i1 2 etc Convenient variables are M equals the quantity of component i to make i 1 2 etc Bi equals the quantity of component ito buy i 1 2 etc Exercise 3 The Janders Company markets various business and engineering products Currently J anders is preparing to introduce two new calculators one for the business market called the Financial Manager and one for the engineering market called the Technician Each calculator has three components a base an electronic cartridge and a face plate or top The same base is used for both calculators but the cartridges and tops are different All components can be manufactured by the company or purchased from outside suppliers The manufacturing costs and purchase costs are summarized below as well as the manufacturing time in minutes for the components Cost per unit To Make To Buy Manufacturing Time r nmnnnnnf regular time in minutes Base 050 0 60 10 minutes Financial cartridge 375 4 00 30 minutes Technician cartridge 3 30 3 90 2 5 minutes Financial op 060 0 65 10 minutes Technician top 0 75 0 78 l 5 minutes Janders forecasters indicate that 3000 Financial Manager calculators and 2000 Technician calculators will be needed However manufacturing capacity is limited The company has 200 hours of regular manufacturing time and 50 hours of overtime that can be scheduled for calculators Overtime involves a premium at the additional cost of 900 per hour The problem for Janders is to determine how many units of each component to manufacture and how many units of each component to purchase in order to minimize total cost page 13 of 24 0 Summarize the problem 0 Define the decision variables precisely 0 Completely specify the LP model page 14 of 24 0 Clearly State the optimal solution Exercise 3 solution sheet 6666667 2000 0 page 15 of 24 Exercise 3 sensitivity report Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable Notice that 1 The range of optimality of the coefficient for overtime is C11 2 5 dollars per hour 2 The shadow price for the manufacturing time constraint 00833 dollars per minute for minutes in the range 10000 to 19000 Converting to hours dCOSTdR7 5 dollars per hour for hours in the range 16667 to 31667 Thus if additional hours of manufacturing time come With a premium on cost per hour then Janders management should be Willing to pay up to 5 additional dollars per hour page 16 of 24 Exercise 3 formulas sheet page 17 of 24 Application type 4 production scheduling problems In production scheduling problems the manager s job is to determine an efficient multiperiod production and inventory schedule that allows the company to meet product demand and other requirements There may be additional decisions to make as well If there is more than one product then products are indexed using i1 2 etc Each period is indexed using j1 2 etc Let X equal the amount of product 139 produced during period j Sij equal the amount of product i stored at the end of period j for all possible i and j Since the current period s demand can be met from the current period s production or from inventory carried over from previous periods there will be constraints of the form Beginning Inventory Current Production Ending Inventory Current Demand for each product and period in the problem Exercise 4 Bollinger Electronics Company produces two different electronic components for a major airplane engine manufacturer The airplane engine manufacturer notifies the Bollinger sales office each quarter of its monthly requirements for components for each of the next three months The monthly requirements for the components may vary considerably depending on the type of engine the airplane engine manufacturer is producing The order for the next three month period is shown below Component April May June 322A 1000 3000 5000 802B 1000 500 3000 After the order is processed a demand statement is sent to the production control department The production control department must then develop a threemonth production plan for the components In arriving at the desired schedule the production manager will want to identify the total production cost and the inventory holding cost Component 322A costs 2000 per unit to produce and component 802B costs 1000 per unit to produce Monthly basis inventory holding costs are 15 of the production cost Machine labor and storage requirements for each component are given below Machine Labor Storage Component hrsunit hrsunit sgrftunit 322A 0 10 0 05 2 802B 0 08 0 07 3 page 18 of 24 Capacities for the next three months are as follows Capacities Machine Labor Storage Month hours hours square feet April 600 300 10000 May 500 300 10000 June 400 300 10000 Further assume there are 500 units of component 322A and 200 units of component 802B inventoried at the beginning of the threemonth period and that Bollinger specifies a minimum inventory level of 400 units of component 322A and 200 units of component 802B at the end of the threemonth period Bollinger would like to determine a three month production schedule that minimizes total cost 0 Summarize the problem 0 Define the decision variables precisely page 19 of 24 0 Completely Specify the LP made page 20 of 24 0 Clearly state the optimal solution 0 If the ending inventory requirement was 200 of each type how much would be saved 0 fan additional 20 hours ofmachine time in June could be obtained at a premium of 500 per hour would it make sense for Bollinger to obtain it If it does make sense find the new cost if not explain why page 21 of 24 Exercise 4 solution sheet page 22 of 24 Exercise 4 sensitivity report Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable Notice that The April demand shadow prices are exactly equal to the costs of producing an item of each type Ane the May demand shadow prices are exactly equal to the costs of producing an item in April and storing that item for one month page 23 of 24 Exercise 4 formulas sheet page 24 of 24 Mathematics for Management Science Notes 09 prepared by Professor Jenny Baglivo Jenny A Baglivo 2002 All rights reserved Decision analysis can be used to determine an optimal strategy when a decision maker is faced with several decision alternatives and an uncertain or risk filled pattern of future events For example the owner of a snow removal business may need to decide now whether or not to update snow removal equipment the decision to update would be wise if the amount of snowfall in the next season is high and would be unwise if the amount of snowfall in the next season is low The components of the problem are as follows 0 Decision alternatives di for il2 M 0 states of nature sj for jl2N Payoffs Vij for il2M jl2N To solve the problem we construct a payoff table and then choose the best alternative There are several different approaches to choosing the best 11 Optimistic Approach Maximax approach 0 For each alternative identify the maximum payoff 0 Choose the alternative with the largest maximum payoff 12 Pessimistic Approach Maximin approach 0 For each alternative identify the minimum payoff 0 Choose the alternative with the largest minimum payoff 13 Minimax Regret Approach 1 For each 1 compute the regret Rij which is the maximum payoff for state j minus Vij 2 For each alternative determine the maximum regret 3 Choose the alternative with the smallest maximum regret page 1 of 18 Example 1 Pittsburgh Development Corporation PDC has purchased land for a luxury riverfront condominium complex The site provides a spectacular view of downtown Pittsburgh and the Golden Triangle where the Allegheny and Monongahela rivers meet to form the Ohio River The individual condominium units will be priced from 300000 to 1200000 depending on the floor the unit is located on the square footage of the unit and optional features such as fireplaces and large balconies The company has had preliminary architectural drawings developed for three different project sizes 0 A small condominium complex with six floors and thirty units 0 A medium condominium complex with twelve floors and sixty units and 0 A large condominium complex with eighteen floors and ninety units The financial success of the project will depend heavily on the decision that PDC makes regarding the size of the condominium project When asked about possible market acceptance of the project management identified two possibilities High market acceptance and hence substantial demand for the units and 0 Low market acceptance and hence a limited demand for the units Further using the best information available management has estimated the payoffs or profits for the PDC condominium project Payoffs in millions of dollars High Acceptance Low Acceptance Build Small 8 7 Build Medium 14 5 Build Large 20 9 PDC would like to determine the best size complex to build page 2 of 18 0 Determine PDC s best strategy using the optimistic pessimistic and minimax regret approaches page 3 of 18 Example 2 Hartsfield International Airport in Atlanta Georgia is one of the busiest airports in the world During the past 30 years the airport has expanded again and again to accomodate the increasing numbers of ights being routed through Atlanta Analysts project that this increase will continue well into the next century However commercial development around the airport prevents it from building additional runways to handle the future air traffic demands As a solution to this problem plans are being developed to build another airport outside the city limits Two possible locations for the new airport have been identified but a final decision on the new location is not expected to be made for another year The Magnolia Inns hotel chain intends to build a new facility near the new airport once its site is determined Barbara Monroe is responsible for real estate acquisition for the company and she faces a difficult decision about where to buy land Currently land values around the two possible sites for the new airport are increasing as investors speculate that property values will increase greatly in the vicinity of the new airport The following table summarizes the current price of each parcel of land the estimated present value of the future cash ows that a hotel would generate at each site if the airport is ultimately located at the site and the present value of the amount for which the company believes it can resell each parcel if the airport is not built at that site Parcel near Tncatinn A Tncatinn B Purchase Price in Millions 18 Present Value in Millions of Future Cash Flow if airport Is built at that site 31 23 Is not built at that site 6 4 The company can buy either site both sites or neither site Barbara must determine which sites if any the company should purchase page 4 of 18 0 Determine the best strategy for Magnolia Inns using the optimistic pessimistic and minimax approaches page 5 of 18 Incorporatng Probabilities For each state of nature let Psj the probability that state j will occur The probabilities you use must satisfy Psl Psz PsN 1 Expected Value 1 EV Approach 1 For each decision alternative calculate EVdj PS Vil PSz Viz PSN ViN 2 Choose the alternative with maximum EV Expected Opportunity Loss 1EOL Approach 1 For each deCiSion alternative calculate EOLdj PS Ril Psz Riz PSN RiN 2 Choose the alternative with the minimum EOL In addition we define the axpected value of perfect information EVPI as follows EVPI EVwPI EVwoPI where 1 The expected value with perfect information EVwPI is the weighted average of the maximum payoffs for each state of natures and 2 The expected value without perfect information EVwoPI is the maximum of the expected values for each decision alternative Notes 1 The EV and EOL approaches always result in the selection of the same decision alternative 2 The value of EOL at the best decision is exactly equal to EVPI page 6 of 18 Example 1 continued Suppose that PDC is optimistic abou the potential for a high demand and assigns PHigh Acceptance 80 and PL0w Acceptance 20 0 Determine the best strategy using the EV and EOL approaches 0 Find the value afEVPI page 7 of 18 Example 2 continued The management at Magnolia Inns assigns the following probabilities to building at the two sites PBuild at A40 PBuild at B60 0 Determine the best Strategy using the EV approach 0 Find the value afEVPI page 8 of 18 We can an 3 Wm sensmvny analysts Wm 5mg h EV appmach and wnh m mus at mm m 5mg 1 p and Skate 2 113 m pmhlzm s m dammm h huge at pmhahumzs m whlch 3m wm d chums each 31mm W Th fullnwmg table shnws m expecmdvahlz anachdzclsmn 31mm mmmm ugh mammal as W Am me An m W m Ev 707 7 5 2n n 1 29 5nxrvmaztaxnfmzmm mawxx 0 Determine the ranges afprababilitiesfar which PDC would choose to build small build medium or build large page 10 of 18 Decision Trees A decision tree is a graphical representation of a decision problem There are three types of nodes in a decision tree 0 Decision nodes where you make a decision 0 Event nodes where an event occurs not under your control 0 Terminal nodes Example 1 Decision tree for the PDC problem incorporating the EV solution Example 1 Pittsburgh Development 80 High Acceptance 8 Build Small 20 Low Acceptance 7 80 High Acceptance 14 Build Medium 142 122 20 Low Acceptance 5 5 o High Acceptance 20 Build Large 20 Low Acceptance 9 page ll of 18 Example 2 Decision tree for the Magnolia Inns problem incorporating the EV solution Example 2 Magnolia Inns 40 Airport Built at A 3i Buy A 60 Airport Built at B 40 Airport Built at A 4 60 Airport Built at B 23 40 Airport Built at A 35 60 Airport Built at B page 12 of 18 Example 3 The Occupational Safety and Health Administration OSHA has recently announced it will award an 85000 research grant to the person or company submitting the best proposal for using wireless communications technology to enhance safety in the coal mining industry Steve Hinton the owner of COMTECH a small communications research firm is considering whether or not to apply for this grant Steve estimates he would spend approximately 5000 preparing his grant proposal and that he has about a 50 50 chance of actually receiving the grant If he is awarded the grant he would then need to decide whether to use microwave cellular or infrared communications technology He has some experience in all three areas but would need to acquire some new equipment depending on which technology is used The cost of the equipment needed for each technology is summarized as Technology Equipment Cost Microwave 4 000 Cellular 5000 Infrared 4 000 In addition to the equipment costs Steve knows he will spend money in research and development RampD to carry out the research proposal but he doesn t exactly know what the RampD costs will be For simplicity Steve estimates the following best case and worst case RampD costs associated with using each technology and he assigns probabilities to each outcome based on his degree of expertise in the area RampD Costs Prnhahi litv Best Case Worst Case Best Case Worst Case Microwave 30000 60000 40 60 Cellular 40000 70000 80 20 Infrared 40000 80000 90 10 Steve needs to synthesize all the factors in the problem to decide whether or not to submit a grant proposal to OSHA page 13 of 18 0 Design a decision tree 40 Best Case Example 5 COMTECH Microwave Worst Case 50 80 Best Case Receive Grant Cellular Worst Case 70 90 Best Case Submit Proposal Infrared 50 Do Not Receive Do Not Submit A 0 0 page 14 of 18 0 Use the EVappraach to solve Steve sproblem page 15 of 18 Example 4 A real estate investor wants to determine whether to buy an existing apartment building or to buy land now with the potential of developing the land later The cost of the apartment building is 800000 If the building is purchased then two states of nature are possible the town s population will grow or not over the next ten years with pertinent information summarized in the following table State of Nature Probability Earnings over 10 year period Population Growth 60 2000000 No Population Growth 40 225000 The cost of the land is 200000 If the land is purchased then the investor will wait three years to determine whether to develop or sell the land If the town exhibits population growth over the first period probability 60 then the investor can build an apartment building at a cost of 800000 or sell the land for 450000 The probabilities for continued growth and earnings if the apartment building is constructed are summarized in the following table State of Nature Probability Earnings over 10 yr period Growth in 2nd Period 80 3000000 No Growth in 2 d 20 700000 Period If the town does not exhibit population growth over the first period probability 40 then the investor can construct a commercial building at a cost of 600000 or sell the land for 210000 The probabilities for continued growth and earnings if the commercial building is constructed are summarized in the following table State of Nature Probability Earnings over 10 yr period Growth in 2nd Period 30 2300000 No Growth in 2 d 70 1000000 Period page 16 of 18 0 Design a decision tree Real Estate Example Growth A l Buy Apt Bdg No Growth l I 80 Growth 2 Build Apartments 50 Growth 1 Sell Land 30 Growth 2 Build Commercial No Growth1 Sell Land page 17 of 18 0 Use the EV approach to solve the investor39sproblem page 18 of 18 Break Even Analysis The purpose of breakeven analysis also called pro t analysis is to determine the number of units of a product the volume to sell that will equate total revenue with total cost The sales volume where this occurs is called the break even point and at this point pro t is zero The breakeven point provides a point of reference in determining how many units need to be sold to ensure a pro t In a breakeven analysis we assume that we can sell everything we produce Volume Cost and Pro t The volume can be expressed as the number of units produced and sold as the dollar volume of sales or as a percentage of total capacity There are two types of costs typically incurred in the production of a product xed costs and variables costs Fixed costs generally remain constant regardless of how many units are produced within a given range Variable costs are on a perunit basis The pro t is the difference between total revenue and total costs Parameters and Decision Variables The parameters are the actual numerical values that are assumed to be constant In a breakeven analysis the parameters are the selling price per unit the xed costs and the variable cost per unit The decision variables are those quantities that can be modi ed by the manager to reach the objective In a breakeven analysis there is only one decision variable 7 the quantity produced which equals quantity sold Equations and Formulas Total Cost xed variable xed volume x variable cost per unit TC cf vcV Total Revenue volume gtlt selling price TR Vp Pro t Total Revenue 7 Total Cost Z Vp 7 57 vcv Western Clothing Company WCC produces denim jeans It costs them 8 to produce each pair of jeans and their xed monthly costs are 10000 They can sell each pair ofjeans for 23 a What is their monthly pro t from the sales of 400 jeans b How many jeans do they need to sell each month in order to breakeven ie when TC TR or when Z 0 c Graph the TC and TR equations on the same graph to show loss pro t and breakeven point Sensitivi Analysis We assumed in this model that the parameters were constant and the only thing that changed is the decision variable ie the number of jeans to produce In sensitivity analysis we analyze how a change in the parameters affects the solution to the model 2 WCC wants their jeans to reach a higherend market and so are proposing to change the look of their jeans by adding fancy customized stitching to their jeans a V If the new stitching will increase the cost to produce each pair by 4 how many pairs would they need to sell in order to breakeven b WCC thinks they can sell the new jeans for 30 per pair If so what would their new breakeven point be 0 V To promote their new look WCC will also need to increase their advertising budget by 3000 How does this affect the breakeven point 9 V Should WCC adopt this new look W The management of WCC has decided to stick with their old look but eXpect that sales volume will stagnate at 550 pairs per month resulting in a net loss each month Suppose that sales volume remains at 550 pairs what would the selling price per pair need to be in order for WCC to breakeven each month n Sticky Note Criterion In this course all problems with conteXt in other words almost all must be answered in their conteXt This means that answers must be given in a complete sentence If you are not sure if your answer is adequate apply the sticky note criterion Ifyour answer is written on a sticky note can it be understood by someone with only a general understanding ofthe problem for example your boss Introduction to Linear Programming Beaver Creek Pottery Beaver Creek Pottery Company is a small crafts operation run by a Native American tribal council The company employs skilled artisans to produce clay bowls and mugs with authentic Native American designs and colors The two primary resources used by the company are special pottery clay and skilled labor Each bowl that is produced requires 1 hour of labor 4 pounds of clay and provides 40 in pro t Each mug requires 2 hour of labor 3 pounds of clay and provides 50 in pro t For the current production period the company has 40 hours of labor and 120 pounds of clay available each day for production Given these limited resources how many bowls and mugs should Beaver Creek produce each day in order to maximize their total pro t Assume that they can sell every bowl and mug that they produce We will formulate this problem as a linear programming model by identifying the objective decision variables and constraints 0 What is Beaver Creek s objective 0 What are the decision variables These are the quantities that will affect Beaver Creek s objective 0 What are the restrictions or constraints that will limit Beaver Creek s pursuit of their objective 0 Organize the parameters ie the actual numerical values in the objective function and constraints in the following table 39 Write out the complete linear programming model Each choice of values for the decision variables ie the number of bowls and mugs to produce yields either a feasible solution ie one that satis es the constraints or an infeasible solution ie one that does not satisfy the constraints Our objective is to nd the feasible solution that yields the highest pro t This is called the optimal solution The objective function value OFV is the value of the objective function at the optimal solution Solving the Beaver Creek Problem Using Graphical Methods The LP model for the Beaver Creek problem is given below Let B the number of bowls to produce and M the number of mugs to produce Maximize Pro t 403 50M st 13 2M 5 40 Labor hours 43 3M 5 120 Clay pounds B M 2 0 The constraints have been graphed and labeled The feasible region is outlined and shaded in What is the purpose of the non negativity constraints Note that the term feasible region is what the textbook refers to as the feasible solution area 4B3MS 120 Feasible Region B2M 40 10 On the next page use each of the two graphical methods to find the solution to this LP model Method 1 Comer Point Method 1 Find the coordinates of all comer points Make sure that you know how to do this yourself Note that the terms comer point and extreme point are both used in the textbook 2 Substitute each pair of coordinates including the origin into the objective function and compare the resulting values of the OFV Since we are looking to maximize the objective function the pro t we chose the largest value for the OFV B M 0 0 0 40 30 0 24 8 3 Give the complete solution to the problem 7 ie the optimal solution and OFV Method 2 Sliding Lines Method 1 Graph a sample objective function line on the feasible region graph by choosing an appropriate value for the profit a suitable multiple of the coefficients Objective function Bintercept M intercept slope 403 50M 2 Keeping your ruler parallel to the sample objective function line move it to the point in the feasible region that is the furthest from the origin 3 Find the coordinates of this point and determine the OFV 4 Give the complete solution to the problem 7 ie optimal solution and OFV Binding ConstraintsI Slack amp Surplus Note that our textbook does not use the terms binding and nonbinding It also treats slack and surplus differently Ifyou are having dif culty with this concept then please come and see me during of ce hours Binding Constraints Recall that the optimal solution will always occur at the intersection of two or more constraint lines Note that the xaXis or yaXis can be one of the constraints Since the optimal solution point is on the line its coordinates will satisfy the constraint In other words the usage LHS equals the availability RHS for those constraints We call these constraints binding Binding constraints are important because they tell us which resources are actually limiting the objective In a maXimize pro t situation where constraints represent resource availability having more of a resource that is binding translates into more pro t In the solution to any LP model there must be at least two binding constraints although one of these can be a nonnegativity constraint Graphically binding constraints correspond to the lines that intersect at the optimal solution Algebraically binding constraints correspond to those where LHS RHS NonBinding Constraints Graphically nonbinding constraints correspond to the lines that do not intersect at the optimal solution Algebraically nonbinding constraints correspond to those where LHS i RHS in other words some resource is left over in a S constraint or where there is a surplus generated in a 2 constraint Slack and Surplus If the constraint is a stype then the difference between LHS and RHS is called slack RHS 7 LHS If the constraint is a ztype then the difference between LHS and RHS is called surplus LHS 7 RHS Examples The graph below shows the feasible region the unshaded area and corner points de ned by the constraints 4x y 2 ll 1 x 2y 2 8 2 2x 2y 5 l6 3 5x 7 5y 5 10 4 The corner points are l 7 2 3 4 2 and 5 3 4xyz11 2x2ys1s Use the constraints and associated feasible region to answer the questions on the following page Suppose the objective function is Maximize x 6y In this case the optimal solution would be 1 7 Answer the following questions based on this objective function a Which constraints are binding iii and 7 Why b Which are nonbinding i 7 and 7 Why c How much slack or surplus does each nonbinding constraint have Constraint if has a slack OR surplus of i777 units circle your choice Constraint if has a slack OR surplus of i777 units circle your choice If the objective function is Minimize 2x 7y then the optimal solution is 4 2 Answer the following questions based on this objective function a Which constraints are binding iii and 7 Why b Which are nonbinding and Why c How much slack or surplus does each nonbinding constraint have Constraint if has a slack OR surplus of i777 units circle your choice Constraint if has a slack OR surplus of i777 units circle your choice A Simple quot 39 39 quot Problem The Fertilizer Problem A farmer is preparing to plant a crop in the spring and needs to fertilize a eld There are two brands of fertilizer to choose from SuperGro and CropQuick Each brand yields a speci c amount of nitrogen and phosphate per bag Nutrient Contribution lbbag Nitrogen Phosphate SuperGro 2 4 CropQuick 4 3 The farmer s eld requires at least 16 pounds of nitrogen and 24 pounds of phosphate SuperGro costs 6 per bag and CropQuick costs 3 The farmer wants to know how many bags of each brand to purchase in order to minimize the total cost of fertilizing Formulate this as a linear programming model by identifying the objective decision variables and constraints 0 What is the farmer s objective 0 What are the decision variables These are the quantities that will affect the farmer s objective 0 What are the restrictions or constraints that will limit the farmer s pursuit of their objective 0 Organize the parameters ie the actual numerical values in the objective function and constraints in the following table 39 Write out the complete linear programming model The constraints have been graphed for you L 4S3C224 2S4C216 1 Outline the feasible region or feasible solution area and shade it in 2 Draw a sample objective function line on the graph and use the Sliding Lines Method to solve this LP model 3 Determine the optimal solution and associated OFV objective function value Answer the neXt questions after having worked through the handout Binding Constraints Slack and Surplus 4 Which constraints are binding and which are nonbinding 5 Does the nonbinding constraint have slack or surplus and how much does it have Creating and Solving Two Variable LP Models Create the Mathematical Model 1 Carefully read and understand the problem 2 Identify the objective decision variables and constraints 3 Formulate a complete linear programming model a Describe the decision variables Let B the number of bowls to produce etc b Determine the objective function Maximize 403 50M c Identify the constraints 43 3M S 120 Remember the nonnegativity constraints Things to Remember A complete LP model requires the three components listed above in that order Remember to describe the decision variables accurately in words Include the word Maximize or Minimize before the objective function Constraints must be linear 7 in other words variables can t be multiplied together there can t be any exponents on the variables and variables can t appear in the denominator Don t forget the nonnegativity constraints Solving an LP Model Graphically 1 Graph the feasible region 2 Find the solution using either the Comer Point Method or the Sliding Lines Method 3 Give your answer in a complete sentence in the context of the problem Note that our textbook does not use these names but does discuss each method at the bottom of page 46 Corner Point Method Determine coordinates of all comer points of the feasible region including the origin if applicable Calculate the value of the objective function at each comer point make a table If the objective is to maximize then the largest value is the objective function value OFV If the objective is to minimize then the smallest value is the OFV The optimal solution is the associated pair of values of the decision variables Sliding Lilies Method Graph a sample objective function line by choosing an appropriate value for the objective function based on the scale of the numbers ie a suitable multiple of the coef cients Keeping the ruler parallel to the sample objective function line move it away from the origin for maximizing or closer to the origin for minimizing until you can determine which comer point is at the extreme edge of the feasible region Determine the coordinates of this point 7 this is the optimal solution Calculate the objective function value OFV by substituting the optimal solution into the objective nction Things to Remember You can use either method unless otherwise speci ed When asked for the complete solution you must describe the optimal solution and the OFV in words and in the context of the problem Your nal answer must be in sentence form Use the sticky note criterion If your answer is written on a sticky note can it be understood by someone with only a basic understanding of the problem for example your boss Not All LP Models Have a Unique Solution In most cases an LP model has a unique solution corresponding to a comer point However there are other possibilities The solution may occur along an edge of the feasible region in nitely many solutions there may not be a solution at all or the solution may be unbounded How to Determine if There are In nitelv Manv S quot Alternate Optimal S l Comer Point Method a The same OFV occurs at more than one comer point b All the points along the edge of the feasible region connecting the two comer points are optimal solutions Sliding Lines Method a The sample objective function line has the same as the slope of one of the constraints ie the optimal solution occurs along an edge of the feasible region b All the points along the edge of the feasible region corresponding to that constraint are optimal solutions Equot How to Determine if There are no S quot Infeasibility An LP model is infeasible if there is no feasible region In other words there are no points in the intersection of the constraints This usually occurs when the constraints have been made too restrictive How to Determine if the Solution is Unbounded An LP model is unbounded if the value of the objective function can be made in nitely large for maximizing or in nitely small for minimizing NB In a properly formulated LP model this situation should never occur Recall that our textbook does not use the terms binding and nonbinding It also treats slack and surplus differently If you are having difficulty with this concept then please come and see me during office hours Binding Constraints A binding constraint is one for which the inequality is actually satis ed at the optimal solution In resource utilization problems this means that usage LHS equals availability RHS Binding constraints are important because they tell us which resources are actually limiting the objective In a maXimize pro t situation where constraints represent resource availability having more of a resource that is binding translates into more pro t In the solution to any LP model there must be at least two binding constraints although one of these can be a nonnegativity constraint Graphically binding constraints correspond to the lines that intersect at the optimal solution quot 39 39 quot binding 39 l 39 to those where LHS RHS Non Binding Constraints Graphically nonbinding constraints correspond to the lines that do not intersect at the optimal solution Algebraically nonbinding constraints correspond to those where LHS 7E RHS in other words some resource is left over in a S constraint or where there is a surplus generated in a 2 constraint Slack and Surplus If the constraint is a stype then the difference between LHS and RHS is called slack RHS 7 LHS If the constraint is a ztype then the difference between LHS and RHS is called surplus LHS 7 RHS 40 41 The Valley Wine Company produces two kinds of wine 7 the Valley Nectar and Valley Red The wines are produced from 64 tons of grapes the company has acquired this season A l000 gallon batch of Nectar requires 4 tons of grapes and a batch of Red requires 8 tons However production is limited by the availability of only 50 cubic yards of storage space for aging and 120 hours of processing time A batch of each type of wine requires 5 cubic yards of storage space The processing time for a batch of Nectar is 15 hours and the processing time for a batch of Red is 8 hours Demand for each type of wine is limited to 7 batches The profit for a batch of Nectar is 9000 and the profit for a batch of Red is 12000 The company wants to determine the number of l000 gallon batches of Nectar and Red to produce in order to maximize profit Formulate a mathematical model and solve From the previous problem a How much processing time will be left unused at the optimal solution b What would be the effect on the optimal solution of increasing the available storage space from 50 to 60 cubic yards 54 Starbright Coffee Shop at the Galleria Mall serves two coffee blends it brews on a daily basis Pomona and Coastal Each is a blend of three highquality coffees from Colombia Kenya and Indonesia The coffee shop has 6 pounds of each of these coffees available each day Each pound of coffee will produce sixteen 16ounce cups of coffee The shop has enough brewing capacity to brew 30 gallons of these two coffee blends each day Pomona is a blend of 20 Columbian 35 Kenyan and 45 Indonesian while Coastal is a blend of 60 Columbian 10 Kenyan and 30 Indonesian The shop sells 15 times more Pomona than Coastal each day Pomona sells for 205 per cup and Coastal sells for 185 per cup The manager wants to know how many cups of each blend to sell each day in order to maximize sales Formulate a mathematical model and solve l Enter the missing values and formulas for this Excel spreadsheet model of the Beaver Creek problem A B C D The Beaver Creek Pottery Company N w h Products Pro t per unit Bowls Mugs Resources 0 Resources Usage Constraint A vailable Left 0 ver O N Clay lbunit Labor hrunit D Production Bowls Mugs N Total Pro t z Enterthe relevant mfmmanenmm the Sulvaquot dxalug bux Equa m 0 ex 0 Mn 0 lame af Ev haman 2H5 subvert m the anstvamts m 3 Chnk un Add m getthefulluvnng Swing bux Whmhxsusedtu Entercunstxzmt Remember m eheek me Assume Lmess Madequot and Assume NunrNeganvequot buxes m the Ophuni39 642mg bux Sin mum Assume man Made D gse nutamene Scahnq A Ssume NanrNegatwe D Shaw Itevatm Besu ts Seavch D New 0 Quadvah t O Cgmuqate 5 Enter the missing values and formulas for this Excel spreadsheet model of the Fertilizer problem A B C D The Fertilizer Problem N w h Products Cost per unit SuperGro CropQuick Nutrients 0 Nutrient Contribution Contribution Constraint Requirement Surplus O N Nitrogen lbunit Phosphate lbunit to Purchase SuperGro CropQuick N Total Cost Sensitivitv Range for Coef cients Beaver Creek Refer to the file Beaver Creek Solution for model and graphical solution to this problem Relationship Between Slope of the Objective Function and Optimal Solution As seen in the graph of the feasible region for this problem if the slope of the objective function changes a little bit then the point 24 8 will still be optimal However if the slope changes too much then the optimal solution will change We want to answer the following question How much can the slope of the objective function change before the optimal solution changes To do this we need to compare the slope of the objective function with the slopes of each of the binding constraints Reorder the slopes from smallest to largest Note that since the slopes are negative the smaller number has the largest negative value Line Slope of Line Slopes Reordered Associated Line Objective Function Labor Clay What is the relationship between the slope of the objective function and the slopes of the binding constraints Write this relationship as an inequality where m is the slope of objective function line This inequality tells us that the slope of the objective function line must fall somewhere between the values of the binding constraints and therefore as long as the slope is between these two values the optimal solution won t change We can describe this result algebraically by writing the objective function in the following general form Max Pro t C193 CMM where CB represents the coef cient of B and CM represents the coef cient of M In this form the slope of the objective function is m ngCM Relating this slope to the slopes of the binding constraints gives us the following important result The current optimal solution will remain optimal provided that 5 ngCMS Sensitivity Range for Each Coef cient Since the slope of the objective function line is actually the ratio of the coef cients of the decision variables we can use the above results to answer the following question I f we keep of the coe icients xed at its current value how much we can change the other coefficient without changing the optimal solution The range of values for which this is true is called the sensitivity range for the chosen coef cient Recall that in the Beaver C reek problem the coe icients represent the profit per unit produced therefore the sensitivity range is a range of profit values for each product To nd the sensitivity range for C3 in the Beaver Creek problem keep CM fixed at 50 and solve the above general inequality for C3 Likewise to nd the sensitivity range for CM keep CB xed at 40 and solve the inequality for CM Since the unknown value is in the denominator your first step should always be to take the reciprocal of the inequality to put the unknown in the numerator When you do this remember to switch the directions of the inequalities We can summarize the sensitivity range for the two coef cients in table form Current Value Coef cient Sensitivity Range for Max Increase Max Decrease of Coef cient Coefficient from Current from Current CB 40 CM 50 This information can also be found in Excel using Solver Compare Solver s Sensitivity Report to the table that we produced above We will discuss how to obtain this report and what the meaning of the other parts of this report in later lectures Adiuslable C ells MicrosoftExcel 120 Sensitivity Report Worksheet L7 Beaver Creek amp Fertilzer templatexlsBeaver Creek complete ReportCreated 81 52008 52428 PM Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease B10 Bowls 24 4O 2666666667 1 3933911 Mugs 8 O 50 30 20 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price RH Side Increase Decrease E6 Labor hrUnit U sage 4O 16 4O 4O 1O E7 Clay lbunit Usaqe 120 6 120 40 60 Sensitivity Range for Constraints Beaver Creek In this section we will look at the effect of a change to RHS of a constraint A common application of this is when a manager wants to know the effect of a change in resource availability Resource Availability Changes Recall that in the Beaver Creek problem the optimal production levels are 24 bowls and 8 mugs resulting in optimal profit level of 1360 The question we want to answer is How will production analprofit levels change ifthere is a change in resource availability I Suppose that additional hours of labor are available How do you think this would affect the solution to the model How can you tell I Do you think that additional pounds of clay would affect production levels Current Situation Recall that the optimal solution isB 24 andM 8 The OFV is 1360 Refer to the Beaver Creek solution handout from last class for the graphical solution Initial Observations A change in resource availability will mean a change to the RHS of one of the constraints in this case the labor constraint This will affect the shape of the feasible region and likely means that the optimal solution will change However as long as the change in the RHS is not too large the change should not affect which constraints are binding and which are nonbinding Our goal will be to determine how much the RHS can change before the relationship between the binding constraints changes Change 1 Available Labor Increases to 60 Hours First observe the effect on the feasible region and optimal solution if available labor increases to 60 Observations I Optimal solution changes to 12 24 but labor 43 3M 3 120 and clay constraints are still binding I Pro t is now 40 X 12 50 x 24 1680 an Optimal solution mcrease of 3 20 12 24 gt A RHS 20 gt A OFV 320 Chan e 2 Available Labor Increases to 80 Hours M Optimal solution Observati0ns 0 40 I Optimal solution changes to 0 40 but labor and clay constraints are still binding I Pro t is now 40 X 0 50 X 40 2000 an increase of 640 gt A RHS 40 gt A OFV 640 Chan e 3 Available Labor Increases to 90 Hours Observations I Optimal solution changes to 0 40 but the binding constraints change They are now clay and a nonnegativity constraint 7 the labor constraint is no longer binding I Pro t is now 40 X 0 50 X 40 2000 an increase of 640 Optimal solution 40 0 40 gt A RHS 50 gt A OFV 640 Change 4 Available Labor Decreases to 30 Hours M Observations I Optimal solution changes to 30 0 but labor 43 3M 3 120 and clay constraints are still binding I Pro t is now 40 X 30 50 X 0 1200 a decrease of 160 gt A RHS 10 gt A OFV 7160 Available Labor Decreases to 20 Hours M Observations I Optimal solution changes to 20 0 but the binding constraints change They are now labor and a nonnegativity constraint 7 the clay constraint is no longer binding I Pro t is now 40 X 20 50 X 0 800 a decrease of 560 4B3M3120 gt A RHS 20 gt A OFV 560 Optimal solution 20 0 Summag of Observations As the available labor hours changes the feasible region changes and so does the optimal solution However as long as labor hours falls between 30 and 80 the labor and clay constraints remain binding This range of values is referred to as the sensitivity range for the labor constraint Making Predictions Although we know that a change in labor hours will affect the optimal solution there is no easy way to predict the new optimal solution without resolving the model However it turns out that we can easily predict the new pro t OFV based on a change in available hours provided that the change falls within the sensitivity range for the constraint To see this the results from this example have been summarized in the following table Fill in the last column by calculating the ratio of the change in pro t by the change in labor A OFV A RHS Observe that this value is constant provided that A RHS falls within the sensitivity range Labor Binding 11151916 A RHS A OFV A OFV hours Constraints SilaSIEZgy from original From from original A RHS 40 labor amp clay Yes 0 1360 0 NA 60 labor amp clay Yes 20 1680 320 80 labor amp clay Yes 40 2000 640 90 clay amp nonneg No 50 2000 640 30 labor amp clay Yes 10 1200 160 20 labor amp nonneg No 20 800 560 Shadow Price The value in the last column shows the change in pro t per unit increase in labor hours 7 in other words it represents the marginal profit relative to labor In the language of linear programming this value is referred to as the shadow price and has a number of useful interpretations We will investigate applications of the shadow price in subsequent lectures C 39 39 quot the Sen itivitv Range for a Constraint and its A 39 t A Shadow Price Recall that we found the boundaries of the sensitivity range for the labor constraint using this procedure I We moved the labor constraint line upwards which also moved the optimal solution point until we arrived at the next comer point in the feasible region We used the coordinates of this comer point to calculate the RHS value of the labor This gave us the upper value of the sensitivity range To get the lower value for the sensitivity range we moved the constraint line downwards to the next corner point and found the RHS value of the constraint We write the sensitivity range for the labor constraint as 30 s ql s 80 We found the shadow price by calculating the difference between the OFV at one end of the range of feasibility and divided this by the difference in the RHS values Sensitivity Range for the Clay Constraint Use the above procedure to determine the sensitivity range and shadow price for the clay constraint Remember our point of reference is the original optimal solution B 24 andM 8 with profit 1360 Comer Point RHS of clay constraint Pro t OFV A OFV Coordinates 43 3M 403 50M A RHS 24 8 120 1360 NA 0 20 60 1000 7360760 6 40 0 160 1600 24040 6 The sensitivity range for the clay 39 t is The shadow price for the clay constraint is Range of Feasibility in Excel Excel provides the shadow price and sensitivity range for each constraint in the Sensitivity Report Compare the sensitivity report for the Beaver Creek problem to the calculations made above Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease B1O Bowls 24 O 40 2666666667 15 519851911 Mugss 8 O 50 3O 20 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price RH Side Increase Decrease E6 Labor hrUnit Usaqe 4O 16 4O 4O 10 SEER Clay lbunit Usage 120 6 120 40 6O Sensitivity Analysis Beaver Creek modi ed The management of Beaver Creek Pottery has incorporated a packaging constraint to the production of bowls and mugs There are 5 hours of packaging available and it takes 12 minutes 02 hours to package each bowl and 6 minutes 01 hour to package each mug With this change the linear programming model is Maximize Pro t 403 50M st 13 2M 5 40 Labor hours 43 3M 5 120 Clay pounds 023 01M s 5 Packaging hours B M 2 0 The problem was solved in Excel and the following Sensitivity Report was generated Use it to answer the following questions Adjustable Cells Final uced Objective Allowable Allowable Cell Name Value Cost Coef cient Increase Decrease 11 Bowls 20 0 40 60 15 B12 Mugs 10 0 50 30 30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price RH Side Increase Decrease W labor hrunit Usage 40 20 40 15 15 E7 clay lbunit usage 110 0 120 1E30 10 magln mrunlt Usage 5 100 5 06 3 1 What is the sensitivity range for the bowls coefficient N What is the sensitivity range for the mugs coefficient 3 What is the sensitivity range for the labor constraint 4 What is the sensitivity range for the packaging constraint 5 What does the 1E30 mean in the Allowable Increase column for the clay constraint 6 What is the sensitivity range for the clay constraint In each of the following questions assume that each change is the only change made from the original problem Use the sensitivity report to find if possible the complete solution In other words give the optimal solution point and the value of the objective function at those values If either or both of these can t be obtained from the sensitivity report state this and explain why not 7 Suppose the profit per bowl increases to 60 9 Suppose the profit per bowl decreases to 20 9 Suppose the profit per mug increases to 90 O Suppose the profit per mug decreases to 20 11 Suppose only 30 hours of labor are available 12 Suppose an additional 40 pounds of clay are available 13 Suppose an additional 2 hours of packaging are available 14 Suppose there are an additional 4 hours of labor and 15 hours of packaging available ll Irwin Textile Mills produces two types of cotton cloth 7 denim and corduroy Corduroy is a heavier grade of cotton cloth and as such requires 75 pounds of raw cotton per yard whereas denim requires 5 pounds of raw cotton per yard A yard of corduroy requires 32 hours of processing time a yard of denim requires 30 hours Although the demand for denim is practically unlimited the maximum demand for corduroy is 510 yards per month The manufacturer has 6500 pounds of cotton and 3000 hours of processing time available each month The manufacturer makes a pro t of 225 per yard of denim and 310 per yard of corduroy The manufacturer wants to know how many yards of each type of cloth to produce to maximize profit The resulting mathematical model is as below Let D the of yards of denim produced and C the of yards of corduroy produced Max 2 25D 3 10C SJ 50D 75C s 6500 cotton lbs 30D 32C s 3000 time hrs C s 510 demand yds D C 2 0 non neg Irwin Textile Mills should produce 456 yards of denim and 510 yards of corduroy for a maximal profit of 2607 12 Apply sensitivity analysis a How much extra cotton and processing time are left over at the optimal solution Is the demand for corduroy met Fquot What is the effect on the complete solution if the profit per yard of denim is increased from 225 to 290 c What would be the effect on the complete solution if demand for corduroy increased to 600 yards per month 700 yards nick Screen Clothin 1 1 1 QuickScreen is a clothing manufacturing company that specializes in producing commemorative shirts immediately following major sporting events such as the World Series Super Bowl and Final Four The company has been contracted to produce a standard set of shirts for the winning team either State University or Tech following a college football bowl game on New Year s Day The items produced include two sweatshirts one with silkscreen printing on the front and one with print on both sides and two Tshirts of the same con guration QuickScreen has to complete all production within 72 hours of the game s conclusion at which time a trailer truck will pick up the shirts thus the company will work around the clock The truck has enough capacity to accommodate 1200 standardsize boxes A standardsize box holds 12 Tshirts and a box of 12 sweatshirts is three times the size of a standard box The company has budgeted 25000 for the production run It has 500 dozen blank sweatshirts and Tshirts each in stock ready for production The resource requirements unit costs and pro t per dozen for each type of shirt are shown in the following table Processjng Time hr39 Cost per Dozen Pro t per Dozen per Dozen Sweatshirt front only 010 36 90 Sweatshirt both sides 025 48 125 Tshirt front only 008 25 45 Tshirt both sides 021 35 65 The company wants to know how many dozen boxes of each type of shirt to produce in order to maximize pro t Suggestion Try to develop this modelfor yourselfat a later time without the aid ofthe solution The linear programming model for this problem is Maximize Pro t s t 90x1 0 10x1 3amp9 36x1 x1 x1 125x2 025x2 3x2 48x2 x2 5 x2 45x3 008x3 x3 25x3 x3 x3 65x4 021x4 s 72 x4 5 L200 35x4 5 25000 5 500 x4 5 500 x4 2 0 processing time hours shipping capacity boxes budget sweatshirt inventory dozens Tshirt inventory dozens The problem was solved in Excel with its associated Sensitivity Report Use it to answer the following questions Adjustable Cells Final Reduced Objective Allowable Howable Cell Name Value Cost Coefficient Increase Decrease Wlm hm mly 17556 115 2 3 6 8 B15 sweatshirts both sides 5778 000 125 1321429 1192308 B16 Tshirts front only 50000 000 45 1E30 411111 5W7 TEl llrts both sides 000 1033 65 1033333 1E30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price RH Side Increase Decrease 1 Based on the Sensitivity report what is the optimal solution and objective function value 2 In the conteXt of this problem does it make sense to produce fractional items Why or why not 3 Give the revised optimal solution and maXimum pro t 4 Suppose the pro t per dozen of the front only sweatshirts increases to 100 Should QuickScreen produce more of these sweatshirts Why or why not Would this change affect pro ts 5 Suppose that management has decided that at least 10 boxes of each type of product must be produced What is the impact of this decision on pro ts or is it impossible to tell without resolving the model 6 We all know that you need to spend money in order to make money Management wants to increase pro ts by investing more money on this job What do you think of this decision 7 The supplier has an additional 180 boxes of T shirts in stock which they can sell to QuickScreen at the regular price but management would need to pay an additional 400 to have the boxes shipped overnight so that they arrive in time for the current production run What would you advise and why 8 A larger truck with a capacity of 1400 standardsize boxes could be rented for 500 more than the cost of the regularsized truck Would you advise QuickScreen to rent this larger truck Explain why Note that the cost of the regularsized truck has been factored into the pro t gures for the clothing 9 The president of QuickScreen asks the production manager why none of the both sided Tshirts are being produced What should the manager s response be What are some things that could be done to ensure that some of these Tshirts would be produced 10 As an outside consultant it is your job to propose changes that will improve pro ts For each proposed change explain the impact on pro ts Can you nd a way to summarize and compare the proposed changes Sensitivity Analysis Summary Overview In Sensitivity Analysis we analyze the effects of changes to one of the problem parameters Sensitivity Analysis is essentially the study of what if scenarios There are two types of changes that we will analyze and interpret Type I A change is made to one of the coef cients of the objective function 1 N If the change is within the sensitivity range for the coefficient then the optimal solution does not change but the objective function value OFV does Compute the new OFV using the new value for the coef cient and the original values for the other coefficients If the change is outside the sensitivity range for the coef cient then we must re solve the problem in its entirety to find the new solution ie resolve to find optimal solution and OFV Type II A change is made to the RHS of one of the constraints A J If the change is within the sensitivity range for the constraint and the constraint is a nonbinding then there is no change to either the optimal solution or the OFV b binding then we can compute the new OFV using New OFV old OFV of unit changes gtlt shadow price Note that in order to determine the new optimal solution we would need to re solve the problem in its entirety ie resolve to find optimal solution If the change is outside the sensitivity range for the constraint then the problem needs to be resolved in its entirety to find the new solution ie resolve to nd optimal solution and OFV Note Sensitivity reports can only be used when a single change to the original problem is made If more than one change is made then the answer CANNOT be determined from the sensitivity report the problem would need to be re solved Future Application Reduced Cost The reduced cost column in the sensitivity report indicates how much the objective function coefficient would need to decrease by in order for that decision variable to change usually from a value of zero If the reduced cost is negative then the objective function coef cient must increase Sensitivity Analysis Practice Tom s Inc produces two types of salsa Western Foods and Mexico City using different blends of whole tomatoes tomato sauce and tomato paste Western Foods Salsa provides a pro t of 100 per jar and Mexico City Salsa provides a pro t of 125 per jar The problem was solved using Excel and the following sensitivity report was generated Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease 5737510 Jars Produced Western Foods 560 0 1 625 6107 C10 Jars Produced Mexico City 240 0 125 015 025 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price RH Side Increase Decrease B15 Whole Tomatoes ounces 4480 0125 4480 1120 160 B16 Tomato Sauce ounces 1920 0 2080 1E30 160 B17 Tomato Paste ounces 1600 01875 1600 40 320 a Suppose the pro t on each jar of Mexico City Salsa increases by 50 Explain and calculate any effects that this would have on the current production levels and pro t 57 The marketing department is considering putting 20 off coupons on each jar of Western Foods Salsa Assuming that each customer would redeem the coupon explain and calculate any effects that this would have on the current production levels and pro t 0 P e Are any constraints nonbinding How do you know What would be the effect of decreasing the number of ounces of whole tomatoes to 4400 What would be the effect of increasing the amount of tomato paste available to 1700 ounces 2 A mathematical model was developed to allow Hart Manufacturing to determine the optimal mix of products that will maximize pro t while meeting the availability of labor in hours in its three departments The problem was solved in Excel and the following Sensitivity Report was generated Note that the unit for the objective function coefficients is per unit Adjustable Cells Final Reduced Objective Allowable Allowable Cost Cell Name Value Coefficient Increase Decrease 12lJ its Produced Product 1 0 1 25 1 1E C12 Units Produced Product 2 375 0 28 32 8 D12 Units Produced Product 3 125 0 30 40 1333 Con strai nts Final Shadow Constraint Allowable Allowable Cell Name Value Price RH Side Increase Decrease 31481917 DeptA hours LHS 3625 0 450 1E30 875 B18 DeptB hours LHS 350 8 350 150 250 B19 Dept C hours LHS 50 40 50 159 15 a What is Hart Manufacturing s optimal pro t b Is a production level of x1 80 x2 20 x3 60 a feasible solution Explainyour reasoning c Suppose the pro t on Product 1 decreases by 1 per unit Explain and calculate any effects that this would have on the current production levels and pro t d Suppose the pro t on Product 1 increases by 2 per unit Explain and calculate any effects that this would have on the current production levels and pro t e Suppose the pro t on Product 2 increases to 35 per unit Explain and calculate any effects that this would have on the current production levels and pro t f Suppose that an additional 12 hours of labor is available in Department B Explain and calculate any effects that this would have on the current production levels and pro t Suppose that 5 hours of additional labor is available How should this additional labor be allocated P Mathematics for Management Science Notes 01 prepared by Professor Jenny Baglivo Jenny A Baglivo 2002 All rights reserved Linear Programming 0 Optimize maximize or minimize a linear function the objective function 0 Subject to a system of linear equations and inequalities the constraints Graphical Solution 1 Plot the boundary line of each constraint 2 Identify the set of points that simultaneously satisfy all the constraints the feasible region 3 Locate the optimal solutions by one of the following methods 0 By level curves curves with constant objective 0 By checking the comers or extreme points Fundamental Theorem of Linear Programm39 If the feasible region is closed and bounded then there will be at least one optimal solution which will occur at one of the comers extreme points Example 1 Galaxy Industries continued Maximize weekly profit by adjusting production level in dozens of Zappers and Space Rays Production must be scheduled so that the weekly supply of plastic and the availability of production time are not exceeded The two marketing department guidelines conceming maximum total production and product mix must be met Steps 1 amp 2 of graphical solution page 1 of 16 an m bun gun mun nu smm Example 2 To feed his stock a farmer can purchase two kinds of feed The farmer has determined that his herd requires 60 84 and 72 units of nutritional elements A B and C respectively per day The contents and cost of a pound of each of the two feeds are given in the following table Nutritional elements unitspound Cost A B C centspound Feed 1 3 7 2 10 Feed 2 2 2 6 4 Obviously the farmer could use only one feed to meet the daily nutritional requirements For example it can easily be seen that 24 lb of the first feed would provide an adequate diet at a daily cost of 240 However the farmer wants to determine the least expensive way of providing an adequate diet by combining the two feeds To do this the farmer should consider all possible diets that satisfy the specified requirements and then select from this set that diet of minimal cost Summam of the problem Definitions of the decision variables page 5 of 16 Linear programming model Graphical Solution Step 1 boundary lines find the intercepts page 6 of 16 SpQO Mxmanvenpnmalsnlnnnns unmmasnmnnmmmgmnmnupmmsnmnnn a xmmesnmmmmmmmm mew Speculum Mnmxszalgglmusn zs Snypnsemaxmepm xmnmnms pm Xuxx W naesnnmunmeammmme mncm spmd Prndnmnnhmxanmsleanm Xumgnnn Hangman X z mu 3km zpmmxunnldllkeymmmndnce mmm mm 2311an mm dam mm Using Excel to Solve Linear Programming Problems Organizing the worksheet 1 Data section summary of information from the problem 2 Model section 0 Changing cells 69 Decision variables 0 Target cell 69 Objective function 0 Constraints sequence of lefthand sides LHS relational symbols 5 2 and righthand sides RHS Using Solver 1 Solver is an option under the Tools menu Tools 9 Solver You may need to reinstall Excel to be sure to include the Solver Addon 2 On first window 0 Set Target Cell address of the objective function 0 Equal to either max or min 0 Subject to constraints choose Add for each new constraint Choose Options check Assume Linear Model OK gets you back to first window Then Solve 55 Creating a Formulas Worksheet 1 Create a copy of your problem worksheet 0 Select Move or Copy Sheet under the Edit menu Edit 9 Move or Copy 0 Check create a copy 0 Select where copy should go and hit OK 2 Working with the copy 0 Choose Preferences under Tools menu Tools 9 Preferences 0 Select the View tab and check the formulas box Printing with row column headings amp gridlines 1 Select Print Preview under the File menu File 9 Print Preview 2 Select the Sheet tab Check gridlines row and column headings Note also 1 Remember that formulas begin with an equal sign 2 To outline a rectangular region of cells Choose Cells under the Format menu Format 9 Cells then the Border tab Now you can create a border 3 To create a rectangular text box with Solver information see examples 0 Be sure the Drawing tool bar is visible View 9 Toolbars 9 Drawing 0 Click on the rectangle icon and then draw the rectangle on the sheet 4 Remember to add shading to model cells as illustrated below page 12 of 16 Example 1 Galaxy Industries BZO changing B18C18 to lt C24C27 8C18 gt 0 or Assume NonNegative Assume Linear Model page 13 of 16 Formulas worksheet page 14 of 16 Example 2 Farmer changing B17C17 to gt D22D24 7818 gt 0 or Assume Nonnegative Assume Linear Model page 15 of 16 Formulas worksheet page 16 of 16 Mathemz ts for Managemmt Science Notes 06 yup eammm magye quotInga Lem Plugnmmllg am Wu me valqu at he dzcman vanahlzs m a max pmyammmg pmhlem an nsmcmd m meg we sayum we have ammgntwwpmgvmmmg pmmm axanILP pmhlzm nthzle ale Marge nnmhexaffeasxhlz means we usua y mlve me Ln 72mmquot mm mm 1 me pmhlzm 3m gee by gummg me nsmconn an Image dzcxsmn vanahlzs and mum me valvzs a me and thn um axe emyexy few feasxh sahmam n 1 best an mlve he pmhlzm by h39ymga pasnh mes smeee c Eanscran 5 ms Eanscran 2 5 ms Nanrneyacxvxw39 IX 2 n x2 2 n meegeee Xe Xe new 1 Th easmxe see in me my pmhlem is me Canadian at 11 pm hghhghmd helaw Th2 easmxe se muw u exmemnpmmemme gay areaphlsthz hmmdary hm Th2 dashzdlmusle3 mm 2 The solution to the ILP problem is x122 and x222 The maximum value of the objective function is 10 OBJECTIVE OBJECTIVE 0 4 3 7 6 10 3 The solution to the LP relaxation problem is x136 13 and x229552 The maximum value of the objective function is 110192 110192 Max 7 4 If you round the solution to the LP relaxation problem then you get an infeasible solution that is one that does not satisfy the constraints Bounding Principles 1 The maximum value for an ILP problem is always less than or equal to the maximum value for the LP relaxation MAX for ILP 5 MAX for LP relaxation 2 The minimum value for an ILP problem is always greater than or equal to the minimum for the LP relaxation MIN for ILP 2 MIN for LP relaxation page 2 of 25 ino ILP problems in Solver Step I Add constraints for each integer decision variable 1 Put the address of the decision variable in the lefthand side 2 Select int where you would normally select a relation 5 2 3 The word integer should appear on the righthand side If nothing appears then you should type integer on the righthand side Step 2 Make appropriate selections in the options window 1 Assume Linear Model 2 Assume Non negative 3 0 Tolerance 4 1000 Iterations or more if needed Comments 39 If one or more variables are integer then Solver switches from the usual linear programming algorithm to one that searches the integer points It uses a technique called branch and bound so that it doesn t have to look at every single integer point before finding the optimal solution This algorithm does not produce sensitivity reports 39 The tolerance default value is 5 If the 5 value stays in effect then Solver will return a solution whose objective function value is within 5 of the best possible For maximization problems the stopping criterion is 095 Maximum for LP relaxation 5 Current objective function value For minimization problems the stopping criterion is Current objective function value 5 105 Minimum for LP relaxation page 3 of 25 Simple example solution sheet 814 Changing Bi 1 Ci 1 to 817818 lt C17D18 BHC11 integer Assume Linear Model Assume Nonnegative 0 Tolerance 1000 Iterations page 4 of 25 Exercise 1 AirExpress is an express shipping service that guarantees overnight delivery of packages anywhere in the continental United States The company has various operations centers called hubs at airports in major cities across the country Packages are received at hubs from other locations and then shipped to intermediate hubs or to their final destinations The manager of the AirExpress hub in Baltimore Maryland is concerned about labor costs at the hub and is interested in determining the most effective way to schedule workers The hub operates seven days a week and the number of packages it handles each day varies Usin historical data on the average number of packages received each day the manager estimates the number of workers needed to handle the packages as IDay ISun IMon ITue IWed Thu IFri Sat I Required I18 27 22 26 25 21 19 The package handlers working for Air Express are unionized and are guaranteed a five day work week with two consecutive days off The base wage for the handlers is 655 per week Because most workers prefer to have Saturday or Sunday off the union has negotiated bonuses of 25 per day for its members who work on these days The possible shifts and salaries for package handlers are The manager wants to keep the total wage expense for the hub as low as possible With this in mind how many package handlers should be assigned to each shift if the manager wants to have a sufficient number of workers available each day 0 Summarize the problem page 5 of 25 0 Define the decision variables precisely 0 Completely specify the LP nwdel page 6 of 25 0 Clearly State the optimal solution Exercise 1 solution sheet Minimize B 22 By Changing B19H19 Subject to 3251332 gt D251D32 B19H19 integer Assume Linear Model Assume Nonnegative 0 Tolerance 1000 Iterations page 7 of 25 Formulas sheet page 8 of 25 Programming with binary 0 1 variables To restrict an integer variable to the values 0 and 1 only add a constraint as follows 1 Put the address of the decision variable in the lefthand side 2 Select bin where you would normally select a relation s 2 3 The word binary should appear on the righthand side If nothing appears then you should type binary on the righthand side The next exercise is an example of a capital budgeting problem where the decision variable equals 1 when a project is selected and equals 0 otherwise Exercise 2 In his position as vice president of research and development RampD for CRT Technologies Mark Schwartz is responsible for evaluating and choosing which RampD projects to support The company received 18 RampD proposals from its scientists and engineers and identified six projects as being consistent with the company s mission However the company does not have the funds available to undertake all six projects Mark must determine which of the projects to select The funding requirements for each project are summarized below along with the expected net present value NPV the company expects each project to generate Expected CaEital Thous Reguired in PV N Project Thous Year 1 Year 2 Year 3 Year 4 Year 5 141 75 1 25 20 15 10 2 187 90 35 0 0 30 3 121 60 15 15 15 15 4 83 30 20 10 5 5 5 265 100 25 20 20 20 6 127 50 20 10 30 40 The company currently has 250000 available to invest in new projects It has budgeted 75000 for continued support for these projects in year 2 and 50000 per year in years 3 4 and 5 Which projects should CRT support in order to maximize total expected NPV 0 Summarize the problem page 9 of 25 0 Define the decision variables precisely 0 Completely specify the LP model 0 Clearly state the optimal solution page 10 of 25 Exercise 2 solutions and formulas sheets page 11 of 25 Changing B17G17 to lt D252D29 B17G17 binary Linear Model Nonnegative Tolerance Iterations The next exercise is an example of a fixed cost problem It is assumed that the cost of production includes a setup cost which is a fixed cost and a variable cost which is directly related to the quantity produced The setup cost is only incurred if you choose to produce the product Products are indexed using i1 2 etc For product i let 1 Xi equal the number of items produced 2 equal 1 when the fixed cost is incurred and 0 otherwise and 3 equal the maximum number that could be produced under given constraints A constraint of the form S assures that 1 when and only when is positive Exercise 3 Remington Manufacturing is planning its next production cycle The company can produce three products each of which must undergo machining grinding and assembly operations The table below summarizes the hours of machining grinding and assembling required for each unit and the total hours of capacity available in the next production cycle Hours Reguired by Onprah39nn Product 1 Product 2 Product 3 Hours Available Machining 2 3 6 1200 Grinding 6 3 4 600 Assembly 5 6 2 800 Products 1 2 and 3 have unit profits of 48 55 and 50 respectively There are also production line setup costs associated with producing each product These costs are 1000 for Product 1 800 for Product 2 and 900 for Product 3 The marketing department believes it can sell all the products produced Therefore the management of Remington wants to determine the most profitable mix of products to produce 0 What are the values 0be M2 and M3 page 12 of 25 0 Define the decision variables precisely 0 Completely specify the LP nwdel page 13 of 25 0 Clearly State the optimal solution Exercise 3 solution sheet page 14 of 25 Changing 818D19 to BZ7B32 lt D27D32 818D18 integer 819D19 binary Linear Model Nonnegative Tolerance 000 Iterations Formulas sheet page 15 of 25 Exercise 4 In the Remington Manufacturing problem suppose that management does not want to produce a product unless it produces at least 70 units of that product The sheet below gives the new optimal solution 133333333 0 How does the original made change page 16 of 25 Exercise 5 The MartinBeck Company operates a plant in St Louis which has an annual production capacity of 30000 units The final product is shipped to regional distribution centers located in Boston Atlanta and Houston Because of an anticipated increase in demand MartinBeck plans to increase capacity by constructing a new plant in one or more of the following cities Detroit Toledo Denver Kansas City The estimated annual fixed cost and the armual capacity for the four proposed plants are as follows The company s longrange planning group has developed the following forecasts of the anticipated annual demand at the distribution centers The shipping cost per unit from each proposed plant to each distribution center is shown in the following table Finally MartinBeck would like to locate a plant in either Detroit or Toledo but not both MartinBeck would like to determine where to locate the new plants and how much should be shipped from each plant to each distribution center in order to minimize the total armual fixed costs and transportation costs Notation Source nodes 1 Detroit 2 Toledo 3 Denver 4 Kansas City 5 St Louis Destination nodes 1 Boston 2 Atlanta 3 Houston page 17 of 25 0 Define the decision variables precisely 0 Completely specify the LP nwdel page 18 of 25 0 Clearly State the optimal solution Exercise 5 solution sheet 0 O 90949E13 O O 5000 Changmg B21 D2S F21 F24 to 211125 BZ7D27 8291D29 B32 32 B21D2S mteger F21F24 bmary Mnear mode1 nonrnegatwe To1erance 1terat1ons page 19 of 25 Formula sheet page 20 of 25 Some modelling tips for binary variables Suppose that 1 when a project is chosen and 0 otherwise for i1 2 n 1 To choose exactly k of the projects add the constraint X1X2Xn k Replace with 2 when you want k or more projects chosen Replace with s when you want k or fewer projects chosen 2 To choose project i only when project j is chosen add the constraint Xi X 3 The conditional constraint has the following effect 39 If Xi equals 1 then it forces to equal 1 39 If X j equals 0 then it forces to equal 0 page 21 of 25 Exercise 6 Office Warehouse OW has been downsizing its operations It is in the process of moving to a much smaller location and reducing the number of different computer products it carries Coming under scrutiny are ten products OW has carried for the past year For each of these products OW has estimated the oor space required for effective display the capital required to restock if the product line is retained and the shortterm loss that OW will incur if the corresponding product is eliminated through liquidation sales etc Cost to Cost to Floor Product Product Manufactured Liquidate Restock Space Number Line by thous thous sqrft 1 Notebook Toshiba 10 15 50 2 Notebook Compaq 8 12 60 3 Compaq 20 25 200 4 PC Packard Bell 12 22 200 5 Macintosh Apple 25 20 145 6 Monitor Packard Bell 4 12 85 7 Monitor Sony 15 13 50 8 Printer Apple 5 14 100 9 Printer HP 18 25 150 10 Printer Epson 6 10 125 quotNotebookquot refers to a notebook computer Office Warehouse wishes to minimize the loss due to the liquidation of product lines subject to the following conditions 1 At least four product lines will be eliminated 2 The remaining products will occupy no more than 600 sqr ft of oor space 3 At most 75000 is to be spent on restocking the product lines 4 If one product from a particular manufacturer is eliminated then all products from that manufacturer will be eliminated 5 At least two of the five computer models Toshiba NB Compaq NB Compaq PC Packard Bell PC Apple Macintosh will continue to be canied by Office Warehouse 6 If the Toshiba notebook computer is to be retained then the Epson line of printers will also be retained page 22 of 25 0 Define the decision variables precisely 0 Completely specify the LP nwdel page 23 of 25 0 Clearly State the optimal solution Exercise 6 solution sheet Changmg BZSK25 to lt D321D34 8351837 D3SD37 B38 gt D3 B39 lt D39 BZSK25 bmary Mnear model nonrnegatwe Tolerance teratwons page 24 of 25 4525K25 2K12BZSK25 page 25 of 25 MT235 Introduction prepared by Professor Jenny Baglivo September 2002 In this brief introduction 1 will discuss the modeling process formulate a twovariable linear programming model and solve the problem graphically References Sections 111 4 and 2122 In making decisions an analyst needs to 0 Evaluate the alternatives and 0 Choose the best course of action Mathematics can help A mathematical model uses 0 Symbols to represent decision variables and 0 Functions to describe a real system or decision problem There are benefits to using models In particular 0 Models are less costly to analyze than real systems 0 Models can be analyzed more rapidly 0 Models facilitate quotwhat ifquot analyses 0 Models provide insight about the real system under study I 1 Modeling Process The modeling process includes the following steps 1 Identify the problem 2 Formulate the model 3 Analyze the model 4 Test the resulm 5 Implement the solution We will work with steps 2 through 4 From an identified problem we will formulate and analyze the model and interpret the resulm We will ask quotwhat ifquot certain aspecm of the problem change analyze the new model and interpret the results The models we will use are often referred to as optimization models I 2 Example Galaxy Industries is an emerging toy manufacturing company that produces two quotspace agequot water guns that are marketed nationwide primarily to discount toy stores Although many parenm object to the potentially violent implications of these products the producm have proven very popular and are in such demand that Galaxy has had no problem selling all the manufactured items Two models Zapper and Space Ray are produced in lom of one dozen each and are made exclusively from a special plastic compound Two of the limiting resources are the 1200 pounds of the special plastic compound and the 40 hours of produc tion time that are available each week Galaxy39s marketing department is more concerned with building a strong customer demand base for the edgling company39s products than with meeting high production quotas Two of its recommendations which Galaxy39s management has already accepted are to limit total weekly production to 800 dozen units and to prevent weekly production of Space Rays from exceeding that of Zappers by more than 450 dozen The following table summarizes the per dozen resource requiremenm and profit values calculated by subtracting variable production costs from their wholesale selling prices for the Zapper and the Space Ray Per dozen Product Prof it Plastic lbs Production Time mins Zapper 5 l 4 Space Ray 8 2 3 Galaxy has reasoned that since the 8 profit per dozen Space Rays exceeds the 5 profit per dozen Zappers by 60 the company could maximize is profit by producing as many Space Rays as possible while remaining within the marketing guidelines and using any remaining resources to produce Zappers As a result Galaxy has produced 0 550 dozen Space Rays and 0 100 dozen Zappers weekly earning a profit of 4900 Would a different production schedule improve company profits Summary of the problem Maximize weekly profit by adjusting production level in dozens of Zappers and Space Rays Production must be sched uled so that the weekly supply of plastic and the availability of production time are not exceeded The two marketing department guidelines concerning maximum total production and product mix must be met Decision variables x1 number of dozen Zappers produced in one week x2 number of dozen Space Rays produced in one week Objective function in dollars Profit 5x1 8x Plastic constraint in pounds x1 2x lt 1200 Production time constraint in minutes 4x1 3x lt 2400 Marketing time constraints in dozens of units x1 x2 lt 800 and x2 lt x1 450 Nonnegativity constraints x1 gt 0 and x2 gt 0 Decision variables and mathematical model x1 number of dozen Zappers produced in one week number of dozen Space Rays produced in one week x2 MAXIMIZE Profit 5 x1 8 x2 SUBJECT TO THE FOLLOWING CONSTAINTS Plastic X1 2 X2 lt 1200 Production Time 4 X1 3 X2 lt 2400 Production Total X1 X2 lt 800 Production MiX X1 X2 lt 450 Nonnegativity X1 gt 0 X2 gt 0 Solution highlights X1 1000 1200 1 The lines in the display are graphs of the linear equations X1 2X2 1200 4x1 3x2 2400 X1 X2 800 X1 X2 450 X1 0 X2 O 2 The shaded area is the solution to the system of linear inequalities X12X2 lt 1200 4X13X2 lt 2400 X1 X2 lt 800 X1 X2 lt 450 X1 gt 0 X2 gt 0 3 The form of the problem maximizing a linear function over a region defined by a system of linear equations and inequalities means that the method of linear programming can be used to find the best combination of Zappers and Space Rays The solution is Produce 0 x1 240 dozen Zappers and 0 x2 480 dozen Space Rays each week The weekly profit will be 5040 This production schedule is an improvement over the current schedule Mathematics for Management Science Notes 10 prepared by Professor Jenny Baglivo Jenny A Baglivo 2002 All rights reserved Working with Joint and Conditional Probabilities Let A and B represent events and let P A probability that A occurs PB probability that B occurs The joint probability is the probability that both events occur P A and B The conditional probability of A given B is the ratio PAIB PA and BPB The conditional probability of B given A is the ratio PBIA PA and BPA EV Approach with Multiple Decisions At some event nodes the expected value is calculated using marginal probabilities for each event EVdi PSlVi1 PSz Viz PSN ViN For example in the investor problem Example of notes09 we computed EVBuy Bldg and EVBuy Land using the marginal probabilities PGrowth and PNo Growth At some event nodes the expected value is calculated using conditional probabilities EVdilA PSllA Vil PszlA Viz PsNIA ViN where A is the initial event For example in the investor problem we computed EVBuild Apt Building Growth in 1 Period using the conditional probabilities PGrowth 2 Growth 1 and PNo Growth 2 Growth 1 Similarly EVBuild Commercial Bldg No Growth in 1st Period was computed using the conditional probabilities PGrowth 2 No Growth 1 and PNo Growth 2 No Growth 1 page 1 of 15 Example 1 Martin s Service Station is considering entering the snowplowing business for the coming winter Martin can purchase either a snowplow blade attachment for the station s pickup truck or a new heavy duty snowplow truck Martin has analyzed the situation and believes that either alternative would be profitable if the snowfall is heavy Smaller profits would result if the snowfall is moderate and losses would result if the snowfall is light The following table gives the payoffs and the probabilities of heavy moderate and light snowfall Winter snowfall Heavy Moderate Light Buy Blade 3500 1000 l500 Buy Truck 7000 2000 9000 Probability 40 30 30 Martin can either purchase the equipment now or wait until the end of September since September weather is a predictor of winter snowfall The following table gives the joint and marginal probabilities for September weather normal or cold and winter snowfall heavy moderate or light If Martin waits until the end of September then the cost of the blade will increase by 200 and the cost of truck will increase by 500 page 2 of 15 0 Design the decision treefar the Buy Later alternative Martin Service Station Buy later alternative Buy Blade PHSINS PMSINS Buy Truck PHSINS PMSINS Buy Later Buy Blade PHSCS PMSICS Buy Truck PHSICS PMSICS page 3 of 15 0 Complete the solution ofMartin Sproblem Buy Blade EVBlade 1000 Buy Now Buy Truck EVTruck 100 Buy Blade 80 EVBadeNS800 Normal Sept emter Buy Truck Buy Lat er EVTruckNS600 Buy Blade 2 0 Cold September EVBadeCS2050 Buy Truck EVTruckICS3400 page 4 0f 15 Decision trees Recall that a decision tree is a graphical representation of a decision analysis problem The trees are used when implementing the EV approach with multiple decisions Over certain branches we use the marginal probabilities PSj for j12N Over other branches we use conditional probabilities PSj IA for j12N where A is an initial event The marginal probabilities are sometimes called prior probabilities and the conditional probabilities are sometimes called posterior probabilities Bayes probability rule Some decision analysis problems give information on the prior probabilities of the states of nature and the conditional probabilities P Al Sj j1 2 N for each possible event A The process of computing 1 Thejoint probabilities PA and Sj 2 PSj PA Sj 2 The probability of event A and 3 The posterior probabilities P Sj A j1 2 N for each possible event A are applications of Bayes probability rule page 5 of 15 Example 2 Colonial Motors CM is trying to determine what size of manufacturing plant to build for a new car it is developing Only two plant sizes are under consideration large and small The cost of building a large plant is 25 million dollars and the cost of building a small plant is 15 million dollars The following table gives the present value of future profits in millions in each case if the demand for the new car is high or low and the probabilities of each future event 710 310 Note that the profits do not take into account the cost of construction CM has the opportunity to conduct a survey at a cost of 50000 to assess consumer attitudes about the new car The results of the survey would indicate either a favorable or unfavorable attitude about the new car Past experience with such surveys were as follows 0 Survey results were favorable in 67 of cases where the eventual demand for the car was high and unfavorable in 17 of cases where the eventual demand was high 0 Survey results were favorable in 29 of cases where the eventual demand for the car was low and unfavorable in 79 of cases where the eventual demand was low Should CM conduct the survey before making the decision to build a large plant or build a small plant page 6 of 15 1 Construct a table of joint probabilities for the survey outcomes favorable and unfavorable and the two states of nature high demand and low demand Use your table to compute Pfavorable and Punfavorable Phigh demand I favorable and Plow demand I favorable and 0 Phigh demand I unfavorable and Plow demand I unfavorable page 7 of 15 b Construct a decision true for the CM problem High Demand Build Large 0 Low Demand No Survey High Demand Build Small 0 Low Demand Build Large I Favorable I Build Small Survey 0 Build Large Unfavorable I Build Small page 8 of 15 High Demand 0 Low Demand A lt High Demand 0 Low Demand 4 lt High Demand 0 Low Demand 4 lt High Demand 4 0 Low Demand A lt 6 Complete the CM decision analysis problem Values in Millions Build Large EV126 No Survey Build Small EV 104 Build Large EV 14195 Favorable Build Small EV 10795 Build Large EV 9395 Unfavorable Build Small EV 9595 page 9 of 15 The axpected value of sample information EV SI is defined as follows EVSI EVWSI EVWOSI where 1 The expected value with sample information EVwSI is defined as follows EVWSI EV with sample information at no cost 2 The expected value without sample information EVwoSI is defined as follows EVWOSI EV without sample information maxiEVdi Further the efficiency of sample information is defined as the ratio EVSI EVPI Example 2 continued 1 EVwoSI EVwoPI max126 104 126 2 EV wSI 3 EVSI EVwSI EVwoSI 4 EV wPI 5 EVPI EVwPI EVwoPI 6 Efficiency EVSIEVPI page 10 of 15 Example 3 Joseph Software Inc J SI has been investigating the possibility of developing a grammar and style checker for use on microcomputers Based on its experience with other software projects J SI estimates that the total cost to develop a prototype is 200000 39 If the performance of the prototype is somewhat better than existing software a moderate success J SI believes that it could sell the rights to the software to a larger software developer for 600000 39 If the performance of the prototype is significantly better than existing software a major success J SI believes that it can sell the software for 12 million 39 If the performance of the prototype does not exceed the performance of existing software a failure J SI will not be able to sell the software and hence will lose all its development costs J SI estimates that the probability of a moderate success is 20 the probability of a major success is 10 and the probability of a failure is 70 J SI has the opportunity to hire a consultant who will tell them YES JSI should develop the prototype or NO JSI should not develop the prototype The consultant uses a variety of indicators to make the recommendation Past experience with completed projects yields the following conditional probabilities 39 The consultant recommended developing prototype software in 60 of cases where the eventual product was a moderate success 39 The consultant recommended developing the prototype in 90 of cases where the eventual product was a major success 39 The consultant recommended developing the prototype in 20 of cases where the eventual product was a failure The consultant s fees are 20000 Should J SI hire the consultant or not page 11 of 15 1 Construct a table of joint probabilities for the answers the consultant would give YES or NO and the states of nature moderate success major success failure Use your table to compute PYES and PNO 0 PModerate Success YES PMajor Success YES PFailure YES 0 PModerate Success NO PMajor Success NO PFailure NO page 12 of 15 1 Construct a decision tree for the J SI problem Do Not Hire Develop Prototype Moderate Success Major Success Do Not Develop FaHure A YES Develop NO Don39t Develop Develop Prototype Do Not Develop Develop Prototype Do Not Develop page 13 of 15 Moderate Success Major Su ccess Moderate Success Major Su ccess 6 Complete the solution of the J SI problem in thousands Develop Prototype Do Not Hire Do Not Develop Develop Prototype EV 2942857 Do Not Develop Develop Prototype EV 1 276923 Do Not Develop page 14 of 15 d Calculate the expected value of sample information EVSI and the efficiency of the sample information EVSIEVPI page 15 of 15 Mathematics for Management Science Notes 03 prepared by Professor Jenny Baglivo Jenny A Baglivo 2002 All rights reserved Additional Exercise 1 Bluegrass Farms located in Lexington Kentucky has been experimenting with a special diet for its racehorses The feed components available for the diet are a standard horse feed product a vitamin enriched oat product and a new vitamin and mineral feed additive The nutritional values in units per pound and the costs for the three feed components are summarized in the table below Feed rnmnnnnn 1 Standard Enriched Oat Additive Ingredient A 080 Ingredient B 100 150 300 Ingredient C 010 060 200 Costlb 025 050 300 The minimal daily diet requirements for each horse are three units of ingredient A six units of ingredient B and four units of ingredient C In addition to control the weight of the horses the total daily feed for a horse should not exceed six pounds Bluegrass Farms wants to determine the minimumicost mix that will satisfy the daily diet requirements Additional Exercise 2 Electronic Communications manufactures portable radio systems that can be used for twoiway communications The company s new product which has a range of up to 25 miles is suitable for a variety of business and personal uses The distribution channels for the new radio are marine equipment distributors business equipm ent distributors national chain retail stores and direct mail Because of differing distribution and promotional costs the profitability of the product will vary with the distribution channel In addition the advertising cost and the personal sales effort required will vary with the distribution channels The table below summarizes the contributions to profit advertising cost and personal sales effort Per Unit Sold Profit Advertising Cost Personal Sales Effort 5 Marine distributors 90 10 2 hours Business distributors 84 8 3 hours National retail stores 70 9 3 hours Direct mail 60 15 0 hours The firm has set the advertising budget at 5000 and a maximum of 1800 hours of sales force time is available for allocation to the sales effort Management also decided to produce exactly six hundred units for the current production period Finally an ongoing contract with the national chain of retail stores requires that at least one hundred fifty units be distributed through this distribution channel Electronic Communications now faces the problem of establishing a distribution strategy for the radios that will maximize overall profitability of the new radio production Decisions must be made about how many units should be allocated to each of the four distribution channels as well as how to allocate the advertising budget and sales force effort to each of the four distribution channels page 1 of 17 Additional Exercise 1 continued 0 Summarize the problem 0 Define the decision variables precisely 0 Completely Specify the LP madel page 2 of 17 0 Clearly state the optimal solution 0 How much of each nutrient will the race horse receive 0 What is the total cost for the standard feed enriched oat feed and additive 0 Are there alternative optimal solutions 0 Suppose the maximum weight was increasedfrom 6pounds to 7pounds Would the total cost increase decrease or stay the same If it changes what is the new total cost page 3 of 17 Additional Exercise 1 solution and sen vity report 9554054054 4 Adjustable Cells Final Reduced Objective Allowable Allowable Cell Value Cost Coefficient Increase Decrease B17 Pounds Standard 3513513514 0 025 1E30 0642857143 15505517 Pounds Enriched Oat 0945945946 0 05 0425 1E30 p7 Pounds Additive 15495519541 0 3 1E30 14782608 Constraints Final Shadow Constraint Allowable Allowable ce Value Prige RH Side I39Lqregie Decrease Ingredient A LHS 3 1216216216 3 0368421053 1857142857 B23 Ingredient B LHS 9554054054 0 6 4 15585524 Ingredient C LHS 4 1959459459 4 0 19 3555545 WeigntLHS 6 09189777178919 6 07877 page 4 of 17 Additional Exercise 2 continued 0 Summarize the problem 0 Define the decision variables precisely 0 Completely Specify the LP madel page 5 of 17
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