Mathematics for Management Science
Mathematics for Management Science MT 235
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Review of Derivatives To date we have studied linear programming problems in which the only mathematical operations on the decision variables are multiplication by a constant the coef cient and the addition and subtraction of variables Decision variables are not raised to a power nor are they multiplied or divided together In nonlinear programming problems NLP both the objective function and the constraints can involve nonlinear functions ofthe decision variables We will study only a very limited subset of nonlinear programming problems those that involve one or two decision variables and a nonlinear objective function but with linear constraints We will solve these problems using two methods algebraically using principles of calculus and later with Excel Derivative Rules In order to solve nonlinear problems using calculus we will need to calculate the derivative of a function Recall the following basic derivative rules for the derivatives of polynomials Rule Function Derivative Derivative of a constant is zero fx c a constant f39X 0 Power rule 1 X X f 39X nx 391 Derivative of a sum is the sum of the derivatives fx gx hx f39X g39X h39x Derivative of a constant times a function is the constant le C9X 1 X 09 X times the derivative of the function Exercise 1 2 Let gx 6x3 LZ 3n3 Calculate g39X 4 x x Exercise 2 Let fx 3x4 4x3 12x2 40 Find the derivative f39X and evaluate f391 f392 and f393 Properties of Derivatives Recall that when the derivative of a function is zero for a speci c value X a there are 3 possibilities 1 The function value attains a maximum at X a 2 The function value attains a minimum at X a 3 The function has an in ection point at the value X a so it is neither a maximum nor minimum Exercise 3 Draw schematic graphs that illustrate each ofthe three different possibilities listed above Exercise 4 Let fx 3x4 4x3 12x2 40 Determine the values ofX for which f39X 0 these are called the critical values off This is the same function given in exercise 2 Exercise 5 Find the critical values of fx 000125x2 2 x Definition Second Derivative Recall that the second derivative offX written fquotX is the derivative ofthe function f39X Exercise 6 2 Let gx 6x3 xjZ n3 Calculate gquotX This is the same function given in exercise 1 x x Properties of the Second Derivative c When fquota gt 0 then the function fX is concave up at the point X a c When fquota lt 0 then the function fX is concave down at the point X a c When fquota 0 then the function fX is neither concave up nor concave down at the point X a Exercise 7 Let fx 3x4 4x3 12x240 At each of the critical values of f X as determined in exercise 4 determine ifthe function is concave up or concave down NonLinear Pro rammin One Variable General Overview To date we have studied linear programming problems ie where the only mathematical operations on the decision variables are multiplication by a constant the coefficient and the addition and subtraction of variables Decision variables are not raised to a power nor are they multiplied or divided together In nonlinear programming problems NLP both the objective function and the constraints can involve nonlinear functions of the decision variables We will study only a very limited subset of nonlinear programming problems those that involve one or two decision variables and a nonlinear objective function but with linear constraints We will solve these problems using two methods algebraically using principles of calculus and later with Excel A typical nonlinear programming problem might look like this Max f X 3X4 4X3 12X2 40 note that the only decision variable is X st X s 4 X 2 3 variables often take on negative values in nonlinear problems In the language of linear programming X is the decision variable and fX is the objective function Constrained vs Unconstrained Functions A basic theorem from calculus tells us that to determine the minimum and maximum values of an unconstrained function Le a function that does not have any constraints we determine the critical points and then check to see ifthe function is concave up or concave down at each critical point In this case we are looking for the local minimum and local maximum values of the function which will give us the global minimum and maximum values For a constrained function we are again looking for the global minimum and global maximum values of the function To do this we determine the critical values and the boundaries of the constraints in the onevariable case this means the endpoints of the interval on which the variable is defined These values are then substituted into the original function NOT the derivative and compared The largest value is the global maximum and the smallest is the global minimum value Procedure for Finding Maximum or Minimum Values of a Constrained Function Use the following procedure to find the maximum and minimum values of the constrained function fX 1 Calculate f39X 2 Set f39X equal to zero and solve for X These are the critical values 3 Determine the value of f X at each critical value and endpoint of the interval 4 The largest of these is the maximum OFV value and the smallest is the minimum OFV value In linear programming language the optimal solution is the value of the decision variable X ie X a and the OFV is the value fa Exercise 1 Maximize fx 3x4 4x3 12x2 40 subject to 3 s X s 4 Recall that we determined the critical values X 2 1 and 0 previously Exercise 2 Minimize fx 000125x2 2 subject to 10 s x s 400 Recall that we determined the x critical value X 100 previously Exercise 3 A manufacturer has been selling 1000 TV sets a week at 450 each A market survey indicates that for every 10 increase in the price of the sets the number of sets sold will decrease by 100 per week Likewise for every 10 decrease in price the number sold will increase by 100 Due to cost restrictions the lowest price the manufacturer can sell each of the TV sets for is 250 a Determine a formula for the weekly revenue b Determine the constraints that are implicitly stated in the context of the problem c Use calculus to determine the best price to sell the TV sets for to maximize revenue d Suppose that the company s weekly cost is 68000 plus 150 for each TV set produced Determine the company s profit formula and use calculus to determine the best price to sell the TV sets to maximize profit Exercise 4 A manufacturer estimates that they can sell 600 digital cameras a month at a price of 80 per camera and 500 cameras a month at a price of 100 per camera Costs are 6000 plus 48 per camera Assume that the price and number sold are linearly related The lowest possible selling price for the cameras is 60 a Express the price as a function of x the number of cameras sold b Determine the constraints that are implicitly stated in the context of the problem 0 Determine a formula forthe monthly pro t d Use calculus to determine the best price to sell the digital cameras at and the number sold that will maximize the company s pro t EOQ Economic Order Quantity Model Suppose a company requires a constant supply of a product over the course of the year Also assume that each new order is received in full when the inventory reaches zero and that there is a fixed cost for each order placed regardless of the number of units ordered In addition there is a holding cost for each unit that is kept in inventory The overall cost will depend on how we order If we place a single order for the entire annual demand then we reduce the number of orders placed but this means that the inventory holding costs will be higher If we order fewer items at a time then inventory costs are low but ordering costs will be higher Therefore we want to determine the optimal number of units of the product to order at a time so that we minimize the total costs associated with the purchase ordering and storage of the product The only decision that needs to be made is how many units to order each time order quantity The total cost will depend on the total demand for the year the purchase cost for each item unit cost the fixed cost to place the order order cost and the cost of holding an item in inventory The cost will also depend on the number of orders per year but this number can be determined from other values We use the following notation D total demand for the year c unit purchase cost forthe item unit cost R cost of placing each order order cost h cost of holding one unit in inventory for an entire year expressed as a percentage of the cost x number of items ordered with each order order quantity Express each of the following using the variables and parameters listed above a Determine the annual purchase cost for the product b Determine the number of orders placed per year 0 v Determine the annual fixed cost for placing orders O V Determine the inventory holding cost for one unit for an entire year in D v Determine the average inventory over the entire year f Determine the total annual inventory holding cost g The objective function is the sum of the purchase cost a fixed order cost c and inventory holding cost f Use the above results to find an expression for the objective function 339 v Determine the interval on which this function is valid Exercise 1 Alan Wang Alan Wang is responsible for purchasing the paper used in all the copy machines and laser printers at the corporate headquarters of MetroBank Alan projects that in the coming year he will need to purchase a total of 24000 boxes of paper which will be used at a fairly steady rate throughout the year Each box of paper costs 35 Alan estimates that it costs 50 each time an order is placed this includes the costs of placing the order and the related costs of shipping and receiving MetroBank assigns a cost of 18 to funds allocated to supplies and inventories because such funds are the lifeline of the bank and could be lent out to credit card customers who are willing to pay this rate on money borrowed from the bank Alan has been placing paper orders once a quarter but he wants to determine if another ordering pattern would be better He wants to determine the most economical order quantity to use in purchasing paper Determine each of the following values D c R h The equation for the total cost and the interval for X are TX Interval a Use calculus to find the optimal reorder size to minimize total cost b How many orders will Alan Wang place the first year under this optimal ordering strategy 0 v How often in days should Alan Wang place an order d How much savings are there with this new ordering strategy Recall that the original ordering strategy was 6000 boxes of paper each quarter We can also use Excel to solve this problem Fill in the following generic EOQ spreadsheet model with the appropriate values and formulas Model Demand per unit Cost cost of unit cost Bounds Decision Variable Order size of costs Purchase Order Storage Total Cost Minimum order size Maximum order size Variation Bulk Discount Pricing When bqu discount pricing is available the EOQ model can still be used but each pricing level scenario needs to be treated as a separate EOQ problem and the optimal ordering strategies are compared to determine the one that is the most cost effective Exercise 21Keith Shoes Keith Shoe Store carries a basic black men s dress shoe They sell 2000 shoes each year at a fairly constant rate Keith s current buying policy is to order 500 pairs every three months It costs Keith 150 to place an order The annual inventory holding cost rate is 25 The supplier offers bqu pricing discounts based on the number of shoes ordered at a time as follows Order Quantity Price per pair 1149 36 150299 34 300 or more 32 With an order quantity of 500 Keith obtains shoes at the lowest possible unit cost 32 per pair Keith wants to know if this is the best ordering strategy a Use Excel to solve the problem in three stages and complete the table with the relevant optimal values from each scenario Scenario Order size Total Cost 1149 36 per pair 150299 34 per pair 300 or more 32 per pair b What is the optimal ordering strategy for Keith Shoes and what is the total cost of this strategy c What are the annual savings of this ordering policy over the policy currently being used by Keith Two Variable NLP Unconstrained The second type of NLP problem we will see involves two decision variables and a nonlinear objective function As in the onevariable case we need to calculate the first derivative to find the critical values and then use the second derivative to determine if the function has a local maximum or local minimum at that point The main difference in the twovariable case is that the derivatives need to be calculated with respect to each variable separately and the second derivative test is slightly more complicated Partial Derivatives Consider the function fx y x2y 4993 Sxy 7x10y2 40 Since there are two variables we can t calculate the derivative so instead we calculate the partial derivatives with respect to each variable The partials are is obtained by taking the derivative of the function with respect to each variable separately while treating the other variable as if it were a constant The notation is fXX y means the partial of fwith respect to X fyX y means the partial of fwith respect to y39 Exercise 1 Let fx y x2y 4993 Sxy 7x10y2 40 Calculate the partial derivatives fXX y and fyX y Critical Fairs and Critical Points If a point X y is such that fXX y O and fyX y 0 then we call X y a critical pair and the corresponding triple X y fX y is a critical point of the function fX y Second Partial Derivatives For a function of twovariables we have two partial derivatives each of which itself is a function of two variables we can actually calculate four second partial derivatives We can take the derivative of each partial derivative with respect to each variable Therefore we take the derivative of the original function twice but it can be with either variable To keep track of which variable and in which order we use two subscripts on the function to indicate the order of the partials fXXX y means the partial of fX with respect to X OR the partial of f with respect to X first and X second fxyX y means the partial of fX with respect to y OR the partial of f with respect to X first and y second fyXX y means the partial of fy with respect to X OR the partial of f with respect to y first and X second fyyX y means the partial of fy with respect to y OR the partial of f with respect to y first and y second Exercise 2 Calculate the second partial derivatives of the function fX y from exercise 1 Discriminant Function We de ne the discriminant function DX y which will be used in the second derivative test DX Y fxxX Y 39 fyyX Y fxyX Y 39 fyxX Y The Second Derivative Test for Unconstrained Functions of Two Variables The second derivative test for two variables has a similar application as in the onevariable case but is complicated by the fact that there are four partial derivatives When X y is a critical pair for fX y in other words fXx y O and fyx y 0 then If DX y gt O and fxxx y lt 0 then there is a local maximum at X y If DX y gt O and fxxx y gt 0 then there is a local minimum at X y If DX y lt 0 then there is a saddle point which is neither a local maximum nor minimum Ifthere is more than one local minimum or maximum value then choose the pair that yields the largest or smallest value for the function as the absolute maximum or minimum Exercise 3 Let fxy x3 3xy y3 Use the second derivative test for two variables to determine whether the points 00 11 1 2 and 2 2 correspond to local minimum local maximum saddle points or none ofthese Exercise 4 Find all critical pairs and critical points of the function fxy 3x2 2yx 6x 2y2 2y 58 Determine where fhas a local minimum local maximum or saddle point Two Variable NLP Constrained When we have an NLP of two variables with a constraint we convert the problem to one involving only a single variable We can also calculate a value called the Lagrange Multiplier that is analogous to the shadow price in an LPP The setting of a two variable constrained NLP is decision variables X y objective function fx y which we will maximize or minimize subject to gx y c c is a constant Exercise 1 Green Lawns Inc provides a lawn fertilizer and weed control service The company provides four treatments of fertilizer and weed control chemicals to its subscribers each year In order to attract new customers Green Lawns has decided to offer a special aeration treatment as a lowcost extra service Management is planning to promote this new service in two media radio and directmail advertising A budget of 3000 is to be used on this promotional campaign over the next quarter Based on past experience in promoting its other services Green Lawns has been able to obtain an estimate of the relationship between sales and the amount spent on promotion in these two media Total sales in thousands of dollars Sx y 2x2 1Oy2 8xy 18x 34y where x thousands of dollars spent on radio advertising y thousands of dollars spent on directmail promotion Green Laws would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the promotional budget Develop a nonlinear model to maximize sales subject to the advertising budget Solve the problem and calculate the Lagrange multiplier Step 1 Find the minimum or maximum of f 1 Use the constraint to rewrite the function fx y in terms of one of the variables Usually this means solving the constraint in terms of yand substituting this into the objective function fx y This gives us a new function hx in terms of x alone 2 Determine associated endpoints for the single variable using the context of the problem 3 Use calculus to find the maximum or minimum value of the function hx 4 Use this value for x to find the associated value for y from the constraint gx y c and then use these to determine the objective function value fx y 5 The optimal solution is the point x y and the objective function value is fx y Two Variable Constrained NLP Lagrange Multipliers The method of Lagrange multipliers is a general mathematical technique that can be used for solving constrained optimization problems consisting of a nonlinear objective function and one or more linear or nonlinear constraint equations In this method the constraints are subtracted from the objective function as multiples of a Lagrange multiplier A We will start with the general setting of a two variable NLP with a single linear constraint ecision variables objective function fx y which will be maximized or minimized subject to ax by c The Lagrangian method involves the following steps 1 Rewrite the constraint in the form ax by c 0 2 Create the Lagrangian function Lx y A fx y Max by c 0 Note that we haven t really changed the objective function since ax by c 0 3 Use the techniques for solving unconstrained NLP In other words calculate the partial derivatives LX Ly and LA set all three equal to zero and solve The general approach to solving the first partials is to eliminate from the first two partials by subtracting a suitable multiple of Ly from LX Use this new equation and LA to solve for either x or y 4 Calculate the second partials LXX Lyy ny LyX and the discriminant to confirm that the critical pair correspondsto a local maximum or local minimum as appropriate 0 Don t forget to use the fact that ny and LyX should be equal as a check Exercise 1 Beaver Creek Pottery Company produces bowls x and mugs y Each bowl takes one hour to produce and each mug takes two hours to produce There are 75 hours of labor available in the next month Beaver Creek s profit function is given by the equation Px y 4x 5y 01x2 02y2 The NLP model is Max Profit Px y 4x 5y 01x2 02y2 st x 2y 75 Rewrite the constraint as x 2y 75 0 The Lagrangian function is Lx y A 4x 5y 01x2 02y2 Ax 2y 75 The first partial derivatives are LX 4 02x 0 Ly5 04y 2 0 Lj x 2y 75 0 Solve these three equations to determine the critical pair x y and the Lagrange multiplier A The second partial derivatives are XX L 04 Ly0 Lyx0 D LXXLyy LXyLyX 02 04 00 008 gt 0 Since D gt 0 and LXX lt 0 the critical pair corresponds to a local maximum Calculate the OFV and give the final answer as a complete sentence in context Interpretation of the Lagrange multiplier The Lagrange multiplier in nonlinear programming is analogous to the shadow price for linear programming The Lagrange multiplier for a constraint gives the approximate change in the objective function associated with the change in the RHS of the constraint The actual change may be more or less than the value of the Lagrange multiplier however the estimate is fairly good as long as the change in the RHS is not too large Note that since nonlinear relationships are involved the value of the multiplier will change as soon as there is even a small change in the RHS Therefore the Lagrange multiplier does not have a range of feasibility and the Lagrange multiplier can only be interpreted as an approximation of the value of additional resources in a nonlinear model A general guideline is that we can estimate the new OFV using the Lagrange multiplier as long as the change to the RHS is relatively small say within 25 of its original value So in the example above we can use the Lagrange multiplier as long as the change to the RHS is 1875 or less a Use the Lagrange multiplier to estimate the change in profit if the number of available hours of labor decreases to 50 b Use the Lagrange multiplier to estimate the change in profit if the number of available hours of labor increases to 81 Note that by actually resolving the problem with 81 hours we would determine that the profit would decrease to 2805 an actual decrease of 1320 Although this is not exactly the same as our Lagrange multiplier estimate it is reasonably close Two Variable NLP with More Than One Constraint For problems involving two variables and more than one constraint we can use Excel to solve Setting up the spreadsheet model is fairly straightforward and Excel s Sensitivity Report will indicate the value of the Lagrange multiplier for each constraint Exercise 2 The XYZ company produces two products The total profit achieved from these products is described by the following equation Total profit in thousands of dollars Px y 02x2 04y2 8x 12y 1500 where X thousands of units of product 1 and y thousands of units of product 2 Every thousand units of product 1 requires one hour in the shipping department and every thousand units of product 2 requires 30 minutes in the shipping department Eighty hours of shipping will be available in the next production period Each unit of product requires two pounds of a special ingredient of which 64000 pounds are available in the next production period The demand for each product is unlimited The NLP model is Max Pro t PX y o2x2 04y2 8X 12y 1500 st X 05y s 80 Shipping hours 2X 2y 5 64 Pounds of special ingredient x y z 0 The model was solved with Excel in the usual way to generate the following Sensitivity Report Adjustable Cells Final Reduced I Cell Name Value Gradient B10 Units produced thousands Product 1 18 0 C10 Units produced thousands Product 2 14 0 Constraints Final Lagrange Cell Name Value Multiplier E15 Shipping hours He 25 0 EJ 7 R 9UUQIEQEN LHS 64 04 1 What is the pro t if an extra 10 hours of shipping is available in the next production period 2 Estimate the pro t ifan extra 10000 pounds of special ingredient are available 3 Estimate the pro t ifan extra 30000 pounds of special ingredient are available Introduction to Decision Analysis Overview Decisionmaking is the process of choosing among different alternatives based on a ranking criteria established by the decisionmaker Most ranking criteria require assigning a numeric value to each decision alternative such as scores grades or preference ratings In business and management the ranking criteria is often based on money highest profit versus lowest cost Decision analysis involves developing and implementing decisionmaking strategies where there is an uncertain or riskfilled pattern of future events The objective is to make the best decision with the current information available In any decision analysis problem there are decision alternatives and statesofnature son The decision alternatives are the possible decisions that the decision maker is faced with The statesof nature are the future events that will affect the decision but that the decision maker has no control over Each decision alternative and stateof nature pair is called a potential outcome Each outcome has a payoff which is the numerical value assigned to that outcome which may or may not be money Notation Decision alternatives d1 d2 d Statesof nature son s1 52 5 Payoffs v for i 1 2 m and j 1 2 n note that v is the payoff for the outcome corresponding to decision alternative d and stateof nature 5 The decision alternatives statesof nature and payoffs can be summarized in a payoff table The DecisionMaking Process The decisionmaking process must take into consideration the possible statesof nature since these are the unknown and uncontrollable events There are two different approaches depending on whether or not the decision maker can assign a probability to each of the statesof nature Decisionmaking strategies are simpler but more risky when the probabilities are not known We will deal with this situation first Decision Making Without Probabilities There are a number of different decisionmaking strategies that can be used when the probability that the stateofnature will occur is not known but we will consider only three kinds the optimistic approach the conservative or pessimistic approach and the minimax regret approach Optimistic Approach Maximax For each decision alternative calculate the maximum payoff Choose the decision alternative with the largest maximum payoff Conservative A roach Maximin For each decision alternative calculate the minimum payoff Choose the decision alternative with the largest minimum payoff Minimax Regret Approach For each stateofnature calculate the maximum payoff Compile the regret table which has the same structure as the payoff table First determine the maximum payoff for each stateof nature column Then for each outcome subtract its payoff from the maximum payoff for that stateof nature This is called the regret or opportunity loss For each decision alternative row determine the maximum regret Choose the decision with the smallest maximum regret Minimax Regretquot means that we are minimizing the maximum regret values for each decision Pittsburgh Development Corporation jwithout probabilitiesj 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 o 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 0 High market acceptance and hence substantial demand forthe 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 net profit for the PDC condominium project Payoff in millions s1 High Acceptance s2 Low Acceptance d1 build small 8 7 d2 build medium 14 5 d3 build large 20 9 PDC would like to determine the best size complex to build 1 Identify the states of nature 2 Identify the decision alternatives 3 Determine the optimal decision using Optimistic and Conservative Approaches 4 Create the Regret Table S1 52 Max d1 d2 d3 5 Determine the optimal decision using the Minimax Regret Approach Magnolia Inns without probabilitiesl 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 accommodate the increasing number of flights 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 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 lnns hotel chain intends to build a new facility near the new airport once its site is determined 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 flow 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 all figures are in millions Parcel near Location A Location B Purchase price 18 12 Present value of future cash flow if airport is built at that site 31 23 a is not built at that site 6 4 The company must decide whether to purchase one both or neither of the sites 1 Identify the states of nature and the decision alternatives l Establish the Payoff Table note that you need to calculate actual payoffs for each decision 3 Determine the optimal decision using Optimistic and Conservative Approaches P Establish the Regret Table 5 Determine the optimal decision using the Minimax Regret Approach WM Decisions based on the three previous methods optimistic conservative amp minimax regret suffer from the weakness that there is no likelihood assigned to each outcome Stronger methods incorporate the probability that each stateofnature will occur thereby allowing us to assign a numerical value to each decision alternative Notation Let Ps the probability that the stateofnature swi occur i 1 2 3 n Since the statesofnature include all possible scenarios we must have the following relationships Ps1 Ps2 Ps 1 Pnot s 1 Ps The two approaches that we will consider are the expected value approach and the expected opportunity loss approach Ex ected Value EV A roach 1 Multiply the payoff for each outcome decision alternative and stateofnature pair by the probability of its stateofnature This is the expected value of that outcome 2 For each decision alternative calculate the sum of the expected values This is the expected value of that decision alternative Symbolically EVd Ps1 v1 Ps2 v2 Ps v where v is payoff 3 Choose the decision with the largest expected value Expected C quot Loss EOL Approach 1 Multiply the regret for each outcome decision alternative and stateofnature pair by the probability of its stateofnature This is the expected opportunity loss of each outcome 2 For each decision alternative calculate the sum of the expected opportunity losses This is the expected opportunity loss for that decision alternative Symbolically EOLd Ps1 r1 Ps2 r2 Ps r where r is regret 3 Choose the decision with the smallest expected opportunity loss Expected Value With Perfect 39 quot EVwPl Defined to be the weighted average of the maximum payoffs for each stateofnature 1 Identify the maximum payoff for each stateofnature 2 Multiply the maximum payoff by its associated probability 3 Calculate the sum Expected Value Without Perfect39 quot EVwoPl Defined to be the maximum of the expected values for each decision alternative this is the same calculation as the EV Approach Ex ected Value of Perfect Information EVPI Defined to be the difference between the expected value with and without perfect information EVPI EVwPI EVwoPI As a verification EVPI should be the same value as that calculated with the EOL Approach Cautionary Note The expected value assigns a numerical value to each decision alternative This allows us to evaluate decision alternatives by comparing their expected values the larger the better However this final number has no tangible significance and should not be interpreted as how much we can expect to receive from this decision 7quot 39 39 l 39 Corporation with probabilities Recall the Pittsburgh Development Corporation PDC condominium project presented earlier Suppose that PDC has gathered more information about the probability of each stateofnature and wants to analyze the situation with this new information PDC is optimistic about the potential for high demand and assigns an 80 probability that acceptance will be high Therefore the probability of low acceptance is 20 Ps1 80 and Ps2 20 1 Determine the best strategy using the Expected Value EV Approach 2 Determine the best strategy using the Expected Opportunity Loss EOL Approach 3 Determine the Expected Value with Perfect Information EVwPl 4 Determine the Expected Value without Perfect Information EVwoPl 5 Determine the Expected Value of Perfect Information EVPI Magnolia Inns with probabilities Recall the Magnolia Inns hotel project presented earlier Management thinks that there is a 40 chance that the airport will be built at location A and a 60 chance that it will be built at location B 1 Determine the best strategy using the Expected Value EV Approach 2 Determine the best strategy using the Expected Opportunity Loss EOL Approach 3 Determine the Expected Value with Perfect Information EVwPl 4 Determine the Expected Value without Perfect Information EVwoPI 5 Determine the Expected Value of Perfect Information EVPI Decision Analysis Decision Trees In more complicated decisionmaking problems involving multiple decision alternatives and statesof nature decision trees can be used to organize and implement the Expected Value Approach A decision tree is a graphical representation ofa decision problem The tree consists of nodes and paths At each node a decision between different alternatives is required or the different options for a stateof nature can happen Each path represents the sequence of a possible outcome Each path has an associated payoff and the path with the maximum payoff is selected Nodes and Paths c There are three different kinds of nodes in a decision tree 1 Decision nodes El where a choice between the decision alternatives must be made 2 Stateof Nature nodes 0 where an event with different possibilities can occur and which we have no control over Also called an Event node A Terminal nodes l where no further decision is required or stateof nature can occur Each terminal node corresponds to a single outcome 0 The rst node is often a decision node corresponding to the rst decision that must be made however ifthe decision is obvious the rst node may be a stateofnature node 0 Paths extend from each node other than a terminal node and correspond to the different decision alternatives or statesof nature that can occur at that node 0 Every path ends at a node either terminal decision or stateof nature node Labeling 0 Each node is assigned a number 0 Each path from a decision node is labeled with a brief description 0 Each path from a stateof nature node is labeled with a brief description and its associated probability Payoffs and Expected Values 0 Most paths have a payoff associated with the decision or stateof nature event at that node This number should also be labeled on the path 0 All paths also have an expected value EV which is determined from the node at the end ofthe path The EV at a stateofnature node is the weighted average ofthe payoffs of all paths from that node The EV at a decision node is the maximum ofthe payoffs of all paths from that node 0 Each path should be labeled with its EV This number should be boxed or otherwise highlighted to indicate that it is the EV and not a payoff Decision Tree Procedure Summary 0 When constructing a decision tree work forwards labeling nodes paths payoffs and probabilities Then work back down the tree calculating EV at each node Remember to calculate the weighted average at each stateof nature node and the maximum EV at each decision node Josh Penske Josh Penske is considering embarking upon a sunken treasure search in an area of Lake Erie where he believes there may lie buried a ship laden with a cargo of precious metals His initial expenditure would be for a sonar search in the area for the existence ofa sunken vessel This search would cost 30000 If the search indicates a vessel is present Josh would then rent deep sea equipment at a cost of 50000 to investigate lfthe sunken vessel proves to be the treasure ship he estimates its value at 500000 otherwise he estimates its value at 35000 Josh believes that there is a 60 chance of nding a vessel in the area and a 15 chance that if a vessel is present it is the precious metal cargo ship a Identify the decision alternatives and statesofnature b Construct a decision tree for this problem lling in probabilities and payoffs c Calculate the expected value at each node d Using the expected value approach state completely Josh s optimal strategy ComTech 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 coalmining industry Steve Hinton the owner of ComTech a small communications research rm is considering whether or not to apply for this grant Steve estimates he would spend approximately 5000 preparing his grant proposal He estimates that his chances of receiving the grant are even 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 equipment needed for each technology is summarized as Technology IEquigment Cost Microwave 4000 Cellular 5000 lnfrared 4000 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 bestcase and worstcase RampD costs associated with using each technology and he assigns probabilities to each outcome based on his degree of expertise in the area R amp D Costs Probability Technology Best Case Worst Best Case Worst Case 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 submit a grant proposal to OSHA a Identify the decision alternatives and statesofnature b Construct a decision tree for the ComTech problem filling in all probabilities and payoffs Complete the tree by calculating the expected value for each path Using the expected value approach state completely ComTech s optimal strategy Decision Analysis Graphical Sensitivity Analysis The Expected Value Approach EV for decisionmaking involves taking the weighted average ofthe payoffs based on the probability of each stateof nature The choice of the decision depends on the actual probabilities assigned to the statesof nature a different assignment of probabilities could yield a different decision Recall the Payofftable forthe PDC problem Payoff in millions 51 High 52 Low Acceptance Acceptance d1 build small 8 7 d2 build 14 5 medium d3 build la e 20 9 fthere is an 80 probability of high acceptance of the project then using the expected value we determine that d3 build large is the best decision However what if the probability of high acceptance is only 60 Which decision is the best Ps1 60 Psz 40 EV d1 d2 d3 As we can see the decision ultimately depends on the probability assigned to each stateof nature Therefore an important question is For what probability values ofa stateof nature would each decision be made We will answer this question in the situation where there are only two statesof nature s1 and 52 as in the PDC and Magnolia Inns examples Since there are only two statesof nature they are complementary in that Ps1 Psz 1 So if we are interested in the probability value of 51 we can write Psz 1 Ps1 Alternatively we can set Ps1 X and then write Ps2 1 X This allows us to write the expected value of each decision in terms ofxalone In the PDC example we express each decision alternative d1 d2 and d3 corresponding to build small medium or large in terms of Ps1 xand Psz 1 x EVd1 8x 71 x 8x 7 7xx 7 EVd214x 51 X14x 5 5x 9x 5 EVd3 20x 91 x 20x 9 9x 29x 9 Since each expected value is expressed in terms of x and each represents the equation ofa line we can graph the expected value and x on the same graph Note that x represents a probability so it can only take on values between 0 and 1 and the y axis will correspond to the expected values Plot the three EV lines on the same graph We determine the points ofintersection in the usual way by setting two lines equal to each other EV 20 0391 072 0394 0395 016 017 0398 0399 1390x How do we interpret this graph Since the probability must be between 0 and 1 we only need to graph for O 3 XS 1 The lines correspond to the expected value for each decision A particular value ofX corresponds to a value for Ps1 since we set Ps1 X We want the largest overall expected value so we choose the decision with the highest expected value for a particular probability in other words the top line ofthe three Which line is on top depends on X in other words the probability of the stateofnature The X coordinates ofthe points of intersection give us the values for Ps1 where the best decision changes For this example we have the following results summarized in table form Best Decision Alternative H hest EV Range ofvalues forX Ps1 Le Probability of high acceptance o s x s 025 d1 025 s X s 07 d2 07 S xs1 d3 Notes At the points of intersection both decisions have the same expected value so either can be chosen In this example if the probability of high acceptance is exactly 25 then either d1 or d2 can be chosen they have the same expected value of725 Likewise if the probability is 70 then either d2 or d3 can be chosen It is possible that a decision may never be chosen if its expected value line is always below one ofthe other lines for O s X s 1 Points of intersection that do not occur on a top line are irrelevant In the PDC example the coordinate 057757 has no bearing on the best decision Alternatively we could have solved this problem using X Ps2 This would yield a different graph but the same summary results Decision Trees A decision tree is a graphical diagram that can be used to analyze a decision situation The decisionmaker computes the expected value of each outcome and makes a decision based on these expected values One of the primary benefits of a decision tree is that it provides a visual representation of the decisionmaking process A decision tree consists of nodes and branches Each node represents a decision alternative or a stateof nature and each branch represents a stage in an outcome Nodes and Branches There are three different kinds of nodes in a decision tree 1 Decision nodes El where a choice between decision alternatives must be made 2 Stateof Nature nodes 0 where stateof nature options can occur Also called Event nodes 3 Terminal nodes l where no further decision alternative or stateof nature occurs A complete path from the start ofthe decision tree to a terminal node corresponds to one possible outcome Every branch ends at a node terminal decision or stateof nature Branches extend from each node other than a terminal node and correspond to the different decision alternatives or statesofnature that can occur at that node For easier reference each node is numbered in sequence P yoffs and Expected Values Each outcome needs to be assigned a payoff In some cases the payoff for an outcome is given in other cases the payoff is calculated from values on intermediate branches Each node has an expected value EV associated with it which is calculated based on payoffs or EV of branches extending from the node The EV at a stateof nature node is the weighted average of the branch payoffs The EV at a decision node is the maximum or minimum when dealing with costs payoff ofthe branches extending from the node Each node should be labeled with its EV This number should be boxed or otherwise highlighted to clearly show that it is an EV and not a payoff Decision Tree Procedure Summary When constructing a decision tree work forwards labeling nodes branches payoffs and probabilities Then work backwards through the tree calculating the EV at each node Example Pittsburgh Development Corporation PDC Recall that PDC wants to build a luxury riverfront condominium complex and needs to determine the best size complex to build Based on previous experience with this type of development they have assigned an 80 probability of high market acceptance The payoff table in millions is Pa off Hi hAcce tance Low Acce tance Small complex 8 7 Medium complex 14 5 Large complex 20 9 Create the decision tree associated with this decision situation MultiStep Decision Making When a decision situation involves only a single decision a decision tree is basically a visual representation of a payoff table In these cases a payoff table is usually a more ef cient decisionmaking tool However when a decision situation involves a series of decisions to be made called sequential decisionmaking a decision tree is far superior to a payoff table PDC Revisited In the PDC example suppose that PDC has the opportunity to conduct a market research survey to help them better identify the chances that the project will have a high or low market acceptance The report will provide one of two results a favorable report a significant number of individuals contacted express interest in purchasing a PDC condominium or an unfavorable report very few express interest PDC s decision is now more complex since they must rst decide whether or not to conduct the survey and then based on these results determine the most appropriate size condominium project to build In order to analyze this decision situation we need to know the probabilities associated with each ofthe different outcome scenarios If the market research survey is conducted the probability of a favorable report is 77 and the probability of an unfavorable report is 23 If the report is favorable the probability of high market acceptance is 94 and the probability of low market acceptance is 6 If the report is unfavorable the probability of high market acceptance is only 35 and the probability of low market acceptance is 65 Conditional Probabilities The latter two probabilities are examples of conditional probabilities the probability depends on some prior event in this case the outcome ofthe market research The notation PH F is used to designate the conditional probability of high market acceptance given that the report was favorable Write out each of the different probabilities involved in this decision situation using correct notation Use H for high acceptance L for low acceptance F for a favorable report U for an unfavorable report Complete the decision tree given on the next page with all relevant payoffs probabilities and expected values Using the results from the decision tree state completely PDC s optimal decision strategy Ex ected Value of Sam le Information EVSI The difference between the expected values at nodes 2 and 5 survey vs no survey provides a measure ofthe value of the market research survey called EVSI For the PDC problem what is the EVSI lfthis market research survey costs 12 million should PDC purchase it PDC Decision Tree Payo s all gures are ins millions ngh SmaH Luvv ngh Favmame Repun Medmm a O MamatResearm Survey Uniavmame Repun Nu MamatResearm Survey Joseph Software Joseph Software Inc JSI has been investigating the possibility of developing a grammar andstyle checker for use on microcomputers Based on its experience with other software projects JSI estimates that the total cost to develop a prototype is 200000 If the performance of the prototype is somewhat better than existing software referred to as a moderate success JSI believes that it could sell the rights to the software to a larger software developer for 600000 lfthe performance ofthe prototype is signi cantly better than existing software referred to as a major success JSI believes that it can sell the software for 12 million However if the performance of the prototype does not exceed the performance of existing software referred to as a failure JSI will not be able to sell the software and hence will lose all its development costs Use the following notation to represent the problem symbolically Fail failure Mod moderate success and Maj major success JSI best estimates of the states of nature are PFail 070 PMod 020 and PMaj 010 Suppose that JSI can hire an independent consultant to review its ideas for the new software For a fee of 20000 the consultant will provide a report that gives two results either positive Pos or negative Neg depending on the consultant s view of the chances of success of the prototype Based on experience with this consultant JSI has assigned the following probabilities PPos 035 PFail Pos 040 PFail Neg 086 PNeg 065 PMod Pos 034 PMod Neg 012 PMaj Pos 026 PMaj Neg 002 a A blank decision tree is given on the next page Complete the decision tree by including all labels payoffs and probabilities Calculate the expected value for each branch b What is the EVSI of the consultant s report 0 Based on the decision tree should JSI hire the consultant Explain d What is JSl s optimal decision strategy using the expected value approach Decision Analysis Computing Branch Probabilities In the previous examples the probabilities were completely speci ed for all stateof nature nodes in the decision tree In some cases we may have probabilities but not the ones that are needed for the decision tree We will investigate two such situations and learn how to calculate the necessary probabilities Given Joint Probabilities Calculate Conditional Probabilities For two events A and B PA n B PBA or PA BPBAPA IPA Recall that PA B is the probability that A occurs given that B has occurred and is called the conditional probability PA r B is the probability that A and B both occur and is called the joint probability ofA and B In a decision tree when the stateofnature A occurs before the stateof nature B we require both PA and PB A in the decision tree However if we know PA r B and PA we can calculate the missing conditional probability Example Martin s Service Station 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 heavyduty snowplow truck Martin has analyzed the situation and believes that either alternative would be pro table if the snowfall is heavy Smaller pro ts would result if the snowfall is moderate and losses would result if the snowfall is light The following table gives the payoffs for each outcome V nter Snowfall Heavy 39 39 quot Light Buy Blade 3500 1000 1500 Buy Truck 7000 2000 9000 Since it is only April and longrange weather forecasts are unreliable the best current predictions are a 40 chance for heavy snowfall 30 for moderate snowfall and 30 chance of light snowfall More reliable results can be obtained by waiting until September since the September weather conditions either normal or unusually cold are a better predictor of winter snowfall levels However the cost of purchasing equipment in September is 400 more than in April The following table shows the joint probabilities of September weather conditions and winter snowfall levels In other words the rst value of 28 corresponds to the probability that September weather is normal and that winter snowfall is heavy V nter Snowfall September Heavy Moderate Light Weather Normal 28 24 28 Unusually Cold 12 6 2 Solving the Problem In order to determine Martin s optimal decision strategy we need to nd the appropriate conditional probabilities and compute expected values using a decision tree The rst decision that Martin needs to make is whether to buy the equipment now or wait until September If he waits then he will then decide which equipment to purchase based on the September weather In either case he will also need to decide which type of equipment to buy blade or truck The probabilities that need to be taken into account involve the uncertain statesofnature September weather conditions and snowfall level and the conditional probability of the snowfall level based on the September weather conditions Let s represent the different components ofthe problem as N normal Sept C cold Sept H heavy snowfall M moderate snowfall L light snowfall Thus we need to determine the following six conditional probabilities PH N PH C PM N PM C PL N PL C We use the fact that we know the joint probabilities to calculate the conditionals For example PH N 028 PN W a Calculate the remaining conditional probabilities PHl N 035 Note that PN PH n N PM n N PL n N and PC PH n C PM n C PL n C Fill in the blank decision tree on the next page with all relevant probabilities and expected values Use your completed tree to answer these questions b What is the EVSI of this decision and what does it correspond to in the context of this problem c Should Martin wait until Septemberto purchase the equipment d Using your decision tree state completely Martin s optimal strategy Martin39s Service Sta on Decision Tree Payo s Hazy smai BW Bade Mcdeaesmdi i ugwt SwoMai Buy NON Hazy smai BW delt Mcdeaesmdi Il ugwt SwoMai Hazy smai Buy Bade Modeaesmai u g1 SwoMai Nma Seuewbe Il Hazy smai BW delt Mcdeaesrmzfdi u g1 SwoMai Buy Lae I Hazy smai BW Bade Mcdeaesrmzfdi I u g1 SwoMai Cdd Seuewbe I 7 Hazy smai BW delt Mcdeaesrmzfdi mama Bayes Theorem Calculating Revised Probabilities Bayes theorem is used to revise previously calculated probabilities when you have new information You can apply Bayes theorem to situations similar to the following The Consumer Electronics Company is considering marketing a new model oftelevision set In the past 40 of the television sets introduced by the company have been successful and 60 have been unsuccessful Before introducing the television set to the marketplace the marketing research department conducts an extensive study and releases a report either favorable or unfavorable In the past 80 of the successful television sets had received a favorable market research report and 30 of the unsuccessful television sets had received a favorable report For this new model of television set under consideration the marketing research department has issued a favorable report What is the probability that the television set will be successful We will use the following notation S TV is successful N TV is not successful F favorable report U unfavorable report The probabilities that we have are PS 40 and PN 60 We also know that 80 of successful television sets had received a favorable market research report in other words PF S 80 Likewise we know that 30 of unsuccessful television sets had received a favorable report or PF N 30 What we want to determine is the probability that the TV set is successful knowing that it PsnF PF received a favorable report or PS F Recall that we know that PSI F PF S PS F Since PFI S 08 we have Per F PFI SPS 0804 032 PF N PN F Also PFI N N 06 03sowe have PN FPFINPN0306018 Furthermore PF PS n F PN n F 032 018 05 m E 064 In other words knowing that PF 05 the TV received a favorable report means that the probability of it being successful is actually 64 instead ofthe original 40 probability that we obtained without a market research survey This allows us to determine that PSI F We can summarize the calculations we made into table form StatesofNature Prior Probs Conditional Probs Joint Probabilities Revised or Posterior Probs Ps PF s PF 5 Ps F Sr S PS PF S PF 8 PS PF S PSl F PF 8 PF S2 N PN PF N PF N PN PF N PN F PF N PF Favorable PF PF 8 PF N We will now apply this idea to a decision analysis application Example Colonial Motors 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 each type of plant is given in the following table as well as the present value PV of future profits depending on whether the demand for the car is high or low PV of Future Profits millions Construction Cost Factory Size High Demand Low Demand millions Large 175 95 25 Small 125 105 15 Based on previous experience Colonial Motors believes that there is a 70 chance that the demand for this new car will be high and a 30 chance that the demand will below In order to get more information about the market conditions Colonial Motors can conduct a market research survey at a cost of 500000 to assess consumer attitudes about the new car The results of the survey will indicate either a favorable or unfavorable attitude about the new car The market survey company has indicated that they have performed 16 similar surveys in the past and have been accurate 13 times with the following breakdown In the 7 cases where the eventual demand for the car was high a favorable market research report had been issued 6 times In the 9 cases where the eventual demand for the car was low an unfavorable market research report had been issued 7 times Colonial Motors wants to know whether or not to pay for the market research survey and in either case which size factory they should build Use H high demand L low demand F favorable survey U unfavorable survey a Determine the revised demand probabilities based on the results of the market research survey Favorable Report Unfavorable Report The completed decision tree is shown on the next page Use it to answer these questions b Should Colonial Motors conduct the market research survey c Determine the optimal decision strategy for Colonial Motors Colonial Motors Decision Tree 70 allfigures are in millions ngDena39d Layaway 0 175 25 am LoNDanatl New 316 El m ngDena39tl Smallme 12 6 15 104 33 LoNDanatl mm 90 ngDena39tl Laysme 175 12667 2 10 LoNDe39na39 657 Famitle 316 Ii El 90 ngDena39tl Smalfamy 12 9 15 108 10 LoNDe39na39 ans 103 9 MI 30 ngDena39tl Laysme 175 2 70 LoNDe39na39 333 thamanle 316 I m ngDena39tl Smalfamy 12 15 70 LoNDe39na39 Payo s 150 70 110 150 70 110 150 70 110 Sensitivity Analysis RMC Inc The Excel sensitivity for a completed LPP model is shown below Use it to answer these questions Excel 110 Sensitivity Report RMC complete Final Reduced Objective Allowable Allowable Final Shadow Constraint Allowable Allowable 1 What are the ranges of optimality for CF and C3 the coefficients of F and S The range of optimality for CF is 24 56 and the range for Cs is 20 50 2 What are the ranges of feasibility RFl RFz and RF3 for each of the 3 constraints The range of feasibility for Material 1 is 14 215 for Material 2 is 4 00 and for Material 3 is 1775 30 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 or alternatively the change in the objective function value If either or both of these can t be obtained from the sensitivity report state this and explain why not 3 Suppose the profit per ton of fuel additive decreases to 35 The new coefficient is inside the range so the optimal solution remains 25 20 and the OFV will decrease by 525 125 4 Suppose the profit per ton of fuel additive increases to 50 The new coefficient is in the range so the optimal solution remains 25 20 The OFV will increase by 1025 250 V39 05 gt1 9 50 Suppose the profit per ton of solvent base increases to 45 The new coefficient is in the range so the optimal solution remains 25 20 The OFV will increase by 1520 300 Suppose the profit per ton of solvent base decreases to 10 The new coefficient is outside the range so the model will need to be resolved for the new optimal solution and object function value Suppose the number of tons of material 1 increases to 25 The new right hand side is outside the range so the model will need to be resolved for the new optimal solution and object function value Suppose the number of tons of material 2 increases to 15 The new right hand side is in the range of feasibility but the Material 2 constraint is nonbinding since the shadow price is 0 Therefore the complete solution will be unchanged Suppose the number of tons of material 3 increases to 25 The new right hand side is in the range of feasibility so the OFV will increase by 4444 1776 However the model will have to be resolved for the optimal solution