-Linearity:the impact of decision variables islinear in constraints and objective funct-Divisibility:noninteger values of decisionvariables are acceptable;
-Certainty:values of parameters are known andconstant; -Nonnegativity:negative values of decisionvariables are unacceptable.
*Tricky Constraints: 1) % of total production x1/(x1+x2+x3)>0.5 3) Units of x must be more than 5 times y x-5y=0
2) “x” must be equal to 2/3 y x-2/3*y=0 4) Transhipment problem for centre a x1a+x2a-x10z-xa11=0
5) Inventory Constraint Beginning Inv+Production-demand-ending inventory=0
1)Assign a variable to each axis2) Plot the constraints (using x intercept-y=0 and Y intercept-x=0) 3)determine feasible region ( above constraints if >= and under if
<=)4)solve for max or min by either i) plotting max/min function and moving it to the last point it is within feasible region OR ii) using the coordinates of each intercept of the
binding constraints and using the highest one.
Reports: 1)Answer: Target cell=Minimized or Maximized Profit, Adjustable cells=optimal solution, Constraints=binding/non binding and amount of slack
2)Sensitivity: Adjustable cells= optimal solution, objective coefficient, allowable inc and dec and reduced cost, Constraints=Final value, shadow price(max price you
are willing to pay for an additional unit) and slack (extra resources available at optimal solution)
*Shadow price’s range= (final value+allowable inc). (final value-allowable decrease)
Sensitivity &What if Analysis
1)Obj. Coefficient Ranging –How much could a single value cjchange without changing the current set of variables and their values in the existing optimal solution ?
*point remains the same but optimal value changes
2) RHS Ranging-What is the range of values for the RHS of each original constraint such that the current set of basic variables (but not theirvalues!) is still optimal ?
Special Case LP’s
1.Transportation Problems: Decision Variables: The amount of goods to be transported from origin i to destination j (x ) , Obj. Function: Minimize transportation costs (sum
of individual shipping costs), Constraints: i)Supply, ii) Demand( # of constraints=#of nodes)*if Supply>demand a dummy variable is added with 0 costs and demand=(supply-
demand) and if supplytaks then slack will show unassigned worker but if worker)
Decision Analysis -The decision alternatives are the different possible strategies or choices the decision maker can employ.
-The states of nature refer to future events which may occur.
-Selection of a decision alternative while a particular state of nature occurs produces a payoff or outcome.
-The decision maker can control the choice of an alternative, but cannot control which state of nature will occur.
1.Clearly define the problem 2.List all possible alternatives
3.Identify all possible outcomes for each alternative 4.Identify the payoff for each alternative and outcome combination
5.Use a decision modeling technique to choose an alternative (Payoff table or Decision tree)
1.Decision making under certainty (know for sure the payoff for every decision)
2.Decision making under uncertainty(do not know the likelihood that a specific outcome will occur. Maximax(highest), Maximin(lowest) ,Criterion of Realism (α
x (Max payoff for alternative)+ (1 –α) x (Min payoff for alternative)) ,Equally Likely (highest average ),Minimax regret ( minimize regret)
3:Decision making under risk (some knowledge regarding the probability of occurrence of each outcome) Expected monetary value ( payoff*prob), Expected opportunity loss,
Expected value of perfect information(EVwPI=Best payoff*prob and EVPI= EVwPI-Max EMV)
Prior Prob=P(B) *Conditional Prob=P(A|B)=Joint Prob= P(A and B)--> P(A and B) / P(B)= Posterior Probability=P(B|A)= P(A|B)*P(B)/ [P(A|B)*P(B) + P(A|C)*P(C)]
Phase 1 : Project Planning: 1.What is the project goal or objective?
2.What are the activities (or tasks) involved?
3.How are activities linked (i.e., precedence relationship) one another?
4.How much time is required for each activity?
5.What resources other than time are required for each activity?
Phase 2: Project Scheduling(Developing a time schedule for each activity and assigning resources to specific
activities) CPM PERT
1.When will the entire project be completed? 1.Optimistic time (a)
2.What is the scheduled start and end time for each activity? 2.Pessimistic time (b)
3.Most likely time (m)
3.Which are the “critical” activities(slack of 0)?- LST-EST=slack
4.Which are the noncritical activities(have slack)
5.How late can noncritical activities be without delaying the project?
6.Allowing for uncertainty, what is the probability of completing the
project by a specific deadline?
Phase 3: Project Controlling (Monitoring of schedules, resources and budget, and obtain feedback to revise the project plan )
1.Is the project on schedule? Ahead or behind?
2.Are costs equal to the budget? Over budget? Under budget?
3.Are there adequate resources to finish on time?
4.What is the best way to reduce project duration at minimum cost?
QUESTION 1: (25 points) SENSITIVITY REPORT
(c) From the optimal solution, what are the sales effort allocations to each of the distribution channels? (4 points)
Solution: Sales effort allocation to each distribution channel is: 1) T:258 units * 2.5 hrs per unit = 645 hours 2)B:
0units*3.75hrsperunit= 0hours 3) C:480 units * 3.8 hrs per unit = 1824 hours 4) I:4262 units * 0.5 per unit = 2131 hours
d) What would need to happen to the Business distribution option unit profits before it would be optimal to acquire an
advertising and sales force allocation of its own? (4 points)
Solution: Unit Profits from the Business distribution option would need to increase by the “Reduced Cost” amount of $31.375
to $94.375 (=63+31.375) before it would be optimal to distribute there and acquire an advertising and sales force allocation.
(e) Given the highly competitive markets for MP3 players in 2005, EPG was considering committing more financial resources
to support the advertising costs for this project. After some discussion, the Board of Directors approved another $12,000 for
this project. What will be the impact on the new optimal profit of these new operating funds for advertising? (
Solution: Since the Shadow Price for Advertising dollars is zero, then all the available funds are not being used up now.
Therefore, it would not be worthwhile for EPG to add more advertising dollars, and the new optimal profit will not change
from these new operating funds.
(f) Given a high volume of sales in early December, Wal-Mart has sent a memo to EPG requesting that the current contract
be renegotiated so that the 480 units be increased by 144 more units. Determine the impact on EPG’s total optimal profit if
the new terms of the contract go into effect. (4 points)
Solution: Given the Shadow Price for Commercial Contract sales of -45.15, then it costs EPG to increase distribution to Wal-
Mart. Thus, only if forced to do so, the optimal objective function value would decrease from $230,878 to $230,878 + (144
units * - 45.15) = $230,878-$6501.60 = $224,376.40. Note also that the decrease of 144 units does not exceed the
“Allowable Increase” of 156 units and therefore the Shadow Price is applicable.
QUESTION 2: BINARY PROBLEM
DWR Ltd has MADE new game consoles: zbox and ybox. Production lines cost $25,000 for zbox and $32,000 for ybox. Each console has profits: $150/unit for zbox and
$210/unit for ybox. DWR Ltd has space for creating 3 production lines capable of manufacturing the consoles and each line can be used to manufacture both consoles at the
same time. However, management has decided that in order to mitigate the risks, only one production line would be open.
zbox can be produced at the rate of 15 units/hr on line 1, 18 units/hr on line 2, and 22 units/hr on line 3. Ybox can be produced at the rate of 20 units/hr, 17 units/hr, and 10
units/hr respectively. Line 1 has 340 hours production capacity, line 2 has 400 hrs, line 3= 380 hours capacity.
Write in mixed binary integer programming problem. DO NOT SOLVE.
Let Xij = the number of game console i produced at on production line j, Where i = 1(zbox), 2 (ybox) and j = 1, 2,3
Yi = 1 if game c