During any period, a potential customer arrives at a certain facility with probability 0.5. If there are already two people at the facility (including the one being served), the potential customer leaves the facility immediately and never returns. However, if there is one person or less, he enters the facility and becomes an actual customer. The manager of the facility has three types of service configurations available. At the beginning of each period, a decision must be made on which configuration to use. If she uses her “off” configuration at a cost of $0/period, no customer is served. If she uses her “slow” configuration at a cost of $3/period and any customers are present, one customer will be served and leave the facility at the end of the period with probability 0.6. If she uses her “fast” configuration at a cost of $9/period and any customers are present, one customer will be served and leave the facility at the end of the period with probability 0.8. The probability of more than one customer arriving or more than one customer being served in a period is zero. If no customer is present at the time of decision, no customer is served in that period (i.e., the arriving customer will have to wait for the next period to enter service). A revenue of $50 is earned when a customer is served and departed. There is no penalty for rejecting a customer. a. Formulate the problem of choosing the service configuration period by period as a Markov decision process. Identify the states and decisions. For each combination of state and decision, find the expected profit (revenue – cost of configuration) incurred during that period. b. Formulate a linear programming problem for finding an optimal stationary deterministic policy maximizing expected profit per period. (For full credit, i) Define the decision variables in the formulation, ii) Use the values of the parameters in the formulation, rather than the parameter names, iii) All constraints must be written explicitly (one-by-one, rather than as a family)) c. Solve the linear programming problem by, e.g., Excel solver. Then, report the optimal stationary policy and the expected profit per period. d. If, instead, we consider the discounted profit problem with ? = 0.9, what would be the optimal stationary policy?
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
During any period, a potential customer arrives at a certain facility with probability 0.5. If there are already two people at the facility (including the one being served), the potential customer leaves the facility immediately and never returns. However, if there is one person or less, he enters the facility and becomes an actual customer. The manager of the facility has three types of service configurations available. At the beginning of each period, a decision must be made on which configuration to use. If she uses her “off” configuration at a cost of $0/period, no customer is served. If she uses her “slow” configuration at a cost of $3/period and any customers are present, one customer will be served and leave the facility at the end of the period with probability 0.6. If she uses her “fast” configuration at a cost
of $9/period and any customers are present, one customer will be served and leave the facility at the end of the period with probability 0.8. The probability of more than one customer arriving or more than one customer being served in a period is zero. If no customer is present at the time of decision, no customer is served in that period (i.e., the arriving customer will have to wait for the next period to enter service). A revenue of $50 is earned when a customer is served and departed. There is no penalty for rejecting a customer.
a. Formulate the problem of choosing the service configuration period by
period as a Markov decision process. Identify the states and decisions. For each combination of state and decision, find the expected profit (revenue – cost of configuration) incurred during that period.
b. Formulate a linear programming problem for finding an optimal stationary deterministic policy maximizing expected profit per period. (For full credit, i) Define the decision variables in the formulation, ii) Use the values of the parameters in the formulation, rather than the parameter names, iii) All constraints must be written explicitly (one-by-one, rather than as a family))
c. Solve the linear programming problem by, e.g., Excel solver. Then, report the optimal stationary policy and the expected profit per period.
d. If, instead, we consider the discounted profit problem with ? = 0.9, what would be the optimal stationary policy?
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