A family operated Ice Cream Parlor has decided to engage in a promotional campaign, offering coupons for free ice cream during the Summer time. They plan to keep the campaign for an extended week, offering coupons from Friday morning until the following Sunday evening. They have estimated that such campaigns may increase the number of clients per day by as much as 20%. The Parlor offers a variety of flavors, each in different sizes or presentations. A large ice cream sells for $2.00, and a small cone for $0.75. Each day a (random) number of families come in to buy ice cream. We know the type of families in terms of size and preference. Thus each client is a group of people that can buy one or more of the products offered.
First Coupon Strategy: A client that spends $3.50 or more can ask for one coupon that entitles the bearer to get two small cones and two large ice creams, and which expires the following day. It is expected that only 50% of eligible clients (families) will actually ask for the coupon, and then only 75% will return to redeem it.
Second Coupon Strategy: A client that spends $2.00 or more can ask for two coupons, each entitling the bearer to get one small cones and one large ice cream, and which expires the following day. It is expected that 62% of the eligible clients will ask for coupons, but only 50% of them will redeem the two coupons and 20% will return to redeem only one coupon.
Chosen Flavor  Probability of Small Cone 
Chocolate  0.35 
Butterscotch Milk  0.75 
Vanilla Almond  0.60 
Small cones are sold for $0.75 and large ice creams for $2.00, regardless of flavor. We assume that the number of families arriving each day (that do not come to redeem coupons) is independent of previous days and follows a Poisson distribution. The average numebr of clients per day is given in the table below, as well as the family size distribution, which is independent of the strategy. By ``family size'' we mean the effective size, that is, the number of people asking for ice creams, or if you prefer, the number of ice creams ordereb by the family, which is the important quantity.


Finally, we consider the following model for coupon redemption. When a family comes back with a coupon, they will sometimes bring in friends and spend some more money. Under Strategy 1,a family that redeems a coupon will get the free cones, or may be buy something else. Under Strategy 2, only families that redeem the two coupons will (possibly) by something else. The table below gives the possible gains and probablities associated with them:
Coupons Redeemed  Expected Gain  Probability 
Strategy 1: one coupon  $5.50  0.85 
Strategy 1: one coupon  $3.50  0.15 
Strategy 2: one coupon  $2.75  1.00 
Strategy 2: two coupons  $5.50  0.85 
Strategy 2: two coupons  $3.50  0.15 
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