Fran's Weekly Health Care Product Sales Optimization

Product Information and Constraints

Fran Farnsworth sells three health care products: Flex, Feelgood, and Firmup. Her costs, sales time, delivery costs, and commission for each product are as follows:

• Flex: $3 cost, 10 minutes to sell, $0.50 delivery cost, $2 commission
• Feelgood: $5 cost, 20 minutes to sell, no delivery cost, $4 commission
• Firmup: $4 cost, 12 minutes to sell, $1 delivery cost, $3 commission

Fran's weekly constraints are:

• Maximum $1500 worth of products
• Maximum $85 delivery expense
• Maximum 40 hours (2400 minutes) for sales

Linear Programming Model

Let x, y, and z represent the number of boxes of Flex, Feelgood, and Firmup sold per week, respectively. Let C be Fran's weekly commission. The LP model is:

Maximize:

C = 2x + 4y + 3z

Subject to:

3x + 5y + 4z ≤ 1500 ($ - worth)

10x + 20y + 12z ≤ 2400 (minutes - sell time)

0.5x + z ≤ 85 ($ - delivery)

x ≥ 0, y ≥ 0, z ≥ 0

Analysis and Recommendations

Optimal Selling Strategy:

• x = 0 (sell no Flex)
• y = 69 (sell 69 boxes of Feelgood)
• z = 85 (sell 85 boxes of Firmup)

Maximum weekly commission: C = $531

Resource Utilization:

Fran will spend 40 hours selling products and use her full $85 delivery allowance.

Product Allowance:

Fran should not request a higher product allowance. She sells $685 worth of products, leaving $815 unspent.

Babysitter:

Fran should hire the babysitter. The shadow price for selling time is $12/hour, exceeding the $8/hour babysitter cost. The allowable increase in selling time is 54.33 hours, so the extra 5 hours is within the limit.

Reduced Delivery Allowance:

If the delivery allowance is reduced to $50, Fran's commission will decrease by $21 ($0.60/unit decrease * $35 decrease). She will need to reduce Firmup sales.

Flex Sales Promotion:

Selling at least 10 boxes of Flex will decrease Fran's weekly commission by $3 (reduced cost of $0.3/box * 10 boxes).

Profit Range Analysis:

The following table shows the profit ranges for each product: