The gardening season is in full swing and many nurseries are planning how they will be…

The gardening season is in full swing and many nurseries are
planning how they will be able to meet customer demands over the
next few months. PlantaLot is a nursery that specializes in
cultivating starter tomato plants for distribution at local garden
centres. They would like to develop a production schedule to
determine the number of starter tomato plants that will need to be
cultivated over the months of July, August, and September.

Based on the previous year’s sales, demand for July, August, and
September are 6,000 starter plants, 8,000 starter plants, and 3,000
starter plants respectively. We can assume that the sales will be
the same as estimated demands in this case. PlantaLot has a
production capacity that allows them to produce up to 7,000 starter
plants each month. Furthermore, PlantaLot has storage capacity for
up to 1,000 starter plants at the end of every month. In June,
PlantaLot produced 5500 starter plants, but only sold 5000. Thus,
they have 500 starter plants in inventory at the start of
July

In addition to the above information, it is desirable to have a
production schedule that maintains the current number of employees
and has the same production quantities every month. But given the
fluctuations from month to month in terms of demand, this MAY NOT
be possible. The nursery’s accounting department estimates that
increasing the production by one starter plant from one month to
the next will cost $0.50 per starter plant and decreasing
production by one starter plant form one month to the next will
cost $0.40 per starter plant.

Ignoring production and inventory costs, formulate the algebraic
linear programming model that will help PlantaLot determine how
many plants to produce in the months of July, August, and September
such that the cost of changing the production level is minimized
while PlantaLot also satisfies monthly demands.

ONLY FORMULATE – DO NOT SOLVE THE PROBLEM FOR AN OPTIMAL
SOLUTION

Question 4 (continued)

Hint: The cost of changing production levels is the cost of the
monthly increases in production plus the cost of monthly decreases
in production.

You should use the following decision variables to formulate your
model, thus there is no need to redefine or define more decision
variables. Let Xi represent the number of tomato starter plants
produced in month i Let Yi represent the number of tomato starter
plants placed in storage at the end of month i Let ei represent the
increase in the total production level in month i Let di represent
the decrease in the total production level in month i where i = 1,
2, and 3 (1 is July; 2 is August, and 3 is September).