Question

metal fabricators, inc. manufacturers gas grill tanks, model # 1420, for four original equipment manufacturers (oems). demand is forecast to be as follows: quarter 1 - 3,100 tanks, quarter 2 - 3,700, quarter 3 - 4,000, and quarter 4 - 3,200. due to a hedging program for sheet steel and increases in international tariffs, production cost per quarter vary as follows: quarter 1 - $25.90 per tank, quarter 2 - $30.80, quarter 3 - $28.50, and quarter 4 - $31.90. due to production contracts with the oems, no shortages are allowed. beginning inventory for quarter 1 is 330 tanks. at the end of each quarter, inventory holding costs are $4.70 per tank. formulate this as a linear optimization model but do not solve. round your answers for the coefficients for objective function to two decimal places and round other answers to the nearest whole number. use a minus sign where appropriate and do not leave any fields blank. if the constant is one or minus one, enter \1\ or \-1\ correspondingly. let: pi = number of units produced in month i, ii = inventory at the end of month i minimal total cost the constraints are as follows: week 1: p1 + + i1 select week 2: i1 + p2 + i2 select week 3: i2 + p3 + i3 select week 4: i3 + p4 + i4 select pt, it ≥ 0 for t = 1, 2, 3

Explanation:

Step1: Define objective function

The objective is to minimize the total cost, which is the sum of production costs and inventory - holding costs. The production cost for each quarter is the cost per unit times the number of units produced, and the inventory - holding cost is the cost per unit in inventory times the number of units in inventory.
The objective function $Z$ is:
$Z = 25.90P_1+30.80P_2 + 28.50P_3+31.90P_4+4.70I_1 + 4.70I_2+4.70I_3+4.70I_4$

Step2: Define constraints

For Quarter 1: The beginning inventory is 330. The sum of beginning inventory and production must be at least equal to the demand (3100) plus the ending inventory. So, $330 + P_1=3100 + I_1$, or $P_1-I_1=3100 - 330=2770$.
For Quarter 2: The beginning inventory is $I_1$. The sum of beginning inventory and production must be at least equal to the demand (3700) plus the ending inventory. So, $I_1+P_2=3700 + I_2$, or $I_1+P_2-I_2=3700$.
For Quarter 3: The beginning inventory is $I_2$. The sum of beginning inventory and production must be at least equal to the demand (4000) plus the ending inventory. So, $I_2+P_3=4000 + I_3$, or $I_2+P_3-I_3=4000$.
For Quarter 4: The beginning inventory is $I_3$. The sum of beginning inventory and production must be at least equal to the demand (3200) plus the ending inventory. So, $I_3+P_4=3200 + I_4$, or $I_3+P_4-I_4=3200$.
Also, $P_t\geq0$ and $I_t\geq0$ for $t = 1,2,3,4$.

Answer:

Objective function: $Z = 25.90P_1+30.80P_2 + 28.50P_3+31.90P_4+4.70I_1 + 4.70I_2+4.70I_3+4.70I_4$
Constraints:
Quarter 1: $P_1 - I_1=2770$
Quarter 2: $I_1+P_2 - I_2=3700$
Quarter 3: $I_2+P_3 - I_3=4000$
Quarter 4: $I_3+P_4 - I_4=3200$
$P_t\geq0$, $I_t\geq0$ for $t = 1,2,3,4$