Question

Question 1 [35 marks]

A foundry that specializes in producing custom blended alloys has received an order for 1 000 kg of an alloy containing at least 5% chromium and not more than 50% iron. Four types of scrap which can be easily acquired can be blended to produce the order. The cost and metal characteristics of the four scrap types are given below: Scrap type
Item	1	2	3	4
Chromium	5%	4%	-	8%
Iron	40%	80%	60%	32%
Cost per kg	R6	R5	R4	R7

The purchasing manager has formulated the following LP model:

Minimise COST = 6M1 + 5M2 + 4M3 + 7M4

subject to

0,05M1 + 0,04M2 + 0,08M4 ≥ 50 (CHRM)

0,40M1 + 0,80M2 + 0,60M3 + 0,32M4 ≤ 500 (IRON)

M1 + M2 + M3 + M4 = 1000 (MASS)

and all variables ≥ 0,

where Mi = number of kg of scrap type i purchased, i=1,2,3,4.

(a) Solve this model using LINDO or SOLVER.

(b) Write down the foundry's optimal purchasing plan and cost.

Based on your LINDO or SOLVER solution answer the following questions by using only the initial printout of the optimal solution. (This means that you may not change the relevant parameters in the model and do reruns.)

(c) How good a deal would the purchasing manager need to get on scrap type 1 before he would be willing to buy it for this order?

(d) Upon further investigation, the purchasing manager finds that scrap type 2 is now being sold at R5,40 per kg. Will the purchasing plan change? By how much will the cost of purchasing the metals increase?

(e) The customer is willing to raise the ceiling on the iron content in order to negotiate a reduction in the price he pays for the order. How should the purchasing manager react to this?

(f) The customer now specifies that the alloy must contain at least 6% chromium. Can the purchasing manager comply with this new specification? Will the price charged for the order change?

Answer 1

(b) Write down the foundry's optimal purchasing plan and cost. 4 PBA4804 OCTOBER/NOVEMBER 2019 PORTFOLIO EXAMINATION [TURN OVER] Based on your LINDO or SOLVER solution answer the following questions by using only the initial printout of the optimal solution. (This means that you may not change the relevant parameters in the model and do reruns.)

Optimal plan:

M1 = 0

M2 = 220.5882 kg

M3 = 264.7059 kg

M4 = 514.7059 kg

Minimized cost = $5764.706

Solver screenshot

Solver formula

Sensitivity report

(c) How good a deal would the purchasing manager need to get on scrap type 1 before he would be willing to buy it for this order?

We can see reduced cost is 0.088235294 for scrap type 1 so the cost needs ton be reduced by 0.088235294 so that purchasing manager would be willing to buy it

(d) Upon further investigation, the purchasing manager finds that scrap type 2 is now being sold at R5,40 per kg. Will the purchasing plan change? By how much will the cost of purchasing the metals increase?

No the purchasing plan will not change as increase of 0.4 is within allowable increase limit for scrap type 2

Yes, cost of purchasing metals will increase by 220.5882*0.4 = 88.23528

(e) The customer is willing to raise the ceiling on the iron content in order to negotiate a reduction in the price he pays for the order. How should the purchasing manager react to this?

Increasing ceiling on the iron content by 1 percentage point will give cost reduction of -1.470588235 up to total 68% of ceiling of iron content which is the allowable increase limit for the Iron constraint(i.e an increase of maximum 18%) so purchasing manager should not reduce price by more than -1.470588235 for per percentage point increase of ceiling